Integrand size = 29, antiderivative size = 476 \[ \int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {2 b^3 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^9 d}+\frac {\left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \text {arctanh}(\cos (c+d x))}{128 a^9 d}-\frac {b \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^8 d}+\frac {\left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac {b \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}-\frac {\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}+\frac {\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^4 b d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d} \] Output:
2*b^3*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^9 /d+1/128*(5*a^8+40*a^6*b^2-240*a^4*b^4+320*a^2*b^6-128*b^8)*arctanh(cos(d* x+c))/a^9/d-1/105*b*(15*a^6-161*a^4*b^2+245*a^2*b^4-105*b^6)*cot(d*x+c)/a^ 8/d+1/128*(5*a^6-88*a^4*b^2+144*a^2*b^4-64*b^6)*cot(d*x+c)*csc(d*x+c)/a^7/ d+1/105*b*(45*a^4-77*a^2*b^2+35*b^4)*cot(d*x+c)*csc(d*x+c)^2/a^6/d-1/192*( 59*a^4-104*a^2*b^2+48*b^4)*cot(d*x+c)*csc(d*x+c)^3/a^5/d-1/4*cot(d*x+c)*cs c(d*x+c)^4/b/d+1/140*(35*a^4-60*a^2*b^2+28*b^4)*cot(d*x+c)*csc(d*x+c)^4/a^ 4/b/d+1/5*a*cot(d*x+c)*csc(d*x+c)^5/b^2/d-1/240*(48*a^4-85*a^2*b^2+40*b^4) *cot(d*x+c)*csc(d*x+c)^5/a^3/b^2/d+1/7*b*cot(d*x+c)*csc(d*x+c)^6/a^2/d-1/8 *cot(d*x+c)*csc(d*x+c)^7/a/d
Time = 5.29 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.25 \[ \int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {1720320 b^3 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+6720 \left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-6720 \left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+a \csc ^8(c+d x) \left (-35 a \left (1765 a^6+680 a^4 b^2-1392 a^2 b^4+960 b^6\right ) \cos (c+d x)-35 \left (895 a^7-904 a^5 b^2+2736 a^3 b^4-1728 a b^6\right ) \cos (3 (c+d x))-13895 a^7 \cos (5 (c+d x))-17080 a^5 b^2 \cos (5 (c+d x))+62160 a^3 b^4 \cos (5 (c+d x))-33600 a b^6 \cos (5 (c+d x))-525 a^7 \cos (7 (c+d x))+9240 a^5 b^2 \cos (7 (c+d x))-15120 a^3 b^4 \cos (7 (c+d x))+6720 a b^6 \cos (7 (c+d x))+13440 a^6 b \sin (2 (c+d x))+88704 a^4 b^3 \sin (2 (c+d x))-174720 a^2 b^5 \sin (2 (c+d x))+94080 b^7 \sin (2 (c+d x))+13440 a^6 b \sin (4 (c+d x))-86912 a^4 b^3 \sin (4 (c+d x))+183680 a^2 b^5 \sin (4 (c+d x))-94080 b^7 \sin (4 (c+d x))+5760 a^6 b \sin (6 (c+d x))+42112 a^4 b^3 \sin (6 (c+d x))-85120 a^2 b^5 \sin (6 (c+d x))+40320 b^7 \sin (6 (c+d x))+960 a^6 b \sin (8 (c+d x))-10304 a^4 b^3 \sin (8 (c+d x))+15680 a^2 b^5 \sin (8 (c+d x))-6720 b^7 \sin (8 (c+d x))\right )}{860160 a^9 d} \] Input:
Integrate[(Cot[c + d*x]^6*Csc[c + d*x]^3)/(a + b*Sin[c + d*x]),x]
Output:
(1720320*b^3*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 6720*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*Lo g[Cos[(c + d*x)/2]] - 6720*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*Log[Sin[(c + d*x)/2]] + a*Csc[c + d*x]^8*(-35*a*(1765*a^6 + 68 0*a^4*b^2 - 1392*a^2*b^4 + 960*b^6)*Cos[c + d*x] - 35*(895*a^7 - 904*a^5*b ^2 + 2736*a^3*b^4 - 1728*a*b^6)*Cos[3*(c + d*x)] - 13895*a^7*Cos[5*(c + d* x)] - 17080*a^5*b^2*Cos[5*(c + d*x)] + 62160*a^3*b^4*Cos[5*(c + d*x)] - 33 600*a*b^6*Cos[5*(c + d*x)] - 525*a^7*Cos[7*(c + d*x)] + 9240*a^5*b^2*Cos[7 *(c + d*x)] - 15120*a^3*b^4*Cos[7*(c + d*x)] + 6720*a*b^6*Cos[7*(c + d*x)] + 13440*a^6*b*Sin[2*(c + d*x)] + 88704*a^4*b^3*Sin[2*(c + d*x)] - 174720* a^2*b^5*Sin[2*(c + d*x)] + 94080*b^7*Sin[2*(c + d*x)] + 13440*a^6*b*Sin[4* (c + d*x)] - 86912*a^4*b^3*Sin[4*(c + d*x)] + 183680*a^2*b^5*Sin[4*(c + d* x)] - 94080*b^7*Sin[4*(c + d*x)] + 5760*a^6*b*Sin[6*(c + d*x)] + 42112*a^4 *b^3*Sin[6*(c + d*x)] - 85120*a^2*b^5*Sin[6*(c + d*x)] + 40320*b^7*Sin[6*( c + d*x)] + 960*a^6*b*Sin[8*(c + d*x)] - 10304*a^4*b^3*Sin[8*(c + d*x)] + 15680*a^2*b^5*Sin[8*(c + d*x)] - 6720*b^7*Sin[8*(c + d*x)]))/(860160*a^9*d )
Time = 4.28 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.14, number of steps used = 28, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.931, Rules used = {3042, 3375, 27, 3042, 3534, 27, 3042, 3534, 25, 3042, 3534, 27, 3042, 3534, 3042, 3534, 25, 3042, 3534, 27, 3042, 3480, 3042, 3139, 1083, 217, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6}{\sin (c+d x)^9 (a+b \sin (c+d x))}dx\) |
