\(\int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx\) [249]

Optimal result

Integrand size = 25, antiderivative size = 308 \[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx=-\frac {6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac {2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}+\frac {6 \cos (c+d x) \cot ^4(c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{5 a d (e \cot (c+d x))^{9/2} \sqrt {\sin (2 c+2 d x)}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)} \] Output:

-6/5*cos(d*x+c)*cot(d*x+c)^3/a/d/(e*cot(d*x+c))^(9/2)-2/15*cot(d*x+c)^3*(5 
-3*sec(d*x+c))/a/d/(e*cot(d*x+c))^(9/2)-6/5*cos(d*x+c)*cot(d*x+c)^4*Ellipt 
icE(cos(c+1/4*Pi+d*x),2^(1/2))/a/d/(e*cot(d*x+c))^(9/2)/sin(2*d*x+2*c)^(1/ 
2)+1/2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)/a/d/(e*cot(d*x+c))^(9/2 
)/tan(d*x+c)^(9/2)+1/2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)/a/d/(e*c 
ot(d*x+c))^(9/2)/tan(d*x+c)^(9/2)-1/2*arctanh(2^(1/2)*tan(d*x+c)^(1/2)/(1+ 
tan(d*x+c)))*2^(1/2)/a/d/(e*cot(d*x+c))^(9/2)/tan(d*x+c)^(9/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 14.51 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx=\frac {\sqrt {e \cot (c+d x)} \left (-8+6 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right ) \cot ^{\frac {3}{2}}(c+d x)-6 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right ) \cot ^{\frac {3}{2}}(c+d x)+8 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},-\tan ^2(c+d x)\right )-8 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\tan ^2(c+d x)\right )+3 \sqrt {2} \cot ^{\frac {3}{2}}(c+d x) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-3 \sqrt {2} \cot ^{\frac {3}{2}}(c+d x) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right ) \sec (c+d x) \left (1+\sqrt {\sec ^2(c+d x)}\right ) \sin ^2\left (\frac {1}{2} (c+d x)\right )}{6 a d e^5} \] Input:

Integrate[1/((e*Cot[c + d*x])^(9/2)*(a + a*Sec[c + d*x])),x]
 

Output:

(Sqrt[e*Cot[c + d*x]]*(-8 + 6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x] 
]]*Cot[c + d*x]^(3/2) - 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*C 
ot[c + d*x]^(3/2) + 8*Hypergeometric2F1[-1/2, 3/4, 7/4, -Tan[c + d*x]^2] - 
 8*Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[c + d*x]^2] + 3*Sqrt[2]*Cot[c + d 
*x]^(3/2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 3*Sqrt[2]*C 
ot[c + d*x]^(3/2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])*Sec[ 
c + d*x]*(1 + Sqrt[Sec[c + d*x]^2])*Sin[(c + d*x)/2]^2)/(6*a*d*e^5)
 

Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.88, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.120, Rules used = {3042, 4388, 3042, 4376, 25, 3042, 4369, 27, 3042, 4372, 3042, 3093, 3042, 3095, 3042, 3052, 3042, 3119, 3957, 266, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \frac {1}{(a \sec (c+d x)+a) (e \cot (c+d x))^{9/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a \sec (c+d x)+a) (e \cot (c+d x))^{9/2}}dx\)

\(\Big \downarrow \) 4388

\(\displaystyle \frac {\int \frac {\tan ^{\frac {9}{2}}(c+d x)}{\sec (c+d x) a+a}dx}{\tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\left (-\cot \left (c+d x+\frac {\pi }{2}\right )\right )^{9/2}}{\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{\tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 4376

\(\displaystyle \frac {\int -\left ((a-a \sec (c+d x)) \tan ^{\frac {5}{2}}(c+d x)\right )dx}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int (a-a \sec (c+d x)) \tan ^{\frac {5}{2}}(c+d x)dx}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \left (-\cot \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 4369

\(\displaystyle -\frac {\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}-\frac {2}{5} \int \frac {1}{2} (5 a-3 a \sec (c+d x)) \sqrt {\tan (c+d x)}dx}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}-\frac {1}{5} \int (5 a-3 a \sec (c+d x)) \sqrt {\tan (c+d x)}dx}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}-\frac {1}{5} \int \sqrt {-\cot \left (c+d x+\frac {\pi }{2}\right )} \left (5 a-3 a \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 4372

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \int \sec (c+d x) \sqrt {\tan (c+d x)}dx-5 a \int \sqrt {\tan (c+d x)}dx\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \int \sec (c+d x) \sqrt {\tan (c+d x)}dx-5 a \int \sqrt {\tan (c+d x)}dx\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 3093

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-2 \int \cos (c+d x) \sqrt {\tan (c+d x)}dx\right )-5 a \int \sqrt {\tan (c+d x)}dx\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-2 \int \frac {\sqrt {\tan (c+d x)}}{\sec (c+d x)}dx\right )-5 a \int \sqrt {\tan (c+d x)}dx\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 3095

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\tan (c+d x)} \int \sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}\right )-5 a \int \sqrt {\tan (c+d x)}dx\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\tan (c+d x)} \int \sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}\right )-5 a \int \sqrt {\tan (c+d x)}dx\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 3052

