\(\int \frac {\sqrt {e \cot (c+d x)}}{a+a \sec (c+d x)} \, dx\) [244]

Optimal result

Integrand size = 25, antiderivative size = 261 \[ \int \frac {\sqrt {e \cot (c+d x)}}{a+a \sec (c+d x)} \, dx=\frac {2 \cot (c+d x) \sqrt {e \cot (c+d x)} (1-\sec (c+d x))}{3 a d}-\frac {\sqrt {e \cot (c+d x)} \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 a d}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {2} a d}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {2} a d}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {2} a d} \] Output:

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

Mathematica [C] (warning: unable to verify)

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

Time = 11.79 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {e \cot (c+d x)}}{a+a \sec (c+d x)} \, dx=-\frac {4 \sqrt {e \cot (c+d x)} \csc (c+d x) \left (\cot ^2(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {1}{2},\frac {1}{4},-\tan ^2(c+d x)\right )+3 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\tan ^2(c+d x)\right )+\cot ^2(c+d x) \left (-1+\operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right )\right ) \left (1+\sqrt {\sec ^2(c+d x)}\right ) \sin ^2\left (\frac {1}{2} (c+d x)\right )}{3 a d} \] Input:

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

Output:

(-4*Sqrt[e*Cot[c + d*x]]*Csc[c + d*x]*(Cot[c + d*x]^2*Hypergeometric2F1[-3 
/4, -1/2, 1/4, -Tan[c + d*x]^2] + 3*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[ 
c + d*x]^2] + Cot[c + d*x]^2*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + 
 d*x]^2]))*(1 + Sqrt[Sec[c + d*x]^2])*Sin[(c + d*x)/2]^2)/(3*a*d)
 

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.94, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.040, Rules used = {3042, 4388, 3042, 4376, 25, 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.

\(\displaystyle \int \frac {\sqrt {e \cot (c+d x)}}{a \sec (c+d x)+a} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {e \cot (c+d x)}}{a \sec (c+d x)+a}dx\)

\(\Big \downarrow \) 4388

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \int \frac {1}{(\sec (c+d x) a+a) \sqrt {\tan (c+d x)}}dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4376

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4370

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4372

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3094

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3053

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3120

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

\(\Big \downarrow \) 3957

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 755

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

-((Sqrt[e*Cot[c + d*x]]*(((-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]*S 
qrt[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)))*Sqrt[Tan[c + d*x]]) 
/a^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 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 = 1.48 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.95

method	result	size
default	\(-\frac {\left (-1+\cos \left (d x +c \right )\right ) \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 )-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 )-8 \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 )+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 )+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 )+2 \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )\right ) \left (1+\cos \left (d x +c \right )\right )^{2} \sec \left (d x +c \right ) \csc \left (d x +c \right )^{2}}{6 a d}\)	\(509\)

Input:

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

Output:

-1/6/a/d*(-1+cos(d*x+c))*(e*cot(d*x+c))^(1/2)*(3*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))-3*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))-8*(-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))+3*(-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))+3*(-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))+2*( 
1-cos(d*x+c))^3*csc(d*x+c)^3-2*csc(d*x+c)+2*cot(d*x+c))*(1+cos(d*x+c))^2*s 
ec(d*x+c)*csc(d*x+c)^2
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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