| \(\Big \downarrow \) 3375 |
| \(\displaystyle \frac {\int \frac {4 \csc ^7(c+d x) \left (-10 \left (28 a^4-49 b^2 a^2+24 b^4\right ) \sin ^2(c+d x)-a b \left (14 a^2-5 b^2\right ) \sin (c+d x)+7 \left (48 a^4-85 b^2 a^2+40 b^4\right )\right )}{a+b \sin (c+d x)}dx}{1120 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\csc ^7(c+d x) \left (-10 \left (28 a^4-49 b^2 a^2+24 b^4\right ) \sin ^2(c+d x)-a b \left (14 a^2-5 b^2\right ) \sin (c+d x)+7 \left (48 a^4-85 b^2 a^2+40 b^4\right )\right )}{a+b \sin (c+d x)}dx}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-10 \left (28 a^4-49 b^2 a^2+24 b^4\right ) \sin (c+d x)^2-a b \left (14 a^2-5 b^2\right ) \sin (c+d x)+7 \left (48 a^4-85 b^2 a^2+40 b^4\right )}{\sin (c+d x)^7 (a+b \sin (c+d x))}dx}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int -\frac {5 \csc ^6(c+d x) \left (a \left (7 a^2+8 b^2\right ) \sin (c+d x) b^2-7 \left (48 a^4-85 b^2 a^2+40 b^4\right ) \sin ^2(c+d x) b+12 \left (35 a^4-60 b^2 a^2+28 b^4\right ) b\right )}{a+b \sin (c+d x)}dx}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {5 \int \frac {\csc ^6(c+d x) \left (a \left (7 a^2+8 b^2\right ) \sin (c+d x) b^2-7 \left (48 a^4-85 b^2 a^2+40 b^4\right ) \sin ^2(c+d x) b+12 \left (35 a^4-60 b^2 a^2+28 b^4\right ) b\right )}{a+b \sin (c+d x)}dx}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {5 \int \frac {a \left (7 a^2+8 b^2\right ) \sin (c+d x) b^2-7 \left (48 a^4-85 b^2 a^2+40 b^4\right ) \sin (c+d x)^2 b+12 \left (35 a^4-60 b^2 a^2+28 b^4\right ) b}{\sin (c+d x)^6 (a+b \sin (c+d x))}dx}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {5 \left (\frac {\int -\frac {\csc ^5(c+d x) \left (-a \left (95 a^2-56 b^2\right ) \sin (c+d x) b^3-48 \left (35 a^4-60 b^2 a^2+28 b^4\right ) \sin ^2(c+d x) b^2+35 \left (59 a^4-104 b^2 a^2+48 b^4\right ) b^2\right )}{a+b \sin (c+d x)}dx}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {5 \left (-\frac {\int \frac {\csc ^5(c+d x) \left (-a \left (95 a^2-56 b^2\right ) \sin (c+d x) b^3-48 \left (35 a^4-60 b^2 a^2+28 b^4\right ) \sin ^2(c+d x) b^2+35 \left (59 a^4-104 b^2 a^2+48 b^4\right ) b^2\right )}{a+b \sin (c+d x)}dx}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {5 \left (-\frac {\int \frac {-a \left (95 a^2-56 b^2\right ) \sin (c+d x) b^3-48 \left (35 a^4-60 b^2 a^2+28 b^4\right ) \sin (c+d x)^2 b^2+35 \left (59 a^4-104 b^2 a^2+48 b^4\right ) b^2}{\sin (c+d x)^5 (a+b \sin (c+d x))}dx}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {5 \left (-\frac {\frac {\int -\frac {3 \csc ^4(c+d x) \left (-35 \left (59 a^4-104 b^2 a^2+48 b^4\right ) \sin ^2(c+d x) b^3+64 \left (45 a^4-77 b^2 a^2+35 b^4\right ) b^3+a \left (175 a^4-200 b^2 a^2+112 b^4\right ) \sin (c+d x) b^2\right )}{a+b \sin (c+d x)}dx}{4 a}-\frac {35 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {5 \left (-\frac {-\frac {3 \int \frac {\csc ^4(c+d x) \left (-35 \left (59 a^4-104 b^2 a^2+48 b^4\right ) \sin ^2(c+d x) b^3+64 \left (45 a^4-77 b^2 a^2+35 b^4\right ) b^3+a \left (175 a^4-200 b^2 a^2+112 b^4\right ) \sin (c+d x) b^2\right )}{a+b \sin (c+d x)}dx}{4 a}-\frac {35 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {5 \left (-\frac {-\frac {3 \int \frac {-35 \left (59 a^4-104 b^2 a^2+48 b^4\right ) \sin (c+d x)^2 b^3+64 \left (45 a^4-77 b^2 a^2+35 b^4\right ) b^3+a \left (175 a^4-200 b^2 a^2+112 b^4\right ) \sin (c+d x) b^2}{\sin (c+d x)^4 (a+b \sin (c+d x))}dx}{4 a}-\frac {35 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {5 \left (-\frac {-\frac {3 \left (\frac {\int \frac {\csc ^3(c+d x) \left (128 \left (45 a^4-77 b^2 a^2+35 b^4\right ) \sin ^2(c+d x) b^4-a \left (435 a^4-1064 b^2 a^2+560 b^4\right ) \sin (c+d x) b^3+105 \left (5 a^6-88 b^2 a^4+144 b^4 a^2-64 b^6\right ) b^2\right )}{a+b \sin (c+d x)}dx}{3 a}-\frac {64 b^3 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {5 \left (-\frac {-\frac {3 \left (\frac {\int \frac {128 \left (45 a^4-77 b^2 a^2+35 b^4\right ) \sin (c+d x)^2 b^4-a \left (435 a^4-1064 b^2 a^2+560 b^4\right ) \sin (c+d x) b^3+105 \left (5 a^6-88 b^2 a^4+144 b^4 a^2-64 b^6\right ) b^2}{\sin (c+d x)^3 (a+b \sin (c+d x))}dx}{3 a}-\frac {64 b^3 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {5 \left (-\frac {-\frac {3 \left (\frac {\frac {\int -\frac {\csc ^2(c+d x) \left (-105 \left (5 a^6-88 b^2 a^4+144 b^4 a^2-64 b^6\right ) \sin ^2(c+d x) b^3+128 \left (15 a^6-161 b^2 a^4+245 b^4 a^2-105 b^6\right ) b^3-a \left (525 a^6+2280 b^2 a^4-4592 b^4 a^2+2240 b^6\right ) \sin (c+d x) b^2\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {105 b^2 \left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {64 b^3 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {5 \left (-\frac {-\frac {3 \left (\frac {-\frac {\int \frac {\csc ^2(c+d x) \left (-105 \left (5 a^6-88 b^2 a^4+144 b^4 a^2-64 b^6\right ) \sin ^2(c+d x) b^3+128 \left (15 a^6-161 b^2 a^4+245 b^4 a^2-105 b^6\right ) b^3-a \left (525 a^6+2280 b^2 a^4-4592 b^4 a^2+2240 b^6\right ) \sin (c+d x) b^2\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {105 b^2 \left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {64 b^3 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {5 \left (-\frac {-\frac {3 \left (\frac {-\frac {\int \frac {-105 \left (5 a^6-88 b^2 a^4+144 b^4 a^2-64 b^6\right ) \sin (c+d x)^2 b^3+128 \left (15 a^6-161 b^2 a^4+245 b^4 a^2-105 b^6\right ) b^3-a \left (525 a^6+2280 b^2 a^4-4592 b^4 a^2+2240 b^6\right ) \sin (c+d x) b^2}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{2 a}-\frac {105 b^2 \left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {64 b^3 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {5 \left (-\frac {-\frac {3 \left (\frac {-\frac {\frac {\int -\frac {105 \csc (c+d x) \left (a \left (5 a^6-88 b^2 a^4+144 b^4 a^2-64 b^6\right ) \sin (c+d x) b^3+\left (5 a^8+40 b^2 a^6-240 b^4 a^4+320 b^6 a^2-128 b^8\right ) b^2\right )}{a+b \sin (c+d x)}dx}{a}-\frac {128 b^3 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {105 b^2 \left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {64 b^3 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {5 \left (-\frac {-\frac {3 \left (\frac {-\frac {-\frac {105 \int \frac {\csc (c+d x) \left (a \left (5 a^6-88 b^2 a^4+144 b^4 a^2-64 b^6\right ) \sin (c+d x) b^3+\left (5 a^8+40 b^2 a^6-240 b^4 a^4+320 b^6 a^2-128 b^8\right ) b^2\right )}{a+b \sin (c+d x)}dx}{a}-\frac {128 b^3 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {105 b^2 \left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {64 b^3 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {5 \left (-\frac {-\frac {3 \left (\frac {-\frac {-\frac {105 \int \frac {a \left (5 a^6-88 b^2 a^4+144 b^4 a^2-64 b^6\right ) \sin (c+d x) b^3+\left (5 a^8+40 b^2 a^6-240 b^4 a^4+320 b^6 a^2-128 b^8\right ) b^2}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {128 b^3 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {105 b^2 \left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {64 b^3 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {-\frac {5 \left (-\frac {-\frac {3 \left (\frac {-\frac {-\frac {105 \left (\frac {b^2 \left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \int \csc (c+d x)dx}{a}-\frac {128 b^5 \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {128 b^3 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {105 b^2 \left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {64 b^3 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {5 \left (-\frac {-\frac {3 \left (\frac {-\frac {-\frac {105 \left (\frac {b^2 \left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \int \csc (c+d x)dx}{a}-\frac {128 b^5 \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {128 b^3 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {105 b^2 \left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {64 b^3 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {-\frac {5 \left (-\frac {-\frac {3 \left (\frac {-\frac {-\frac {105 \left (\frac {b^2 \left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \int \csc (c+d x)dx}{a}-\frac {256 b^5 \left (a^2-b^2\right )^3 \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}-\frac {128 b^3 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {105 b^2 \left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {64 b^3 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {5 \left (-\frac {-\frac {3 \left (\frac {-\frac {-\frac {105 \left (\frac {512 b^5 \left (a^2-b^2\right )^3 \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}+\frac {b^2 \left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \int \csc (c+d x)dx}{a}\right )}{a}-\frac {128 b^3 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {105 b^2 \left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {64 b^3 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {5 \left (-\frac {-\frac {3 \left (\frac {-\frac {-\frac {105 \left (\frac {b^2 \left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \int \csc (c+d x)dx}{a}-\frac {256 b^5 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}\right )}{a}-\frac {128 b^3 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {105 b^2 \left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {64 b^3 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}\right )}{6 a}-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}}{280 a^2 b^2}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {-\frac {7 \left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {5 \left (-\frac {12 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {-\frac {35 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {3 \left (\frac {-\frac {105 b^2 \left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {-\frac {128 b^3 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}-\frac {105 \left (-\frac {256 b^5 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}-\frac {b^2 \left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \text {arctanh}(\cos (c+d x))}{a d}\right )}{a}}{2 a}}{3 a}-\frac {64 b^3 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}}{5 a}\right )}{6 a}}{280 a^2 b^2}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}\) |
Input:
Int[(Cot[c + d*x]^6*Csc[c + d*x]^3)/(a + b*Sin[c + d*x]),x]
Output:
-1/4*(Cot[c + d*x]*Csc[c + d*x]^4)/(b*d) + (a*Cot[c + d*x]*Csc[c + d*x]^5) /(5*b^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^6)/(7*a^2*d) - (Cot[c + d*x]*Csc [c + d*x]^7)/(8*a*d) + ((-7*(48*a^4 - 85*a^2*b^2 + 40*b^4)*Cot[c + d*x]*Cs c[c + d*x]^5)/(6*a*d) - (5*((-12*b*(35*a^4 - 60*a^2*b^2 + 28*b^4)*Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d) - ((-35*b^2*(59*a^4 - 104*a^2*b^2 + 48*b^4)*C ot[c + d*x]*Csc[c + d*x]^3)/(4*a*d) - (3*((-64*b^3*(45*a^4 - 77*a^2*b^2 + 35*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d) + (-1/2*((-105*((-256*b^5*(a^ 2 - b^2)^(5/2)*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/( a*d) - (b^2*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*Arc Tanh[Cos[c + d*x]])/(a*d)))/a - (128*b^3*(15*a^6 - 161*a^4*b^2 + 245*a^2*b ^4 - 105*b^6)*Cot[c + d*x])/(a*d))/a - (105*b^2*(5*a^6 - 88*a^4*b^2 + 144* a^2*b^4 - 64*b^6)*Cot[c + d*x]*Csc[c + d*x])/(2*a*d))/(3*a)))/(4*a))/(5*a) ))/(6*a))/(280*a^2*b^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[cos[(e_.) + (f_.)*(x_)]^6*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[Cos[e + f*x]*(d*Sin[ e + f*x])^(n + 1)*((a + b*Sin[e + f*x])^(m + 1)/(a*d*f*(n + 1))), x] + (-Si mp[b*(m + n + 2)*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*((a + b*Sin[e + f*x] )^(m + 1)/(a^2*d^2*f*(n + 1)*(n + 2))), x] - Simp[a*(n + 5)*Cos[e + f*x]*(d *Sin[e + f*x])^(n + 3)*((a + b*Sin[e + f*x])^(m + 1)/(b^2*d^3*f*(m + n + 5) *(m + n + 6))), x] + Simp[Cos[e + f*x]*(d*Sin[e + f*x])^(n + 4)*((a + b*Sin [e + f*x])^(m + 1)/(b*d^4*f*(m + n + 6))), x] + Simp[1/(a^2*b^2*d^2*(n + 1) *(n + 2)*(m + n + 5)*(m + n + 6)) Int[(d*Sin[e + f*x])^(n + 2)*(a + b*Sin [e + f*x])^m*Simp[a^4*(n + 1)*(n + 2)*(n + 3)*(n + 5) - a^2*b^2*(n + 2)*(2* n + 1)*(m + n + 5)*(m + n + 6) + b^4*(m + n + 2)*(m + n + 3)*(m + n + 