\(\int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{3/2}} \, dx\) [313]

Optimal result

Integrand size = 25, antiderivative size = 723 \[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{3/2}} \, dx=\frac {a \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2-b^2\right ) d e^{3/2}}-\frac {b^2 \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a \left (a^2-b^2\right ) d e^{3/2}}-\frac {a \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2-b^2\right ) d e^{3/2}}+\frac {b^2 \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a \left (a^2-b^2\right ) d e^{3/2}}+\frac {a \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}+\sqrt {e} \tan (c+d x)}\right )}{\sqrt {2} \left (a^2-b^2\right ) d e^{3/2}}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}+\sqrt {e} \tan (c+d x)}\right )}{\sqrt {2} a \left (a^2-b^2\right ) d e^{3/2}}-\frac {2 (a-b \sec (c+d x))}{\left (a^2-b^2\right ) d e \sqrt {e \tan (c+d x)}}+\frac {2 \sqrt {2} b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right ),-1\right ) \sqrt {e \tan (c+d x)}}{a (a-b)^{3/2} (a+b)^{3/2} d e^2 \sqrt {\sin (c+d x)}}-\frac {2 \sqrt {2} b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right ),-1\right ) \sqrt {e \tan (c+d x)}}{a (a-b)^{3/2} (a+b)^{3/2} d e^2 \sqrt {\sin (c+d x)}}+\frac {2 b \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{\left (a^2-b^2\right ) d e^2 \sqrt {\sin (2 c+2 d x)}}-\frac {2 b \cos (c+d x) (e \tan (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e^3} \] Output:

1/2*a*arctan(1-2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))*2^(1/2)/(a^2-b^2)/d/e 
^(3/2)-1/2*b^2*arctan(1-2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))*2^(1/2)/a/(a 
^2-b^2)/d/e^(3/2)-1/2*a*arctan(1+2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))*2^( 
1/2)/(a^2-b^2)/d/e^(3/2)+1/2*b^2*arctan(1+2^(1/2)*(e*tan(d*x+c))^(1/2)/e^( 
1/2))*2^(1/2)/a/(a^2-b^2)/d/e^(3/2)+1/2*a*arctanh(2^(1/2)*(e*tan(d*x+c))^( 
1/2)/(e^(1/2)+e^(1/2)*tan(d*x+c)))*2^(1/2)/(a^2-b^2)/d/e^(3/2)-1/2*b^2*arc 
tanh(2^(1/2)*(e*tan(d*x+c))^(1/2)/(e^(1/2)+e^(1/2)*tan(d*x+c)))*2^(1/2)/a/ 
(a^2-b^2)/d/e^(3/2)-2*(a-b*sec(d*x+c))/(a^2-b^2)/d/e/(e*tan(d*x+c))^(1/2)+ 
2*2^(1/2)*b^3*cos(d*x+c)^(1/2)*EllipticPi(sin(d*x+c)^(1/2)/(1+cos(d*x+c))^ 
(1/2),-(a-b)^(1/2)/(a+b)^(1/2),I)*(e*tan(d*x+c))^(1/2)/a/(a-b)^(3/2)/(a+b) 
^(3/2)/d/e^2/sin(d*x+c)^(1/2)-2*2^(1/2)*b^3*cos(d*x+c)^(1/2)*EllipticPi(si 
n(d*x+c)^(1/2)/(1+cos(d*x+c))^(1/2),(a-b)^(1/2)/(a+b)^(1/2),I)*(e*tan(d*x+ 
c))^(1/2)/a/(a-b)^(3/2)/(a+b)^(3/2)/d/e^2/sin(d*x+c)^(1/2)-2*b*cos(d*x+c)* 
EllipticE(cos(c+1/4*Pi+d*x),2^(1/2))*(e*tan(d*x+c))^(1/2)/(a^2-b^2)/d/e^2/ 
sin(2*d*x+2*c)^(1/2)-2*b*cos(d*x+c)*(e*tan(d*x+c))^(3/2)/(a^2-b^2)/d/e^3
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 22.49 (sec) , antiderivative size = 1571, normalized size of antiderivative = 2.17 \[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{3/2}} \, dx =\text {Too large to display} \] Input:

Integrate[1/((a + b*Sec[c + d*x])*(e*Tan[c + d*x])^(3/2)),x]
 

Output:

