Integrand size = 25, antiderivative size = 820 \[ \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^3} \, dx=\frac {9 b^{7/2} \left (11 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 ) \sqrt [4]{\sec ^2(e+f x)}}{8 \left (a^2+b^2\right )^{17/4} d^2 f \sqrt {d \sec (e+f x)}}-\frac {9 b^{7/2} \left (11 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 ) \sqrt [4]{\sec ^2(e+f x)}}{8 \left (a^2+b^2\right )^{17/4} d^2 f \sqrt {d \sec (e+f x)}}+\frac {9 a b^2 \left (11 a^2-2 b^2\right ) E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{4 \left (a^2+b^2\right )^4 d^2 f \sqrt {d \sec (e+f x)}}+\frac {3 a \left (8 a^2-109 b^2\right ) E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{20 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}-\frac {9 a b^2 \left (11 a^2-2 b^2\right ) \tan (e+f x)}{4 \left (a^2+b^2\right )^4 d^2 f \sqrt {d \sec (e+f x)}}-\frac {9 a b^3 \left (11 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 ) \sqrt [4]{\sec ^2(e+f x)} \sqrt {-\tan ^2(e+f x)}}{8 \left (a^2+b^2\right )^{9/2} d^2 f \sqrt {d \sec (e+f x)}}+\frac {9 a b^3 \left (11 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 ) \sqrt [4]{\sec ^2(e+f x)} \sqrt {-\tan ^2(e+f x)}}{8 \left (a^2+b^2\right )^{9/2} d^2 f \sqrt {d \sec (e+f x)}}+\frac {b \left (4 a^2-9 b^2\right )}{10 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac {a b \left (8 a^2-109 b^2\right )}{20 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}+\frac {9 \left (11 a^2-2 b^2\right ) \left (b^3+a b^2 \tan (e+f x)\right )}{4 \left (a^2+b^2\right )^4 d^2 f \sqrt {d \sec (e+f x)}} \] Output:
9/8*b^(7/2)*(11*a^2-2*b^2)*arctan(b^(1/2)*(sec(f*x+e)^2)^(1/4)/(a^2+b^2)^( 1/4))*(sec(f*x+e)^2)^(1/4)/(a^2+b^2)^(17/4)/d^2/f/(d*sec(f*x+e))^(1/2)-9/8 *b^(7/2)*(11*a^2-2*b^2)*arctanh(b^(1/2)*(sec(f*x+e)^2)^(1/4)/(a^2+b^2)^(1/ 4))*(sec(f*x+e)^2)^(1/4)/(a^2+b^2)^(17/4)/d^2/f/(d*sec(f*x+e))^(1/2)+9/4*a *b^2*(11*a^2-2*b^2)*EllipticE(sin(1/2*arctan(tan(f*x+e))),2^(1/2))*(sec(f* x+e)^2)^(1/4)/(a^2+b^2)^4/d^2/f/(d*sec(f*x+e))^(1/2)+3/20*a*(8*a^2-109*b^2 )*EllipticE(sin(1/2*arctan(tan(f*x+e))),2^(1/2))*(sec(f*x+e)^2)^(1/4)/(a^2 +b^2)^3/d^2/f/(d*sec(f*x+e))^(1/2)-9/4*a*b^2*(11*a^2-2*b^2)*tan(f*x+e)/(a^ 2+b^2)^4/d^2/f/(d*sec(f*x+e))^(1/2)-9/8*a*b^3*(11*a^2-2*b^2)*cot(f*x+e)*El lipticPi((sec(f*x+e)^2)^(1/4),-b/(a^2+b^2)^(1/2),I)*(sec(f*x+e)^2)^(1/4)*( -tan(f*x+e)^2)^(1/2)/(a^2+b^2)^(9/2)/d^2/f/(d*sec(f*x+e))^(1/2)+9/8*a*b^3* (11*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)*(sec(f*x+e)^2)^(1/4)*(-tan(f*x+e)^2)^(1/2)/(a^2+b^2)^(9/2)/d^2/f/(d*s ec(f*x+e))^(1/2)+1/10*b*(4*a^2-9*b^2)/(a^2+b^2)^2/d^2/f/(d*sec(f*x+e))^(1/ 2)/(a+b*tan(f*x+e))^2+2/5*cos(f*x+e)^2*(b+a*tan(f*x+e))/(a^2+b^2)/d^2/f/(d *sec(f*x+e))^(1/2)/(a+b*tan(f*x+e))^2+1/20*a*b*(8*a^2-109*b^2)/(a^2+b^2)^3 /d^2/f/(d*sec(f*x+e))^(1/2)/(a+b*tan(f*x+e))+9/4*(11*a^2-2*b^2)*(b^3+a*b^2 *tan(f*x+e))/(a^2+b^2)^4/d^2/f/(d*sec(f*x+e))^(1/2)
Result contains complex when optimal does not.