5)*(m + n + 6) + a*b*m*(a^2*(n + 1)*(n + 2) - b^2*(m + n + 5)*(m + n + 6))*Sin[e + f*x] - (a^4*(n + 1)*(n + 2)*(4 + n)*(n + 5) + b^4*(m + n + 2)*(m + n + 4 )*(m + n + 5)*(m + n + 6) - a^2*b^2*(n + 1)*(n + 2)*(m + n + 5)*(2*n + 2*m + 13))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, m, n}, x] && Ne Q[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && NeQ[n, -1] && NeQ[n, -2] && NeQ[m + n + 5, 0] && NeQ[m + n + 6, 0] && !IGtQ[m, 0]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b - a*B)/(b*c - a*d) Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ (b*c - a*d) Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 1.79 (sec) , antiderivative size = 728, normalized size of antiderivative = 1.53
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a^{7}}{8}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} a^{6} b}{7}-\frac {2 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a^{5} b^{2}}{3}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{6} b -\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{4} b^{3}}{5}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{7}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{5} b^{2}+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3} b^{4}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{6} b +\frac {56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{4} b^{3}}{3}-\frac {32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2} b^{5}}{3}+2 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+30 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{5} b^{2}-64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{3} b^{4}+32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a \,b^{6}+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{6} b -176 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} b^{3}+288 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{5}-128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{7}}{256 a^{8}}-\frac {1}{2048 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}-\frac {-4 a^{2}+4 b^{2}}{1536 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {4 a^{4}-24 a^{2} b^{2}+16 b^{4}}{1024 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {4 a^{6}+60 a^{4} b^{2}-128 a^{2} b^{4}+64 b^{6}}{512 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-10 a^{8}-80 a^{6} b^{2}+480 a^{4} b^{4}-640 a^{2} b^{6}+256 b^{8}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256 a^{9}}+\frac {b}{896 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {b \left (5 a^{2}-4 b^{2}\right )}{640 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {b \left (9 a^{4}-28 a^{2} b^{2}+16 b^{4}\right )}{384 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (5 a^{6}-88 a^{4} b^{2}+144 a^{2} b^{4}-64 b^{6}\right )}{128 a^{8} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b^{3} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{9} \sqrt {a^{2}-b^{2}}}}{d}\) | \(728\) |
| default | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a^{7}}{8}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} a^{6} b}{7}-\frac {2 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a^{5} b^{2}}{3}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{6} b -\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{4} b^{3}}{5}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{7}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{5} b^{2}+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3} b^{4}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{6} b +\frac {56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{4} b^{3}}{3}-\frac {32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2} b^{5}}{3}+2 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+30 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{5} b^{2}-64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{3} b^{4}+32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a \,b^{6}+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{6} b -176 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} b^{3}+288 