Integrand size = 29, antiderivative size = 303 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {3 b \left (a^2+2 b^2\right ) \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a \left (2 a^2+21 b^2\right ) \cot (c+d x)}{35 d}-\frac {b \left (105 a^4-116 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{560 a^2 d}-\frac {\left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{140 a d}+\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac {\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d} \]
-3/16*b*(a^2+2*b^2)*arctanh(cos(d*x+c))/d-1/35*a*(2*a^2+21*b^2)*cot(d*x+c) /d-1/560*b*(105*a^4-116*a^2*b^2+12*b^4)*cot(d*x+c)*csc(d*x+c)/a^2/d-1/140* (4*a^4-19*a^2*b^2+2*b^4)*cot(d*x+c)*csc(d*x+c)^2/a/d+1/280*b*(53*a^2-6*b^2 )*cot(d*x+c)*csc(d*x+c)^3*(a+b*sin(d*x+c))^2/a^2/d+1/35*(8*a^2-b^2)*cot(d* x+c)*csc(d*x+c)^4*(a+b*sin(d*x+c))^3/a^2/d+1/14*b*cot(d*x+c)*csc(d*x+c)^5* (a+b*sin(d*x+c))^4/a^2/d-1/7*cot(d*x+c)*csc(d*x+c)^6*(a+b*sin(d*x+c))^4/a/ d
Time = 1.38 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.07 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {56 a \left (14 a^2-3 b^2\right ) \cos (3 (c+d x)) \csc ^7(c+d x)+112 a^3 \cos (5 (c+d x)) \csc ^7(c+d x)-504 a b^2 \cos (5 (c+d x)) \csc ^7(c+d x)-16 a^3 \cos (7 (c+d x)) \csc ^7(c+d x)-168 a b^2 \cos (7 (c+d x)) \csc ^7(c+d x)+3360 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+6720 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-3360 a^2 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-6720 b^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+70 \cot (c+d x) \csc ^6(c+d x) \left (12 a \left (2 a^2+b^2\right )+b \left (31 a^2-18 b^2\right ) \sin (c+d x)\right )+1540 a^2 b \csc ^7(c+d x) \sin (4 (c+d x))+840 b^3 \csc ^7(c+d x) \sin (4 (c+d x))+105 a^2 b \csc ^7(c+d x) \sin (6 (c+d x))-350 b^3 \csc ^7(c+d x) \sin (6 (c+d x))}{17920 d} \]
-1/17920*(56*a*(14*a^2 - 3*b^2)*Cos[3*(c + d*x)]*Csc[c + d*x]^7 + 112*a^3* Cos[5*(c + d*x)]*Csc[c + d*x]^7 - 504*a*b^2*Cos[5*(c + d*x)]*Csc[c + d*x]^ 7 - 16*a^3*Cos[7*(c + d*x)]*Csc[c + d*x]^7 - 168*a*b^2*Cos[7*(c + d*x)]*Cs c[c + d*x]^7 + 3360*a^2*b*Log[Cos[(c + d*x)/2]] + 6720*b^3*Log[Cos[(c + d* x)/2]] - 3360*a^2*b*Log[Sin[(c + d*x)/2]] - 6720*b^3*Log[Sin[(c + d*x)/2]] + 70*Cot[c + d*x]*Csc[c + d*x]^6*(12*a*(2*a^2 + b^2) + b*(31*a^2 - 18*b^2 )*Sin[c + d*x]) + 1540*a^2*b*Csc[c + d*x]^7*Sin[4*(c + d*x)] + 840*b^3*Csc [c + d*x]^7*Sin[4*(c + d*x)] + 105*a^2*b*Csc[c + d*x]^7*Sin[6*(c + d*x)] - 350*b^3*Csc[c + d*x]^7*Sin[6*(c + d*x)])/d
Time = 2.09 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.04, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.655, Rules used = {3042, 3372, 27, 3042, 3526, 3042, 3526, 25, 3042, 3510, 27, 3042, 3500, 3042, 3227, 3042, 4254, 24, 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.
