Integrand size = 25, antiderivative size = 566 \[ \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=\frac {\left (3 a^2-2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{8 \sqrt {b} \left (a^2+b^2\right )^{9/4} f \sec ^2(e+f x)^{3/4}}-\frac {\left (3 a^2-2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{8 \sqrt {b} \left (a^2+b^2\right )^{9/4} f \sec ^2(e+f x)^{3/4}}-\frac {5 a E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac {5 a \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{4 \left (a^2+b^2\right )^2 f}-\frac {a \left (3 a^2-2 b^2\right ) \cot (e+f x) \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{8 b \left (a^2+b^2\right )^{5/2} f \sec ^2(e+f x)^{3/4}}+\frac {a \left (3 a^2-2 b^2\right ) \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{8 b \left (a^2+b^2\right )^{5/2} f \sec ^2(e+f x)^{3/4}}-\frac {b (d \sec (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {5 a b (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))} \]
-5/4*a*(cos(1/2*arctan(tan(f*x+e)))^2)^(1/2)/cos(1/2*arctan(tan(f*x+e)))*E llipticE(sin(1/2*arctan(tan(f*x+e))),2^(1/2))*(d*sec(f*x+e))^(3/2)/(a^2+b^ 2)^2/f/(sec(f*x+e)^2)^(3/4)+5/4*a*cos(f*x+e)*(d*sec(f*x+e))^(3/2)*sin(f*x+ e)/(a^2+b^2)^2/f+1/8*(3*a^2-2*b^2)*arctan((sec(f*x+e)^2)^(1/4)*b^(1/2)/(a^ 2+b^2)^(1/4))*(d*sec(f*x+e))^(3/2)/(a^2+b^2)^(9/4)/f/(sec(f*x+e)^2)^(3/4)/ b^(1/2)-1/8*(3*a^2-2*b^2)*arctanh((sec(f*x+e)^2)^(1/4)*b^(1/2)/(a^2+b^2)^( 1/4))*(d*sec(f*x+e))^(3/2)/(a^2+b^2)^(9/4)/f/(sec(f*x+e)^2)^(3/4)/b^(1/2)- 1/8*a*(3*a^2-2*b^2)*cot(f*x+e)*EllipticPi((sec(f*x+e)^2)^(1/4),-b/(a^2+b^2 )^(1/2),I)*(d*sec(f*x+e))^(3/2)*(-tan(f*x+e)^2)^(1/2)/b/(a^2+b^2)^(5/2)/f/ (sec(f*x+e)^2)^(3/4)+1/8*a*(3*a^2-2*b^2)*cot(f*x+e)*EllipticPi((sec(f*x+e) ^2)^(1/4),b/(a^2+b^2)^(1/2),I)*(d*sec(f*x+e))^(3/2)*(-tan(f*x+e)^2)^(1/2)/ b/(a^2+b^2)^(5/2)/f/(sec(f*x+e)^2)^(3/4)-1/2*b*(d*sec(f*x+e))^(3/2)/(a^2+b ^2)/f/(a+b*tan(f*x+e))^2-5/4*a*b*(d*sec(f*x+e))^(3/2)/(a^2+b^2)^2/f/(a+b*t an(f*x+e))
Result contains complex when optimal does not.
Time = 35.27 (sec) , antiderivative size = 14364, normalized size of antiderivative = 25.38 \[ \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=\text {Result too large to show} \]
Time = 0.84 (sec) , antiderivative size = 413, normalized size of antiderivative = 0.73, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 3994, 498, 27, 688, 27, 25, 719, 225, 212, 504, 310, 353, 73, 827, 218, 221, 993, 1537, 412}
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.
