\(\int \frac {(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx\) [243]

Optimal result

Integrand size = 25, antiderivative size = 341 \[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx=\frac {2 \cot (c+d x) (e \cot (c+d x))^{3/2} (1-\sec (c+d x))}{5 a d}-\frac {2 (e \cot (c+d x))^{3/2} (5-3 \sec (c+d x)) \tan (c+d x)}{5 a d}+\frac {6 (e \cot (c+d x))^{3/2} E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sin (c+d x) \tan (c+d x)}{5 a d \sqrt {\sin (2 c+2 d x)}}+\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} a d}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} a d}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right ) (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} a d}-\frac {6 (e \cot (c+d x))^{3/2} \sin (c+d x) \tan ^2(c+d x)}{5 a d} \] Output:

2/5*cot(d*x+c)*(e*cot(d*x+c))^(3/2)*(1-sec(d*x+c))/a/d-2/5*(e*cot(d*x+c))^ 
(3/2)*(5-3*sec(d*x+c))*tan(d*x+c)/a/d-6/5*(e*cot(d*x+c))^(3/2)*EllipticE(c 
os(c+1/4*Pi+d*x),2^(1/2))*sin(d*x+c)*tan(d*x+c)/a/d/sin(2*d*x+2*c)^(1/2)-1 
/2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*(e*cot(d*x+c))^(3/2)*tan(d*x+c)^(3/ 
2)*2^(1/2)/a/d-1/2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*(e*cot(d*x+c))^(3/2) 
*tan(d*x+c)^(3/2)*2^(1/2)/a/d+1/2*arctanh(2^(1/2)*tan(d*x+c)^(1/2)/(1+tan( 
d*x+c)))*(e*cot(d*x+c))^(3/2)*tan(d*x+c)^(3/2)*2^(1/2)/a/d-6/5*(e*cot(d*x+ 
c))^(3/2)*sin(d*x+c)*tan(d*x+c)^2/a/d
 

Mathematica [C] (warning: unable to verify)

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

Time = 13.63 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.93 \[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx=-\frac {e \sqrt {e \cot (c+d x)} \left (30 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right ) \cot ^{\frac {3}{2}}(c+d x)-30 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right ) \cot ^{\frac {3}{2}}(c+d x)+120 \cot ^2(c+d x)-24 \cot ^4(c+d x)+24 \cot ^4(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {1}{2},-\frac {1}{4},-\tan ^2(c+d x)\right )-120 \cot ^2(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{4},\frac {3}{4},-\tan ^2(c+d x)\right )-40 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\tan ^2(c+d x)\right )+15 \sqrt {2} \cot ^{\frac {3}{2}}(c+d x) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-15 \sqrt {2} \cot ^{\frac {3}{2}}(c+d x) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right ) \sec (c+d x) \left (1+\sqrt {\sec ^2(c+d x)}\right ) \sin ^2\left (\frac {1}{2} (c+d x)\right )}{30 a d} \] Input:

Integrate[(e*Cot[c + d*x])^(3/2)/(a + a*Sec[c + d*x]),x]
 

Output:

-1/30*(e*Sqrt[e*Cot[c + d*x]]*(30*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + 
d*x]]]*Cot[c + d*x]^(3/2) - 30*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x 
]]]*Cot[c + d*x]^(3/2) + 120*Cot[c + d*x]^2 - 24*Cot[c + d*x]^4 + 24*Cot[c 
 + d*x]^4*Hypergeometric2F1[-5/4, -1/2, -1/4, -Tan[c + d*x]^2] - 120*Cot[c 
 + d*x]^2*Hypergeometric2F1[-1/2, -1/4, 3/4, -Tan[c + d*x]^2] - 40*Hyperge 
ometric2F1[1/2, 3/4, 7/4, -Tan[c + d*x]^2] + 15*Sqrt[2]*Cot[c + d*x]^(3/2) 
*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 15*Sqrt[2]*Cot[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)/(a*d)
 

Rubi [F]

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.