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} \int \sqrt {\sin (2 c+2 d x)}dx}{\sqrt {\sin (2 c+2 d x)}}\right )-5 a \int \sqrt {\tan (c+d x)}dx\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} \int \sqrt {\sin (2 c+2 d x)}dx}{\sqrt {\sin (2 c+2 d x)}}\right )-5 a \int \sqrt {\tan (c+d x)}dx\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 3119

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )-5 a \int \sqrt {\tan (c+d x)}dx\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 3957

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )-\frac {5 a \int \frac {\sqrt {\tan (c+d x)}}{\tan ^2(c+d x)+1}d\tan (c+d x)}{d}\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 266

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )-\frac {10 a \int \frac {\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 826

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )-\frac {10 a \left (\frac {1}{2} \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 1476

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )-\frac {10 a \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )-\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 1082

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )-\frac {10 a \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )-\frac {10 a \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 1479

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )-\frac {10 a \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )-\frac {10 a \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )-\frac {10 a \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

\(\Big \downarrow \) 1103

\(\displaystyle -\frac {\frac {1}{5} \left (3 a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )-\frac {10 a \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}\right )+\frac {2 \tan ^{\frac {3}{2}}(c+d x) (5 a-3 a \sec (c+d x))}{15 d}}{a^2 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\)

Input:

Int[1/((e*Cot[c + d*x])^(9/2)*(a + a*Sec[c + d*x])),x]
 

Output:

-(((2*(5*a - 3*a*Sec[c + d*x])*Tan[c + d*x]^(3/2))/(15*d) + ((-10*a*((-(Ar 
cTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Ta 
n[c + d*x]]]/Sqrt[2])/2 + (Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d* 
x]]/(2*Sqrt[2]) - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sq 
rt[2]))/2))/d + 3*a*((-2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[Ta 
n[c + d*x]])/(d*Sqrt[Sin[2*c + 2*d*x]]) + (2*Cos[c + d*x]*Tan[c + d*x]^(3/ 
2))/d))/5)/(a^2*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x]^(9/2)))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3052

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]] 
, x_Symbol] :> Simp[Sqrt[a*Sin[e + f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e 
 + 2*f*x]])   Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f}, x]
 

rule 3093

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_), x_Symbol] :> Simp[a^2*(a*Sec[e + f*x])^(m - 2)*((b*Tan[e + f*x])^(n + 
1)/(b*f*(m + n - 1))), x] + Simp[a^2*((m - 2)/(m + n - 1))   Int[(a*Sec[e + 
 f*x])^(m - 2)*(b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && ( 
GtQ[m, 1] || (EqQ[m, 1] && EqQ[n, 1/2])) && NeQ[m + n - 1, 0] && IntegersQ[ 
2*m, 2*n]
 

rule 3095

Int[Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]]/sec[(e_.) + (f_.)*(x_)], x_Symbol] 
:> Simp[Sqrt[Cos[e + f*x]]*(Sqrt[b*Tan[e + f*x]]/Sqrt[Sin[e + f*x]])   Int[ 
Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]], x], x] /; FreeQ[{b, e, f}, x]
 

rule 3119

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* 
(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

rule 3957

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d   Subst[Int 
[x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && 
!IntegerQ[n]
 

rule 4369

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( 
a_)), x_Symbol] :> Simp[(-e)*(e*Cot[c + d*x])^(m - 1)*((a*m + b*(m - 1)*Csc 
[c + d*x])/(d*m*(m - 1))), x] - Simp[e^2/m   Int[(e*Cot[c + d*x])^(m - 2)*( 
a*m + b*(m - 1)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m 
, 1]
 

rule 4372

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + 
(a_)), x_Symbol] :> Simp[a   Int[(e*Cot[c + d*x])^m, x], x] + Simp[b   Int[ 
(e*Cot[c + d*x])^m*Csc[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, m}, x]
 

rule 4376

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( 
a_))^(n_), x_Symbol] :> Simp[a^(2*n)/e^(2*n)   Int[(e*Cot[c + d*x])^(m + 2* 
n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[a 
^2 - b^2, 0] && ILtQ[n, 0]
 

rule 4388

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*((a_) + (b_.)*sec[(c_.) + (d_.)*(x 
_)])^(n_.), x_Symbol] :> Simp[(e*Cot[c + d*x])^m*Tan[c + d*x]^m   Int[(a + 
b*Sec[c + d*x])^n/Tan[c + d*x]^m, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] 
 &&  !IntegerQ[m]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.45 (sec) , antiderivative size = 722, normalized size of antiderivative = 2.34