| \(\displaystyle \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a+b \sin (c+d x))^3}{\sin (c+d x)^8}dx\) |
| \(\Big \downarrow \) 3372 |
| \(\displaystyle -\frac {\int 3 \csc ^6(c+d x) (a+b \sin (c+d x))^3 \left (-\left (\left (14 a^2-b^2\right ) \sin ^2(c+d x)\right )+a b \sin (c+d x)+2 \left (8 a^2-b^2\right )\right )dx}{42 a^2}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \csc ^6(c+d x) (a+b \sin (c+d x))^3 \left (-\left (\left (14 a^2-b^2\right ) \sin ^2(c+d x)\right )+a b \sin (c+d x)+2 \left (8 a^2-b^2\right )\right )dx}{14 a^2}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {(a+b \sin (c+d x))^3 \left (-\left (\left (14 a^2-b^2\right ) \sin (c+d x)^2\right )+a b \sin (c+d x)+2 \left (8 a^2-b^2\right )\right )}{\sin (c+d x)^6}dx}{14 a^2}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\frac {1}{5} \int \csc ^5(c+d x) (a+b \sin (c+d x))^2 \left (-3 b \left (18 a^2-b^2\right ) \sin ^2(c+d x)-2 a \left (3 a^2-b^2\right ) \sin (c+d x)+b \left (53 a^2-6 b^2\right )\right )dx-\frac {2 \left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}}{14 a^2}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{5} \int \frac {(a+b \sin (c+d x))^2 \left (-3 b \left (18 a^2-b^2\right ) \sin (c+d x)^2-2 a \left (3 a^2-b^2\right ) \sin (c+d x)+b \left (53 a^2-6 b^2\right )\right )}{\sin (c+d x)^5}dx-\frac {2 \left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}}{14 a^2}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \int -\csc ^4(c+d x) (a+b \sin (c+d x)) \left (b^2 \left (163 a^2-6 b^2\right ) \sin ^2(c+d x)+a b \left (81 a^2-2 b^2\right ) \sin (c+d x)+6 \left (4 a^4-19 b^2 a^2+2 b^4\right )\right )dx-\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}\right )-\frac {2 \left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}}{14 a^2}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{5} \left (-\frac {1}{4} \int \csc ^4(c+d x) (a+b \sin (c+d x)) \left (b^2 \left (163 a^2-6 b^2\right ) \sin ^2(c+d x)+a b \left (81 a^2-2 b^2\right ) \sin (c+d x)+6 \left (4 a^4-19 b^2 a^2+2 b^4\right )\right )dx-\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}\right )-\frac {2 \left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}}{14 a^2}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{5} \left (-\frac {1}{4} \int \frac {(a+b \sin (c+d x)) \left (b^2 \left (163 a^2-6 b^2\right ) \sin (c+d x)^2+a b \left (81 a^2-2 b^2\right ) \sin (c+d x)+6 \left (4 a^4-19 b^2 a^2+2 b^4\right )\right )}{\sin (c+d x)^4}dx-\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}\right )-\frac {2 \left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}}{14 a^2}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int -3 \csc ^3(c+d x) \left (8 \left (2 a^2+21 b^2\right ) \sin (c+d x) a^3+b^3 \left (163 a^2-6 b^2\right ) \sin ^2(c+d x)+b \left (105 a^4-116 b^2 a^2+12 b^4\right )\right )dx+\frac {2 a \left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}\right )-\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}\right )-\frac {2 \left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}}{14 a^2}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {2 a \left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-\int \csc ^3(c+d x) \left (8 \left (2 a^2+21 b^2\right ) \sin (c+d x) a^3+b^3 \left (163 a^2-6 b^2\right ) \sin ^2(c+d x)+b \left (105 a^4-116 b^2 a^2+12 b^4\right )\right )dx\right )-\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}\right )-\frac {2 \left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}}{14 a^2}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {2 a \left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-\int \frac {8 \left (2 a^2+21 b^2\right ) \sin (c+d x) a^3+b^3 \left (163 a^2-6 b^2\right ) \sin (c+d x)^2+b \left (105 a^4-116 b^2 a^2+12 b^4\right )}{\sin (c+d x)^3}dx\right )-\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}\right )-\frac {2 \left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}}{14 a^2}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \left (-\frac {1}{2} \int \csc ^2(c+d x) \left (16 \left (2 a^2+21 b^2\right ) a^3+105 b \left (a^2+2 b^2\right ) \sin (c+d x) a^2\right )dx+\frac {2 a \left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}+\frac {b \left (105 a^4-116 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}\right )-\frac {2 \left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}}{14 a^2}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \left (-\frac {1}{2} \int \frac {16 \left (2 a^2+21 b^2\right ) a^3+105 b \left (a^2+2 b^2\right ) \sin (c+d x) a^2}{\sin (c+d