Integrand size = 25, antiderivative size = 422 \[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \, dx=-\frac {b^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{\left (a^2+b^2\right )^{7/4} f (d \sec (e+f x))^{3/2}}-\frac {b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{\left (a^2+b^2\right )^{7/4} f (d \sec (e+f x))^{3/2}}+\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sec ^2(e+f x)^{3/4}}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2}}+\frac {a b^2 \cot (e+f x) \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}}{\left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac {a b^2 \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}}{\left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2}} \]
-b^(5/2)*arctan((sec(f*x+e)^2)^(1/4)*b^(1/2)/(a^2+b^2)^(1/4))*(sec(f*x+e)^ 2)^(3/4)/(a^2+b^2)^(7/4)/f/(d*sec(f*x+e))^(3/2)-b^(5/2)*arctanh((sec(f*x+e )^2)^(1/4)*b^(1/2)/(a^2+b^2)^(1/4))*(sec(f*x+e)^2)^(3/4)/(a^2+b^2)^(7/4)/f /(d*sec(f*x+e))^(3/2)+2/3*a*(cos(1/2*arctan(tan(f*x+e)))^2)^(1/2)/cos(1/2* arctan(tan(f*x+e)))*EllipticF(sin(1/2*arctan(tan(f*x+e))),2^(1/2))*(sec(f* x+e)^2)^(3/4)/(a^2+b^2)/f/(d*sec(f*x+e))^(3/2)+a*b^2*cot(f*x+e)*EllipticPi ((sec(f*x+e)^2)^(1/4),-b/(a^2+b^2)^(1/2),I)*(sec(f*x+e)^2)^(3/4)*(-tan(f*x +e)^2)^(1/2)/(a^2+b^2)^2/f/(d*sec(f*x+e))^(3/2)+a*b^2*cot(f*x+e)*EllipticP i((sec(f*x+e)^2)^(1/4),b/(a^2+b^2)^(1/2),I)*(sec(f*x+e)^2)^(3/4)*(-tan(f*x +e)^2)^(1/2)/(a^2+b^2)^2/f/(d*sec(f*x+e))^(3/2)+2/3*(b+a*tan(f*x+e))/(a^2+ b^2)/f/(d*sec(f*x+e))^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 7.17 (sec) , antiderivative size = 418, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \, dx=\frac {a^2 b \sec ^2(e+f x)+b^3 \sec ^2(e+f x)+a^2 b \cos (2 (e+f x)) \sec ^2(e+f x)+b^3 \cos (2 (e+f x)) \sec ^2(e+f x)-3 b^{5/2} \sqrt [4]{a^2+b^2} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}-3 b^{5/2} \sqrt [4]{a^2+b^2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}+2 a^3 \tan (e+f x)+2 a b^2 \tan (e+f x)+a \left (a^2+b^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{3/4} \tan (e+f x)+3 a b^2 \cot (e+f x) \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}+3 a b^2 \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}} \]
(a^2*b*Sec[e + f*x]^2 + b^3*Sec[e + f*x]^2 + a^2*b*Cos[2*(e + f*x)]*Sec[e + f*x]^2 + b^3*Cos[2*(e + f*x)]*Sec[e + f*x]^2 - 3*b^(5/2)*(a^2 + b^2)^(1/ 4)*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x ]^2)^(3/4) - 3*b^(5/2)*(a^2 + b^2)^(1/4)*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2) ^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(3/4) + 2*a^3*Tan[e + f*x] + 2 *a*b^2*Tan[e + f*x] + a*(a^2 + b^2)*Hypergeometric2F1[1/2, 3/4, 3/2, -Tan[ e + f*x]^2]*(Sec[e + f*x]^2)^(3/4)*Tan[e + f*x] + 3*a*b^2*Cot[e + f*x]*Ell ipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(3/4)*Sqrt[-Tan[e + f*x]^2] + 3*a*b^2*Cot[e + f*x]*EllipticPi[b/S qrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(3/4) *Sqrt[-Tan[e + f*x]^2])/(3*(a^2 + b^2)^2*f*(d*Sec[e + f*x])^(3/2))
Time = 0.72 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.77, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.720, Rules used = {3042, 3994, 496, 27, 719, 229, 504, 312, 118, 25, 353, 73, 756, 218, 221, 925, 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.
