Integrand size = 29, antiderivative size = 238 \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 b^4 \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d}-\frac {b \left (a^4+4 a^2 b^2-8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}+\frac {\left (2 a^4+5 a^2 b^2-15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d} \] Output:
-2*b^4*(a^2-b^2)^(1/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^ 6/d-1/8*b*(a^4+4*a^2*b^2-8*b^4)*arctanh(cos(d*x+c))/a^6/d+1/15*(2*a^4+5*a^ 2*b^2-15*b^4)*cot(d*x+c)/a^5/d-1/8*b*(a^2-4*b^2)*cot(d*x+c)*csc(d*x+c)/a^4 /d+1/15*(a^2-5*b^2)*cot(d*x+c)*csc(d*x+c)^2/a^3/d+1/4*b*cot(d*x+c)*csc(d*x +c)^3/a^2/d-1/5*cot(d*x+c)*csc(d*x+c)^4/a/d
Leaf count is larger than twice the leaf count of optimal. \(506\) vs. \(2(238)=476\).
Time = 3.39 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.13 \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-1920 b^4 \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+32 \left (2 a^5+5 a^3 b^2-15 a b^4\right ) \cot \left (\frac {1}{2} (c+d x)\right )-30 a^4 b \csc ^2\left (\frac {1}{2} (c+d x)\right )+120 a^2 b^3 \csc ^2\left (\frac {1}{2} (c+d x)\right )+15 a^4 b \csc ^4\left (\frac {1}{2} (c+d x)\right )-120 a^4 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-480 a^2 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+960 b^5 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+120 a^4 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+480 a^2 b^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-960 b^5 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+30 a^4 b \sec ^2\left (\frac {1}{2} (c+d x)\right )-120 a^2 b^3 \sec ^2\left (\frac {1}{2} (c+d x)\right )-15 a^4 b \sec ^4\left (\frac {1}{2} (c+d x)\right )-16 a^5 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+320 a^3 b^2 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+a^5 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-20 a^3 b^2 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-3 a^5 \csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-64 a^5 \tan \left (\frac {1}{2} (c+d x)\right )-160 a^3 b^2 \tan \left (\frac {1}{2} (c+d x)\right )+480 a b^4 \tan \left (\frac {1}{2} (c+d x)\right )+6 a^5 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{960 a^6 d} \] Input:
Integrate[(Cot[c + d*x]^2*Csc[c + d*x]^4)/(a + b*Sin[c + d*x]),x]
Output:
(-1920*b^4*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2] ] + 32*(2*a^5 + 5*a^3*b^2 - 15*a*b^4)*Cot[(c + d*x)/2] - 30*a^4*b*Csc[(c + d*x)/2]^2 + 120*a^2*b^3*Csc[(c + d*x)/2]^2 + 15*a^4*b*Csc[(c + d*x)/2]^4 - 120*a^4*b*Log[Cos[(c + d*x)/2]] - 480*a^2*b^3*Log[Cos[(c + d*x)/2]] + 96 0*b^5*Log[Cos[(c + d*x)/2]] + 120*a^4*b*Log[Sin[(c + d*x)/2]] + 480*a^2*b^ 3*Log[Sin[(c + d*x)/2]] - 960*b^5*Log[Sin[(c + d*x)/2]] + 30*a^4*b*Sec[(c + d*x)/2]^2 - 120*a^2*b^3*Sec[(c + d*x)/2]^2 - 15*a^4*b*Sec[(c + d*x)/2]^4 - 16*a^5*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 320*a^3*b^2*Csc[c + d*x]^3*S in[(c + d*x)/2]^4 + a^5*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 20*a^3*b^2*Csc[( c + d*x)/2]^4*Sin[c + d*x] - 3*a^5*Csc[(c + d*x)/2]^6*Sin[c + d*x] - 64*a^ 5*Tan[(c + d*x)/2] - 160*a^3*b^2*Tan[(c + d*x)/2] + 480*a*b^4*Tan[(c + d*x )/2] + 6*a^5*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(960*a^6*d)
Time = 2.24 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.17, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.793, Rules used = {3042, 3368, 3042, 3535, 25, 3042, 3534, 3042, 3534, 25, 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.
