\(\int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))} \, dx\) [642]

Optimal result

Integrand size = 25, antiderivative size = 579 \[ \int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))} \, dx=\frac {3 b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}}-\frac {\sqrt {3} b^{4/3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right ) \sqrt [6]{\sec ^2(e+f x)}}{2 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac {\sqrt {3} b^{4/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right ) \sqrt [6]{\sec ^2(e+f x)}}{2 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}-\frac {b^{4/3} \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sqrt [6]{\sec ^2(e+f x)}}{\left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac {b^{4/3} \log \left (\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sqrt [6]{\sec ^2(e+f x)}}{4 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}-\frac {b^{4/3} \log \left (\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sqrt [6]{\sec ^2(e+f x)}}{4 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan (e+f x)}{a f \sqrt [3]{d \sec (e+f x)}} \] Output:

3*b/(a^2+b^2)/f/(d*sec(f*x+e))^(1/3)+1/2*3^(1/2)*b^(4/3)*arctan(-1/3*3^(1/ 
2)+2/3*b^(1/3)*(sec(f*x+e)^2)^(1/6)*3^(1/2)/(a^2+b^2)^(1/6))*(sec(f*x+e)^2 
)^(1/6)/(a^2+b^2)^(7/6)/f/(d*sec(f*x+e))^(1/3)+1/2*3^(1/2)*b^(4/3)*arctan( 
1/3*3^(1/2)+2/3*b^(1/3)*(sec(f*x+e)^2)^(1/6)*3^(1/2)/(a^2+b^2)^(1/6))*(sec 
(f*x+e)^2)^(1/6)/(a^2+b^2)^(7/6)/f/(d*sec(f*x+e))^(1/3)-b^(4/3)*arctanh(b^ 
(1/3)*(sec(f*x+e)^2)^(1/6)/(a^2+b^2)^(1/6))*(sec(f*x+e)^2)^(1/6)/(a^2+b^2) 
^(7/6)/f/(d*sec(f*x+e))^(1/3)+1/4*b^(4/3)*ln((a^2+b^2)^(1/3)-b^(1/3)*(a^2+ 
b^2)^(1/6)*(sec(f*x+e)^2)^(1/6)+b^(2/3)*(sec(f*x+e)^2)^(1/3))*(sec(f*x+e)^ 
2)^(1/6)/(a^2+b^2)^(7/6)/f/(d*sec(f*x+e))^(1/3)-1/4*b^(4/3)*ln((a^2+b^2)^( 
1/3)+b^(1/3)*(a^2+b^2)^(1/6)*(sec(f*x+e)^2)^(1/6)+b^(2/3)*(sec(f*x+e)^2)^( 
1/3))*(sec(f*x+e)^2)^(1/6)/(a^2+b^2)^(7/6)/f/(d*sec(f*x+e))^(1/3)+AppellF1 
(1/2,1,7/6,3/2,b^2*tan(f*x+e)^2/a^2,-tan(f*x+e)^2)*(sec(f*x+e)^2)^(1/6)*ta 
n(f*x+e)/a/f/(d*sec(f*x+e))^(1/3)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 48.79 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))} \, dx=-\frac {60 d \operatorname {AppellF1}\left (\frac {7}{3},\frac {7}{6},\frac {7}{6},\frac {10}{3},\frac {a-i b}{a+b \tan (e+f x)},\frac {a+i b}{a+b \tan (e+f x)}\right ) (a \cos (e+f x)+b \sin (e+f x))}{7 b f (d \sec (e+f x))^{4/3} \left (7 (a+i b) \operatorname {AppellF1}\left (\frac {10}{3},\frac {7}{6},\frac {13}{6},\frac {13}{3},\frac {a-i b}{a+b \tan (e+f x)},\frac {a+i b}{a+b \tan (e+f x)}\right )+7 (a-i b) \operatorname {AppellF1}\left (\frac {10}{3},\frac {13}{6},\frac {7}{6},\frac {13}{3},\frac {a-i b}{a+b \tan (e+f x)},\frac {a+i b}{a+b \tan (e+f x)}\right )+20 \operatorname {AppellF1}\left (\frac {7}{3},\frac {7}{6},\frac {7}{6},\frac {10}{3},\frac {a-i b}{a+b \tan (e+f x)},\frac {a+i b}{a+b \tan (e+f x)}\right ) (a+b \tan (e+f x))\right )} \] Input:

