Integrand size = 25, antiderivative size = 480 \[ \int \frac {(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^2} \, dx=-\frac {3 a d^2 \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{2 b^{5/2} \sqrt [4]{a^2+b^2} f \sec ^2(e+f x)^{3/4}}+\frac {3 a d^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{2 b^{5/2} \sqrt [4]{a^2+b^2} f \sec ^2(e+f x)^{3/4}}-\frac {3 d^2 E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{b^2 f \sec ^2(e+f x)^{3/4}}+\frac {3 d^2 \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{b^2 f}+\frac {3 a^2 d^2 \cot (e+f x) \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{2 b^3 \sqrt {a^2+b^2} f \sec ^2(e+f x)^{3/4}}-\frac {3 a^2 d^2 \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{2 b^3 \sqrt {a^2+b^2} f \sec ^2(e+f x)^{3/4}}-\frac {d^2 (d \sec (e+f x))^{3/2}}{b f (a+b \tan (e+f x))} \]
-3/2*a*d^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)/b^(5/2)/(a^2+b^2)^(1/4)/f/(sec(f*x+e)^2)^(3/4)+3/2*a*d^2*arctan h((sec(f*x+e)^2)^(1/4)*b^(1/2)/(a^2+b^2)^(1/4))*(d*sec(f*x+e))^(3/2)/b^(5/ 2)/(a^2+b^2)^(1/4)/f/(sec(f*x+e)^2)^(3/4)-3*d^2*(cos(1/2*arctan(tan(f*x+e) ))^2)^(1/2)/cos(1/2*arctan(tan(f*x+e)))*EllipticE(sin(1/2*arctan(tan(f*x+e ))),2^(1/2))*(d*sec(f*x+e))^(3/2)/b^2/f/(sec(f*x+e)^2)^(3/4)+3*d^2*cos(f*x +e)*(d*sec(f*x+e))^(3/2)*sin(f*x+e)/b^2/f+3/2*a^2*d^2*cot(f*x+e)*EllipticP i((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^3/f/(sec(f*x+e)^2)^(3/4)/(a^2+b^2)^(1/2)-3/2*a^2*d^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^3/f/(sec(f*x+e)^2)^(3/4)/(a^2+b^2)^(1/2)-d^2 *(d*sec(f*x+e))^(3/2)/b/f/(a+b*tan(f*x+e))
Result contains complex when optimal does not.
Time = 29.03 (sec) , antiderivative size = 1129, normalized size of antiderivative = 2.35 \[ \int \frac {(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^2} \, dx =\text {Too large to display} \]
(Cos[e + f*x]*(d*Sec[e + f*x])^(7/2)*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*( (3*Cos[e + f*x])/(a*b) + (3*Sin[e + f*x])/b^2 - 1/(b*(a*Cos[e + f*x] + b*S in[e + f*x]))))/(f*(a + b*Tan[e + f*x])^2) + (3*(d*Sec[e + f*x])^(7/2)*(a* Cos[e + f*x] + b*Sin[e + f*x])^2*(-((a*EllipticE[ArcSin[Tan[(e + f*x)/2]], -1]*Sqrt[1 + Tan[(e + f*x)/2]^2])/Sqrt[1 - Tan[(e + f*x)/2]^2]) + (2*a*El lipticF[ArcSin[Tan[(e + f*x)/2]], -1]*Sqrt[1 + Tan[(e + f*x)/2]^2])/Sqrt[1 - Tan[(e + f*x)/2]^2] + (-2*Sqrt[2]*a*b*Sqrt[a^2 + b^2]*EllipticF[ArcSin[ Sqrt[((1 + I)*(1 + Tan[(e + f*x)/2]))/(I + Tan[(e + f*x)/2])]/Sqrt[2]], 2] *Sqrt[-((1 + I*Tan[(e + f*x)/2])/(I + Tan[(e + f*x)/2]))] + Sqrt[2]*a^2*Sq rt[a^2 + b^2]*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[((1 + I)*(1 + Tan[(e + f*x)/2]))/(I + Tan[(e + f*x)/2])]/Sqrt[2]], 2]*Sqrt[-((1 + I*Tan[(e + f*x)/2])/(I + Tan[(e + f* x)/2]))] + a^2*(a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[((1 + I)*(1 + Tan[(e + f*x)/2]))/(I + Tan[(e + f*x)/2])]/Sqrt[2]], 2]*Sqrt[-((2 + (2*I)* Tan[(e + f*x)/2])/(I + Tan[(e + f*x)/2]))] - a^3*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[((1 + I)* (1 + Tan[(e + f*x)/2]))/(I + Tan[(e + f*x)/2])]/Sqrt[2]], 2]*Sqrt[-((2 + ( 2*I)*Tan[(e + f*x)/2])/(I + Tan[(e + f*x)/2]))] - I*a^2*b*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sq...
Time = 0.69 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.68, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 3994, 492, 605, 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.