| \(\displaystyle \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^2}{\sin (c+d x)^6 (a+b \sin (c+d x))}dx\) |
| \(\Big \downarrow \) 3368 |
| \(\displaystyle \int \frac {\left (1-\sin ^2(c+d x)\right ) \csc ^6(c+d x)}{a+b \sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1-\sin (c+d x)^2}{\sin (c+d x)^6 (a+b \sin (c+d x))}dx\) |
| \(\Big \downarrow \) 3535 |
| \(\displaystyle \frac {\int -\frac {\csc ^5(c+d x) \left (-4 b \sin ^2(c+d x)+a \sin (c+d x)+5 b\right )}{a+b \sin (c+d x)}dx}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\csc ^5(c+d x) \left (-4 b \sin ^2(c+d x)+a \sin (c+d x)+5 b\right )}{a+b \sin (c+d x)}dx}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {-4 b \sin (c+d x)^2+a \sin (c+d x)+5 b}{\sin (c+d x)^5 (a+b \sin (c+d x))}dx}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {\int \frac {\csc ^4(c+d x) \left (15 b^2 \sin ^2(c+d x)-a b \sin (c+d x)+4 \left (a^2-5 b^2\right )\right )}{a+b \sin (c+d x)}dx}{4 a}-\frac {5 b \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {15 b^2 \sin (c+d x)^2-a b \sin (c+d x)+4 \left (a^2-5 b^2\right )}{\sin (c+d x)^4 (a+b \sin (c+d x))}dx}{4 a}-\frac {5 b \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {\frac {\int -\frac {\csc ^3(c+d x) \left (-8 b \left (a^2-5 b^2\right ) \sin ^2(c+d x)-a \left (8 a^2+5 b^2\right ) \sin (c+d x)+15 b \left (a^2-4 b^2\right )\right )}{a+b \sin (c+d x)}dx}{3 a}-\frac {4 \left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {5 b \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {-\frac {\int \frac {\csc ^3(c+d x) \left (-8 b \left (a^2-5 b^2\right ) \sin ^2(c+d x)-a \left (8 a^2+5 b^2\right ) \sin (c+d x)+15 b \left (a^2-4 b^2\right )\right )}{a+b \sin (c+d x)}dx}{3 a}-\frac {4 \left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {5 b \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {-\frac {\int \frac {-8 b \left (a^2-5 b^2\right ) \sin (c+d x)^2-a \left (8 a^2+5 b^2\right ) \sin (c+d x)+15 b \left (a^2-4 b^2\right )}{\sin (c+d x)^3 (a+b \sin (c+d x))}dx}{3 a}-\frac {4 \left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {5 b \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {-\frac {\frac {\int -\frac {\csc ^2(c+d x) \left (-15 b^2 \left (a^2-4 b^2\right ) \sin ^2(c+d x)+a b \left (a^2-20 b^2\right ) \sin (c+d x)+8 \left (2 a^4+5 b^2 a^2-15 b^4\right )\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {15 b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 \left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {5 b \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {-\frac {-\frac {\int \frac {\csc ^2(c+d x) \left (-15 b^2 \left (a^2-4 b^2\right ) \sin ^2(c+d x)+a b \left (a^2-20 b^2\right ) \sin (c+d x)+8 \left (2 a^4+5 b^2 a^2-15 b^4\right )\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {15 b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 \left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {5 b \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {-\frac {-\frac {\int \frac {-15 b^2 \left (a^2-4 b^2\right ) \sin (c+d x)^2+a b \left (a^2-20 b^2\right ) \sin (c+d x)+8 \left (2 a^4+5 b^2 a^2-15 b^4\right )}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{2 a}-\frac {15 b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 \left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {5 b \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {-\frac {-\frac {\frac {\int -\frac {15 \csc (c+d x) \left (a \left (a^2-4 b^2\right ) \sin (c+d x) b^2+\left (a^4+4 b^2 a^2-8 b^4\right ) b\right )}{a+b \sin (c+d x)}dx}{a}-\frac {8 \left (2 a^4+5 a^2 b^2-15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 \left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {5 b \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {-\frac {-\frac {-\frac {15 \int \frac {\csc (c+d x) \left (a \left (a^2-4 b^2\right ) \sin (c+d x) b^2+\left (a^4+4 b^2 a^2-8 b^4\right ) b\right )}{a+b \sin (c+d x)}dx}{a}-\frac {8 \left (2 a^4+5 a^2 b^2-15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 \left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {5 b \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {-\frac {-\frac {-\frac {15 \int \frac {a \left (a^2-4 b^2\right ) \sin (c+d x) b^2+\left (a^4+4 b^2 a^2-8 b^4\right ) b}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {8 \left (2 a^4+5 a^2 b^2-15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 \left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {5 b \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle -\frac {\frac {-\frac {-\frac {-\frac {15 \left (\frac {b \left (a^4+4 a^2 b^2-8 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {8 b^4 \left (a^2-b^2\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {8 \left (2 a^4+5 a^2 