Integrand size = 23, antiderivative size = 191 \[ \int \frac {\cot ^4(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx=\frac {45 \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{8 a^{5/2} f}-\frac {4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{a^{5/2} f}-\frac {19 \cot (e+f x)}{8 a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {13 \cot (e+f x) \csc (e+f x)}{12 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a^2 f \sqrt {a+a \sin (e+f x)}} \] Output:
45/8*arctanh(a^(1/2)*cos(f*x+e)/(a+a*sin(f*x+e))^(1/2))/a^(5/2)/f-4*arctan h(1/2*a^(1/2)*cos(f*x+e)*2^(1/2)/(a+a*sin(f*x+e))^(1/2))*2^(1/2)/a^(5/2)/f -19/8*cot(f*x+e)/a^2/f/(a+a*sin(f*x+e))^(1/2)+13/12*cot(f*x+e)*csc(f*x+e)/ a^2/f/(a+a*sin(f*x+e))^(1/2)-1/3*cot(f*x+e)*csc(f*x+e)^2/a^2/f/(a+a*sin(f* x+e))^(1/2)
Result contains complex when optimal does not.
Time = 3.32 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.74 \[ \int \frac {\cot ^4(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left ((1536+1536 i) (-1)^{3/4} \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right )-\frac {8 \csc ^9\left (\frac {1}{2} (e+f x)\right ) \left (396 \cos \left (\frac {1}{2} (e+f x)\right )-218 \cos \left (\frac {3}{2} (e+f x)\right )-114 \cos \left (\frac {5}{2} (e+f x)\right )-396 \sin \left (\frac {1}{2} (e+f x)\right )-405 \log \left (1+\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)+405 \log \left (1-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)-218 \sin \left (\frac {3}{2} (e+f x)\right )+114 \sin \left (\frac {5}{2} (e+f x)\right )+135 \log \left (1+\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (3 (e+f x))-135 \log \left (1-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (3 (e+f x))\right )}{\left (\csc ^2\left (\frac {1}{4} (e+f x)\right )-\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )^3}\right )}{192 f (a (1+\sin (e+f x)))^{5/2}} \] Input:
Integrate[Cot[e + f*x]^4/(a + a*Sin[e + f*x])^(5/2),x]
Output:
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*((1536 + 1536*I)*(-1)^(3/4)*ArcTa nh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])] - (8*Csc[(e + f*x)/2]^9 *(396*Cos[(e + f*x)/2] - 218*Cos[(3*(e + f*x))/2] - 114*Cos[(5*(e + f*x))/ 2] - 396*Sin[(e + f*x)/2] - 405*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2 ]]*Sin[e + f*x] + 405*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] - 218*Sin[(3*(e + f*x))/2] + 114*Sin[(5*(e + f*x))/2] + 135*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] - 135*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[3*(e + f*x)]))/(Csc[(e + f*x)/4]^2 - Se c[(e + f*x)/4]^2)^3))/(192*f*(a*(1 + Sin[e + f*x]))^(5/2))
Time = 2.99 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.97, number of steps used = 30, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.261, Rules used = {3042, 3196, 3042, 3258, 3042, 3463, 27, 3042, 3464, 3042, 3128, 219, 3252, 219, 3523, 27, 3042, 3463, 27, 3042, 3463, 27, 3042, 3464, 3042, 3128, 219, 3252, 219}
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.