Time = 29.14 (sec) , antiderivative size = 15481, normalized size of antiderivative = 18.88 \[ \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^3} \, dx=\text {Result too large to show} \] Input:
Integrate[1/((d*Sec[e + f*x])^(5/2)*(a + b*Tan[e + f*x])^3),x]
Output:
Result too large to show
Time = 1.16 (sec) , antiderivative size = 626, normalized size of antiderivative = 0.76, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.960, Rules used = {3042, 3994, 496, 27, 686, 27, 25, 688, 27, 688, 27, 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.
| \(\displaystyle \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^3}dx\) |
| \(\Big \downarrow \) 3994 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \int \frac {1}{(a+b \tan (e+f x))^3 \left (\tan ^2(e+f x)+1\right )^{9/4}}d(b \tan (e+f x))}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 496 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}-\frac {2 b^2 \int -\frac {3 \left (\frac {a^2}{b^2}+3\right ) b^2+7 a \tan (e+f x) b}{2 b^2 (a+b \tan (e+f x))^3 \left (\tan ^2(e+f x)+1\right )^{5/4}}d(b \tan (e+f x))}{5 \left (a^2+b^2\right )}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\int \frac {3 \left (a^2+3 b^2\right )+7 a b \tan (e+f x)}{(a+b \tan (e+f x))^3 \left (\tan ^2(e+f x)+1\right )^{5/4}}d(b \tan (e+f x))}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {2 b^4 \int -\frac {3 \left (\left (-\frac {a^4}{b^4}-\frac {12 a^2}{b^2}+15\right ) b^4+a \left (3 a^2+16 b^2\right ) \tan (e+f x) b\right )}{2 b^4 (a+b \tan (e+f x))^3 \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))}{a^2+b^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {3 \int -\frac {a^4+12 b^2 a^2-b \left (3 a^2+16 b^2\right ) \tan (e+f x) a-15 b^4}{(a+b \tan (e+f x))^3 \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))}{a^2+b^2}+\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {3 \int \frac {a^4+12 b^2 a^2-b \left (3 a^2+16 b^2\right ) \tan (e+f x) a-15 b^4}{(a+b \tan (e+f x))^3 \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))}{a^2+b^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {3 \left (-\frac {b^2 \int -\frac {4 a b^2 \left (\frac {a^4}{b^2}+9 a^2-31 b^2\right )-b \left (4 a^4+28 b^2 a^2-15 b^4\right ) \tan (e+f x)}{2 b^2 (a+b \tan (e+f x))^2 \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))}{2 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+28 a^2 b^2-15 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{a^2+b^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {3 \left (\frac {\int \frac {4 a \left (a^4+9 b^2 a^2-31 b^4\right )-b \left (4 a^4+28 b^2 a^2-15 b^4\right ) \tan (e+f x)}{(a+b \tan (e+f x))^2 \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+28 a^2 b^2-15 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{a^2+b^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {3 \left (\frac {-\frac {b^2 \int -\frac {\left (\frac {8 a^6}{b^2}+64 a^4-304 b^2 a^2+30 b^4\right ) b^2+a \left (8 a^4+64 b^2 a^2-139 b^4\right ) \tan (e+f x) b}{2 b^2 (a+b \tan (e+f x)) \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))}{a^2+b^2}-\frac {a b^2 \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+28 a^2 b^2-15 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{a^2+b^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {3 \left (\frac {\frac {\int \frac {2 \left (4 a^6+32 b^2 a^4-152 b^4 a^2+15 b^6\right )+a b \left (8 a^4+64 b^2 a^2-139 b^4\right ) \tan (e+f x)}{(a+b \tan (e+f x)) \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))}{2 \left (a^2+b^2\right )}-\frac {a b^2 \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+28 a^2 b^2-15 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{a^2+b^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {3 \left (\frac {\frac {a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \int \frac {1}{\sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))-15 b^4 \left (11 a^2-2 b^2\right ) \int \frac {1}{(a+b \tan (e+f x)) \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))}{2 \left (a^2+b^2\right )}-\frac {a b^2 \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+28 a^2 b^2-15 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{a^2+b^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 225 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {3 \left (\frac {\frac {a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-\int \frac {1}{\left (\tan ^2(e+f x)+1\right )^{5/4}}d(b \tan (e+f x))\right )-15 b^4 \left (11 a^2-2 b^2\right ) \int \frac {1}{(a+b \tan (e+f x)) \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))}{2 \left (a^2+b^2\right )}-\frac {a b^2 \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+28 a^2 b^2-15 