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Examples from Timofeev (1948) Ch. 5, divided into sections kindly provided by clicliclic vel Martin on sci.math.symbolic

fricas
-- Timofeev (
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)set break resume
-- Problems 1 - 3 (p. 202-203)
integrate(cos(x)^2, x)
 (1)
Type: Union(Expression(Integer),...)
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integrate(cos(x)^3, x)
 (2)
Type: Union(Expression(Integer),...)
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integrate(sin(x)^4, x)
 (3)
Type: Union(Expression(Integer),...)
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-- Problems 4 - 7 (p. 208)
integrate(cos(x)^6, x)
 (4)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^8, x)
 (5)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(1/4*%pi+1/2*x)^4, x)
 (6)
Type: Union(Expression(Integer),...)
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integrate(sin(-1/12*%pi+3*x)^3, x)
 (7)
Type: Union(Expression(Integer),...)
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-- Problems 8 - 11 (p. 211)
integrate(1/sin(x)^6, x)
 (8)
Type: Union(Expression(Integer),...)
fricas
integrate(csc(x)^7, x)
 (9)
Type: Union(Expression(Integer),...)
fricas
integrate(1/cos(x)^12, x)
 (10)
Type: Union(Expression(Integer),...)
fricas
integrate(1/cos(1/4*%pi+3*x)^3, x)
 (11)
Type: Union(Expression(Integer),...)
fricas
-- Problems 12 - 14 (p. 213)
integrate(tan(x)^6, x)
 (12)
Type: Union(Expression(Integer),...)
fricas
integrate(1/tan(x)^5, x)
 (13)
Type: Union(Expression(Integer),...)
fricas
integrate(cot(-3/4*%pi+1/3*x)^4, x)
 (14)
Type: Union(Expression(Integer),...)
fricas
-- Problems 15 - 20 (p. 219-220)
integrate(cos(x)^6*sin(x)^4, x)
 (15)
Type: Union(Expression(Integer),...)
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integrate(cos(x)^6*sin(x)^7, x)
 (16)
Type: Union(Expression(Integer),...)
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integrate(sin(x)^11/cos(x), x)
 (17)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(cos(x)^6*sin(x)^6), x)
 (18)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)^(2*m)*sin(x)^(2*m), x)
 (19)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(cos(1/4*%pi+2*x)*sin(1/4*%pi+2*x)^3), x)
 (20)
Type: Union(Expression(Integer),...)
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-- Problems 21 - 29 (p. 223)
integrate(sec(x)^2*tan(x)^2, x)
 (21)
Type: Union(Expression(Integer),...)
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integrate(cot(x)^3*csc(x), x)
 (22)
Type: Union(Expression(Integer),...)
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integrate(sec(x)^3*tan(x), x)
 (23)
Type: Union(Expression(Integer),...)
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integrate(cot(x)^2*csc(x)^3, x)
 (24)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)^3/sin(x)^7, x)
 (25)
Type: Union(Expression(Integer),...)
fricas
integrate(sqrt(sec(x)^3)*tan(x)^5, x)
 (26)
Type: Union(Expression(Integer),...)
fricas
integrate(sec(x)^4*tan(x)^(3/2), x)
 (27)
Type: Union(Expression(Integer),...)
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integrate(cot(x)^4*csc(x)^3, x)
 (28)
Type: Union(Expression(Integer),...)
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)set output tex off

