\(\int \frac {\tan ^4(e+f x)}{(a+b \sin ^2(e+f x))^{5/2}} \, dx\) [469]

Optimal result

Integrand size = 25, antiderivative size = 348 \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {(5 a-3 b) b \cos (e+f x) \sin (e+f x)}{3 (a+b)^3 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {8 (a-b) b \cos (e+f x) \sin (e+f x)}{3 (a+b)^4 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {8 (a-b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 (a+b)^4 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {(5 a-3 b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 (a+b)^3 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (2 a-b) \tan (e+f x)}{3 (a+b)^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\sec ^2(e+f x) \tan (e+f x)}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \] Output:

1/3*(5*a-3*b)*b*cos(f*x+e)*sin(f*x+e)/(a+b)^3/f/(a+b*sin(f*x+e)^2)^(3/2)+8 
/3*(a-b)*b*cos(f*x+e)*sin(f*x+e)/(a+b)^4/f/(a+b*sin(f*x+e)^2)^(1/2)+8/3*(a 
-b)*(cos(f*x+e)^2)^(1/2)*EllipticE(sin(f*x+e),(-b/a)^(1/2))*sec(f*x+e)*(a+ 
b*sin(f*x+e)^2)^(1/2)/(a+b)^4/f/(1+b*sin(f*x+e)^2/a)^(1/2)-1/3*(5*a-3*b)*( 
cos(f*x+e)^2)^(1/2)*EllipticF(sin(f*x+e),(-b/a)^(1/2))*sec(f*x+e)*(1+b*sin 
(f*x+e)^2/a)^(1/2)/(a+b)^3/f/(a+b*sin(f*x+e)^2)^(1/2)-2/3*(2*a-b)*tan(f*x+ 
e)/(a+b)^2/f/(a+b*sin(f*x+e)^2)^(3/2)+1/3*sec(f*x+e)^2*tan(f*x+e)/(a+b)/f/ 
(a+b*sin(f*x+e)^2)^(3/2)
 

Mathematica [A] (verified)

Time = 4.43 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.68 \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {2 a b \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} \left (8 a (a-b) E\left (e+f x\left |-\frac {b}{a}\right .\right )+\left (-5 a^2-2 a b+3 b^2\right ) \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )\right )+\sqrt {2} b \left (2 a b (a+b) \sin (2 (e+f x))+4 (a-b) b (2 a+b-b \cos (2 (e+f x))) \sin (2 (e+f x))-4 (a-b) (2 a+b-b \cos (2 (e+f x)))^2 \tan (e+f x)+(a+b) (2 a+b-b \cos (2 (e+f x)))^2 \sec ^2(e+f x) \tan (e+f x)\right )}{6 b (a+b)^4 f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \] Input:

Integrate[Tan[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(5/2),x]
 

Output:

(2*a*b*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*(8*a*(a - b)*EllipticE[e + 
 f*x, -(b/a)] + (-5*a^2 - 2*a*b + 3*b^2)*EllipticF[e + f*x, -(b/a)]) + Sqr 
t[2]*b*(2*a*b*(a + b)*Sin[2*(e + f*x)] + 4*(a - b)*b*(2*a + b - b*Cos[2*(e 
 + f*x)])*Sin[2*(e + f*x)] - 4*(a - b)*(2*a + b - b*Cos[2*(e + f*x)])^2*Ta 
n[e + f*x] + (a + b)*(2*a + b - b*Cos[2*(e + f*x)])^2*Sec[e + f*x]^2*Tan[e 
 + f*x]))/(6*b*(a + b)^4*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.09, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 3675, 372, 402, 27, 402, 25, 27, 402, 25, 27, 399, 323, 321, 330, 327}

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.

\(\displaystyle \int \frac {\tan ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\tan (e+f x)^4}{\left (a+b \sin (e+f x)^2\right )^{5/2}}dx\)

\(\Big \downarrow \) 3675

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \int \frac {\sin ^4(e+f x)}{\left (1-\sin ^2(e+f x)\right )^{5/2} \left (b \sin ^2(e+f x)+a\right )^{5/2}}d\sin (e+f x)}{f}\)