x)^2}dx+\frac {2 a \left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}+\frac {b \left (105 a^4-116 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}\right )-\frac {2 \left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}}{14 a^2}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \left (-105 a^2 b \left (a^2+2 b^2\right ) \int \csc (c+d x)dx-16 a^3 \left (2 a^2+21 b^2\right ) \int \csc ^2(c+d x)dx\right )+\frac {2 a \left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}+\frac {b \left (105 a^4-116 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}\right )-\frac {2 \left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}}{14 a^2}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \left (-105 a^2 b \left (a^2+2 b^2\right ) \int \csc (c+d x)dx-16 a^3 \left (2 a^2+21 b^2\right ) \int \csc (c+d x)^2dx\right )+\frac {2 a \left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}+\frac {b \left (105 a^4-116 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}\right )-\frac {2 \left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}}{14 a^2}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {16 a^3 \left (2 a^2+21 b^2\right ) \int 1d\cot (c+d x)}{d}-105 a^2 b \left (a^2+2 b^2\right ) \int \csc (c+d x)dx\right )+\frac {2 a \left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}+\frac {b \left (105 a^4-116 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}\right )-\frac {2 \left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}}{14 a^2}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {16 a^3 \left (2 a^2+21 b^2\right ) \cot (c+d x)}{d}-105 a^2 b \left (a^2+2 b^2\right ) \int \csc (c+d x)dx\right )+\frac {2 a \left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}+\frac {b \left (105 a^4-116 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}\right )-\frac {2 \left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}}{14 a^2}+\frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {2 a \left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}+\frac {b \left (105 a^4-116 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}+\frac {1}{2} \left (\frac {105 a^2 b \left (a^2+2 b^2\right ) \text {arctanh}(\cos (c+d x))}{d}+\frac {16 a^3 \left (2 a^2+21 b^2\right ) \cot (c+d x)}{d}\right )\right )-\frac {b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}\right )-\frac {2 \left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}}{14 a^2}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\) |
(b*Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^4)/(14*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^4)/(7*a*d) - ((-2*(8*a^2 - b^2 )*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(5*d) + ((((105*a^2* b*(a^2 + 2*b^2)*ArcTanh[Cos[c + d*x]])/d + (16*a^3*(2*a^2 + 21*b^2)*Cot[c + d*x])/d)/2 + (b*(105*a^4 - 116*a^2*b^2 + 12*b^4)*Cot[c + d*x]*Csc[c + d* x])/(2*d) + (2*a*(4*a^4 - 19*a^2*b^2 + 2*b^4)*Cot[c + d*x]*Csc[c + d*x]^2) /d)/4 - (b*(53*a^2 - 6*b^2)*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x ])^2)/(4*d))/5)/(14*a^2)
3.12.25.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[Cos[e + f*x]*(a + b* Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*d*f*(n + 1))), x] + (-Si mp[b*(m + n + 2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x] )^(n + 2)/(a^2*d^2*f*(n + 1)*(n + 2))), x] - Simp[1/(a^2*d^2*(n + 1)*(n + 2 )) Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n + 2)*Simp[a^2*n*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*m*Sin[e + f*x] - (a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n]) && !m < -1 && LtQ[n, -1] && (LtQ[n, -2] || EqQ[m + n + 4, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((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)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
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[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.68 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.80
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35 \sin \left (d x +c \right )^{5}}\right )+3 a^{2} b \left (-\frac {\cos ^{5}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {3 a \,b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}+b^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(243\) |
| default | \(\frac {a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35 \sin \left (d x +c \right )^{5}}\right )+3 a^{2} b \left (-\frac {\cos ^{5}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {3 a \,b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}+b^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(243\) |