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {-\frac {2 \sin \left (d x +c \right ) \cos \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \left (i \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \left (15 \cot \left (d x +c \right )+15 \csc \left (d x +c \right )\right )+i \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \left (-15 \cot \left (d x +c \right )-15 \csc \left (d x +c \right )\right )+\sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \left (-36 \cot \left (d x +c \right )-36 \csc \left (d x +c \right )\right )+\sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \operatorname {EllipticF}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (18 \csc \left (d x +c \right )+18 \cot \left (d x +c \right )\right )+\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \left (-15 \cot \left (d x +c \right )-15 \csc \left (d x +c \right )\right )+\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \left (-15 \cot \left (d x +c \right )-15 \csc \left (d x +c \right )\right )+56 \cot \left (d x +c \right )-48 \csc \left (d x +c \right )-20 \sec \left (d x +c \right ) \csc \left (d x +c \right )+12 \sec \left (d x +c \right )^{2} \csc \left (d x +c \right )\right )}{60 a d \sqrt {e \cot \left (d x +c \right )}\, \sqrt {-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, e^{4}}\)	\(722\)

Input:

int(1/(e*cot(d*x+c))^(9/2)/(a+a*sec(d*x+c)),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

1/60/a/d*2^(1/2)*(-2*sin(d*x+c)*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)/(e*cot( 
d*x+c))^(1/2)/(-sin(d*x+c)*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)/e^4*(I*(-cot 
(d*x+c)+csc(d*x+c)+1)^(1/2)*(2*cot(d*x+c)-2*csc(d*x+c)+2)^(1/2)*(-csc(d*x+ 
c)+cot(d*x+c))^(1/2)*EllipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2-1/2*I 
,1/2*2^(1/2))*(15*cot(d*x+c)+15*csc(d*x+c))+I*(-cot(d*x+c)+csc(d*x+c)+1)^( 
1/2)*(2*cot(d*x+c)-2*csc(d*x+c)+2)^(1/2)*(-csc(d*x+c)+cot(d*x+c))^(1/2)*El 
lipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-15*cot( 
d*x+c)-15*csc(d*x+c))+(2*cot(d*x+c)-2*csc(d*x+c)+2)^(1/2)*(-csc(d*x+c)+cot 
(d*x+c))^(1/2)*EllipticE((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2*2^(1/2))*(-c 
ot(d*x+c)+csc(d*x+c)+1)^(1/2)*(-36*cot(d*x+c)-36*csc(d*x+c))+(2*cot(d*x+c) 
-2*csc(d*x+c)+2)^(1/2)*(-csc(d*x+c)+cot(d*x+c))^(1/2)*(-cot(d*x+c)+csc(d*x 
+c)+1)^(1/2)*EllipticF((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2*2^(1/2))*(18*c 
sc(d*x+c)+18*cot(d*x+c))+(-cot(d*x+c)+csc(d*x+c)+1)^(1/2)*(2*cot(d*x+c)-2* 
csc(d*x+c)+2)^(1/2)*(-csc(d*x+c)+cot(d*x+c))^(1/2)*EllipticPi((-cot(d*x+c) 
+csc(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-15*cot(d*x+c)-15*csc(d*x+c)) 
+(-cot(d*x+c)+csc(d*x+c)+1)^(1/2)*(2*cot(d*x+c)-2*csc(d*x+c)+2)^(1/2)*(-cs 
c(d*x+c)+cot(d*x+c))^(1/2)*EllipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2 
+1/2*I,1/2*2^(1/2))*(-15*cot(d*x+c)-15*csc(d*x+c))+56*cot(d*x+c)-48*csc(d* 
x+c)-20*sec(d*x+c)*csc(d*x+c)+12*sec(d*x+c)^2*csc(d*x+c))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx=\text {Timed out} \] Input:

integrate(1/(e*cot(d*x+c))^(9/2)/(a+a*sec(d*x+c)),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx=\text {Timed out} \] Input:

integrate(1/(e*cot(d*x+c))**(9/2)/(a+a*sec(d*x+c)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\left (e \cot \left (d x + c\right )\right )^{\frac {9}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \] Input:

integrate(1/(e*cot(d*x+c))^(9/2)/(a+a*sec(d*x+c)),x, algorithm="maxima")
 

Output:

integrate(1/((e*cot(d*x + c))^(9/2)*(a*sec(d*x + c) + a)), x)
 

Giac [F]

\[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\left (e \cot \left (d x + c\right )\right )^{\frac {9}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \] Input:

integrate(1/(e*cot(d*x+c))^(9/2)/(a+a*sec(d*x+c)),x, algorithm="giac")
 

Output:

integrate(1/((e*cot(d*x + c))^(9/2)*(a*sec(d*x + c) + a)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx=\int \frac {\cos \left (c+d\,x\right )}{a\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{9/2}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \] Input:

int(1/((e*cot(c + d*x))^(9/2)*(a + a/cos(c + d*x))),x)
 

Output:

int(cos(c + d*x)/(a*(e*cot(c + d*x))^(9/2)*(cos(c + d*x) + 1)), x)
 

Reduce [F]

\[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )^{5} \sec \left (d x +c \right )+\cot \left (d x +c \right )^{5}}d x \right )}{a \,e^{5}} \] Input:

int(1/(e*cot(d*x+c))^(9/2)/(a+a*sec(d*x+c)),x)
 

Output:

(sqrt(e)*int(sqrt(cot(c + d*x))/(cot(c + d*x)**5*sec(c + d*x) + cot(c + d* 
x)**5),x))/(a*e**5)