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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),...)
fricas
integrate(cos(x)^3, x)
 (2)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^4, x)
 (3)
Type: Union(Expression(Integer),...)
fricas
-- 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),...)
fricas
integrate(sin(-1/12*%pi+3*x)^3, x)
 (7)
Type: Union(Expression(Integer),...)
fricas
-- 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),...)
fricas
integrate(cos(x)^6*sin(x)^7, x)
 (16)
Type: Union(Expression(Integer),...)
fricas
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),...)
fricas
-- Problems 21 - 29 (p. 223)
integrate(sec(x)^2*tan(x)^2, x)
 (21)
Type: Union(Expression(Integer),...)
fricas
integrate(cot(x)^3*csc(x), x)
 (22)
Type: Union(Expression(Integer),...)
fricas
integrate(sec(x)^3*tan(x), x)
 (23)
Type: Union(Expression(Integer),...)
fricas
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),...)
fricas
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)
2x + %pi        2x + %pi
sin(--------) + cos(--------) + 1
2x + %pi 4            4               4
- cos(--------) log(---------------------------------)
4                     2x + %pi
cos(--------) + 1
4
+
2x + %pi        2x + %pi
sin(--------) - cos(--------) - 1
2x + %pi 4            4               4
cos(--------) log(---------------------------------)
4                     2x + %pi
cos(--------) + 1
4
+
2x + %pi 2         2x + %pi
(- cos(--------)  + 2)sin(--------)
4                  4
/
2x + %pi 4
4cos(--------)
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),...)
fricas
integrate((4-3*cos(x))*(1-1/2*sin(x))^4, x)
 (30)
Type: Union(Expression(Integer),...)
fricas
integrate((1/2-3*cot(x))*(3-2*cot(x))^3, x)
 (31)
Type: Union(Expression(Integer),...)
fricas
-- 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),...)
fricas
integrate(cos(x)*cos(4*x), x)
 (34)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(4*x)/cos(x)^5, x)
 (35)
Type: Union(Expression(Integer),...)
fricas
-- Problems 37 - 39 (p. 233)
integrate(cos(x)^4*cos(4*x), x)
 (36)
Type: Union(Expression(Integer),...)
fricas
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),...)
fricas
-- Problems 40 - 49 (p. 254-255)
integrate(cos(x)/(sin(x)*(2+sin(2*x))), x)
 (39)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)^2/(cos(3*x)*sin(x)), x)
 (40)
Type: Union(Expression(Integer),...)
fricas
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)
(43)
-
atan
+-+
(104765547927309152411486\|3  - 181459251892870078413729)sin(x)
+
+-+
(18277149216643579325555\|3  - 31656951060734835109266)cos(x)
+
+-+
18277149216643579325555\|3  - 31656951060734835109266
/
+-+
(63313902121469670218532\|3  - 109662895299861475953330)cos(x)
+
+-+
63313902121469670218532\|3  - 109662895299861475953330
/
+-+
\|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),...)
fricas
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),...)
fricas
integrate((5-tan(x)-6*tan(x)^2)/(1+3*tan(x))^3, x)
 (47)
Type: Union(Expression(Integer),...)
fricas
-- Problems 50 - 56 (p. 260)
integrate(cos(x)^2/cos(3*x), x)
 (48)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)/cos(2*x), x)
 (49)
Type: Union(Expression(Integer),...)
fricas
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),...)
fricas
-- Problems 57 - 61 (p. 266)
integrate(sqrt(1+sin(2*x)), x)
 (55)
Type: Union(Expression(Integer),...)
fricas
integrate(1/sqrt(1+cos(2*x)), x)
 (56)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(1-cos(3*x))^(3/2), x)
 (57)
Type: Union(Expression(Integer),...)
fricas
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),...)