((b + a*Cos[c + d*x])*Sec[c + d*x]*((-2*(b - a*Cos[c + d*x])*Csc[c + d*x]) 
/(-a^2 + b^2) + (2*b*Sin[c + d*x])/(-a^2 + b^2))*Tan[c + d*x]^2)/(d*(a + b 
*Sec[c + d*x])*(e*Tan[c + d*x])^(3/2)) + ((b + a*Cos[c + d*x])*Sec[c + d*x 
]*Tan[c + d*x]^(3/2)*(-1/12*((-a^2 + 3*b^2)*Sec[c + d*x]*(6*Sqrt[2]*(a^2 - 
 b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 6*Sqrt[2]*a^2*ArcTan[1 + Sq 
rt[2]*Sqrt[Tan[c + d*x]]] + 6*Sqrt[2]*b^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + 
d*x]]] - (6 + 6*I)*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 - ((1 + I)*Sqrt[b]*S 
qrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)] + (6 + 6*I)*Sqrt[b]*(a^2 - b^2)^(3/4 
)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)] - 3*S 
qrt[2]*a^2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 3*Sqrt[2]* 
b^2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 3*Sqrt[2]*a^2*Log 
[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - 3*Sqrt[2]*b^2*Log[1 + Sq 
rt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + (3 + 3*I)*Sqrt[b]*(a^2 - b^2)^( 
3/4)*Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + 
d*x]] + I*b*Tan[c + d*x]] - (3 + 3*I)*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a 
^2 - b^2] + (1 + I)*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + d*x]] + I*b*Tan 
[c + d*x]] + 8*a*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[c + d*x]^2, (b^2*Tan[c 
+ d*x]^2)/(a^2 - b^2)]*Tan[c + d*x]^(3/2))*(a + b*Sqrt[1 + Tan[c + d*x]^2] 
))/((a^3 - a*b^2)*(b + a*Cos[c + d*x])*(1 + Tan[c + d*x]^2)) + (b*Cos[2*(c 
 + d*x)]*Sec[c + d*x]^2*(-84*Sqrt[2]*b*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + ...
 

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4381

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4370

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4372

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3093

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3095

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3052

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

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

\(\Big \downarrow \) 3957

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 826

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

\(\Big \downarrow \) 4378

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3212

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3209

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

Input:

Int[1/((a + b*Sec[c + d*x])*(e*Tan[c + d*x])^(3/2)),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 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 3209

Int[1/(((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(g_)*tan[(e_.) + (f_.)*( 
x_)]]), x_Symbol] :> Simp[Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*Sqrt[g*Tan 
[e + f*x]])   Int[Sqrt[Cos[e + f*x]]/(Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x 
])), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
 

rule 3212

Int[(cot[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_.), x_Symbol] :> Simp[g^(2*IntPart[p])*(g*Cot[e + f*x])^FracPart[p] 
*(g*Tan[e + f*x])^FracPart[p]   Int[(a + b*Sin[e + f*x])^m/(g*Tan[e + f*x]) 
^p, x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] &&  !IntegerQ[p]
 

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 4378

Int[Sqrt[cot[(c_.) + (d_.)*(x_)]*(e_.)]/(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a 
_)), x_Symbol] :> Simp[1/a   Int[Sqrt[e*Cot[c + d*x]], x], x] - Simp[b/a 
Int[Sqrt[e*Cot[c + d*x]]/(b + a*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, 
 e}, x] && NeQ[a^2 - b^2, 0]
 

rule 4381

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)/(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( 
a_)), x_Symbol] :> Simp[1/(a^2 - b^2)   Int[(e*Cot[c + d*x])^m*(a - b*Csc[c 
 + d*x]), x], x] + Simp[b^2/(e^2*(a^2 - b^2))   Int[(e*Cot[c + d*x])^(m + 2 
)/(a + b*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^ 
2, 0] && ILtQ[m + 1/2, 0]
 

Maple [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2683 vs. \(2 (615 ) = 1230\).

Time = 1.45 (sec) , antiderivative size = 2684, normalized size of antiderivative = 3.71

method	result	size
default	\(\text {Expression too large to display}\)	\(2684\)

Input:

int(1/(a+b*sec(d*x+c))/(e*tan(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-2/d*2^(1/2)*b/((a^2-b^2)^(1/2)-a+b)/((a^2-b^2)^(1/2)+a-b)/(a-b)/a/(a+b)*( 
1-cos(d*x+c))*(-2*sin(d*x+c)*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*(2*a*b^2-4 
*a^2*b+2*a^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))*a^3+4*a^2*b*(1-cos(d*x+c))^2*csc(d*x+c)^ 
2-2*a*b^2*(1-cos(d*x+c))^2*csc(d*x+c)^2+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)*Ellipt 
icPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*a*b^2-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))*a^2*b-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)+cs 
c(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*a*b^2+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))*a^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))*b^3+(a^2-b^2)^(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)*(-csc(d*x+c)+cot(d*x+c))^(1/2)*EllipticP 
i((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),(a-b)/(a-b+((a+b)*(a-b))^(1/2)),1/2*...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*sec(d*x+c))/(e*tan(d*x+c))^(3/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate(1/(a+b*sec(d*x+c))/(e*tan(d*x+c))**(3/2),x)
 

Output:

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

Maxima [F]

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

integrate(1/(a+b*sec(d*x+c))/(e*tan(d*x+c))^(3/2),x, algorithm="maxima")
 

Output:

integrate(1/((b*sec(d*x + c) + a)*(e*tan(d*x + c))^(3/2)), x)
 

Giac [F]

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

integrate(1/(a+b*sec(d*x+c))/(e*tan(d*x+c))^(3/2),x, algorithm="giac")
 

Output:

integrate(1/((b*sec(d*x + c) + a)*(e*tan(d*x + c))^(3/2)), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int(1/(a+b*sec(d*x+c))/(e*tan(d*x+c))^(3/2),x)
 

Output:

(sqrt(e)*int(sqrt(tan(c + d*x))/(sec(c + d*x)*tan(c + d*x)**2*b + tan(c + 
d*x)**2*a),x))/e**2