\(\Big \downarrow \) 372

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{3 (a+b) \left (1-\sin ^2(e+f x)\right )^{3/2} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (2 a-b) \sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {3 \left (\frac {\frac {8 b (a-b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{(a+b) \sqrt {a+b \sin ^2(e+f x)}}-\frac {\int -\frac {a \left (8 (a-b) b \sin ^2(e+f x)+(a-3 b) (3 a-b)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a (a+b)}}{3 (a+b)}+\frac {b (5 a-3 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{a+b}}{3 (a+b)}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{3 (a+b) \left (1-\sin ^2(e+f x)\right )^{3/2} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (2 a-b) \sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {3 \left (\frac {\frac {\int \frac {a \left (8 (a-b) b \sin ^2(e+f x)+(a-3 b) (3 a-b)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a (a+b)}+\frac {8 b (a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{(a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 (a+b)}+\frac {b (5 a-3 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{a+b}}{3 (a+b)}\right )}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{3 (a+b) \left (1-\sin ^2(e+f x)\right )^{3/2} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (2 a-b) \sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {3 \left (\frac {\frac {\int \frac {8 (a-b) b \sin ^2(e+f x)+(a-3 b) (3 a-b)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a+b}+\frac {8 b (a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{(a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 (a+b)}+\frac {b (5 a-3 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{a+b}}{3 (a+b)}\right )}{f}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{3 (a+b) \left (1-\sin ^2(e+f x)\right )^{3/2} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (2 a-b) \sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {3 \left (\frac {\frac {8 (a-b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-(5 a-3 b) (a+b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a+b}+\frac {8 b (a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{(a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 (a+b)}+\frac {b (5 a-3 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{a+b}}{3 (a+b)}\right )}{f}\)

\(\Big \downarrow \) 323

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{3 (a+b) \left (1-\sin ^2(e+f x)\right )^{3/2} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (2 a-b) \sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {3 \left (\frac {\frac {8 (a-b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-\frac {(5 a-3 b) (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}d\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}}{a+b}+\frac {8 b (a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{(a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 (a+b)}+\frac {b (5 a-3 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{a+b}}{3 (a+b)}\right )}{f}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{3 (a+b) \left (1-\sin ^2(e+f x)\right )^{3/2} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (2 a-b) \sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {3 \left (\frac {\frac {8 (a-b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-\frac {(5 a-3 b) (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a+b}+\frac {8 b (a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{(a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 (a+b)}+\frac {b (5 a-3 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{a+b}}{3 (a+b)}\right )}{f}\)

\(\Big \downarrow \) 330

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{3 (a+b) \left (1-\sin ^2(e+f x)\right )^{3/2} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (2 a-b) \sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {3 \left (\frac {\frac {\frac {8 (a-b) \sqrt {a+b \sin ^2(e+f x)} \int \frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {(5 a-3 b) (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a+b}+\frac {8 b (a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{(a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 (a+b)}+\frac {b (5 a-3 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{a+b}}{3 (a+b)}\right )}{f}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{3 (a+b) \left (1-\sin ^2(e+f x)\right )^{3/2} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (2 a-b) \sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {3 \left (\frac {\frac {\frac {8 (a-b) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {(5 a-3 b) (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a+b}+\frac {8 b (a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{(a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 (a+b)}+\frac {b (5 a-3 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{a+b}}{3 (a+b)}\right )}{f}\)

Input:

Int[Tan[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(5/2),x]
 

Output:

(Sqrt[Cos[e + f*x]^2]*Sec[e + f*x]*(Sin[e + f*x]/(3*(a + b)*(1 - Sin[e + f 
*x]^2)^(3/2)*(a + b*Sin[e + f*x]^2)^(3/2)) - ((2*(2*a - b)*Sin[e + f*x])/( 
(a + b)*Sqrt[1 - Sin[e + f*x]^2]*(a + b*Sin[e + f*x]^2)^(3/2)) - (3*(((5*a 
 - 3*b)*b*Sin[e + f*x]*Sqrt[1 - Sin[e + f*x]^2])/(3*(a + b)*(a + b*Sin[e + 
 f*x]^2)^(3/2)) + ((8*(a - b)*b*Sin[e + f*x]*Sqrt[1 - Sin[e + f*x]^2])/((a 
 + b)*Sqrt[a + b*Sin[e + f*x]^2]) + ((8*(a - b)*EllipticE[ArcSin[Sin[e + f 
*x]], -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/Sqrt[1 + (b*Sin[e + f*x]^2)/a] - 
 ((5*a - 3*b)*(a + b)*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sqrt[1 + (b* 
Sin[e + f*x]^2)/a])/Sqrt[a + b*Sin[e + f*x]^2])/(a + b))/(3*(a + b))))/(a 
+ b))/(3*(a + b))))/f
 

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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 323

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( 
d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 330

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]   Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ[a, 
0]
 

rule 372

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 
)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 
))   Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + 
 (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, 
e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a 
, b, c, d, e, m, 2, p, q, x]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3675

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^ 
(m_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff^(m + 1 
)*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x]))   Subst[Int[x^m*((a + b*ff^2*x^2) 
^p/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b 
, e, f, p}, x] && IntegerQ[m/2] &&  !IntegerQ[p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(666\) vs. \(2(318)=636\).

