\(\int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{5/2}} \, dx\) [314]

Optimal result
Mathematica [C] (warning: unable to verify)
Rubi [F]
Maple [B] (warning: unable to verify)
Fricas [F(-1)]
Sympy [F]
Maxima [F]
Giac [F]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 25, antiderivative size = 696 \[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{5/2}} \, dx=\frac {a \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2-b^2\right ) d e^{5/2}}-\frac {b^2 \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a \left (a^2-b^2\right ) d e^{5/2}}-\frac {a \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2-b^2\right ) d e^{5/2}}+\frac {b^2 \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a \left (a^2-b^2\right ) d e^{5/2}}-\frac {a \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}+\sqrt {e} \tan (c+d x)}\right )}{\sqrt {2} \left (a^2-b^2\right ) d e^{5/2}}+\frac {b^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}+\sqrt {e} \tan (c+d x)}\right )}{\sqrt {2} a \left (a^2-b^2\right ) d e^{5/2}}-\frac {2 (a-b \sec (c+d x))}{3 \left (a^2-b^2\right ) d e (e \tan (c+d x))^{3/2}}-\frac {2 \sqrt {2} b^3 \operatorname {EllipticPi}\left (\frac {b}{a-\sqrt {a^2-b^2}},\arcsin \left (\frac {\sqrt {-\cos (c+d x)}}{\sqrt {1+\sin (c+d x)}}\right ),-1\right ) \sqrt {\sin (c+d x)}}{a \left (a^2-b^2\right )^{3/2} d e^2 \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}+\frac {2 \sqrt {2} b^3 \operatorname {EllipticPi}\left (\frac {b}{a+\sqrt {a^2-b^2}},\arcsin \left (\frac {\sqrt {-\cos (c+d x)}}{\sqrt {1+\sin (c+d x)}}\right ),-1\right ) \sqrt {\sin (c+d x)}}{a \left (a^2-b^2\right )^{3/2} d e^2 \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}+\frac {b \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 \left (a^2-b^2\right ) d e^2 \sqrt {e \tan (c+d x)}} \] 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 
^(5/2)-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^(5/2)-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^(5/2)+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^(5/2)-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^(5/2)+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^(5/2)-2/3*(a-b*sec(d*x+c))/(a^2-b^2)/d/e/(e*tan(d*x+c))^(3/2 
)-2*2^(1/2)*b^3*EllipticPi((-cos(d*x+c))^(1/2)/(1+sin(d*x+c))^(1/2),b/(a-( 
a^2-b^2)^(1/2)),I)*sin(d*x+c)^(1/2)/a/(a^2-b^2)^(3/2)/d/e^2/(-cos(d*x+c))^ 
(1/2)/(e*tan(d*x+c))^(1/2)+2*2^(1/2)*b^3*EllipticPi((-cos(d*x+c))^(1/2)/(1 
+sin(d*x+c))^(1/2),b/(a+(a^2-b^2)^(1/2)),I)*sin(d*x+c)^(1/2)/a/(a^2-b^2)^( 
3/2)/d/e^2/(-cos(d*x+c))^(1/2)/(e*tan(d*x+c))^(1/2)+1/3*b*InverseJacobiAM( 
c-1/4*Pi+d*x,2^(1/2))*sec(d*x+c)*sin(2*d*x+2*c)^(1/2)/(a^2-b^2)/d/e^2/(e*t 
an(d*x+c))^(1/2)
 

Mathematica [C] (warning: unable to verify)

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

Time = 33.14 (sec) , antiderivative size = 2169, normalized size of antiderivative = 3.12 \[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{5/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[1/((a + b*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2)),x]
 

Output:

((b + a*Cos[c + d*x])*((2*a)/(3*(a^2 - b^2)) - (2*(-a + b*Cos[c + d*x])*Cs 
c[c + d*x]^2)/(3*(-a^2 + b^2)))*Sec[c + d*x]*Tan[c + d*x]^3)/(d*(a + b*Sec 
[c + d*x])*(e*Tan[c + d*x])^(5/2)) - ((b + a*Cos[c + d*x])*Sec[c + d*x]*Ta 
n[c + d*x]^(5/2)*((2*(3*a^2 - 5*b^2)*Sec[c + d*x]^3*(a + b*Sqrt[1 + Tan[c 
+ d*x]^2])*(((-1/8 + I/8)*a*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Tan[c + d* 
x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Tan[c + d*x]] 
)/(a^2 - b^2)^(1/4)] + 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]] - 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]]))/(Sqrt 
[b]*(a^2 - b^2)^(3/4)) + (5*b*(-a^2 + b^2)*AppellF1[1/4, -1/2, 1, 5/4, -Ta 
n[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Tan[c + d*x]]*Sqrt[1 
+ Tan[c + d*x]^2])/((5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, -Tan[c + d* 
x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 
9/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (a^2 - b^2)*Appe 
llF1[5/4, 1/2, 1, 9/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]) 
*Tan[c + d*x]^2)*(a^2 - b^2*(1 + Tan[c + d*x]^2)))))/((b + a*Cos[c + d*x]) 
*(1 + Tan[c + d*x]^2)^2) + (8*a*b*Sec[c + d*x]^2*(a + b*Sqrt[1 + Tan[c + d 
*x]^2])*((Sqrt[b]*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Tan[c + d*x]])/(-a^ 
2 + b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Tan[c + d*x]])/(-a^2 
+ b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[b]*(-a^2 + b^2)^(1/...
 

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.

\(\displaystyle \int \frac {1}{(e \tan (c+d x))^{5/2} (a+b \sec (c+d x))} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4381

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4370

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4372

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3094

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3053

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3120

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

\(\Big \downarrow \) 3957

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 755

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

\(\Big \downarrow \) 4380

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3212

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3208

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3386

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

Input:

Int[1/((a + b*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2)),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 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 3208
Int[Sqrt[(g_.)*tan[(e_.) + (f_.)*(x_)]]/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_ 
)]), x_Symbol] :> Simp[Sqrt[Cos[e + f*x]]*(Sqrt[g*Tan[e + f*x]]/Sqrt[Sin[e 
+ f*x]])   Int[Sqrt[Sin[e + f*x]]/(Sqrt[Cos[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 3386
Int[Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[cos[(e_.) + (f_.)*(x_)]]*((a_ 
) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> With[{q = Rt[-a^2 + b^2, 
2]}, Simp[2*Sqrt[2]*d*((b + q)/(f*q))   Subst[Int[1/((d*(b + q) + a*x^2)*Sq 
rt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]], x] - 
 Simp[2*Sqrt[2]*d*((b - q)/(f*q))   Subst[Int[1/((d*(b - q) + a*x^2)*Sqrt[1 
 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]], x]] /; F 
reeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
 

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 4380
Int[1/(Sqrt[cot[(c_.) + (d_.)*(x_)]*(e_.)]*(csc[(c_.) + (d_.)*(x_)]*(b_.) + 
 (a_))), x_Symbol] :> Simp[1/a   Int[1/Sqrt[e*Cot[c + d*x]], x], x] - Simp[ 
b/a   Int[1/(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. 6738 vs. \(2 (593 ) = 1186\).

Time = 2.14 (sec) , antiderivative size = 6739, normalized size of antiderivative = 9.68

method result size
default \(\text {Expression too large to display}\) \(6739\)

Input:

int(1/(a+b*sec(d*x+c))/(e*tan(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*sec(d*x+c))/(e*tan(d*x+c))^(5/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate(1/(a+b*sec(d*x+c))/(e*tan(d*x+c))**(5/2),x)
 

Output:

Integral(1/((e*tan(c + d*x))**(5/2)*(a + b*sec(c + d*x))), x)
 

Maxima [F]

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

integrate(1/(a+b*sec(d*x+c))/(e*tan(d*x+c))^(5/2),x, algorithm="maxima")
 

Output:

integrate(1/((b*sec(d*x + c) + a)*(e*tan(d*x + c))^(5/2)), x)
 

Giac [F]

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

integrate(1/(a+b*sec(d*x+c))/(e*tan(d*x+c))^(5/2),x, algorithm="giac")
 

Output:

integrate(1/((b*sec(d*x + c) + a)*(e*tan(d*x + c))^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{5/2}} \, dx=\int \frac {\cos \left (c+d\,x\right )}{{\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}\,\left (b+a\,\cos \left (c+d\,x\right )\right )} \,d x \] Input:

int(1/((e*tan(c + d*x))^(5/2)*(a + b/cos(c + d*x))),x)
 

Output:

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

Reduce [F]

\[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{5/2}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\tan \left (d x +c \right )}}{\sec \left (d x +c \right ) \tan \left (d x +c \right )^{3} b +\tan \left (d x +c \right )^{3} a}d x \right )}{e^{3}} \] Input:

int(1/(a+b*sec(d*x+c))/(e*tan(d*x+c))^(5/2),x)
 

Output:

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