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

Output:

(-60*d*AppellF1[7/3, 7/6, 7/6, 10/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + 
I*b)/(a + b*Tan[e + f*x])]*(a*Cos[e + f*x] + b*Sin[e + f*x]))/(7*b*f*(d*Se 
c[e + f*x])^(4/3)*(7*(a + I*b)*AppellF1[10/3, 7/6, 13/6, 13/3, (a - I*b)/( 
a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])] + 7*(a - I*b)*AppellF 
1[10/3, 13/6, 7/6, 13/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b* 
Tan[e + f*x])] + 20*AppellF1[7/3, 7/6, 7/6, 10/3, (a - I*b)/(a + b*Tan[e + 
 f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(a + b*Tan[e + f*x])))
 

Rubi [A] (warning: unable to verify)

Time = 0.78 (sec) , antiderivative size = 423, normalized size of antiderivative = 0.73, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 3994, 504, 333, 353, 61, 73, 825, 27, 221, 1142, 25, 27, 1082, 217, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))}dx\)

\(\Big \downarrow \) 3994

\(\displaystyle \frac {\sqrt [6]{\sec ^2(e+f x)} \int \frac {1}{(a+b \tan (e+f x)) \left (\tan ^2(e+f x)+1\right )^{7/6}}d(b \tan (e+f x))}{b f \sqrt [3]{d \sec (e+f x)}}\)

\(\Big \downarrow \) 504

\(\displaystyle \frac {\sqrt [6]{\sec ^2(e+f x)} \left (a \int \frac {1}{\left (\tan ^2(e+f x)+1\right )^{7/6} \left (a^2-b^2 \tan ^2(e+f x)\right )}d(b \tan (e+f x))-\int \frac {b \tan (e+f x)}{\left (\tan ^2(e+f x)+1\right )^{7/6} \left (a^2-b^2 \tan ^2(e+f x)\right )}d(b \tan (e+f x))\right )}{b f \sqrt [3]{d \sec (e+f x)}}\)

\(\Big \downarrow \) 333

\(\displaystyle \frac {\sqrt [6]{\sec ^2(e+f x)} \left (\frac {b \tan (e+f x) \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a}-\int \frac {b \tan (e+f x)}{\left (\tan ^2(e+f x)+1\right )^{7/6} \left (a^2-b^2 \tan ^2(e+f x)\right )}d(b \tan (e+f x))\right )}{b f \sqrt [3]{d \sec (e+f x)}}\)

\(\Big \downarrow \) 353

\(\displaystyle \frac {\sqrt [6]{\sec ^2(e+f x)} \left (\frac {b \tan (e+f x) \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a}-\frac {1}{2} \int \frac {1}{\left (\frac {\tan (e+f x)}{b}+1\right )^{7/6} \left (a^2-b^2 \tan ^2(e+f x)\right )}d\left (b^2 \tan ^2(e+f x)\right )\right )}{b f \sqrt [3]{d \sec (e+f x)}}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {\sqrt [6]{\sec ^2(e+f x)} \left (\frac {1}{2} \left (\frac {6 b^2}{\left (a^2+b^2\right ) \sqrt [6]{\frac {\tan (e+f x)}{b}+1}}-\frac {b^2 \int \frac {1}{\sqrt [6]{\frac {\tan (e+f x)}{b}+1} \left (a^2-b^2 \tan ^2(e+f x)\right )}d\left (b^2 \tan ^2(e+f x)\right )}{a^2+b^2}\right )+\frac {b \tan (e+f x) \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a}\right )}{b f \sqrt [3]{d \sec (e+f x)}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\sqrt [6]{\sec ^2(e+f x)} \left (\frac {1}{2} \left (\frac {6 b^2}{\left (a^2+b^2\right ) \sqrt [6]{\frac {\tan (e+f x)}{b}+1}}-\frac {6 b^4 \int \frac {b^4 \tan ^4(e+f x)}{-\tan ^6(e+f x) b^8+b^2+a^2}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}}{a^2+b^2}\right )+\frac {b \tan (e+f x) \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a}\right )}{b f \sqrt [3]{d \sec (e+f x)}}\)