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)set output algebra on
integrate(sec(1/4*%pi+1/2*x)^3*tan(1/4*%pi+1/2*x)^2, x)
(29)
2 x + %pi 4        2 x + %pi
- cos(---------) log(sin(---------) + 1)
4                  4
+
2 x + %pi 4          2 x + %pi
cos(---------) log(- sin(---------) + 1)
4                    4
+
2 x + %pi 2         2 x + %pi
(- 2 cos(---------)  + 4)sin(---------)
4                   4
/
2 x + %pi 4
8 cos(---------)
4
Type: Union(Expression(Integer),...)
fricas
)set output tex on

fricas
)set output algebra off
-- Problems 30 - 32 (p. 228)
integrate((1+cot(x)^3)*(a*sec(x)^2-sin(2*x)), x)
 (29)
Type: Union(Expression(Integer),...)
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integrate((4-3*cos(x))*(1-1/2*sin(x))^4, x)
 (30)
Type: Union(Expression(Integer),...)
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integrate((1/2-3*cot(x))*(3-2*cot(x))^3, x)
 (31)
Type: Union(Expression(Integer),...)
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-- Problems 33 - 36 (p. 229)
integrate(cos(5*x)/cos(x)^5, x)
 (32)
Type: Union(Expression(Integer),...)
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integrate(cos(4*x)/cos(x), x)
 (33)
Type: Union(Expression(Integer),...)
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integrate(cos(x)*cos(4*x), x)
 (34)
Type: Union(Expression(Integer),...)
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integrate(cos(4*x)/cos(x)^5, x)
 (35)
Type: Union(Expression(Integer),...)
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-- Problems 37 - 39 (p. 233)
integrate(cos(x)^4*cos(4*x), x)
 (36)
Type: Union(Expression(Integer),...)
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integrate(cos(5*x)/sin(x)^5, x)
 (37)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(4*x)/sin(x)^4, x)
 (38)
Type: Union(Expression(Integer),...)
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-- Problems 40 - 49 (p. 254-255)
integrate(cos(x)/(sin(x)*(2+sin(2*x))), x)
 (39)
Type: Union(Expression(Integer),...)
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integrate(cos(x)^2/(cos(3*x)*sin(x)), x)
 (40)
Type: Union(Expression(Integer),...)
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integrate(sin(2*x)/(cos(x)^4+sin(x)^4), x)
 (41)
Type: Union(Expression(Integer),...)
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)set output tex off

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)set output algebra on
integrate(1/(4+sin(x)+cos(x)*sqrt(3)), x)
+-+                       +-+
2 \|3 sin(x) + 6 cos(x) + 2 \|3
atan(--------------------------------)
+-+
3 \|3 sin(x) - 3 cos(x)
(43)  - --------------------------------------
+-+
2 \|3
Type: Union(Expression(Integer),...)
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)set output tex on

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)set output algebra off
integrate(1/(3+4*cos(x)+4*sin(x)), x)
 (42)
Type: Union(Expression(Integer),...)
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integrate(1/(4-3*cos(x)^2+5*sin(x)^2), x)
 (43)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(4+4*cot(x)+tan(x)), x)
 (44)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(2*sec(x)+sin(x))^2, x)
 (45)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(cos(x)+2*sec(x))^2, x)
 (46)
Type: Union(Expression(Integer),...)
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integrate((5-tan(x)-6*tan(x)^2)/(1+3*tan(x))^3, x)
 (47)
Type: Union(Expression(Integer),...)
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-- Problems 50 - 56 (p. 260)
integrate(cos(x)^2/cos(3*x), x)
 (48)
Type: Union(Expression(Integer),...)
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integrate(sin(x)/cos(2*x), x)
 (49)
Type: Union(Expression(Integer),...)
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integrate(sin(x)^2/cos(2*x), x)
 (50)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^3/cos(3*x), x)
 (51)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)/sin(3*x), x)
 (52)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)/sin(4*x), x)
 (53)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^3/sin(4*x), x)
 (54)
Type: Union(Expression(Integer),...)
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-- Problems 57 - 61 (p. 266)
integrate(sqrt(1+sin(2*x)), x)
 (55)
Type: Union(Expression(Integer),...)
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integrate(1/sqrt(1+cos(2*x)), x)
 (56)
Type: Union(Expression(Integer),...)
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integrate(1/(1-cos(3*x))^(3/2), x)
 (57)
Type: Union(Expression(Integer),...)
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integrate((1-sin(2/3*x))^(5/2), x)
 (58)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)*(-cos(x)^2+2*(1+2*sin(x))^(1/4))/(1+2*sin(x))^(3/2), x)
 (59)
Type: Union(Expression(Integer),...)
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-- Problems 62 - 66 (p. 268)
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)set output tex off