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{5}-128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{7}}{256 a^{8}}-\frac {1}{2048 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}-\frac {-4 a^{2}+4 b^{2}}{1536 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {4 a^{4}-24 a^{2} b^{2}+16 b^{4}}{1024 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {4 a^{6}+60 a^{4} b^{2}-128 a^{2} b^{4}+64 b^{6}}{512 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-10 a^{8}-80 a^{6} b^{2}+480 a^{4} b^{4}-640 a^{2} b^{6}+256 b^{8}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256 a^{9}}+\frac {b}{896 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {b \left (5 a^{2}-4 b^{2}\right )}{640 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {b \left (9 a^{4}-28 a^{2} b^{2}+16 b^{4}\right )}{384 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (5 a^{6}-88 a^{4} b^{2}+144 a^{2} b^{4}-64 b^{6}\right )}{128 a^{8} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b^{3} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{9} \sqrt {a^{2}-b^{2}}}}{d}\) | \(728\) |
| risch | \(\text {Expression too large to display}\) | \(1576\) |
Input:
int(cot(d*x+c)^6*csc(d*x+c)^3/(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(1/256/a^8*(1/8*tan(1/2*d*x+1/2*c)^8*a^7-2/7*tan(1/2*d*x+1/2*c)^7*a^6* b-2/3*a^7*tan(1/2*d*x+1/2*c)^6+2/3*tan(1/2*d*x+1/2*c)^6*a^5*b^2+2*tan(1/2* d*x+1/2*c)^5*a^6*b-8/5*tan(1/2*d*x+1/2*c)^5*a^4*b^3+tan(1/2*d*x+1/2*c)^4*a ^7-6*tan(1/2*d*x+1/2*c)^4*a^5*b^2+4*tan(1/2*d*x+1/2*c)^4*a^3*b^4-6*tan(1/2 *d*x+1/2*c)^3*a^6*b+56/3*tan(1/2*d*x+1/2*c)^3*a^4*b^3-32/3*tan(1/2*d*x+1/2 *c)^3*a^2*b^5+2*a^7*tan(1/2*d*x+1/2*c)^2+30*tan(1/2*d*x+1/2*c)^2*a^5*b^2-6 4*tan(1/2*d*x+1/2*c)^2*a^3*b^4+32*tan(1/2*d*x+1/2*c)^2*a*b^6+10*tan(1/2*d* x+1/2*c)*a^6*b-176*tan(1/2*d*x+1/2*c)*a^4*b^3+288*tan(1/2*d*x+1/2*c)*a^2*b ^5-128*tan(1/2*d*x+1/2*c)*b^7)-1/2048/a/tan(1/2*d*x+1/2*c)^8-1/1536*(-4*a^ 2+4*b^2)/a^3/tan(1/2*d*x+1/2*c)^6-1/1024*(4*a^4-24*a^2*b^2+16*b^4)/a^5/tan (1/2*d*x+1/2*c)^4-1/512*(4*a^6+60*a^4*b^2-128*a^2*b^4+64*b^6)/a^7/tan(1/2* d*x+1/2*c)^2+1/256/a^9*(-10*a^8-80*a^6*b^2+480*a^4*b^4-640*a^2*b^6+256*b^8 )*ln(tan(1/2*d*x+1/2*c))+1/896/a^2*b/tan(1/2*d*x+1/2*c)^7-1/640/a^4*b*(5*a ^2-4*b^2)/tan(1/2*d*x+1/2*c)^5+1/384/a^6*b*(9*a^4-28*a^2*b^2+16*b^4)/tan(1 /2*d*x+1/2*c)^3-1/128*b*(5*a^6-88*a^4*b^2+144*a^2*b^4-64*b^6)/a^8/tan(1/2* d*x+1/2*c)+2*b^3*(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/a^9/(a^2-b^2)^(1/2)*arctan( 1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 999 vs. \(2 (449) = 898\).
Time = 1.56 (sec) , antiderivative size = 2082, normalized size of antiderivative = 4.37 \[ \int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="fricas" )
Output:
[-1/26880*(210*(5*a^8 - 88*a^6*b^2 + 144*a^4*b^4 - 64*a^2*b^6)*cos(d*x + c )^7 + 70*(73*a^8 + 584*a^6*b^2 - 1200*a^4*b^4 + 576*a^2*b^6)*cos(d*x + c)^ 5 - 70*(55*a^8 + 440*a^6*b^2 - 1104*a^4*b^4 + 576*a^2*b^6)*cos(d*x + c)^3 - 13440*((a^4*b^3 - 2*a^2*b^5 + b^7)*cos(d*x + c)^8 + a^4*b^3 - 2*a^2*b^5 + b^7 - 4*(a^4*b^3 - 2*a^2*b^5 + b^7)*cos(d*x + c)^6 + 6*(a^4*b^3 - 2*a^2* b^5 + b^7)*cos(d*x + c)^4 - 4*(a^4*b^3 - 2*a^2*b^5 + b^7)*cos(d*x + c)^2)* sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) + 210*(5*a^8 + 40*a^6*b^2 - 112*a^4*b^4 + 64*a^2*b^6)*cos(d*x + c) - 105*((5*a^8 + 40*a ^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*cos(d*x + c)^8 + 5*a^8 + 40* a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8 - 4*(5*a^8 + 40*a^6*b^2 - 24 0*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*cos(d*x + c)^6 + 6*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*cos(d*x + c)^4 - 4*(5*a^8 + 40*a^6* b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) + 105*((5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128* b^8)*cos(d*x + c)^8 + 5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128 *b^8 - 4*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*cos(d* x + c)^6 + 6*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*co s(d*x + c)^4 - 4*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*...