\(\Big \downarrow \) 825

\(\displaystyle \frac {\sqrt [6]{\sec ^2(e+f x)} \left (\frac {1}{2} \left (\frac {6 b^2}{\left (a^2+b^2\right ) \sqrt [6]{\frac {\tan (e+f x)}{b}+1}}-\frac {6 b^4 \left (\frac {\int \frac {1}{\sqrt [3]{a^2+b^2}-b^{8/3} \tan ^2(e+f x)}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}}{3 b^{4/3}}+\frac {\int -\frac {\tan (e+f x) b^{4/3}+\sqrt [6]{a^2+b^2}}{2 \left (\tan ^2(e+f x) b^{8/3}-\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}\right )}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}}{3 b^{4/3} \sqrt [6]{a^2+b^2}}+\frac {\int -\frac {\sqrt [6]{a^2+b^2}-b^{4/3} \tan (e+f x)}{2 \left (\tan ^2(e+f x) b^{8/3}+\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}\right )}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}}{3 b^{4/3} \sqrt [6]{a^2+b^2}}\right )}{a^2+b^2}\right )+\frac {b \tan (e+f x) \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a}\right )}{b f \sqrt [3]{d \sec (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt [6]{\sec ^2(e+f x)} \left (\frac {1}{2} \left (\frac {6 b^2}{\left (a^2+b^2\right ) \sqrt [6]{\frac {\tan (e+f x)}{b}+1}}-\frac {6 b^4 \left (\frac {\int \frac {1}{\sqrt [3]{a^2+b^2}-b^{8/3} \tan ^2(e+f x)}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}}{3 b^{4/3}}-\frac {\int \frac {\tan (e+f x) b^{4/3}+\sqrt [6]{a^2+b^2}}{\tan ^2(e+f x) b^{8/3}-\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}}{6 b^{4/3} \sqrt [6]{a^2+b^2}}-\frac {\int \frac {\sqrt [6]{a^2+b^2}-b^{4/3} \tan (e+f x)}{\tan ^2(e+f x) b^{8/3}+\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}}{6 b^{4/3} \sqrt [6]{a^2+b^2}}\right )}{a^2+b^2}\right )+\frac {b \tan (e+f x) \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a}\right )}{b f \sqrt [3]{d \sec (e+f x)}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt [6]{\sec ^2(e+f x)} \left (\frac {1}{2} \left (\frac {6 b^2}{\left (a^2+b^2\right ) \sqrt [6]{\frac {\tan (e+f x)}{b}+1}}-\frac {6 b^4 \left (-\frac {\int \frac {\tan (e+f x) b^{4/3}+\sqrt [6]{a^2+b^2}}{\tan ^2(e+f x) b^{8/3}-\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}}{6 b^{4/3} \sqrt [6]{a^2+b^2}}-\frac {\int \frac {\sqrt [6]{a^2+b^2}-b^{4/3} \tan (e+f x)}{\tan ^2(e+f x) b^{8/3}+\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}}{6 b^{4/3} \sqrt [6]{a^2+b^2}}+\frac {\text {arctanh}\left (\frac {b^{4/3} \tan (e+f x)}{\sqrt [6]{a^2+b^2}}\right )}{3 b^{5/3} \sqrt [6]{a^2+b^2}}\right )}{a^2+b^2}\right )+\frac {b \tan (e+f x) \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a}\right )}{b f \sqrt [3]{d \sec (e+f x)}}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {\sqrt [6]{\sec ^2(e+f x)} \left (\frac {1}{2} \left (\frac {6 b^2}{\left (a^2+b^2\right ) \sqrt [6]{\frac {\tan (e+f x)}{b}+1}}-\frac {6 b^4 \left (-\frac {\frac {3}{2} \sqrt [6]{a^2+b^2} \int \frac {1}{\tan ^2(e+f x) b^{8/3}-\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [6]{a^2+b^2}-2 b^{4/3} \tan (e+f x)\right )}{\tan ^2(e+f x) b^{8/3}-\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}}{2 \sqrt [3]{b}}}{6 