\(\displaystyle \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3}dx\) |
\(\Big \downarrow \) 3994 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \int \frac {1}{(a+b \tan (e+f x))^3 \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 498 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \left (-\frac {\int -\frac {4 a-b \tan (e+f x)}{2 (a+b \tan (e+f x))^2 \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))}{2 \left (a^2+b^2\right )}-\frac {b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \left (\frac {\int \frac {4 a-b \tan (e+f x)}{(a+b \tan (e+f x))^2 \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 688 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \left (\frac {-\frac {b^2 \int \frac {\left (2-\frac {8 a^2}{b^2}\right ) b^2-5 a b \tan (e+f x)}{2 b^2 (a+b \tan (e+f x)) \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))}{a^2+b^2}-\frac {5 a b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \left (\frac {-\frac {\int -\frac {2 \left (4 a^2-b^2\right )+5 a b \tan (e+f x)}{(a+b \tan (e+f x)) \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))}{2 \left (a^2+b^2\right )}-\frac {5 a b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \left (\frac {\frac {\int \frac {2 \left (4 a^2-b^2\right )+5 a b \tan (e+f x)}{(a+b \tan (e+f x)) \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))}{2 \left (a^2+b^2\right )}-\frac {5 a b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 719 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \left (\frac {\frac {\left (3 a^2-2 b^2\right ) \int \frac {1}{(a+b \tan (e+f x)) \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))+5 a \int \frac {1}{\sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))}{2 \left (a^2+b^2\right )}-\frac {5 a b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 225 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \left (\frac {\frac {\left (3 a^2-2 b^2\right ) \int \frac {1}{(a+b \tan (e+f x)) \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))+5 a \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-\int \frac {1}{\left (\tan ^2(e+f x)+1\right )^{5/4}}d(b \tan (e+f x))\right )}{2 \left (a^2+b^2\right )}-\frac {5 a b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 212 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \left (\frac {\frac {\left (3 a^2-2 b^2\right ) \int \frac {1}{(a+b \tan (e+f x)) \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))+5 a \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )}{2 \left (a^2+b^2\right )}-\frac {5 a b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 504 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \left (\frac {\frac {\left (3 a^2-2 b^2\right ) \left (a \int \frac {1}{\sqrt [4]{\tan ^2(e+f x)+1} \left (a^2-b^2 \tan ^2(e+f x)\right )}d(b \tan (e+f x))-\int \frac {b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1} \left (a^2-b^2 \tan ^2(e+f x)\right )}d(b \tan (e+f x))\right )+5 a \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )}{2 \left (a^2+b^2\right )}-\frac {5 a b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 310 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \left (\frac {\frac {\left (3 a^2-2 b^2\right ) \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {b^2 \tan ^2(e+f x)}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\tan ^2(e+f x)+1}}{b}-\int \frac {b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1} \left (a^2-b^2 \tan ^2(e+f x)\right )}d(b \tan (e+f x))\right )+5 a \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )}{2 \left (a^2+b^2\right )}-\frac {5 a b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 353 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \left (\frac {\frac {\left (3 a^2-2 b^2\right ) \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {b^2 \tan ^2(e+f x)}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\tan ^2(e+f x)+1}}{b}-\frac {1}{2} \int \frac {1}{\sqrt [4]{\frac {\tan (e+f x)}{b}+1} \left (a^2-b^2 \tan ^2(e+f x)\right )}d\left (b^2 \tan ^2(e+f x)\right )\right )+5 a \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )}{2 \left (a^2+b^2\right )}-\frac {5 a b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \left (\frac {\frac {\left (3 a^2-2 b^2\right ) \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {b^2 \tan ^2(e+f x)}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\tan ^2(e+f x)+1}}{b}-2 b^2 \int \frac {\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{-\tan ^4(e+f x) b^6+b^2+a^2}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}\right )+5 a \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )}{2 \left (a^2+b^2\right )}-\frac {5 a b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \left (\frac {\frac {\left (3 a^2-2 b^2\right ) \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {b^2 \tan ^2(e+f x)}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\tan ^2(e+f x)+1}}{b}-2 b^2 \left (\frac {\int \frac {1}{\sqrt {a^2+b^2}-b^3 \tan ^2(e+f x)}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{2 b}-\frac {\int \frac {1}{\tan ^2(e+f x) b^3+\sqrt {a^2+b^2}}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{2 b}\right )\right )+5 a \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )}{2 \left (a^2+b^2\right )}-\frac {5 a b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \left (\frac {\frac {\left (3 a^2-2 b^2\right ) \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {b^2 \tan ^2(e+f x)}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\tan ^2(e+f x)+1}}{b}-2 b^2 \left (\frac {\int \frac {1}{\sqrt {a^2+b^2}-b^3 \tan ^2(e+f x)}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{2 b}-\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}\right )\right )+5 a \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )}{2 \left (a^2+b^2\right )}-\frac {5 a b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \left (\frac {\frac {\left (3 a^2-2 b^2\right ) \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {b^2 \tan ^2(e+f x)}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\tan ^2(e+f x)+1}}{b}-2 b^2 \left (\frac {\text {arctanh}\left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}-\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}\right )\right )+5 a \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )}{2 \left (a^2+b^2\right )}-\frac {5 a b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 993 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \left (\frac {\frac {\left (3 a^2-2 b^2\right ) \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \left (\frac {1}{2} b \int \frac {1}{\left (\sqrt {a^2+b^2}-b^3 \tan ^2(e+f x)\right ) \sqrt {1-b^4 \tan ^4(e+f x)}}d\sqrt [4]{\tan ^2(e+f x)+1}-\frac {1}{2} b \int \frac {1}{\left (\tan ^2(e+f x) b^3+\sqrt {a^2+b^2}\right ) \sqrt {1-b^4 \tan ^4(e+f x)}}d\sqrt [4]{\tan ^2(e+f x)+1}\right )}{b}-2 b^2 \left (\frac {\text {arctanh}\left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}-\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}\right )\right )+5 a \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )}{2 \left (a^2+b^2\right )}-\frac {5 a b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 1537 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \left (\frac {\frac {\left (3 a^2-2 b^2\right ) \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \left (\frac {1}{2} b \int \frac {1}{\sqrt {1-b^2 \tan ^2(e+f x)} \sqrt {b^2 \tan ^2(e+f x)+1} \left (\sqrt {a^2+b^2}-b^3 \tan ^2(e+f x)\right )}d\sqrt [4]{\tan ^2(e+f x)+1}-\frac {1}{2} b \int \frac {1}{\sqrt {1-b^2 \tan ^2(e+f x)} \sqrt {b^2 \tan ^2(e+f x)+1} \left (\tan ^2(e+f x) b^3+\sqrt {a^2+b^2}\right )}d\sqrt [4]{\tan ^2(e+f x)+1}\right )}{b}-2 b^2 \left (\frac {\text {arctanh}\left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}-\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}\right )\right )+5 a \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )}{2 \left (a^2+b^2\right )}-\frac {5 a b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle \frac {(d \sec (e+f x))^{3/2} \left (\frac {\frac {\left (3 a^2-2 b^2\right ) \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \left (\frac {b \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\tan ^2(e+f x)+1}\right ),-1\right )}{2 \sqrt {a^2+b^2}}-\frac {b \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\tan ^2(e+f x)+1}\right ),-1\right )}{2 \sqrt {a^2+b^2}}\right )}{b}-2 b^2 \left (\frac {\text {arctanh}\left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}-\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}\right )\right )+5 a \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )}{2 \left (a^2+b^2\right )}-\frac {5 a b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
((d*Sec[e + f*x])^(3/2)*(-1/2*(b^2*(1 + Tan[e + f*x]^2)^(3/4))/((a^2 + b^2 )*(a + b*Tan[e + f*x])^2) + ((-5*a*b^2*(1 + Tan[e + f*x]^2)^(3/4))/((a^2 + b^2)*(a + b*Tan[e + f*x])) + ((3*a^2 - 2*b^2)*(-2*b^2*(-1/2*ArcTan[(b^(3/ 2)*Tan[e + f*x])/(a^2 + b^2)^(1/4)]/(b^(3/2)*(a^2 + b^2)^(1/4)) + ArcTanh[ (b^(3/2)*Tan[e + f*x])/(a^2 + b^2)^(1/4)]/(2*b^(3/2)*(a^2 + b^2)^(1/4))) + (2*a*Cot[e + f*x]*(-1/2*(b*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(1 + T an[e + f*x]^2)^(1/4)], -1])/Sqrt[a^2 + b^2] + (b*EllipticPi[b/Sqrt[a^2 + b ^2], ArcSin[(1 + Tan[e + f*x]^2)^(1/4)], -1])/(2*Sqrt[a^2 + b^2]))*Sqrt[-T an[e + f*x]^2])/b) + 5*a*(-2*b*EllipticE[ArcTan[Tan[e + f*x]]/2, 2] + (2*b *Tan[e + f*x])/(1 + Tan[e + f*x]^2)^(1/4)))/(2*(a^2 + b^2)))/(4*(a^2 + b^2 ))))/(b*f*(Sec[e + f*x]^2)^(3/4))
3.7.19.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) )*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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]
Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[2*(x/(a + b*x^2)^(1/4)) , x] - Simp[a Int[1/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b}, x] && GtQ[ a, 0] && PosQ[b/a]
Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Sim p[2*(Sqrt[(-b)*(x^2/a)]/x) Subst[Int[x^2/(Sqrt[1 - x^4/a]*(b*c - a*d + d* x^4)), x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
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]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S imp[b/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p*(c*(n + 1) - d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[n , -1] && ((LtQ[n, -1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]) || (SumSimp lerQ[n, 1] && IntegerQ[p]) || ILtQ[Simplify[n + 2*p + 3], 0])
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]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* b) Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b) Int[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[(-a)*c, 2]}, Simp[Sqrt[-c] Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqr t[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] & & GtQ[a, 0] && LtQ[c, 0]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 60918 vs. \(2 (515 ) = 1030\).
Time = 13.11 (sec) , antiderivative size = 60919, normalized size of antiderivative = 107.63
Timed out. \[ \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=\text {Timed out} \]
\[ \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=\int \frac {\left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\left (a + b \tan {\left (e + f x \right )}\right )^{3}}\, dx \]
Timed out. \[ \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=\text {Timed out} \]
\[ \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{{\left (b \tan \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=\int \frac {{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3} \,d x \]