\(\displaystyle \int \frac {(e \cot (c+d x))^{3/2}}{a \sec (c+d x)+a} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(e \cot (c+d x))^{3/2}}{a \sec (c+d x)+a}dx\)

\(\Big \downarrow \) 4388

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4376

\(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \int -\frac {a-a \sec (c+d x)}{\tan ^{\frac {7}{2}}(c+d x)}dx}{a^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \int \frac {a-a \sec (c+d x)}{\tan ^{\frac {7}{2}}(c+d x)}dx}{a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \int \frac {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}{\left (-\cot \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx}{a^2}\)

\(\Big \downarrow \) 4370

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4370

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4372

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3093

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3095

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3052

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

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

\(\Big \downarrow \) 3957

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 826

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

Input:

Int[(e*Cot[c + d*x])^(3/2)/(a + a*Sec[c + d*x]),x]
 

Output:

$Aborted
 

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 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 4370

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

Result contains complex when optimal does not.

Time = 1.22 (sec) , antiderivative size = 667, normalized size of antiderivative = 1.96

method	result	size
default	\(-\frac {\sqrt {2}\, e \sqrt {e \cot \left (d x +c \right )}\, \left (1-\cos \left (d x +c \right )\right ) \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 (5+5 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 )-5 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 )+12 \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 {EllipticE}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-6 \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 {EllipticF}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-5 \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 )-5 \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 (1-\cos \left (d x +c \right )\right )^{4} \csc \left (d x +c \right )^{4}-6 \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}\right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )}{5 a d \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{2} \sqrt {-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}\)	\(667\)

Input:

int((e*cot(d*x+c))^(3/2)/(a+a*sec(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

-1/5/a/d*2^(1/2)*e*(e*cot(d*x+c))^(1/2)*(1-cos(d*x+c))*(-2*sin(d*x+c)*cos( 
d*x+c)/(1+cos(d*x+c))^2)^(1/2)*(5+5*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))-5*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))+12*(-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)*EllipticE((-cot(d*x+c)+csc(d*x+c)+1)^(1 
/2),1/2*2^(1/2))-6*(-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)*EllipticF((-cot(d*x+c)+csc(d* 
x+c)+1)^(1/2),1/2*2^(1/2))-5*(-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))-5*(-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))+(1-co 
s(d*x+c))^4*csc(d*x+c)^4-6*(1-cos(d*x+c))^2*csc(d*x+c)^2)/((1-cos(d*x+c))^ 
2*csc(d*x+c)^2-1)^2/(-sin(d*x+c)*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*cot(d* 
x+c)*csc(d*x+c)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((e*cot(d*x+c))^(3/2)/(a+a*sec(d*x+c)),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \] Input:

integrate((e*cot(d*x+c))**(3/2)/(a+a*sec(d*x+c)),x)
 

Output:

Integral((e*cot(c + d*x))**(3/2)/(sec(c + d*x) + 1), x)/a
 

Maxima [F(-1)]

Timed out. \[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((e*cot(d*x+c))^(3/2)/(a+a*sec(d*x+c)),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx=\int { \frac {\left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}}{a \sec \left (d x + c\right ) + a} \,d x } \] Input:

integrate((e*cot(d*x+c))^(3/2)/(a+a*sec(d*x+c)),x, algorithm="giac")
 

Output:

integrate((e*cot(d*x + c))^(3/2)/(a*sec(d*x + c) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{a\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \] Input:

int((e*cot(c + d*x))^(3/2)/(a + a/cos(c + d*x)),x)
 

Output:

int((cos(c + d*x)*(e*cot(c + d*x))^(3/2))/(a*(cos(c + d*x) + 1)), x)
 

Reduce [F]

\[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )}{\sec \left (d x +c \right )+1}d x \right ) e}{a} \] Input:

int((e*cot(d*x+c))^(3/2)/(a+a*sec(d*x+c)),x)
 

Output:

(sqrt(e)*int((sqrt(cot(c + d*x))*cot(c + d*x))/(sec(c + d*x) + 1),x)*e)/a