b^2-15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 \left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {5 b \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {-\frac {-\frac {-\frac {15 \left (\frac {b \left (a^4+4 a^2 b^2-8 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {8 b^4 \left (a^2-b^2\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {8 \left (2 a^4+5 a^2 b^2-15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 \left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {5 b \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {\frac {-\frac {-\frac {-\frac {15 \left (\frac {b \left (a^4+4 a^2 b^2-8 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {16 b^4 \left (a^2-b^2\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}-\frac {8 \left (2 a^4+5 a^2 b^2-15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 \left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {5 b \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {\frac {-\frac {-\frac {-\frac {15 \left (\frac {32 b^4 \left (a^2-b^2\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}+\frac {b \left (a^4+4 a^2 b^2-8 b^4\right ) \int \csc (c+d x)dx}{a}\right )}{a}-\frac {8 \left (2 a^4+5 a^2 b^2-15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 \left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {5 b \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {-\frac {-\frac {-\frac {15 \left (\frac {b \left (a^4+4 a^2 b^2-8 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {16 b^4 \sqrt {a^2-b^2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}\right )}{a}-\frac {8 \left (2 a^4+5 a^2 b^2-15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 \left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {5 b \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {\frac {-\frac {4 \left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}-\frac {-\frac {15 b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {-\frac {15 \left (-\frac {16 b^4 \sqrt {a^2-b^2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}-\frac {b \left (a^4+4 a^2 b^2-8 b^4\right ) \text {arctanh}(\cos (c+d x))}{a d}\right )}{a}-\frac {8 \left (2 a^4+5 a^2 b^2-15 b^4\right ) \cot (c+d x)}{a d}}{2 a}}{3 a}}{4 a}-\frac {5 b \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{5 a}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\) |
Input:
Int[(Cot[c + d*x]^2*Csc[c + d*x]^4)/(a + b*Sin[c + d*x]),x]
Output:
-1/5*(Cot[c + d*x]*Csc[c + d*x]^4)/(a*d) - ((-5*b*Cot[c + d*x]*Csc[c + d*x ]^3)/(4*a*d) + ((-4*(a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d) - ( -1/2*((-15*((-16*b^4*Sqrt[a^2 - b^2]*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2])/( 2*Sqrt[a^2 - b^2])])/(a*d) - (b*(a^4 + 4*a^2*b^2 - 8*b^4)*ArcTanh[Cos[c + d*x]])/(a*d)))/a - (8*(2*a^4 + 5*a^2*b^2 - 15*b^4)*Cot[c + d*x])/(a*d))/a - (15*b*(a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a*d))/(3*a))/(4*a))/(5 *a)
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_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n }, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])
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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 + a^2*C))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*S in[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[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d *(A*b^2 + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, 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 = 1.13 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.41
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{4}}{5}-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3}}{2}+\frac {a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\frac {4 a^{2} b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-4 a \,b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{2}+16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4}}{32 a^{5}}-\frac {2 b^{4} \sqrt {a^{2}-b^{2}}\, \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{6}}-\frac {1}{160 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {a^{2}+4 b^{2}}{96 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-2 a^{4}-4 a^{2} b^{2}+16 b^{4}}{32 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{64 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {b^{3}}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (a^{4}+4 a^{2} b^{2}-8 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{6}}}{d}\) | \(336\) |