| \(\displaystyle \int \frac {\cot ^4(e+f x)}{(a \sin (e+f x)+a)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (e+f x)^4 (a \sin (e+f x)+a)^{5/2}}dx\) |
| \(\Big \downarrow \) 3196 |
| \(\displaystyle \frac {\int \frac {\csc ^4(e+f x) \left (\sin ^2(e+f x)+1\right )}{\sqrt {\sin (e+f x) a+a}}dx}{a^2}-\frac {2 \int \frac {\csc ^3(e+f x)}{\sqrt {\sin (e+f x) a+a}}dx}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (e+f x)^2+1}{\sin (e+f x)^4 \sqrt {\sin (e+f x) a+a}}dx}{a^2}-\frac {2 \int \frac {1}{\sin (e+f x)^3 \sqrt {\sin (e+f x) a+a}}dx}{a^2}\) |
| \(\Big \downarrow \) 3258 |
| \(\displaystyle \frac {\int \frac {\sin (e+f x)^2+1}{\sin (e+f x)^4 \sqrt {\sin (e+f x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {\int \frac {\csc ^2(e+f x) (a-3 a \sin (e+f x))}{\sqrt {\sin (e+f x) a+a}}dx}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (e+f x)^2+1}{\sin (e+f x)^4 \sqrt {\sin (e+f x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {\int \frac {a-3 a \sin (e+f x)}{\sin (e+f x)^2 \sqrt {\sin (e+f x) a+a}}dx}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle \frac {\int \frac {\sin (e+f x)^2+1}{\sin (e+f x)^4 \sqrt {\sin (e+f x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {\frac {\int -\frac {\csc (e+f x) \left (7 a^2-a^2 \sin (e+f x)\right )}{2 \sqrt {\sin (e+f x) a+a}}dx}{a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sin (e+f x)^2+1}{\sin (e+f x)^4 \sqrt {\sin (e+f x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {-\frac {\int \frac {\csc (e+f x) \left (7 a^2-a^2 \sin (e+f x)\right )}{\sqrt {\sin (e+f x) a+a}}dx}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (e+f x)^2+1}{\sin (e+f x)^4 \sqrt {\sin (e+f x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {-\frac {\int \frac {7 a^2-a^2 \sin (e+f x)}{\sin (e+f x) \sqrt {\sin (e+f x) a+a}}dx}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3464 |
| \(\displaystyle \frac {\int \frac {\sin (e+f x)^2+1}{\sin (e+f x)^4 \sqrt {\sin (e+f x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {-\frac {7 a \int \csc (e+f x) \sqrt {\sin (e+f x) a+a}dx-8 a^2 \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (e+f x)^2+1}{\sin (e+f x)^4 \sqrt {\sin (e+f x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {-\frac {7 a \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x)}dx-8 a^2 \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {\int \frac {\sin (e+f x)^2+1}{\sin (e+f x)^4 \sqrt {\sin (e+f x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {-\frac {\frac {16 a^2 \int \frac {1}{2 a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f}+7 a \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x)}dx}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\int \frac {\sin (e+f x)^2+1}{\sin (e+f x)^4 \sqrt {\sin (e+f x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {-\frac {7 a \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x)}dx+\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {\int \frac {\sin (e+f x)^2+1}{\sin (e+f x)^4 \sqrt {\sin (e+f x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {14 a^2 \int \frac {1}{a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f}}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\int \frac {\sin (e+f x)^2+1}{\sin (e+f x)^4 \sqrt {\sin (e+f x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {14 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {\frac {\int -\frac {\csc ^3(e+f x) (a-11 a \sin (e+f x))}{2 \sqrt {\sin (e+f x) a+a}}dx}{3 a}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}}{a^2}-\frac {2 \left (-\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {14 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\csc ^3(e+f x) (a-11 a \sin (e+f x))}{\sqrt {\sin (e+f x) a+a}}dx}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}}{a^2}-\frac {2 \left (-\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {14 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {a-11 a \sin (e+f x)}{\sin (e+f x)^3 \sqrt {\sin (e+f x) a+a}}dx}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}}{a^2}-\frac {2 \left (-\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {14 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {3 \csc ^2(e+f x) \left (15 a^2-a^2 \sin (e+f x)\right )}{2 \sqrt {\sin (e+f x) a+a}}dx}{2 a}-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}}{a^2}-\frac {2 \left (-\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {14 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {3 \int \frac {\csc ^2(e+f x) \left (15 a^2-a^2 \sin (e+f x)\right )}{\sqrt {\sin (e+f x) a+a}}dx}{4 a}-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}}{a^2}-\frac {2 \left (-\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {14 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {3 \int \frac {15 a^2-a^2 \sin (e+f x)}{\sin (e+f x)^2 \sqrt {\sin (e+f x) a+a}}dx}{4 a}-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}}{a^2}-\frac {2 \left (-\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {14 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (\frac {\int -\frac {\csc (e+f x) \left (17 a^3-15 a^3 \sin (e+f x)\right )}{2 \sqrt {\sin (e+f x) a+a}}dx}{a}-\frac {15 a^2 \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )}{4 a}-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}}{a^2}-\frac {2 \left (-\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {14 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {\int \frac {\csc (e+f x) \left (17 a^3-15 a^3 \sin (e+f x)\right )}{\sqrt {\sin (e+f x) a+a}}dx}{2 a}-\frac {15 a^2 \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )}{4 a}-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}}{a^2}-\frac {2 \left (-\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {14 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {\int \frac {17 a^3-15 a^3 \sin (e+f x)}{\sin (e+f x) \sqrt {\sin (e+f x) a+a}}dx}{2 a}-\frac {15 a^2 \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )}{4 a}-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}}{a^2}-\frac {2 \left (-\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {14 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3464 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {17 a^2 \int \csc (e+f