fricas
-- 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) |------  + 2sin(x) + 2cos(x)         +------+
|             \|cos(x)                          +-+ |sin(x)
|------------------------------------------  + \|2  |------  - 1
\|                  cos(x)                          \|cos(x)
+
-
+-+
4\|2
*
1
atan(---------------------------------------------------------------)
+----------------------------------------+
|            +------+
|  +-+       |sin(x)
|2\|2 cos(x) |------  + 2sin(x) + 2cos(x)         +------+
|           \|cos(x)                          +-+ |sin(x)
|----------------------------------------  + \|2  |------  + 1
\|                 cos(x)                         \|cos(x)
+
+------+
+-+       |sin(x)
2\|2 cos(x) |------  + 2sin(x) + 2cos(x)
+-+               \|cos(x)
- \|2 log(----------------------------------------)
cos(x)
+
+------+
+-+       |sin(x)
- 2\|2 cos(x) |------  + 2sin(x) + 2cos(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+-------+
2log(\|tan(5x)   + 1) - log(- \|tan(5x)   + tan(5x)\|tan(5x)  + 1)
+
3+-------+2
+-+     2\|tan(5x)   - 1
2\|3 atan(----------------)
+-+
\|3
/
20
Type: Union(Expression(Integer),...)
fricas
integrate(1/(4+3*tan(2*x))^(3/2), x)
(64)
-
+-------------------+
+-+ |3sin(2x) + 4cos(2x)
36\|5  |-------------------
\|      cos(2x)
*
atan
+-+
125\|5
/
+--+
\|10
*
ROOT
+-------------------+
+-+ +--+        |3sin(2x) + 4cos(2x)
- 9375\|5 \|10 cos(2x) |-------------------
\|      cos(2x)
+
46875sin(2x) + 140625cos(2x)
/
cos(2x)
+
+-------------------+
+--+ |3sin(2x) + 4cos(2x)        +-+
125\|10  |-------------------  - 375\|5
\|      cos(2x)
+
-
+-------------------+
+-+ |3sin(2x) + 4cos(2x)
36\|5  |-------------------
\|      cos(2x)
*
atan
+-+
125\|5
/
+--+
\|10
*
ROOT
+-------------------+
+-+ +--+        |3sin(2x) + 4cos(2x)
9375\|5 \|10 cos(2x) |-------------------
\|      cos(2x)
+
46875sin(2x) + 140625cos(2x)
/
cos(2x)
+
+-------------------+
+--+ |3sin(2x) + 4cos(2x)        +-+
125\|10  |-------------------  + 375\|5
\|      cos(2x)
+
+-------------------+
+-+ |3sin(2x) + 4cos(2x)
13\|5  |-------------------
\|      cos(2x)
*
log
+-------------------+
+-+ +--+        |3sin(2x) + 4cos(2x)
9375\|5 \|10 cos(2x) |-------------------  + 46875sin(2x)
\|      cos(2x)
+
140625cos(2x)
/
cos(2x)
+
-
+-------------------+
+-+ |3sin(2x) + 4cos(2x)
13\|5  |-------------------
\|      cos(2x)
*
log
+-------------------+
+-+ +--+        |3sin(2x) + 4cos(2x)
- 9375\|5 \|10 cos(2x) |-------------------  + 46875sin(2x)
\|      cos(2x)
+
140625cos(2x)
/
cos(2x)
+
+--+
- 60\|10
/
+-------------------+
+--+ |3sin(2x) + 4cos(2x)
500\|10  |-------------------
\|      cos(2x)
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)
+-------------------+
|- 3sin(x) + 4cos(x)
(12sin(x) - 32cos(x)) |-------------------
\|       cos(x)
+
2
(- 3sin(x) + 4cos(x))log(cos(x) )
+
2
- 24cos(x)sin(x) + 7cos(x)  + 9
(3sin(x) - 4cos(x))log(-------------------------------)
4
/
18sin(x) - 24cos(x)
Type: Union(Expression(Integer),...)