Time = 7.41 (sec) , antiderivative size = 667, normalized size of antiderivative = 1.92

method	result	size
default	\(-\frac {-8 \sqrt {-b \cos \left (f x +e \right )^{4}+\left (a +b \right ) \cos \left (f x +e \right )^{2}}\, b^{2} \left (a -b \right ) \cos \left (f x +e \right )^{6} \sin \left (f x +e \right )+\sqrt {-b \cos \left (f x +e \right )^{4}+\left (a +b \right ) \cos \left (f x +e \right )^{2}}\, b \left (13 a^{2}+2 a b -11 b^{2}\right ) \cos \left (f x +e \right )^{4} \sin \left (f x +e \right )+\sqrt {-b \cos \left (f x +e \right )^{4}+\left (a +b \right ) \cos \left (f x +e \right )^{2}}\, \sqrt {-\frac {b \cos \left (f x +e \right )^{2}}{a}+\frac {a +b}{a}}\, \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, b \left (5 \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}+2 \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b -3 \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}-8 \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}+8 \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b \right ) \cos \left (f x +e \right )^{4}-2 \sqrt {-b \cos \left (f x +e \right )^{4}+\left (a +b \right ) \cos \left (f x +e \right )^{2}}\, \left (2 a^{3}+3 a^{2} b -b^{3}\right ) \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )-\sqrt {-b \cos \left (f x +e \right )^{4}+\left (a +b \right ) \cos \left (f x +e \right )^{2}}\, \sqrt {-\frac {b \cos \left (f x +e \right )^{2}}{a}+\frac {a +b}{a}}\, \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \left (5 \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+7 \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b -\operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}-3 \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{3}-8 \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+8 \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}\right ) \cos \left (f x +e \right )^{2}+\sqrt {-b \cos \left (f x +e \right )^{4}+\left (a +b \right ) \cos \left (f x +e \right )^{2}}\, \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right ) \sin \left (f x +e \right )}{3 \left (1+\sin \left (f x +e \right )\right ) \sqrt {-\left (a +b \sin \left (f x +e \right )^{2}\right ) \left (\sin \left (f x +e \right )-1\right ) \left (1+\sin \left (f x +e \right )\right )}\, \left (\sin \left (f x +e \right )-1\right ) \left (a +b \sin \left (f x +e \right )^{2}\right )^{\frac {3}{2}} \left (a +b \right )^{4} \cos \left (f x +e \right ) f}\)	\(667\)

Input:

int(tan(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/3*(-8*(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*b^2*(a-b)*cos(f*x+e)^6 
*sin(f*x+e)+(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*b*(13*a^2+2*a*b-11* 
b^2)*cos(f*x+e)^4*sin(f*x+e)+(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*(- 
b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*(cos(f*x+e)^2)^(1/2)*b*(5*EllipticF(sin(f* 
x+e),(-b/a)^(1/2))*a^2+2*EllipticF(sin(f*x+e),(-b/a)^(1/2))*a*b-3*Elliptic 
F(sin(f*x+e),(-b/a)^(1/2))*b^2-8*EllipticE(sin(f*x+e),(-b/a)^(1/2))*a^2+8* 
EllipticE(sin(f*x+e),(-b/a)^(1/2))*a*b)*cos(f*x+e)^4-2*(-b*cos(f*x+e)^4+(a 
+b)*cos(f*x+e)^2)^(1/2)*(2*a^3+3*a^2*b-b^3)*cos(f*x+e)^2*sin(f*x+e)-(-b*co 
s(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*(co 
s(f*x+e)^2)^(1/2)*(5*EllipticF(sin(f*x+e),(-b/a)^(1/2))*a^3+7*EllipticF(si 
n(f*x+e),(-b/a)^(1/2))*a^2*b-EllipticF(sin(f*x+e),(-b/a)^(1/2))*a*b^2-3*El 
lipticF(sin(f*x+e),(-b/a)^(1/2))*b^3-8*EllipticE(sin(f*x+e),(-b/a)^(1/2))* 
a^3+8*EllipticE(sin(f*x+e),(-b/a)^(1/2))*a*b^2)*cos(f*x+e)^2+(-b*cos(f*x+e 
)^4+(a+b)*cos(f*x+e)^2)^(1/2)*(a^3+3*a^2*b+3*a*b^2+b^3)*sin(f*x+e))/(1+sin 
(f*x+e))/(-(a+b*sin(f*x+e)^2)*(sin(f*x+e)-1)*(1+sin(f*x+e)))^(1/2)/(sin(f* 
x+e)-1)/(a+b*sin(f*x+e)^2)^(3/2)/(a+b)^4/cos(f*x+e)/f
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 1730, normalized size of antiderivative = 4.97 \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(tan(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="fricas")
 