| parallelrisch | \(\frac {\frac {3 b \left (a^{2}+2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {3 \left (a^{3} \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (d x +c \right )+\frac {7 \cos \left (3 d x +3 c \right )}{15}+\frac {\cos \left (5 d x +5 c \right )}{15}-\frac {\cos \left (7 d x +7 c \right )}{105}\right ) \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {13 b \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (d x +c \right )+\frac {47 \cos \left (3 d x +3 c \right )}{78}+\frac {\cos \left (5 d x +5 c \right )}{26}\right ) a^{2} \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+16 b^{2} \left (\cos \left (d x +c \right )+\frac {\cos \left (3 d x +3 c \right )}{2}+\frac {\cos \left (5 d x +5 c \right )}{10}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) a \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+8 b^{3} \left (\cos \left (d x +c \right )+\frac {5 \cos \left (3 d x +3 c \right )}{3}\right )\right ) \left (\csc ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4096}}{d}\) | \(253\) |
| risch | \(-\frac {-2240 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-448 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-224 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+1680 i a \,b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-3360 i a \,b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+3696 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+5040 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-6720 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-672 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+350 b^{3} {\mathrm e}^{13 i \left (d x +c \right )}-840 b^{3} {\mathrm e}^{11 i \left (d x +c \right )}-630 b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+336 i a \,b^{2}+630 b^{3} {\mathrm e}^{9 i \left (d x +c \right )}+32 i a^{3}+840 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-350 b^{3} {\mathrm e}^{i \left (d x +c \right )}-105 a^{2} b \,{\mathrm e}^{13 i \left (d x +c \right )}-1540 a^{2} b \,{\mathrm e}^{11 i \left (d x +c \right )}-1085 a^{2} b \,{\mathrm e}^{9 i \left (d x +c \right )}-1120 i a^{3} {\mathrm e}^{10 i \left (d x +c \right )}+1085 a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}-1120 i a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+1540 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+105 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}}{280 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}-\frac {3 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}-\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}+\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}\) | \(461\) |
1/d*(a^3*(-1/7/sin(d*x+c)^7*cos(d*x+c)^5-2/35/sin(d*x+c)^5*cos(d*x+c)^5)+3 *a^2*b*(-1/6/sin(d*x+c)^6*cos(d*x+c)^5-1/24/sin(d*x+c)^4*cos(d*x+c)^5+1/48 /sin(d*x+c)^2*cos(d*x+c)^5+1/48*cos(d*x+c)^3+1/16*cos(d*x+c)+1/16*ln(csc(d *x+c)-cot(d*x+c)))-3/5*a*b^2/sin(d*x+c)^5*cos(d*x+c)^5+b^3*(-1/4/sin(d*x+c )^4*cos(d*x+c)^5+1/8/sin(d*x+c)^2*cos(d*x+c)^5+1/8*cos(d*x+c)^3+3/8*cos(d* x+c)+3/8*ln(csc(d*x+c)-cot(d*x+c))))
Time = 0.31 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.15 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {32 \, {\left (2 \, a^{3} + 21 \, a b^{2}\right )} \cos \left (d x + c\right )^{7} - 224 \, {\left (a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \, {\left ({\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{4} - a^{2} b - 2 \, b^{3} + 3 \, {\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 105 \, {\left ({\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{4} - a^{2} b - 2 \, b^{3} + 3 \, {\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 70 \, {\left ({\left (3 \, a^{2} b - 10 \, b^{3}\right )} \cos \left (d x + c\right )^{5} + 8 \, {\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1120 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
-1/1120*(32*(2*a^3 + 21*a*b^2)*cos(d*x + c)^7 - 224*(a^3 + 3*a*b^2)*cos(d* x + c)^5 + 105*((a^2*b + 2*b^3)*cos(d*x + c)^6 - 3*(a^2*b + 2*b^3)*cos(d*x + c)^4 - a^2*b - 2*b^3 + 3*(a^2*b + 2*b^3)*cos(d*x + c)^2)*log(1/2*cos(d* x + c) + 1/2)*sin(d*x + c) - 105*((a^2*b + 2*b^3)*cos(d*x + c)^6 - 3*(a^2* b + 2*b^3)*cos(d*x + c)^4 - a^2*b - 2*b^3 + 3*(a^2*b + 2*b^3)*cos(d*x + c) ^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 70*((3*a^2*b - 10*b^3)*cos (d*x + c)^5 + 8*(a^2*b + 2*b^3)*cos(d*x + c)^3 - 3*(a^2*b + 2*b^3)*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)*sin(d*x + c))