Timed out. \[ \int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**6*csc(d*x+c)**3/(a+b*sin(d*x+c)),x)
Output:
Timed out
Exception generated. \[ \int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="maxima" )
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 948 vs. \(2 (449) = 898\).
Time = 0.28 (sec) , antiderivative size = 948, normalized size of antiderivative = 1.99 \[ \int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="giac")
Output:
1/215040*((105*a^7*tan(1/2*d*x + 1/2*c)^8 - 240*a^6*b*tan(1/2*d*x + 1/2*c) ^7 - 560*a^7*tan(1/2*d*x + 1/2*c)^6 + 560*a^5*b^2*tan(1/2*d*x + 1/2*c)^6 + 1680*a^6*b*tan(1/2*d*x + 1/2*c)^5 - 1344*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 + 840*a^7*tan(1/2*d*x + 1/2*c)^4 - 5040*a^5*b^2*tan(1/2*d*x + 1/2*c)^4 + 33 60*a^3*b^4*tan(1/2*d*x + 1/2*c)^4 - 5040*a^6*b*tan(1/2*d*x + 1/2*c)^3 + 15 680*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 - 8960*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 + 1680*a^7*tan(1/2*d*x + 1/2*c)^2 + 25200*a^5*b^2*tan(1/2*d*x + 1/2*c)^2 - 53760*a^3*b^4*tan(1/2*d*x + 1/2*c)^2 + 26880*a*b^6*tan(1/2*d*x + 1/2*c)^2 + 8400*a^6*b*tan(1/2*d*x + 1/2*c) - 147840*a^4*b^3*tan(1/2*d*x + 1/2*c) + 241920*a^2*b^5*tan(1/2*d*x + 1/2*c) - 107520*b^7*tan(1/2*d*x + 1/2*c))/a^8 - 1680*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*log(abs (tan(1/2*d*x + 1/2*c)))/a^9 + 430080*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^ 9)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2* c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*a^9) + (22830*a^8*tan(1/2*d*x + 1/2*c)^8 + 182640*a^6*b^2*tan(1/2*d*x + 1/2*c)^8 - 1095840*a^4*b^4*tan(1/ 2*d*x + 1/2*c)^8 + 1461120*a^2*b^6*tan(1/2*d*x + 1/2*c)^8 - 584448*b^8*tan (1/2*d*x + 1/2*c)^8 - 8400*a^7*b*tan(1/2*d*x + 1/2*c)^7 + 147840*a^5*b^3*t an(1/2*d*x + 1/2*c)^7 - 241920*a^3*b^5*tan(1/2*d*x + 1/2*c)^7 + 107520*a*b ^7*tan(1/2*d*x + 1/2*c)^7 - 1680*a^8*tan(1/2*d*x + 1/2*c)^6 - 25200*a^6*b^ 2*tan(1/2*d*x + 1/2*c)^6 + 53760*a^4*b^4*tan(1/2*d*x + 1/2*c)^6 - 26880...