Output:

-1/3*(4*(2*((-I*a*b^4 + I*b^5)*cos(f*x + e)^7 + 2*(I*a^2*b^3 - I*b^5)*cos( 
f*x + e)^5 + (-I*a^3*b^2 - I*a^2*b^3 + I*a*b^4 + I*b^5)*cos(f*x + e)^3)*sq 
rt(-b)*sqrt((a^2 + a*b)/b^2) + ((-2*I*a^2*b^3 + I*a*b^4 + I*b^5)*cos(f*x + 
 e)^7 + 2*(2*I*a^3*b^2 + I*a^2*b^3 - 2*I*a*b^4 - I*b^5)*cos(f*x + e)^5 + ( 
-2*I*a^4*b - 3*I*a^3*b^2 + I*a^2*b^3 + 3*I*a*b^4 + I*b^5)*cos(f*x + e)^3)* 
sqrt(-b))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin( 
sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) + I*sin(f*x + 
e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) + 
 4*(2*((I*a*b^4 - I*b^5)*cos(f*x + e)^7 + 2*(-I*a^2*b^3 + I*b^5)*cos(f*x + 
 e)^5 + (I*a^3*b^2 + I*a^2*b^3 - I*a*b^4 - I*b^5)*cos(f*x + e)^3)*sqrt(-b) 
*sqrt((a^2 + a*b)/b^2) + ((2*I*a^2*b^3 - I*a*b^4 - I*b^5)*cos(f*x + e)^7 + 
 2*(-2*I*a^3*b^2 - I*a^2*b^3 + 2*I*a*b^4 + I*b^5)*cos(f*x + e)^5 + (2*I*a^ 
4*b + 3*I*a^3*b^2 - I*a^2*b^3 - 3*I*a*b^4 - I*b^5)*cos(f*x + e)^3)*sqrt(-b 
))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sqrt((2 
*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) - I*sin(f*x + e))), ( 
8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) - (2*((- 
3*I*a^2*b^3 + 2*I*a*b^4 + 5*I*b^5)*cos(f*x + e)^7 - 2*(-3*I*a^3*b^2 - I*a^ 
2*b^3 + 7*I*a*b^4 + 5*I*b^5)*cos(f*x + e)^5 + (-3*I*a^4*b - 4*I*a^3*b^2 + 
6*I*a^2*b^3 + 12*I*a*b^4 + 5*I*b^5)*cos(f*x + e)^3)*sqrt(-b)*sqrt((a^2 + a 
*b)/b^2) - ((-6*I*a^3*b^2 + 17*I*a^2*b^3 + 4*I*a*b^4 - 3*I*b^5)*cos(f*x...
 

Sympy [F]

\[ \int \frac {\tan ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\tan ^{4}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(tan(f*x+e)**4/(a+b*sin(f*x+e)**2)**(5/2),x)
 

Output:

Integral(tan(e + f*x)**4/(a + b*sin(e + f*x)**2)**(5/2), x)
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(tan(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {\tan ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\tan \left (f x + e\right )^{4}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(tan(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="giac")
 

Output:

integrate(tan(f*x + e)^4/(b*sin(f*x + e)^2 + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {tan}\left (e+f\,x\right )}^4}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \] Input:

int(tan(e + f*x)^4/(a + b*sin(e + f*x)^2)^(5/2),x)
 

Output:

int(tan(e + f*x)^4/(a + b*sin(e + f*x)^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\tan ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\sqrt {\sin \left (f x +e \right )^{2} b +a}\, \tan \left (f x +e \right )^{4}}{\sin \left (f x +e \right )^{6} b^{3}+3 \sin \left (f x +e \right )^{4} a \,b^{2}+3 \sin \left (f x +e \right )^{2} a^{2} b +a^{3}}d x \] Input:

int(tan(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x)
 

Output:

int((sqrt(sin(e + f*x)**2*b + a)*tan(e + f*x)**4)/(sin(e + f*x)**6*b**3 + 
3*sin(e + f*x)**4*a*b**2 + 3*sin(e + f*x)**2*a**2*b + a**3),x)