Timed out. \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.69 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {35 \, a^{2} b {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 70 \, b^{3} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {672 \, a b^{2}}{\tan \left (d x + c\right )^{5}} - \frac {32 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{3}}{\tan \left (d x + c\right )^{7}}}{1120 \, d} \]
1/1120*(35*a^2*b*(2*(3*cos(d*x + c)^5 + 8*cos(d*x + c)^3 - 3*cos(d*x + c)) /(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1) - 3*log(cos(d* x + c) + 1) + 3*log(cos(d*x + c) - 1)) - 70*b^3*(2*(5*cos(d*x + c)^3 - 3*c os(d*x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)) - 672*a*b^2/tan(d*x + c)^5 - 32*(7*tan(d*x + c)^2 + 5)*a^3/tan(d*x + c)^7)/d
Time = 0.46 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.50 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 35 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 84 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 70 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 420 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 560 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 840 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 840 \, {\left (a^{2} b + 2 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2178 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4356 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 840 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 105 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 560 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 420 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 105 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 70 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 84 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{4480 \, d} \]
1/4480*(5*a^3*tan(1/2*d*x + 1/2*c)^7 + 35*a^2*b*tan(1/2*d*x + 1/2*c)^6 - 7 *a^3*tan(1/2*d*x + 1/2*c)^5 + 84*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 105*a^2*b* tan(1/2*d*x + 1/2*c)^4 + 70*b^3*tan(1/2*d*x + 1/2*c)^4 - 35*a^3*tan(1/2*d* x + 1/2*c)^3 - 420*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 105*a^2*b*tan(1/2*d*x + 1/2*c)^2 - 560*b^3*tan(1/2*d*x + 1/2*c)^2 + 105*a^3*tan(1/2*d*x + 1/2*c) + 840*a*b^2*tan(1/2*d*x + 1/2*c) + 840*(a^2*b + 2*b^3)*log(abs(tan(1/2*d*x + 1/2*c))) - (2178*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 4356*b^3*tan(1/2*d*x + 1 /2*c)^7 + 105*a^3*tan(1/2*d*x + 1/2*c)^6 + 840*a*b^2*tan(1/2*d*x + 1/2*c)^ 6 - 105*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 560*b^3*tan(1/2*d*x + 1/2*c)^5 - 35 *a^3*tan(1/2*d*x + 1/2*c)^4 - 420*a*b^2*tan(1/2*d*x + 1/2*c)^4 - 105*a^2*b *tan(1/2*d*x + 1/2*c)^3 + 70*b^3*tan(1/2*d*x + 1/2*c)^3 - 7*a^3*tan(1/2*d* x + 1/2*c)^2 + 84*a*b^2*tan(1/2*d*x + 1/2*c)^2 + 35*a^2*b*tan(1/2*d*x + 1/ 2*c) + 5*a^3)/tan(1/2*d*x + 1/2*c)^7)/d
Time = 10.59 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.18 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {a^3}{128}+\frac {3\,a\,b^2}{32}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,a\,b^2}{160}-\frac {a^3}{640}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {3\,a^2\,b}{128}+\frac {b^3}{8}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {3\,a^2\,b}{128}-\frac {b^3}{64}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^2\,b}{16}+\frac {3\,b^3}{8}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {12\,a\,b^2}{5}-\frac {a^3}{5}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^3+12\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (3\,a^3+24\,a\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,a^2\,b-2\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (3\,a^2\,b+16\,b^3\right )+\frac {a^3}{7}+a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a^3}{128}+\frac {3\,a\,b^2}{16}\right )}{d}+\frac {a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{128\,d} \]
(a^3*tan(c/2 + (d*x)/2)^7)/(896*d) - (tan(c/2 + (d*x)/2)^3*((3*a*b^2)/32 + a^3/128))/d + (tan(c/2 + (d*x)/2)^5*((3*a*b^2)/160 - a^3/640))/d - (tan(c /2 + (d*x)/2)^2*((3*a^2*b)/128 + b^3/8))/d - (tan(c/2 + (d*x)/2)^4*((3*a^2 *b)/128 - b^3/64))/d + (log(tan(c/2 + (d*x)/2))*((3*a^2*b)/16 + (3*b^3)/8) )/d - (tan(c/2 + (d*x)/2)^2*((12*a*b^2)/5 - a^3/5) - tan(c/2 + (d*x)/2)^4* (12*a*b^2 + a^3) + tan(c/2 + (d*x)/2)^6*(24*a*b^2 + 3*a^3) - tan(c/2 + (d* x)/2)^3*(3*a^2*b - 2*b^3) - tan(c/2 + (d*x)/2)^5*(3*a^2*b + 16*b^3) + a^3/ 7 + a^2*b*tan(c/2 + (d*x)/2))/(128*d*tan(c/2 + (d*x)/2)^7) + (tan(c/2 + (d *x)/2)*((3*a*b^2)/16 + (3*a^3)/128))/d + (a^2*b*tan(c/2 + (d*x)/2)^6)/(128 *d)