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)set output algebra on
integrate(sqrt(tan(x)), x)
(62)
-
+-+
4 \|2
*
atan
1
/
+---------------------------------------------+
|               +------+
|     +-+       |sin(x)
|- 2 \|2 cos(x) |------  + 2 sin(x) + 2 cos(x)
|              \|cos(x)
|---------------------------------------------
\|                    cos(x)
+
+------+
+-+ |sin(x)
\|2  |------  - 1
\|cos(x)
+
-
+-+
4 \|2
*
atan
1
/
+-------------------------------------------+
|             +------+
|   +-+       |sin(x)
|2 \|2 cos(x) |------  + 2 sin(x) + 2 cos(x)
|            \|cos(x)
|-------------------------------------------
\|                   cos(x)
+
+------+
+-+ |sin(x)
\|2  |------  + 1
\|cos(x)
+
+------+
+-+       |sin(x)
2 \|2 cos(x) |------  + 2 sin(x) + 2 cos(x)
+-+                \|cos(x)
- \|2 log(-------------------------------------------)
cos(x)
+
+------+
+-+       |sin(x)
- 2 \|2 cos(x) |------  + 2 sin(x) + 2 cos(x)
+-+                  \|cos(x)
\|2 log(---------------------------------------------)
cos(x)
/
4
Type: Union(Expression(Integer),...)
fricas
integrate(1/tan(5*x)^(1/3), x)
(63)
3+--------+2              3+--------+2           3+--------+
2 log(\|tan(5 x)   + 1) - log(- \|tan(5 x)   + tan(5 x)\|tan(5 x)  + 1)
+
3+--------+2
+-+     2 \|tan(5 x)   - 1
2 \|3 atan(------------------)
+-+
\|3
/
20
Type: Union(Expression(Integer),...)
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integrate(1/(4+3*tan(2*x))^(3/2), x)
(64)
-
+-----------------------+
+-+ |3 sin(2 x) + 4 cos(2 x)
36 \|5  |-----------------------
\|        cos(2 x)
*
atan
+-+
125 \|5
/
+--+
\|10
*
ROOT
+-----------------------+
+-+ +--+         |3 sin(2 x) + 4 cos(2 x)
- 9375 \|5 \|10 cos(2 x) |-----------------------
\|        cos(2 x)
+
46875 sin(2 x) + 140625 cos(2 x)
/
cos(2 x)
+
+-----------------------+
+--+ |3 sin(2 x) + 4 cos(2 x)         +-+
125 \|10  |-----------------------  - 375 \|5
\|        cos(2 x)
+
-
+-----------------------+
+-+ |3 sin(2 x) + 4 cos(2 x)
36 \|5  |-----------------------
\|        cos(2 x)
*
atan
+-+
125 \|5
/
+--+
\|10
*
ROOT
+-----------------------+
+-+ +--+         |3 sin(2 x) + 4 cos(2 x)
9375 \|5 \|10 cos(2 x) |-----------------------
\|        cos(2 x)
+
46875 sin(2 x) + 140625 cos(2 x)
/
cos(2 x)
+
+-----------------------+
+--+ |3 sin(2 x) + 4 cos(2 x)         +-+
125 \|10  |-----------------------  + 375 \|5
\|        cos(2 x)
+
+-----------------------+
+-+ |3 sin(2 x) + 4 cos(2 x)
13 \|5  |-----------------------
\|        cos(2 x)
*
log
+-----------------------+
+-+ +--+         |3 sin(2 x) + 4 cos(2 x)
9375 \|5 \|10 cos(2 x) |-----------------------
\|        cos(2 x)
+
46875 sin(2 x) + 140625 cos(2 x)
/
