\(\int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx\) [233]

Optimal result

Integrand size = 23, antiderivative size = 259 \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=-\frac {2 (e \cot (c+d x))^{5/2} (a+a \sec (c+d x)) \tan (c+d x)}{3 d}-\frac {a (e \cot (c+d x))^{5/2} \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)} \tan ^2(c+d x)}{3 d}+\frac {a \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}{\sqrt {2} d}-\frac {a \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}{\sqrt {2} d}-\frac {a \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right ) (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}{\sqrt {2} d} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 14.06 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.71 \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=-\frac {a (e \cot (c+d x))^{5/2} \sec (c+d x) \left (\sqrt {\cot (c+d x)} \left (4 (1+\cos (c+d x)) \cot (c+d x)-3 \arcsin (\cos (c+d x)-\sin (c+d x)) \sqrt {\sin (2 (c+d x))}+3 \log \left (\cos (c+d x)+\sin (c+d x)+\sqrt {\sin (2 (c+d x))}\right ) \sqrt {\sin (2 (c+d x))}\right )+2 \sqrt [4]{-1} \sqrt {\csc ^2(c+d x)} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\cot (c+d x)}\right ),-1\right ) \sin (2 (c+d x))\right )}{6 d \cot ^{\frac {5}{2}}(c+d x)} \] Input:

Integrate[(e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x]),x]
 

Output:

-1/6*(a*(e*Cot[c + d*x])^(5/2)*Sec[c + d*x]*(Sqrt[Cot[c + d*x]]*(4*(1 + Co 
s[c + d*x])*Cot[c + d*x] - 3*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[ 
2*(c + d*x)]] + 3*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]] 
]*Sqrt[Sin[2*(c + d*x)]]) + 2*(-1)^(1/4)*Sqrt[Csc[c + d*x]^2]*EllipticF[I* 
ArcSinh[(-1)^(1/4)*Sqrt[Cot[c + d*x]]], -1]*Sin[2*(c + d*x)]))/(d*Cot[c + 
d*x]^(5/2))
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.93, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4388, 3042, 4370, 27, 3042, 4372, 3042, 3094, 3042, 3053, 3042, 3120, 3957, 266, 755, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4388

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4370

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4372

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3094

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3053

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3120

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

\(\Big \downarrow \) 3957

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

\(\Big \downarrow \) 266

\(\displaystyle \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \left (\frac {1}{3} \left (-\frac {6 a \int \frac {1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {2 (a \sec (c+d x)+a)}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 755

\(\displaystyle \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \left (\frac {1}{3} \left (-\frac {6 a \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {2 (a \sec (c+d x)+a)}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 1476

\(\displaystyle \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \left (\frac {1}{3} \left (-\frac {6 a \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {2 (a \sec (c+d x)+a)}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \left (\frac {1}{3} \left (-\frac {6 a \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {2 (a \sec (c+d x)+a)}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \left (\frac {1}{3} \left (-\frac {6 a \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {2 (a \sec (c+d x)+a)}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 1479

\(\displaystyle \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \left (\frac {1}{3} \left (-\frac {6 a \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {2 (a \sec (c+d x)+a)}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \left (\frac {1}{3} \left (-\frac {6 a \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {2 (a \sec (c+d x)+a)}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \left (\frac {1}{3} \left (-\frac {6 a \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {2 (a \sec (c+d x)+a)}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \left (\frac {1}{3} \left (-\frac {6 a \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {2 (a \sec (c+d x)+a)}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\)

Input:

Int[(e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x]),x]
 

Output:

(e*Cot[c + d*x])^(5/2)*(((-6*a*((-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/ 
Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2])/2 + (-1/2*Log[1 
 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sq 
rt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]))/2))/d - (a*EllipticF[c - Pi/ 
4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(d*Sqrt[Tan[c + d*x]]))/3 
 - (2*(a + a*Sec[c + d*x]))/(3*d*Tan[c + d*x]^(3/2)))*Tan[c + d*x]^(5/2)
 

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 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   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 3053

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_ 
)]]), x_Symbol] :> Simp[Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b 
*Cos[e + f*x]])   Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f 
}, x]
 

rule 3094

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

rule 3120

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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 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 [A] (verified)

Time = 2.28 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.05

method	result	size
parts	\(-\frac {2 a e \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-\frac {e^{2} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}-\frac {a \,e^{2} \sqrt {e \cot \left (d x +c \right )}\, \left (\sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \left (1+\sec \left (d x +c \right )\right ) \operatorname {EllipticF}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+2 \csc \left (d x +c \right )\right )}{3 d}\)	\(272\)
default	\(-\frac {a \left (1+\cos \left (d x +c \right )\right ) e^{2} \sqrt {e \cot \left (d x +c \right )}\, \left (3 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 ) \sin \left (d x +c \right )-3 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 ) \sin \left (d x +c \right )+3 \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 ) \sin \left (d x +c \right )+3 \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 ) \sin \left (d x +c \right )-4 \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 ) \sin \left (d x +c \right )+4 \cos \left (d x +c \right )\right ) \sec \left (d x +c \right ) \csc \left (d x +c \right )}{6 d}\)	\(498\)

Input:

int((e*cot(d*x+c))^(5/2)*(a+a*sec(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

-2*a/d*e*(1/3*(e*cot(d*x+c))^(3/2)-1/8*e^2/(e^2)^(1/4)*2^(1/2)*(ln((e*cot( 
d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c) 
+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/( 
e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d* 
x+c))^(1/2)+1)))-1/3*a/d*e^2*(e*cot(d*x+c))^(1/2)*((-csc(d*x+c)+cot(d*x+c) 
)^(1/2)*(-cot(d*x+c)+csc(d*x+c)+1)^(1/2)*(2*cot(d*x+c)-2*csc(d*x+c)+2)^(1/ 
2)*(1+sec(d*x+c))*EllipticF((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2*2^(1/2))+ 
2*csc(d*x+c))
 

Fricas [F(-1)]

Timed out. \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=\text {Timed out} \] Input:

integrate((e*cot(d*x+c))^(5/2)*(a+a*sec(d*x+c)),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=\text {Timed out} \] Input:

integrate((e*cot(d*x+c))**(5/2)*(a+a*sec(d*x+c)),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*cot(d*x+c))^(5/2)*(a+a*sec(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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

integrate((e*cot(d*x+c))^(5/2)*(a+a*sec(d*x+c)),x, algorithm="giac")
 

Output:

integrate((e*cot(d*x + c))^(5/2)*(a*sec(d*x + c) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=\int {\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}\,\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right ) \,d x \] Input:

int((e*cot(c + d*x))^(5/2)*(a + a/cos(c + d*x)),x)
                                                                                    
                                                                                    
 

Output:

int((e*cot(c + d*x))^(5/2)*(a + a/cos(c + d*x)), x)
 

Reduce [F]

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

int((e*cot(d*x+c))^(5/2)*(a+a*sec(d*x+c)),x)
 

Output:

sqrt(e)*a*e**2*(int(sqrt(cot(c + d*x))*cot(c + d*x)**2*sec(c + d*x),x) + i 
nt(sqrt(cot(c + d*x))*cot(c + d*x)**2,x))