| \(\displaystyle \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}dx\) |
| \(\Big \downarrow \) 3994 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \int \frac {1}{(a+b \tan (e+f x)) \left (\tan ^2(e+f x)+1\right )^{7/4}}d(b \tan (e+f x))}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 496 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {2 \left (a b \tan (e+f x)+b^2\right )}{3 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}-\frac {2 b^2 \int -\frac {\left (\frac {a^2}{b^2}+3\right ) b^2+a \tan (e+f x) b}{2 b^2 (a+b \tan (e+f x)) \left (\tan ^2(e+f x)+1\right )^{3/4}}d(b \tan (e+f x))}{3 \left (a^2+b^2\right )}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {\int \frac {a^2+b \tan (e+f x) a+3 b^2}{(a+b \tan (e+f x)) \left (\tan ^2(e+f x)+1\right )^{3/4}}d(b \tan (e+f x))}{3 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{3 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {3 b^2 \int \frac {1}{(a+b \tan (e+f x)) \left (\tan ^2(e+f x)+1\right )^{3/4}}d(b \tan (e+f x))+a \int \frac {1}{\left (\tan ^2(e+f x)+1\right )^{3/4}}d(b \tan (e+f x))}{3 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{3 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 229 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {3 b^2 \int \frac {1}{(a+b \tan (e+f x)) \left (\tan ^2(e+f x)+1\right )^{3/4}}d(b \tan (e+f x))+2 a b \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )}{3 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{3 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 504 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {3 b^2 \left (a \int \frac {1}{\left (\tan ^2(e+f x)+1\right )^{3/4} \left (a^2-b^2 \tan ^2(e+f x)\right )}d(b \tan (e+f x))-\int \frac {b \tan (e+f x)}{\left (\tan ^2(e+f x)+1\right )^{3/4} \left (a^2-b^2 \tan ^2(e+f x)\right )}d(b \tan (e+f x))\right )+2 a b \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )}{3 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{3 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 312 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {3 b^2 \left (\frac {a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {1}{\sqrt {-\frac {\tan (e+f x)}{b}} \left (\frac {\tan (e+f x)}{b}+1\right )^{3/4} \left (a^2-b^2 \tan ^2(e+f x)\right )}d\left (b^2 \tan ^2(e+f x)\right )}{2 b}-\int \frac {b \tan (e+f x)}{\left (\tan ^2(e+f x)+1\right )^{3/4} \left (a^2-b^2 \tan ^2(e+f x)\right )}d(b \tan (e+f x))\right )+2 a b \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )}{3 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{3 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 118 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {3 b^2 \left (-\int \frac {b \tan (e+f x)}{\left (\tan ^2(e+f x)+1\right )^{3/4} \left (a^2-b^2 \tan ^2(e+f x)\right )}d(b \tan (e+f x))-\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int -\frac {1}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{b}\right )+2 a b \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )}{3 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{3 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {3 b^2 \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {1}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{b}-\int \frac {b \tan (e+f x)}{\left (\tan ^2(e+f x)+1\right )^{3/4} \left (a^2-b^2 \tan ^2(e+f x)\right )}d(b \tan (e+f x))\right )+2 a b \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )}{3 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{3 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {3 b^2 \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {1}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{b}-\frac {1}{2} \int \frac {1}{\left (\frac {\tan (e+f x)}{b}+1\right )^{3/4} \left (a^2-b^2 \tan ^2(e+f x)\right )}d\left (b^2 \tan ^2(e+f x)\right )\right )+2 a b \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )}{3 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{3 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {3 b^2 \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {1}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{b}-2 b^2 \int \frac {1}{-\tan ^4(e+f x) b^6+b^2+a^2}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}\right )+2 a b \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )}{3 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{3 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {3 b^2 \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {1}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{b}-2 b^2 \left (\frac {\int \frac {1}{\sqrt {a^2+b^2}-b^3 \tan ^2(e+f x)}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{2 \sqrt {a^2+b^2}}+\frac {\int \frac {1}{\tan ^2(e+f x) b^3+\sqrt {a^2+b^2}}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{2 \sqrt {a^2+b^2}}\right )\right )+2 a b \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )}{3 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{3 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {3 b^2 \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {1}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{b}-2 b^2 \left (\frac {\int \frac {1}{\sqrt {a^2+b^2}-b^3 \tan ^2(e+f x)}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{2 \sqrt {a^2+b^2}}+\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 \sqrt {b} \left (a^2+b^2\right )^{3/4}}\right )\right )+2 a b \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )}{3 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{3 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {3 b^2 \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {1}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{b}-2 