b^{4/3} \sqrt [6]{a^2+b^2}}-\frac {\frac {3}{2} \sqrt [6]{a^2+b^2} \int \frac {1}{\tan ^2(e+f x) b^{8/3}+\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}-\frac {\int \frac {\sqrt [3]{b} \left (2 \tan (e+f x) b^{4/3}+\sqrt [6]{a^2+b^2}\right )}{\tan ^2(e+f x) b^{8/3}+\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}}{2 \sqrt [3]{b}}}{6 b^{4/3} \sqrt [6]{a^2+b^2}}+\frac {\text {arctanh}\left (\frac {b^{4/3} \tan (e+f x)}{\sqrt [6]{a^2+b^2}}\right )}{3 b^{5/3} \sqrt [6]{a^2+b^2}}\right )}{a^2+b^2}\right )+\frac {b \tan (e+f x) \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a}\right )}{b f \sqrt [3]{d \sec (e+f x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt [6]{\sec ^2(e+f x)} \left (\frac {1}{2} \left (\frac {6 b^2}{\left (a^2+b^2\right ) \sqrt [6]{\frac {\tan (e+f x)}{b}+1}}-\frac {6 b^4 \left (-\frac {\frac {3}{2} \sqrt [6]{a^2+b^2} \int \frac {1}{\tan ^2(e+f x) b^{8/3}-\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [6]{a^2+b^2}-2 b^{4/3} \tan (e+f x)\right )}{\tan ^2(e+f x) b^{8/3}-\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}}{2 \sqrt [3]{b}}}{6 b^{4/3} \sqrt [6]{a^2+b^2}}-\frac {\frac {3}{2} \sqrt [6]{a^2+b^2} \int \frac {1}{\tan ^2(e+f x) b^{8/3}+\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}-\frac {\int \frac {\sqrt [3]{b} \left (2 \tan (e+f x) b^{4/3}+\sqrt [6]{a^2+b^2}\right )}{\tan ^2(e+f x) b^{8/3}+\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}}{2 \sqrt [3]{b}}}{6 b^{4/3} \sqrt [6]{a^2+b^2}}+\frac {\text {arctanh}\left (\frac {b^{4/3} \tan (e+f x)}{\sqrt [6]{a^2+b^2}}\right )}{3 b^{5/3} \sqrt [6]{a^2+b^2}}\right )}{a^2+b^2}\right )+\frac {b \tan (e+f x) \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a}\right )}{b f \sqrt [3]{d \sec (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt [6]{\sec ^2(e+f x)} \left (\frac {1}{2} \left (\frac {6 b^2}{\left (a^2+b^2\right ) \sqrt [6]{\frac {\tan (e+f x)}{b}+1}}-\frac {6 b^4 \left (-\frac {\frac {3}{2} \sqrt [6]{a^2+b^2} \int \frac {1}{\tan ^2(e+f x) b^{8/3}-\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}-\frac {1}{2} \int \frac {\sqrt [6]{a^2+b^2}-2 b^{4/3} \tan (e+f x)}{\tan ^2(e+f x) b^{8/3}-\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}}{6 b^{4/3} \sqrt [6]{a^2+b^2}}-\frac {\frac {3}{2} \sqrt [6]{a^2+b^2} \int \frac {1}{\tan ^2(e+f x) b^{8/3}+\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}-\frac {1}{2} \int \frac {2 \tan (e+f x) b^{4/3}+\sqrt [6]{a^2+b^2}}{\tan ^2(e+f x) b^{8/3}+\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}}{6 b^{4/3} \sqrt [6]{a^2+b^2}}+\frac {\text {arctanh}\left (\frac {b^{4/3} \tan (e+f x)}{\sqrt [6]{a^2+b^2}}\right )}{3 b^{5/3} \sqrt [6]{a^2+b^2}}\right )}{a^2+b^2}\right )+\frac {b \tan (e+f x) \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a}\right )}{b f \sqrt [3]{d \sec (e+f x)}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\sqrt [6]{\sec ^2(e+f x)} \left (\frac {1}{2} \left (\frac {6 b^2}{\left (a^2+b^2\right ) \sqrt [6]{\frac {\tan (e+f x)}{b}+1}}-\frac {6 b^4 \left (-\frac {\frac {3 \int \frac {1}{\frac {2 b^{4/3} \tan (e+f x)}{\sqrt [6]{a^2+b^2}}-4}d\left (1-\frac {2 b^{4/3} \tan (e+f x)}{\sqrt [6]{a^2+b^2}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [6]{a^2+b^2}-2 b^{4/3} \tan (e+f x)}{\tan ^2(e+f x) b^{8/3}-\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}}{6 b^{4/3} \sqrt [6]{a^2+b^2}}-\frac {-\frac {1}{2} \int \frac {2 \tan (e+f x) b^{4/3}+\sqrt [6]{a^2+b^2}}{\tan ^2(e+f x) b^{8/3}+\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}-\frac {3 \int \frac {1}{-\frac {2 \tan (e+f x) b^{4/3}}{\sqrt [6]{a^2+b^2}}-4}d\left (\frac {2 \tan (e+f x) b^{4/3}}{\sqrt [6]{a^2+b^2}}+1\right )}{\sqrt [3]{b}}}{6 b^{4/3} \sqrt [6]{a^2+b^2}}+\frac {\text {arctanh}\left (\frac {b^{4/3} \tan (e+f x)}{\sqrt [6]{a^2+b^2}}\right )}{3 b^{5/3} \sqrt [6]{a^2+b^2}}\right )}{a^2+b^2}\right )+\frac {b \tan (e+f x) \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a}\right )}{b f \sqrt [3]{d \sec (e+f x)}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\sqrt [6]{\sec ^2(e+f x)} \left (\frac {1}{2} \left (\frac {6 b^2}{\left (a^2+b^2\right ) \sqrt [6]{\frac {\tan (e+f x)}{b}+1}}-\frac {6 b^4 \left (-\frac {-\frac {1}{2} \int \frac {\sqrt [6]{a^2+b^2}-2 b^{4/3} \tan (e+f x)}{\tan ^2(e+f x) b^{8/3}-\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 b^{4/3} \tan (e+f x)}{\sqrt [6]{a^2+b^2}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{6 b^{4/3} \sqrt [6]{a^2+b^2}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 b^{4/3} \tan (e+f x)}{\sqrt [6]{a^2+b^2}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {2 \tan (e+f x) b^{4/3}+\sqrt [6]{a^2+b^2}}{\tan ^2(e+f x) b^{8/3}+\sqrt [6]{a^2+b^2} \tan (e+f x) b^{4/3}+\sqrt [3]{a^2+b^2}}d\sqrt [6]{\frac {\tan (e+f x)}{b}+1}}{6 b^{4/3} \sqrt [6]{a^2+b^2}}+\frac {\text {arctanh}\left (\frac {b^{4/3} \tan (e+f x)}{\sqrt [6]{a^2+b^2}}\right )}{3 b^{5/3} \sqrt [6]{a^2+b^2}}\right )}{a^2+b^2}\right )+\frac {b \tan (e+f x) \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a}\right )}{b f \sqrt [3]{d \sec (e+f x)}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\sqrt [6]{\sec ^2(e+f x)} \left (\frac {b \tan (e+f x) \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a}+\frac {1}{2} \left (\frac {6 b^2}{\left (a^2+b^2\right ) \sqrt [6]{\frac {\tan (e+f x)}{b}+1}}-\frac {6 b^4 \left (-\frac {\frac {\log \left (\sqrt [3]{a^2+b^2}-b^{4/3} \sqrt [6]{a^2+b^2} \tan (e+f x)+b^{8/3} \tan ^2(e+f x)\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 b^{4/3} \tan (e+f x)}{\sqrt [6]{a^2+b^2}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{6 b^{4/3} \sqrt [6]{a^2+b^2}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 b^{4/3} \tan (e+f x)}{\sqrt [6]{a^2+b^2}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a^2+b^2}+b^{4/3} \sqrt [6]{a^2+b^2} \tan (e+f x)+b^{8/3} \tan ^2(e+f x)\right )}{2 \sqrt [3]{b}}}{6 b^{4/3} \sqrt [6]{a^2+b^2}}+\frac {\text {arctanh}\left (\frac {b^{4/3} \tan (e+f x)}{\sqrt [6]{a^2+b^2}}\right )}{3 b^{5/3} \sqrt [6]{a^2+b^2}}\right )}{a^2+b^2}\right )\right )}{b f \sqrt [3]{d \sec (e+f x)}}\)