x) \sqrt {\sin (e+f x) a+a}dx-32 a^3 \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{2 a}-\frac {15 a^2 \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )}{4 a}-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}}{a^2}-\frac {2 \left (-\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {14 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {17 a^2 \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x)}dx-32 a^3 \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{2 a}-\frac {15 a^2 \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )}{4 a}-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}}{a^2}-\frac {2 \left (-\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {14 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {17 a^2 \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x)}dx+\frac {64 a^3 \int \frac {1}{2 a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f}}{2 a}-\frac {15 a^2 \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )}{4 a}-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}}{a^2}-\frac {2 \left (-\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {14 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {17 a^2 \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x)}dx+\frac {32 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {15 a^2 \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )}{4 a}-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}}{a^2}-\frac {2 \left (-\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {14 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {\frac {32 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {34 a^3 \int \frac {1}{a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f}}{2 a}-\frac {15 a^2 \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )}{4 a}-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}}{a^2}-\frac {2 \left (-\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {14 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {\frac {32 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {34 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {15 a^2 \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )}{4 a}-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}}{a^2}-\frac {2 \left (-\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {14 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{4 a}-\frac {\cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
Input:
Int[Cot[e + f*x]^4/(a + a*Sin[e + f*x])^(5/2),x]
Output:
(-2*(-1/2*(Cot[e + f*x]*Csc[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]) - (-1/2 *((-14*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/f + (8*Sqrt[2]*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*S in[e + f*x]])])/f)/a - (a*Cot[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]))/(4*a )))/a^2 + (-1/3*(Cot[e + f*x]*Csc[e + f*x]^2)/(f*Sqrt[a + a*Sin[e + f*x]]) - (-1/2*(a*Cot[e + f*x]*Csc[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]) - (3*( -1/2*((-34*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]] ])/f + (32*Sqrt[2]*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/f)/a - (15*a^2*Cot[e + f*x])/(f*Sqrt[a + a*Sin[e + f* x]])))/(4*a))/(6*a))/a^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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^4, x_Symbol] :> Simp[-2/(a*b) Int[(a + b*Sin[e + f*x])^(m + 2)/Sin[e + f*x]^ 3, x], x] + Simp[1/a^2 Int[(a + b*Sin[e + f*x])^(m + 2)*((1 + Sin[e + f*x ]^2)/Sin[e + f*x]^4), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && LtQ[m, -1]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)]), x_Symbol] :> Simp[-2*(b/f) Subst[Int[1/(b*c + a*d - d*x^2), x], x, b*(Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/Sqrt[(a_) + (b_.)*sin[(e_. ) + (f_.)*(x_)]], x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((c + d*Sin[e + f*x]) ^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x] - Simp[1/(2* b*(n + 1)*(c^2 - d^2)) Int[(c + d*Sin[e + f*x])^(n + 1)*(Simp[a*d - 2*b*c *(n + 1) + b*d*(2*n + 3)*Sin[e + f*x], x]/Sqrt[a + b*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)) + b*(B*c - A*d)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && Eq Q[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || EqQ[m + 1/2, 0])
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(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/Sqrt[a + b*Sin[e + f*x]], x], x] + Simp[(B*c - A*d)/(b*c - a*d) Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[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_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + 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/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a *d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n + 2) + C* (c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
Time = 0.32 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(-\frac {\left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}\, \left (-135 a^{5} \operatorname {arctanh}\left (\frac {\sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}}{\sqrt {a}}\right ) \sin \left (f x +e \right )^{3}+96 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{5} \sin \left (f x +e \right )^{3}+57 a^{\frac {5}{2}} \left (-a \left (-1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}}-88 a^{\frac {7}{2}} \left (-a \left (-1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}}+39 a^{\frac {9}{2}} \sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}\right )}{24 a^{\frac {15}{2}} \sin \left (f x +e \right )^{3} \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}\) | \(182\) |
Input:
int(cot(f*x+e)^4/(a+sin(f*x+e)*a)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/24/a^(15/2)*(1+sin(f*x+e))*(-a*(-1+sin(f*x+e)))^(1/2)*(-135*a^5*arctanh ((-a*(-1+sin(f*x+e)))^(1/2)/a^(1/2))*sin(f*x+e)^3+96*2^(1/2)*arctanh(1/2*( -a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*a^5*sin(f*x+e)^3+57*a^(5/2)*(-a *(-1+sin(f*x+e)))^(5/2)-88*a^(7/2)*(-a*(-1+sin(f*x+e)))^(3/2)+39*a^(9/2)*( -a*(-1+sin(f*x+e)))^(1/2))/sin(f*x+e)^3/cos(f*x+e)/(a+sin(f*x+e)*a)^(1/2)/ f
Leaf count of result is larger than twice the leaf count of optimal. 564 vs. \(2 (162) = 324\).