b^2 \left (\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 \sqrt {b} \left (a^2+b^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 \sqrt {b} \left (a^2+b^2\right )^{3/4}}\right )\right )+2 a b \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )}{3 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{3 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {3 b^2 \left (-\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \left (-\frac {b^2 \int \frac {1}{\left (1-\frac {b^3 \tan ^2(e+f x)}{\sqrt {a^2+b^2}}\right ) \sqrt {1-b^4 \tan ^4(e+f x)}}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{2 \left (a^2+b^2\right )}-\frac {b^2 \int \frac {1}{\left (\frac {\tan ^2(e+f x) b^3}{\sqrt {a^2+b^2}}+1\right ) \sqrt {1-b^4 \tan ^4(e+f x)}}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{2 \left (a^2+b^2\right )}\right )}{b}-2 b^2 \left (\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 \sqrt {b} \left (a^2+b^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 \sqrt {b} \left (a^2+b^2\right )^{3/4}}\right )\right )+2 a b \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )}{3 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{3 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 1537 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {3 b^2 \left (-\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \left (-\frac {b^2 \int \frac {1}{\left (1-\frac {b^3 \tan ^2(e+f x)}{\sqrt {a^2+b^2}}\right ) \sqrt {1-\sqrt [4]{\frac {\tan (e+f x)}{b}+1}} \sqrt {\sqrt [4]{\frac {\tan (e+f x)}{b}+1}+1}}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{2 \left (a^2+b^2\right )}-\frac {b^2 \int \frac {1}{\left (\frac {\tan ^2(e+f x) b^3}{\sqrt {a^2+b^2}}+1\right ) \sqrt {1-\sqrt [4]{\frac {\tan (e+f x)}{b}+1}} \sqrt {\sqrt [4]{\frac {\tan (e+f x)}{b}+1}+1}}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{2 \left (a^2+b^2\right )}\right )}{b}-2 b^2 \left (\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 \sqrt {b} \left (a^2+b^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 \sqrt {b} \left (a^2+b^2\right )^{3/4}}\right )\right )+2 a b \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )}{3 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{3 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {\sec ^2(e+f x)^{3/4} \left (\frac {3 b^2 \left (-\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \left (-\frac {b^2 \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\frac {\tan (e+f x)}{b}+1}\right ),-1\right )}{2 \left (a^2+b^2\right )}-\frac {b^2 \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\frac {\tan (e+f x)}{b}+1}\right ),-1\right )}{2 \left (a^2+b^2\right )}\right )}{b}-2 b^2 \left (\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 \sqrt {b} \left (a^2+b^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 \sqrt {b} \left (a^2+b^2\right )^{3/4}}\right )\right )+2 a b \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )}{3 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{3 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}\right )}{b f (d \sec (e+f x))^{3/2}}\) |
((Sec[e + f*x]^2)^(3/4)*((2*(b^2 + a*b*Tan[e + f*x]))/(3*(a^2 + b^2)*(1 + Tan[e + f*x]^2)^(3/4)) + (2*a*b*EllipticF[ArcTan[Tan[e + f*x]]/2, 2] + 3*b ^2*(-2*b^2*(ArcTan[(b^(3/2)*Tan[e + f*x])/(a^2 + b^2)^(1/4)]/(2*Sqrt[b]*(a ^2 + b^2)^(3/4)) + ArcTanh[(b^(3/2)*Tan[e + f*x])/(a^2 + b^2)^(1/4)]/(2*Sq rt[b]*(a^2 + b^2)^(3/4))) - (2*a*Cot[e + f*x]*(-1/2*(b^2*EllipticPi[-(b/Sq rt[a^2 + b^2]), ArcSin[(1 + Tan[e + f*x]/b)^(1/4)], -1])/(a^2 + b^2) - (b^ 2*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(1 + Tan[e + f*x]/b)^(1/4)], -1])/( 2*(a^2 + b^2)))*Sqrt[-Tan[e + f*x]^2])/b))/(3*(a^2 + b^2))))/(b*f*(d*Sec[e + f*x])^(3/2))
3.7.8.3.1 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[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^( 3/4)), x_] :> Simp[-4 Subst[Int[1/((b*e - a*f - b*x^4)*Sqrt[c - d*(e/f) + d*(x^4/f)]), x], x, (e + f*x)^(1/4)], x] /; FreeQ[{a, b, c, d, e, f}, x] & & GtQ[-f/(d*e - c*f), 0]
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)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) )*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[1/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Sim p[Sqrt[(-b)*(x^2/a)]/(2*x) Subst[Int[1/(Sqrt[(-b)*(x/a)]*(a + b*x)^(3/4)* (c + d*x)), x], x, x^2], 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[ (-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 *p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad raticQ[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[((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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 1/(2*c) Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[1/(2 *c) Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), 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. 6051 vs. \(2 (391 ) = 782\).
Time = 10.47 (sec) , antiderivative size = 6052, normalized size of antiderivative = 14.34
Timed out. \[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \, dx=\int \frac {1}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (a + b \tan {\left (e + f x \right )}\right )}\, dx \]
\[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \, dx=\int { \frac {1}{\left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \tan \left (f x + e\right ) + a\right )}} \,d x } \]
\[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \, dx=\int { \frac {1}{\left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \tan \left (f x + e\right ) + a\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \, dx=\int \frac {1}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )} \,d x \]