Time = 20.24 (sec) , antiderivative size = 1861, normalized size of antiderivative = 3.91 \[ \int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:
int(cot(c + d*x)^6/(sin(c + d*x)^3*(a + b*sin(c + d*x))),x)
Output:
tan(c/2 + (d*x)/2)^8/(2048*a*d) + (tan(c/2 + (d*x)/2)^5*(b/(640*a^2) + (2* b*(1/(64*a) - b^2/(64*a^3)))/(5*a)))/d - (tan(c/2 + (d*x)/2)^3*(b/(384*a^2 ) - (2*b*(b^2/(64*a^3) - 1/(64*a) + (2*b*(b/(128*a^2) + (2*b*(1/(64*a) - b ^2/(64*a^3)))/a))/a))/(3*a) + (2*b*(1/(64*a) - b^2/(64*a^3)))/(3*a)))/d + (tan(c/2 + (d*x)/2)^2*(1/(128*a) + b^2/(128*a^3) + (b*(b/(128*a^2) + (2*b* (1/(64*a) - b^2/(64*a^3)))/a))/a + (b*(b/(128*a^2) - (2*b*(b^2/(64*a^3) - 1/(64*a) + (2*b*(b/(128*a^2) + (2*b*(1/(64*a) - b^2/(64*a^3)))/a))/a))/a + (2*b*(1/(64*a) - b^2/(64*a^3)))/a))/a))/d + (tan(c/2 + (d*x)/2)*(b/(128*a ^2) - (2*b*(b^2/(64*a^3) - 1/(64*a) + (2*b*(b/(128*a^2) + (2*b*(1/(64*a) - b^2/(64*a^3)))/a))/a))/a - (2*b*(1/(64*a) + b^2/(64*a^3) + (2*b*(b/(128*a ^2) + (2*b*(1/(64*a) - b^2/(64*a^3)))/a))/a + (2*b*(b/(128*a^2) - (2*b*(b^ 2/(64*a^3) - 1/(64*a) + (2*b*(b/(128*a^2) + (2*b*(1/(64*a) - b^2/(64*a^3)) )/a))/a))/a + (2*b*(1/(64*a) - b^2/(64*a^3)))/a))/a))/a + (2*b*(1/(64*a) - b^2/(64*a^3)))/a))/d - (tan(c/2 + (d*x)/2)^6*(1/(384*a) - b^2/(384*a^3))) /d - (tan(c/2 + (d*x)/2)^4*(b^2/(256*a^3) - 1/(256*a) + (b*(b/(128*a^2) + (2*b*(1/(64*a) - b^2/(64*a^3)))/a))/(2*a)))/d - (log(tan(c/2 + (d*x)/2))*( 5*a^8 - 128*b^8 + 320*a^2*b^6 - 240*a^4*b^4 + 40*a^6*b^2))/(128*a^9*d) - ( cot(c/2 + (d*x)/2)^8*(tan(c/2 + (d*x)/2)^3*(2*a^6*b - (8*a^4*b^3)/5) - tan (c/2 + (d*x)/2)^5*(6*a^6*b + (32*a^2*b^5)/3 - (56*a^4*b^3)/3) + tan(c/2 + (d*x)/2)^6*(32*a*b^6 + 2*a^7 - 64*a^3*b^4 + 30*a^5*b^2) + tan(c/2 + (d*...
Time = 0.26 (sec) , antiderivative size = 705, normalized size of antiderivative = 1.48 \[ \int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx =\text {Too large to display} \] Input:
int(cot(d*x+c)^6*csc(d*x+c)^3/(a+b*sin(d*x+c)),x)
Output:
(26880*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))* sin(c + d*x)**8*a**4*b**3 - 53760*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2) *a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**8*a**2*b**5 + 26880*sqrt(a**2 - b **2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**8*b**7 - 1920*cos(c + d*x)*sin(c + d*x)**7*a**7*b + 20608*cos(c + d*x)*sin(c + d *x)**7*a**5*b**3 - 31360*cos(c + d*x)*sin(c + d*x)**7*a**3*b**5 + 13440*co s(c + d*x)*sin(c + d*x)**7*a*b**7 + 525*cos(c + d*x)*sin(c + d*x)**6*a**8 - 9240*cos(c + d*x)*sin(c + d*x)**6*a**6*b**2 + 15120*cos(c + d*x)*sin(c + d*x)**6*a**4*b**4 - 6720*cos(c + d*x)*sin(c + d*x)**6*a**2*b**6 + 5760*co s(c + d*x)*sin(c + d*x)**5*a**7*b - 9856*cos(c + d*x)*sin(c + d*x)**5*a**5 *b**3 + 4480*cos(c + d*x)*sin(c + d*x)**5*a**3*b**5 - 4130*cos(c + d*x)*si n(c + d*x)**4*a**8 + 7280*cos(c + d*x)*sin(c + d*x)**4*a**6*b**2 - 3360*co s(c + d*x)*sin(c + d*x)**4*a**4*b**4 - 5760*cos(c + d*x)*sin(c + d*x)**3*a **7*b + 2688*cos(c + d*x)*sin(c + d*x)**3*a**5*b**3 + 4760*cos(c + d*x)*si n(c + d*x)**2*a**8 - 2240*cos(c + d*x)*sin(c + d*x)**2*a**6*b**2 + 1920*co s(c + d*x)*sin(c + d*x)*a**7*b - 1680*cos(c + d*x)*a**8 - 525*log(tan((c + d*x)/2))*sin(c + d*x)**8*a**8 - 4200*log(tan((c + d*x)/2))*sin(c + d*x)** 8*a**6*b**2 + 25200*log(tan((c + d*x)/2))*sin(c + d*x)**8*a**4*b**4 - 3360 0*log(tan((c + d*x)/2))*sin(c + d*x)**8*a**2*b**6 + 13440*log(tan((c + d*x )/2))*sin(c + d*x)**8*b**8)/(13440*sin(c + d*x)**8*a**9*d)