| default | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{4}}{5}-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3}}{2}+\frac {a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\frac {4 a^{2} b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-4 a \,b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{2}+16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4}}{32 a^{5}}-\frac {2 b^{4} \sqrt {a^{2}-b^{2}}\, \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{6}}-\frac {1}{160 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {a^{2}+4 b^{2}}{96 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-2 a^{4}-4 a^{2} b^{2}+16 b^{4}}{32 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{64 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {b^{3}}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (a^{4}+4 a^{2} b^{2}-8 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{6}}}{d}\) | \(336\) |
| risch | \(\frac {-240 i a^{4} {\mathrm e}^{6 i \left (d x +c \right )}+120 i a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+15 a^{3} b \,{\mathrm e}^{9 i \left (d x +c \right )}-60 a \,b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-80 i a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-720 i b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+480 i b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+90 a^{3} b \,{\mathrm e}^{7 i \left (d x +c \right )}+120 b^{3} a \,{\mathrm e}^{7 i \left (d x +c \right )}+40 i a^{2} b^{2}-240 i a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-80 i a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+16 i a^{4}-120 i b^{4}+480 i b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-90 b \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-120 b^{3} a \,{\mathrm e}^{3 i \left (d x +c \right )}-120 i b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-80 i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+160 i a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-15 b \,a^{3} {\mathrm e}^{i \left (d x +c \right )}+60 b^{3} a \,{\mathrm e}^{i \left (d x +c \right )}}{60 d \,a^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {\sqrt {-a^{2}+b^{2}}\, b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{6}}+\frac {\sqrt {-a^{2}+b^{2}}\, b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{6}}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 a^{2} d}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{4}}-\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{6}}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 a^{2} d}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{4}}+\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{6}}\) | \(582\) |
Input:
int(cot(d*x+c)^2*csc(d*x+c)^4/(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(1/32/a^5*(1/5*tan(1/2*d*x+1/2*c)^5*a^4-1/2*b*tan(1/2*d*x+1/2*c)^4*a^3 +1/3*a^4*tan(1/2*d*x+1/2*c)^3+4/3*a^2*b^2*tan(1/2*d*x+1/2*c)^3-4*a*b^3*tan (1/2*d*x+1/2*c)^2-2*a^4*tan(1/2*d*x+1/2*c)-4*tan(1/2*d*x+1/2*c)*a^2*b^2+16 *tan(1/2*d*x+1/2*c)*b^4)-2*b^4*(a^2-b^2)^(1/2)/a^6*arctan(1/2*(2*a*tan(1/2 *d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))-1/160/a/tan(1/2*d*x+1/2*c)^5-1/96/a^3*(a ^2+4*b^2)/tan(1/2*d*x+1/2*c)^3-1/32*(-2*a^4-4*a^2*b^2+16*b^4)/a^5/tan(1/2* d*x+1/2*c)+1/64/a^2*b/tan(1/2*d*x+1/2*c)^4+1/8/a^4*b^3/tan(1/2*d*x+1/2*c)^ 2+1/8/a^6*b*(a^4+4*a^2*b^2-8*b^4)*ln(tan(1/2*d*x+1/2*c)))
Time = 0.23 (sec) , antiderivative size = 959, normalized size of antiderivative = 4.03 \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^4/(a+b*sin(d*x+c)),x, algorithm="fricas" )
Output:
[-1/240*(240*a*b^4*cos(d*x + c) - 16*(2*a^5 + 5*a^3*b^2 - 15*a*b^4)*cos(d* x + c)^5 + 80*(a^5 + a^3*b^2 - 6*a*b^4)*cos(d*x + c)^3 - 120*(b^4*cos(d*x + c)^4 - 2*b^4*cos(d*x + c)^2 + b^4)*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)*c os(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))*sin(d*x + c) + 15*(a^4*b + 4*a^2*b^3 - 8*b^5 + (a^4 *b + 4*a^2*b^3 - 8*b^5)*cos(d*x + c)^4 - 2*(a^4*b + 4*a^2*b^3 - 8*b^5)*cos (d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 15*(a^4*b + 4*a^2* b^3 - 8*b^5 + (a^4*b + 4*a^2*b^3 - 8*b^5)*cos(d*x + c)^4 - 2*(a^4*b + 4*a^ 2*b^3 - 8*b^5)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 30*((a^4*b - 4*a^2*b^3)*cos(d*x + c)^3 + (a^4*b + 4*a^2*b^3)*cos(d*x + c) )*sin(d*x + c))/((a^6*d*cos(d*x + c)^4 - 2*a^6*d*cos(d*x + c)^2 + a^6*d)*s in(d*x + c)), -1/240*(240*a*b^4*cos(d*x + c) - 16*(2*a^5 + 5*a^3*b^2 - 15* a*b^4)*cos(d*x + c)^5 + 80*(a^5 + a^3*b^2 - 6*a*b^4)*cos(d*x + c)^3 - 240* (b^4*cos(d*x + c)^4 - 2*b^4*cos(d*x + c)^2 + b^4)*sqrt(a^2 - b^2)*arctan(- (a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c)))*sin(d*x + c) + 15*(a^ 4*b + 4*a^2*b^3 - 8*b^5 + (a^4*b + 4*a^2*b^3 - 8*b^5)*cos(d*x + c)^4 - 2*( a^4*b + 4*a^2*b^3 - 8*b^5)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2)*sin (d*x + c) - 15*(a^4*b + 4*a^2*b^3 - 8*b^5 + (a^4*b + 4*a^2*b^3 - 8*b^5)*co s(d*x + c)^4 - 2*(a^4*b + 4*a^2*b^3 - 8*b^5)*cos(d*x + c)^2)*log(-1/2*c...