fricas
integrate(tan(x)/(-1+sqrt(tan(x)))^2, x)
(66)
+------+
(- \|tan(x)  + 1)
*
+------------------------------------------------------+
|        2                               2
\|- 3%%BS1  + (- 2%%BS0 - 2)%%BS1 - 3%%BS0  - 2%%BS0 - 1
+
+------+                           +------+
(- \|tan(x)  + 1)%%BS1 + (- %%BS0 - 1)\|tan(x)  + %%BS0 + 1
*
log
((32%%BS0 + 4)%%BS1 + 4%%BS0 + 2)
*
+------------------------------------------------------+
|        2                               2
\|- 3%%BS1  + (- 2%%BS0 - 2)%%BS1 - 3%%BS0  - 2%%BS0 - 1
+
2             2
(- 32%%BS0 - 4)%%BS1  + (- 32%%BS0  - 32%%BS0 - 2)%%BS1
+
+------+         2
6\|tan(x)  - 4%%BS0  - 2%%BS0 - 3
+
+------+
(\|tan(x)  - 1)
*
+------------------------------------------------------+
|        2                               2
\|- 3%%BS1  + (- 2%%BS0 - 2)%%BS1 - 3%%BS0  - 2%%BS0 - 1
+
+------+                           +------+
(- \|tan(x)  + 1)%%BS1 + (- %%BS0 - 1)\|tan(x)  + %%BS0 + 1
*
log
((- 32%%BS0 - 4)%%BS1 - 4%%BS0 - 2)
*
+------------------------------------------------------+
|        2                               2
\|- 3%%BS1  + (- 2%%BS0 - 2)%%BS1 - 3%%BS0  - 2%%BS0 - 1
+
2             2
(- 32%%BS0 - 4)%%BS1  + (- 32%%BS0  - 32%%BS0 - 2)%%BS1
+
+------+         2
6\|tan(x)  - 4%%BS0  - 2%%BS0 - 3
+
+------+
(2\|tan(x)  - 2)%%BS1
*
log
2           2                         +------+
(64%%BS0 + 8)%%BS1  + (64%%BS0  + 64%%BS0 + 4)%%BS1 + 6\|tan(x)
+
3          2
64%%BS0  + 64%%BS0  + 32%%BS0 - 1
+
+------+
(2%%BS0\|tan(x)  - 2%%BS0)
*
+------+          3          2
log(6\|tan(x)  - 64%%BS0  - 56%%BS0  - 28%%BS0 + 7)
+
+------+          +------+
(2\|tan(x)  - 2)log(\|tan(x)  - 1) - 2
/
+------+
2\|tan(x)  - 2
Type: Union(Expression(Integer),...)
fricas
)set output tex on

fricas
)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             +-------------+
((16cos(x)  + 4)sin(x) - 16cos(x)  + 20cos(x))\|2cos(x)sin(x)
+
4           2
16cos(x)sin(x) - 32cos(x)  + 32cos(x)  + 1
+
log
2                      3             +-------------+
((- 16cos(x)  - 4)sin(x) + 16cos(x)  - 20cos(x))\|2cos(x)sin(x)
+
4           2
16cos(x)sin(x) - 32cos(x)  + 32cos(x)  + 1
+
+-------------+
(8sin(x) + 8cos(x))\|2cos(x)sin(x)
/
16
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^7/sin(2*x)^(7/2), x)
(71)
3                        +-------------+
5cos(x) log((2sin(x) + 2cos(x))\|2cos(x)sin(x)  + 4cos(x)sin(x) + 1)
+
3                          +-------------+
- 5cos(x) log((- 2sin(x) - 2cos(x))\|2cos(x)sin(x)  + 4cos(x)sin(x) + 1)
+
+-------------+
3     2\|2cos(x)sin(x)  + sin(x) + cos(x)
- 10cos(x) atan(-----------------------------------)
sin(x) - cos(x)
+
+-------------+
3     2\|2cos(x)sin(x)  - sin(x) - cos(x)
- 10cos(x) atan(-----------------------------------)
sin(x) - cos(x)
+
2      +-------------+
(- 48cos(x)  + 8)\|2cos(x)sin(x)
/
3
320cos(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
integrate(cos(x)^2*(cos(2*x)-3*tan(x))/((sin(x)^2-sin(2*x))*sin(2*x)^(5/2)), x)
>> Error detected within library code:
Imaginary part is nonzero. Cannot retract.
Continuing to read the file...