cos(2 x)
+
-
+-----------------------+
+-+ |3 sin(2 x) + 4 cos(2 x)
13 \|5  |-----------------------
\|        cos(2 x)
*
log
+-----------------------+
+-+ +--+         |3 sin(2 x) + 4 cos(2 x)
- 9375 \|5 \|10 cos(2 x) |-----------------------
\|        cos(2 x)
+
46875 sin(2 x) + 140625 cos(2 x)
/
cos(2 x)
+
+--+
- 60 \|10
/
+-----------------------+
+--+ |3 sin(2 x) + 4 cos(2 x)
500 \|10  |-----------------------
\|        cos(2 x)
Type: Union(Expression(Integer),...)
fricas
integrate((-sqrt(4-3*tan(x))+3*tan(x))/(cos(x)^2*(4-3*tan(x))^(3/2)), x)
(65)
+---------------------+
|- 3 sin(x) + 4 cos(x)
(12 sin(x) - 32 cos(x)) |---------------------
\|        cos(x)
+
2
(- 3 sin(x) + 4 cos(x))log(cos(x) )
+
2
- 24 cos(x)sin(x) + 7 cos(x)  + 9
(3 sin(x) - 4 cos(x))log(---------------------------------)
4
/
18 sin(x) - 24 cos(x)
Type: Union(Expression(Integer),...)
fricas
integrate(tan(x)/(-1+sqrt(tan(x)))^2, x)
(66)
+------+
(- \|tan(x)  + 1)
*
+----------------------------------------------------------+
|         2                                 2
\|- 3 %%CZ1  + (- 2 %%CZ0 - 2)%%CZ1 - 3 %%CZ0  - 2 %%CZ0 - 1
+
+------+                           +------+
(- \|tan(x)  + 1)%%CZ1 + (- %%CZ0 - 1)\|tan(x)  + %%CZ0 + 1
*
log
((32 %%CZ0 + 4)%%CZ1 + 4 %%CZ0 + 2)
*
+----------------------------------------------------------+
|         2                                 2
\|- 3 %%CZ1  + (- 2 %%CZ0 - 2)%%CZ1 - 3 %%CZ0  - 2 %%CZ0 - 1
+
2              2
(- 32 %%CZ0 - 4)%%CZ1  + (- 32 %%CZ0  - 32 %%CZ0 - 2)%%CZ1
+
+------+          2
6 \|tan(x)  - 4 %%CZ0  - 2 %%CZ0 - 3
+
+------+
(\|tan(x)  - 1)
*
+----------------------------------------------------------+
|         2                                 2
\|- 3 %%CZ1  + (- 2 %%CZ0 - 2)%%CZ1 - 3 %%CZ0  - 2 %%CZ0 - 1
+
+------+                           +------+
(- \|tan(x)  + 1)%%CZ1 + (- %%CZ0 - 1)\|tan(x)  + %%CZ0 + 1
*
log
((- 32 %%CZ0 - 4)%%CZ1 - 4 %%CZ0 - 2)
*
+----------------------------------------------------------+
|         2                                 2
\|- 3 %%CZ1  + (- 2 %%CZ0 - 2)%%CZ1 - 3 %%CZ0  - 2 %%CZ0 - 1
+
2              2
(- 32 %%CZ0 - 4)%%CZ1  + (- 32 %%CZ0  - 32 %%CZ0 - 2)%%CZ1
+
+------+          2
6 \|tan(x)  - 4 %%CZ0  - 2 %%CZ0 - 3
+
+------+
(2 \|tan(x)  - 2)%%CZ1
*
log
2            2
(64 %%CZ0 + 8)%%CZ1  + (64 %%CZ0  + 64 %%CZ0 + 4)%%CZ1
+
+------+           3           2
6 \|tan(x)  + 64 %%CZ0  + 64 %%CZ0  + 32 %%CZ0 - 1
+
+------+
(2 %%CZ0\|tan(x)  - 2 %%CZ0)
*
+------+           3           2
log(6 \|tan(x)  - 64 %%CZ0  - 56 %%CZ0  - 28 %%CZ0 + 7)
+
+------+          +------+
(2 \|tan(x)  - 2)log(\|tan(x)  - 1) - 2
/
+------+
2 \|tan(x)  - 2
Type: Union(Expression(Integer),...)
fricas
)set output tex on