Time = 0.11 (sec) , antiderivative size = 564, normalized size of antiderivative = 2.95 \[ \int \frac {\cot ^4(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate(cot(f*x+e)^4/(a+a*sin(f*x+e))^(5/2),x, algorithm="fricas")
Output:
1/96*(135*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 - (cos(f*x + e)^3 + cos(f*x + e)^2 - cos(f*x + e) - 1)*sin(f*x + e) + 1)*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 + 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)*sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1)) + 192*sqrt(2)*(a*cos(f*x + e)^4 - 2*a*cos(f*x + e)^2 - (a*cos(f*x + e )^3 + a*cos(f*x + e)^2 - a*cos(f*x + e) - a)*sin(f*x + e) + a)*log(-(cos(f *x + e)^2 - (cos(f*x + e) - 2)*sin(f*x + e) - 2*sqrt(2)*sqrt(a*sin(f*x + e ) + a)*(cos(f*x + e) - sin(f*x + e) + 1)/sqrt(a) + 3*cos(f*x + e) + 2)/(co s(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(a ) + 4*(57*cos(f*x + e)^3 + 83*cos(f*x + e)^2 - (57*cos(f*x + e)^2 - 26*cos (f*x + e) - 91)*sin(f*x + e) - 65*cos(f*x + e) - 91)*sqrt(a*sin(f*x + e) + a))/(a^3*f*cos(f*x + e)^4 - 2*a^3*f*cos(f*x + e)^2 + a^3*f - (a^3*f*cos(f *x + e)^3 + a^3*f*cos(f*x + e)^2 - a^3*f*cos(f*x + e) - a^3*f)*sin(f*x + e ))
\[ \int \frac {\cot ^4(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx=\int \frac {\cot ^{4}{\left (e + f x \right )}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(cot(f*x+e)**4/(a+a*sin(f*x+e))**(5/2),x)
Output:
Integral(cot(e + f*x)**4/(a*(sin(e + f*x) + 1))**(5/2), x)
Timed out. \[ \int \frac {\cot ^4(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(cot(f*x+e)^4/(a+a*sin(f*x+e))^(5/2),x, algorithm="maxima")
Output:
Timed out
Time = 0.17 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.26 \[ \int \frac {\cot ^4(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx=\frac {\sqrt {2} \sqrt {a} {\left (\frac {135 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {192 \, \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {192 \, \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {4 \, {\left (228 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 176 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 39 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )}}{96 \, f} \] Input:
integrate(cot(f*x+e)^4/(a+a*sin(f*x+e))^(5/2),x, algorithm="giac")
Output:
1/96*sqrt(2)*sqrt(a)*(135*sqrt(2)*log(abs(-2*sqrt(2) + 4*sin(-1/4*pi + 1/2 *f*x + 1/2*e))/abs(2*sqrt(2) + 4*sin(-1/4*pi + 1/2*f*x + 1/2*e)))/(a^3*sgn (cos(-1/4*pi + 1/2*f*x + 1/2*e))) + 192*log(sin(-1/4*pi + 1/2*f*x + 1/2*e) + 1)/(a^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) - 192*log(-sin(-1/4*pi + 1 /2*f*x + 1/2*e) + 1)/(a^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) - 4*(228*si n(-1/4*pi + 1/2*f*x + 1/2*e)^5 - 176*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 + 39 *sin(-1/4*pi + 1/2*f*x + 1/2*e))/((2*sin(-1/4*pi + 1/2*f*x + 1/2*e)^2 - 1) ^3*a^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))))/f
Timed out. \[ \int \frac {\cot ^4(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx=\int \frac {{\mathrm {cot}\left (e+f\,x\right )}^4}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \] Input:
int(cot(e + f*x)^4/(a + a*sin(e + f*x))^(5/2),x)
Output:
int(cot(e + f*x)^4/(a + a*sin(e + f*x))^(5/2), x)
\[ \int \frac {\cot ^4(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \cot \left (f x +e \right )^{4}}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x \right )}{a^{3}} \] Input:
int(cot(f*x+e)^4/(a+a*sin(f*x+e))^(5/2),x)
Output:
(sqrt(a)*int((sqrt(sin(e + f*x) + 1)*cot(e + f*x)**4)/(sin(e + f*x)**3 + 3 *sin(e + f*x)**2 + 3*sin(e + f*x) + 1),x))/a**3