\[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\cot ^{2}{\left (c + d x \right )} \csc ^{4}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \] Input:
integrate(cot(d*x+c)**2*csc(d*x+c)**4/(a+b*sin(d*x+c)),x)
Output:
Integral(cot(c + d*x)**2*csc(c + d*x)**4/(a + b*sin(c + d*x)), x)
Exception generated. \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^4/(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. 444 vs. \(2 (221) = 442\).
Time = 0.15 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.87 \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {6 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 10 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 60 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 120 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 480 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{5}} + \frac {120 \, {\left (a^{4} b + 4 \, a^{2} b^{3} - 8 \, b^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{6}} - \frac {1920 \, {\left (a^{2} b^{4} - b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{6}} - \frac {274 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1096 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2192 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 480 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 40 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a^{5}}{a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^4/(a+b*sin(d*x+c)),x, algorithm="giac")
Output:
1/960*((6*a^4*tan(1/2*d*x + 1/2*c)^5 - 15*a^3*b*tan(1/2*d*x + 1/2*c)^4 + 1 0*a^4*tan(1/2*d*x + 1/2*c)^3 + 40*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 120*a*b ^3*tan(1/2*d*x + 1/2*c)^2 - 60*a^4*tan(1/2*d*x + 1/2*c) - 120*a^2*b^2*tan( 1/2*d*x + 1/2*c) + 480*b^4*tan(1/2*d*x + 1/2*c))/a^5 + 120*(a^4*b + 4*a^2* b^3 - 8*b^5)*log(abs(tan(1/2*d*x + 1/2*c)))/a^6 - 1920*(a^2*b^4 - b^6)*(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^6) - (274*a^4*b*tan(1/2*d*x + 1/2*c )^5 + 1096*a^2*b^3*tan(1/2*d*x + 1/2*c)^5 - 2192*b^5*tan(1/2*d*x + 1/2*c)^ 5 - 60*a^5*tan(1/2*d*x + 1/2*c)^4 - 120*a^3*b^2*tan(1/2*d*x + 1/2*c)^4 + 4 80*a*b^4*tan(1/2*d*x + 1/2*c)^4 - 120*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 10* a^5*tan(1/2*d*x + 1/2*c)^2 + 40*a^3*b^2*tan(1/2*d*x + 1/2*c)^2 - 15*a^4*b* tan(1/2*d*x + 1/2*c) + 6*a^5)/(a^6*tan(1/2*d*x + 1/2*c)^5))/d
Time = 19.95 (sec) , antiderivative size = 1007, normalized size of antiderivative = 4.23 \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:
int(cot(c + d*x)^2/(sin(c + d*x)^4*(a + b*sin(c + d*x))),x)
Output:
tan(c/2 + (d*x)/2)^5/(160*a*d) + (tan(c/2 + (d*x)/2)^2*(b/(32*a^2) - (b*(1 /(32*a) + b^2/(8*a^3)))/a))/d - (tan(c/2 + (d*x)/2)*(1/(16*a) + b^2/(8*a^3 ) + (2*b*(b/(16*a^2) - (2*b*(1/(32*a) + b^2/(8*a^3)))/a))/a))/d + (tan(c/2 + (d*x)/2)^3*(1/(96*a) + b^2/(24*a^3)))/d + (log(tan(c/2 + (d*x)/2))*((a^ 4*b)/8 - b^5 + (a^2*b^3)/2))/(a^6*d) - (b*tan(c/2 + (d*x)/2)^4)/(64*a^2*d) + (tan(c/2 + (d*x)/2)^4*(2*a^4 - 16*b^4 + 4*a^2*b^2) - a^4/5 - tan(c/2 + (d*x)/2)^2*(a^4/3 + (4*a^2*b^2)/3) + (a^3*b*tan(c/2 + (d*x)/2))/2 + 4*a*b^ 3*tan(c/2 + (d*x)/2)^3)/(32*a^5*d*tan(c/2 + (d*x)/2)^5) - (b^4*atan(((b^4* (b^2 - a^2)^(1/2)*((tan(c/2 + (d*x)/2)*(a^10*b + 32*a^4*b^7 - 32*a^6*b^5 + 2*a^8*b^3))/(4*a^9) - (12*a^8*b^4 - 16*a^6*b^6 + a^10*b^2)/(4*a^10) + (b^ 4*(2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^12 - 32*a^10*b^2))/(4*a^9))*(b^2 - a^2)^(1/2))/a^6)*1i)/a^6 - (b^4*(b^2 - a^2)^(1/2)*((12*a^8*b^4 - 16*a^6*b^ 6 + a^10*b^2)/(4*a^10) - (tan(c/2 + (d*x)/2)*(a^10*b + 32*a^4*b^7 - 32*a^6 *b^5 + 2*a^8*b^3))/(4*a^9) + (b^4*(2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^12 - 32*a^10*b^2))/(4*a^9))*(b^2 - a^2)^(1/2))/a^6)*1i)/a^6)/((8*b^11 - 12*a^ 2*b^9 + 3*a^4*b^7 + a^6*b^5)/(2*a^10) + (tan(c/2 + (d*x)/2)*(8*b^10 - 10*a ^2*b^8 + 2*a^4*b^6))/(2*a^9) + (b^4*(b^2 - a^2)^(1/2)*((tan(c/2 + (d*x)/2) *(a^10*b + 32*a^4*b^7 - 32*a^6*b^5 + 2*a^8*b^3))/(4*a^9) - (12*a^8*b^4 - 1 6*a^6*b^6 + a^10*b^2)/(4*a^10) + (b^4*(2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a ^12 - 32*a^10*b^2))/(4*a^9))*(b^2 - a^2)^(1/2))/a^6))/a^6 + (b^4*(b^2 -...
Time = 0.19 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.32 \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-240 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \sin \left (d x +c \right )^{5} b^{4}+16 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{5}+40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{3} b^{2}-120 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a \,b^{4}-15 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{4} b +60 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2} b^{3}+8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{5}-40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{3} b^{2}+30 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{4} b -24 \cos \left (d x +c \right ) a^{5}+15 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{5} a^{4} b +60 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{5} a^{2} b^{3}-120 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{5} b^{5}}{120 \sin \left (d x +c \right )^{5} a^{6} d} \] Input:
int(cot(d*x+c)^2*csc(d*x+c)^4/(a+b*sin(d*x+c)),x)
Output:
( - 240*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2)) *sin(c + d*x)**5*b**4 + 16*cos(c + d*x)*sin(c + d*x)**4*a**5 + 40*cos(c + d*x)*sin(c + d*x)**4*a**3*b**2 - 120*cos(c + d*x)*sin(c + d*x)**4*a*b**4 - 15*cos(c + d*x)*sin(c + d*x)**3*a**4*b + 60*cos(c + d*x)*sin(c + d*x)**3* a**2*b**3 + 8*cos(c + d*x)*sin(c + d*x)**2*a**5 - 40*cos(c + d*x)*sin(c + d*x)**2*a**3*b**2 + 30*cos(c + d*x)*sin(c + d*x)*a**4*b - 24*cos(c + d*x)* a**5 + 15*log(tan((c + d*x)/2))*sin(c + d*x)**5*a**4*b + 60*log(tan((c + d *x)/2))*sin(c + d*x)**5*a**2*b**3 - 120*log(tan((c + d*x)/2))*sin(c + d*x) **5*b**5)/(120*sin(c + d*x)**5*a**6*d)