\(\displaystyle \int \frac {(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^2}dx\) |
\(\Big \downarrow \) 3994 |
\(\displaystyle \frac {d^2 (d \sec (e+f x))^{3/2} \int \frac {\left (\tan ^2(e+f x)+1\right )^{3/4}}{(a+b \tan (e+f x))^2}d(b \tan (e+f x))}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 492 |
\(\displaystyle \frac {d^2 (d \sec (e+f x))^{3/2} \left (\frac {3 \int \frac {b \tan (e+f x)}{(a+b \tan (e+f x)) \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))}{2 b^2}-\frac {\left (\tan ^2(e+f x)+1\right )^{3/4}}{a+b \tan (e+f x)}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 605 |
\(\displaystyle \frac {d^2 (d \sec (e+f x))^{3/2} \left (\frac {3 \left (\int \frac {1}{\sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))-a \int \frac {1}{(a+b \tan (e+f x)) \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))\right )}{2 b^2}-\frac {\left (\tan ^2(e+f x)+1\right )^{3/4}}{a+b \tan (e+f x)}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 225 |
\(\displaystyle \frac {d^2 (d \sec (e+f x))^{3/2} \left (\frac {3 \left (-a \int \frac {1}{(a+b \tan (e+f x)) \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))-\int \frac {1}{\left (\tan ^2(e+f x)+1\right )^{5/4}}d(b \tan (e+f x))+\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}\right )}{2 b^2}-\frac {\left (\tan ^2(e+f x)+1\right )^{3/4}}{a+b \tan (e+f x)}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 212 |
\(\displaystyle \frac {d^2 (d \sec (e+f x))^{3/2} \left (\frac {3 \left (-a \int \frac {1}{(a+b \tan (e+f x)) \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )+\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}\right )}{2 b^2}-\frac {\left (\tan ^2(e+f x)+1\right )^{3/4}}{a+b \tan (e+f x)}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 504 |
\(\displaystyle \frac {d^2 (d \sec (e+f x))^{3/2} \left (\frac {3 \left (-a \left (a \int \frac {1}{\sqrt [4]{\tan ^2(e+f x)+1} \left (a^2-b^2 \tan ^2(e+f x)\right )}d(b \tan (e+f x))-\int \frac {b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1} \left (a^2-b^2 \tan ^2(e+f x)\right )}d(b \tan (e+f x))\right )-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )+\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}\right )}{2 b^2}-\frac {\left (\tan ^2(e+f x)+1\right )^{3/4}}{a+b \tan (e+f x)}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 310 |
\(\displaystyle \frac {d^2 (d \sec (e+f x))^{3/2} \left (\frac {3 \left (-a \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {b^2 \tan ^2(e+f x)}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\tan ^2(e+f x)+1}}{b}-\int \frac {b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1} \left (a^2-b^2 \tan ^2(e+f x)\right )}d(b \tan (e+f x))\right )-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )+\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}\right )}{2 b^2}-\frac {\left (\tan ^2(e+f x)+1\right )^{3/4}}{a+b \tan (e+f x)}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 353 |
\(\displaystyle \frac {d^2 (d \sec (e+f x))^{3/2} \left (\frac {3 \left (-a \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {b^2 \tan ^2(e+f x)}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\tan ^2(e+f x)+1}}{b}-\frac {1}{2} \int \frac {1}{\sqrt [4]{\frac {\tan (e+f x)}{b}+1} \left (a^2-b^2 \tan ^2(e+f x)\right )}d\left (b^2 \tan ^2(e+f x)\right )\right )-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )+\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}\right )}{2 b^2}-\frac {\left (\tan ^2(e+f x)+1\right )^{3/4}}{a+b \tan (e+f x)}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {d^2 (d \sec (e+f x))^{3/2} \left (\frac {3 \left (-a \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {b^2 \tan ^2(e+f x)}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\tan ^2(e+f x)+1}}{b}-2 b^2 \int \frac {\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{-\tan ^4(e+f x) b^6+b^2+a^2}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}\right )-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )+\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}\right )}{2 b^2}-\frac {\left (\tan ^2(e+f x)+1\right )^{3/4}}{a+b \tan (e+f x)}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \frac {d^2 (d \sec (e+f x))^{3/2} \left (\frac {3 \left (-a \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {b^2 \tan ^2(e+f x)}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\tan ^2(e+f x)+1}}{b}-2 b^2 \left (\frac {\int \frac {1}{\sqrt {a^2+b^2}-b^3 \tan ^2(e+f x)}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{2 b}-\frac {\int \frac {1}{\tan ^2(e+f x) b^3+\sqrt {a^2+b^2}}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{2 b}\right )\right )-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )+\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}\right )}{2 b^2}-\frac {\left (\tan ^2(e+f x)+1\right )^{3/4}}{a+b \tan (e+f x)}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {d^2 (d \sec (e+f x))^{3/2} \left (\frac {3 \left (-a \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {b^2 \tan ^2(e+f x)}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\tan ^2(e+f x)+1}}{b}-2 b^2 \left (\frac {\int \frac {1}{\sqrt {a^2+b^2}-b^3 \tan ^2(e+f x)}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{2 b}-\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}\right )\right )-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )+\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}\right )}{2 b^2}-\frac {\left (\tan ^2(e+f x)+1\right )^{3/4}}{a+b \tan (e+f x)}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {d^2 (d \sec (e+f x))^{3/2} \left (\frac {3 \left (-a \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {b^2 \tan ^2(e+f x)}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\tan ^2(e+f x)+1}}{b}-2 b^2 \left (\frac {\text {arctanh}\left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}-\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}\right )\right )-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )+\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}\right )}{2 b^2}-\frac {\left (\tan ^2(e+f x)+1\right )^{3/4}}{a+b \tan (e+f x)}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 993 |
\(\displaystyle \frac {d^2 (d \sec (e+f x))^{3/2} \left (\frac {3 \left (-a \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \left (\frac {1}{2} b \int \frac {1}{\left (\sqrt {a^2+b^2}-b^3 \tan ^2(e+f x)\right ) \sqrt {1-b^4 \tan ^4(e+f x)}}d\sqrt [4]{\tan ^2(e+f x)+1}-\frac {1}{2} b \int \frac {1}{\left (\tan ^2(e+f x) b^3+\sqrt {a^2+b^2}\right ) \sqrt {1-b^4 \tan ^4(e+f x)}}d\sqrt [4]{\tan ^2(e+f x)+1}\right )}{b}-2 b^2 \left (\frac {\text {arctanh}\left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}-\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}\right )\right )-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )+\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}\right )}{2 b^2}-\frac {\left (\tan ^2(e+f x)+1\right )^{3/4}}{a+b \tan (e+f x)}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 1537 |
\(\displaystyle \frac {d^2 (d \sec (e+f x))^{3/2} \left (\frac {3 \left (-a \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \left (\frac {1}{2} b \int \frac {1}{\sqrt {1-b^2 \tan ^2(e+f x)} \sqrt {b^2 \tan ^2(e+f x)+1} \left (\sqrt {a^2+b^2}-b^3 \tan ^2(e+f x)\right )}d\sqrt [4]{\tan ^2(e+f x)+1}-\frac {1}{2} b \int \frac {1}{\sqrt {1-b^2 \tan ^2(e+f x)} \sqrt {b^2 \tan ^2(e+f x)+1} \left (\tan ^2(e+f x) b^3+\sqrt {a^2+b^2}\right )}d\sqrt [4]{\tan ^2(e+f x)+1}\right )}{b}-2 b^2 \left (\frac {\text {arctanh}\left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}-\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}\right )\right )-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )+\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}\right )}{2 b^2}-\frac {\left (\tan ^2(e+f x)+1\right )^{3/4}}{a+b \tan (e+f x)}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle \frac {d^2 (d \sec (e+f x))^{3/2} \left (\frac {3 \left (-a \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \left (\frac {b \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\tan ^2(e+f x)+1}\right ),-1\right )}{2 \sqrt {a^2+b^2}}-\frac {b \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\tan ^2(e+f x)+1}\right ),-1\right )}{2 \sqrt {a^2+b^2}}\right )}{b}-2 b^2 \left (\frac {\text {arctanh}\left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}-\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}\right )\right )-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )+\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}\right )}{2 b^2}-\frac {\left (\tan ^2(e+f x)+1\right )^{3/4}}{a+b \tan (e+f x)}\right )}{b f \sec ^2(e+f x)^{3/4}}\) |
(d^2*(d*Sec[e + f*x])^(3/2)*(-((1 + Tan[e + f*x]^2)^(3/4)/(a + b*Tan[e + f *x])) + (3*(-2*b*EllipticE[ArcTan[Tan[e + f*x]]/2, 2] + (2*b*Tan[e + f*x]) /(1 + Tan[e + f*x]^2)^(1/4) - a*(-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 + Tan[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[-Tan[e + f*x]^2]) /b)))/(2*b^2)))/(b*f*(Sec[e + f*x]^2)^(3/4))
3.7.10.3.1 Defintions of rubi rules used
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[ (c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 1))), x] - Simp[2*b*(p/(d*(n + 1)) ) Int[x*(c + d*x)^(n + 1)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[n, -1]) && NeQ[n, -1] && !IL tQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
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[((x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)), x_Symbol] :> Simp[1/d Int[x^(m - 1)*(a + b*x^2)^p, x], x] - Simp[c/d Int[x^(m - 1 )*((a + b*x^2)^p/(c + d*x)), x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && LtQ[-1, p, 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. 33210 vs. \(2 (439 ) = 878\).
Time = 132.90 (sec) , antiderivative size = 33211, normalized size of antiderivative = 69.19
\[ \int \frac {(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^2} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}}}{{\left (b \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
integral(sqrt(d*sec(f*x + e))*d^3*sec(f*x + e)^3/(b^2*tan(f*x + e)^2 + 2*a *b*tan(f*x + e) + a^2), x)
Timed out. \[ \int \frac {(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^2} \, dx=\text {Timed out} \]
\[ \int \frac {(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^2} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}}}{{\left (b \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^2} \, dx=\int \frac {{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{7/2}}{{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2} \,d x \]