Input:

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

Output:

((Sec[e + f*x]^2)^(1/6)*((b*AppellF1[1/2, 1, 7/6, 3/2, (b^2*Tan[e + f*x]^2 
)/a^2, -Tan[e + f*x]^2]*Tan[e + f*x])/a + ((-6*b^4*(ArcTanh[(b^(4/3)*Tan[e 
 + f*x])/(a^2 + b^2)^(1/6)]/(3*b^(5/3)*(a^2 + b^2)^(1/6)) - (-((Sqrt[3]*Ar 
cTan[(1 - (2*b^(4/3)*Tan[e + f*x])/(a^2 + b^2)^(1/6))/Sqrt[3]])/b^(1/3)) + 
 Log[(a^2 + b^2)^(1/3) - b^(4/3)*(a^2 + b^2)^(1/6)*Tan[e + f*x] + b^(8/3)* 
Tan[e + f*x]^2]/(2*b^(1/3)))/(6*b^(4/3)*(a^2 + b^2)^(1/6)) - ((Sqrt[3]*Arc 
Tan[(1 + (2*b^(4/3)*Tan[e + f*x])/(a^2 + b^2)^(1/6))/Sqrt[3]])/b^(1/3) - L 
og[(a^2 + b^2)^(1/3) + b^(4/3)*(a^2 + b^2)^(1/6)*Tan[e + f*x] + b^(8/3)*Ta 
n[e + f*x]^2]/(2*b^(1/3)))/(6*b^(4/3)*(a^2 + b^2)^(1/6))))/(a^2 + b^2) + ( 
6*b^2)/((a^2 + b^2)*(1 + Tan[e + f*x]/b)^(1/6)))/2))/(b*f*(d*Sec[e + f*x]) 
^(1/3))
 

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 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, 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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 333

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F 
reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 
0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 353

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ 
{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
 

rule 504

Int[((a_) + (b_.)*(x_)^2)^(p_)/((c_) + (d_.)*(x_)), x_Symbol] :> Simp[c   I 
nt[(a + b*x^2)^p/(c^2 - d^2*x^2), x], x] - Simp[d   Int[x*((a + b*x^2)^p/(c 
^2 - d^2*x^2)), x], x] /; FreeQ[{a, b, c, d, p}, x]
 

rule 825

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator 
[Rt[-a/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r*Cos[2*k 
*m*(Pi/n)] - s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[2*k*(Pi/n)]*x + 
s^2*x^2), x] + Int[(r*Cos[2*k*m*(Pi/n)] + s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 
 + 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x]; 2*(r^(m + 2)/(a*n*s^m))   Int[1/ 
(r^2 - s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m))   Sum[u, {k, 1, (n - 2)/4}], 
x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1 
] && NegQ[a/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 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 3042

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

rule 3994

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( 
x_)])^(n_), x_Symbol] :> Simp[d^(2*IntPart[m/2])*((d*Sec[e + f*x])^(2*FracP 
art[m/2])/(b*f*(Sec[e + f*x]^2)^FracPart[m/2]))   Subst[Int[(a + x)^n*(1 + 
x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, 
n}, x] && NeQ[a^2 + b^2, 0] &&  !IntegerQ[m] && IntegerQ[n]
 

Maple [F]

\[\int \frac {1}{\left (d \sec \left (f x +e \right )\right )^{\frac {1}{3}} \left (a +b \tan \left (f x +e \right )\right )}d x\]

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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