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)set output algebra off
-- Problems 67 - 75 (p. 272-273)
integrate(sin(x)/sqrt(sin(2*x)), x)
 (60)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)/sqrt(sin(2*x)), x)
 (61)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)*sqrt(sin(2*x)), x)
 (62)
Type: Union(Expression(Integer),...)
fricas
)set output tex off

fricas
)set output algebra on
integrate((cos(x)-sin(x))*sqrt(sin(2*x)), x)
(70)
-
log
2                       3
((16 cos(x)  + 4)sin(x) - 16 cos(x)  + 20 cos(x))
*
+--------------+
\|2 cos(x)sin(x)
+
4            2
16 cos(x)sin(x) - 32 cos(x)  + 32 cos(x)  + 1
+
log
2                       3              +--------------+
((- 16 cos(x)  - 4)sin(x) + 16 cos(x)  - 20 cos(x))\|2 cos(x)sin(x)
+
4            2
16 cos(x)sin(x) - 32 cos(x)  + 32 cos(x)  + 1
+
+--------------+
(8 sin(x) + 8 cos(x))\|2 cos(x)sin(x)
/
16
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^7/sin(2*x)^(7/2), x)
(71)
3                          +--------------+
5 cos(x) log((2 sin(x) + 2 cos(x))\|2 cos(x)sin(x)  + 4 cos(x)sin(x) + 1)
+
-
3
5 cos(x)
*
+--------------+
log((- 2 sin(x) - 2 cos(x))\|2 cos(x)sin(x)  + 4 cos(x)sin(x) + 1)
+
+--------------+
3     (sin(x) + cos(x))\|2 cos(x)sin(x)  + 1
5 cos(x) atan(--------------------------------------)
+--------------+
(sin(x) - cos(x))\|2 cos(x)sin(x)
+
+--------------+
3     (sin(x) + cos(x))\|2 cos(x)sin(x)  - 1
- 5 cos(x) atan(--------------------------------------)
+--------------+
(sin(x) - cos(x))\|2 cos(x)sin(x)
+
+--------------+           2
3      (sin(x) - cos(x))\|2 cos(x)sin(x)  + 2 cos(x)  - 1
- 5 cos(x) atan(---------------------------------------------------)
+--------------+
(sin(x) + cos(x))\|2 cos(x)sin(x)  - 2 cos(x)sin(x)
+
+--------------+           2
3      (sin(x) - cos(x))\|2 cos(x)sin(x)  - 2 cos(x)  + 1
5 cos(x) atan(---------------------------------------------------)
+--------------+
(sin(x) + cos(x))\|2 cos(x)sin(x)  + 2 cos(x)sin(x)
+
2      +--------------+
(- 48 cos(x)  + 8)\|2 cos(x)sin(x)
/
3
320 cos(x)
Type: Union(Expression(Integer),...)
fricas
)set output algebra off

fricas
)set output tex on
integrate(cos(x)^7/sin(2*x)^(7/2), x)
 (63)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(2*x)^(3/2)/sin(x)^5, x)
 (64)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(cos(x)^3*sqrt(sin(2*x))), x)
 (65)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(sin(x)*sin(2*x)^(3/2)), x)
 (66)
Type: Union(Expression(Integer),...)
fricas
-- Fails, nonelementary
integrate(cos(x)^2*(cos(2*x)-3*tan(x))/((sin(x)^2-sin(2*x))*sin(2*x)^(5/2)), x)
 (67)
Type: Union(Expression(Integer),...)