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{a^2+b^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 212 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {3 \left (\frac {\frac {a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )-15 b^4 \left (11 a^2-2 b^2\right ) \int \frac {1}{(a+b \tan (e+f x)) \sqrt [4]{\tan ^2(e+f x)+1}}d(b \tan (e+f x))}{2 \left (a^2+b^2\right )}-\frac {a b^2 \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+28 a^2 b^2-15 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{a^2+b^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 504 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {3 \left (\frac {\frac {a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )-15 b^4 \left (11 a^2-2 b^2\right ) \left (a \int \frac {1}{\sqrt [4]{\tan ^2(e+f x)+1} \left (a^2-b^2 \tan ^2(e+f x)\right )}d(b \tan (e+f x))-\int \frac {b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1} \left (a^2-b^2 \tan ^2(e+f x)\right )}d(b \tan (e+f x))\right )}{2 \left (a^2+b^2\right )}-\frac {a b^2 \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+28 a^2 b^2-15 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{a^2+b^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 310 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {3 \left (\frac {\frac {a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )-15 b^4 \left (11 a^2-2 b^2\right ) \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {b^2 \tan ^2(e+f x)}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\tan ^2(e+f x)+1}}{b}-\int \frac {b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1} \left (a^2-b^2 \tan ^2(e+f x)\right )}d(b \tan (e+f x))\right )}{2 \left (a^2+b^2\right )}-\frac {a b^2 \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+28 a^2 b^2-15 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{a^2+b^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {3 \left (\frac {\frac {a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )-15 b^4 \left (11 a^2-2 b^2\right ) \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {b^2 \tan ^2(e+f x)}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\tan ^2(e+f x)+1}}{b}-\frac {1}{2} \int \frac {1}{\sqrt [4]{\frac {\tan (e+f x)}{b}+1} \left (a^2-b^2 \tan ^2(e+f x)\right )}d\left (b^2 \tan ^2(e+f x)\right )\right )}{2 \left (a^2+b^2\right )}-\frac {a b^2 \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+28 a^2 b^2-15 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{a^2+b^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {3 \left (\frac {\frac {a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )-15 b^4 \left (11 a^2-2 b^2\right ) \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {b^2 \tan ^2(e+f x)}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\tan ^2(e+f x)+1}}{b}-2 b^2 \int \frac {\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{-\tan ^4(e+f x) b^6+b^2+a^2}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}\right )}{2 \left (a^2+b^2\right )}-\frac {a b^2 \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+28 a^2 b^2-15 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{a^2+b^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {3 \left (\frac {\frac {a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )-15 b^4 \left (11 a^2-2 b^2\right ) \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {b^2 \tan ^2(e+f x)}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\tan ^2(e+f x)+1}}{b}-2 b^2 \left (\frac {\int \frac {1}{\sqrt {a^2+b^2}-b^3 \tan ^2(e+f x)}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{2 b}-\frac {\int \frac {1}{\tan ^2(e+f x) b^3+\sqrt {a^2+b^2}}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{2 b}\right )\right )}{2 \left (a^2+b^2\right )}-\frac {a b^2 \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+28 a^2 b^2-15 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{a^2+b^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {3 \left (\frac {\frac {a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )-15 b^4 \left (11 a^2-2 b^2\right ) \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {b^2 \tan ^2(e+f x)}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\tan ^2(e+f x)+1}}{b}-2 b^2 \left (\frac {\int \frac {1}{\sqrt {a^2+b^2}-b^3 \tan ^2(e+f x)}d\sqrt [4]{\frac {\tan (e+f x)}{b}+1}}{2 b}-\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}\right )\right )}{2 \left (a^2+b^2\right )}-\frac {a b^2 \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+28 a^2 b^2-15 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{a^2+b^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {3 \left (\frac {\frac {a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )-15 b^4 \left (11 a^2-2 b^2\right ) \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \int \frac {b^2 \tan ^2(e+f x)}{\sqrt {1-b^4 \tan ^4(e+f x)} \left (-b^4 \tan ^4(e+f x)+\frac {a^2}{b^2}+1\right )}d\sqrt [4]{\tan ^2(e+f x)+1}}{b}-2 b^2 \left (\frac {\text {arctanh}\left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}-\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}\right )\right )}{2 \left (a^2+b^2\right )}-\frac {a b^2 \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+28 a^2 b^2-15 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{a^2+b^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 993 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {3 \left (\frac {\frac {a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )-15 b^4 \left (11 a^2-2 b^2\right ) \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \left (\frac {1}{2} b \int \frac {1}{\left (\sqrt {a^2+b^2}-b^3 \tan ^2(e+f x)\right ) \sqrt {1-b^4 \tan ^4(e+f x)}}d\sqrt [4]{\tan ^2(e+f x)+1}-\frac {1}{2} b \int \frac {1}{\left (\tan ^2(e+f x) b^3+\sqrt {a^2+b^2}\right ) \sqrt {1-b^4 \tan ^4(e+f x)}}d\sqrt [4]{\tan ^2(e+f x)+1}\right )}{b}-2 b^2 \left (\frac {\text {arctanh}\left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}-\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}\right )\right )}{2 \left (a^2+b^2\right )}-\frac {a b^2 \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+28 a^2 b^2-15 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{a^2+b^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 1537 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {3 \left (\frac {\frac {a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )-15 b^4 \left (11 a^2-2 b^2\right ) \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \left (\frac {1}{2} b \int \frac {1}{\sqrt {1-b^2 \tan ^2(e+f x)} \sqrt {b^2 \tan ^2(e+f x)+1} \left (\sqrt {a^2+b^2}-b^3 \tan ^2(e+f x)\right )}d\sqrt [4]{\tan ^2(e+f x)+1}-\frac {1}{2} b \int \frac {1}{\sqrt {1-b^2 \tan ^2(e+f x)} \sqrt {b^2 \tan ^2(e+f x)+1} \left (\tan ^2(e+f x) b^3+\sqrt {a^2+b^2}\right )}d\sqrt [4]{\tan ^2(e+f x)+1}\right )}{b}-2 b^2 \left (\frac {\text {arctanh}\left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}-\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}\right )\right )}{2 \left (a^2+b^2\right )}-\frac {a b^2 \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+28 a^2 b^2-15 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{a^2+b^2}}{5 \left (a^2+b^2\right )}+\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {\sqrt [4]{\sec ^2(e+f x)} \left (\frac {2 \left (a b \tan (e+f x)+b^2\right )}{5 \left (a^2+b^2\right ) \left (\tan ^2(e+f x)+1\right )^{5/4} (a+b \tan (e+f x))^2}+\frac {\frac {2 \left (a b \left (3 a^2+16 b^2\right ) \tan (e+f x)+b^4 \left (9-\frac {4 a^2}{b^2}\right )\right )}{\left (a^2+b^2\right ) \sqrt [4]{\tan ^2(e+f x)+1} (a+b \tan (e+f x))^2}-\frac {3 \left (\frac {\frac {a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\frac {2 b \tan (e+f x)}{\sqrt [4]{\tan ^2(e+f x)+1}}-2 b E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )\right )-15 b^4 \left (11 a^2-2 b^2\right ) \left (\frac {2 a \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \left (\frac {b \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\tan ^2(e+f x)+1}\right ),-1\right )}{2 \sqrt {a^2+b^2}}-\frac {b \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\tan ^2(e+f x)+1}\right ),-1\right )}{2 \sqrt {a^2+b^2}}\right )}{b}-2 b^2 \left (\frac {\text {arctanh}\left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}-\frac {\arctan \left (\frac {b^{3/2} \tan (e+f x)}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{3/2} \sqrt [4]{a^2+b^2}}\right )\right )}{2 \left (a^2+b^2\right )}-\frac {a b^2 \left (8 a^4+64 a^2 b^2-139 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+28 a^2 b^2-15 b^4\right ) \left (\tan ^2(e+f x)+1\right )^{3/4}}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\right )}{a^2+b^2}}{5 \left (a^2+b^2\right )}\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\) |
Input:
Int[1/((d*Sec[e + f*x])^(5/2)*(a + b*Tan[e + f*x])^3),x]
Output:
((Sec[e + f*x]^2)^(1/4)*((2*(b^2 + a*b*Tan[e + f*x]))/(5*(a^2 + b^2)*(a + b*Tan[e + f*x])^2*(1 + Tan[e + f*x]^2)^(5/4)) + ((2*((9 - (4*a^2)/b^2)*b^4 + a*b*(3*a^2 + 16*b^2)*Tan[e + f*x]))/((a^2 + b^2)*(a + b*Tan[e + f*x])^2 *(1 + Tan[e + f*x]^2)^(1/4)) - (3*(-1/2*(b^2*(4*a^4 + 28*a^2*b^2 - 15*b^4) *(1 + Tan[e + f*x]^2)^(3/4))/((a^2 + b^2)*(a + b*Tan[e + f*x])^2) + (-((a* b^2*(8*a^4 + 64*a^2*b^2 - 139*b^4)*(1 + Tan[e + f*x]^2)^(3/4))/((a^2 + b^2 )*(a + b*Tan[e + f*x]))) + (-15*b^4*(11*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)) + Ar cTanh[(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]))*S qrt[-Tan[e + f*x]^2])/b) + a*(8*a^4 + 64*a^2*b^2 - 139*b^4)*(-2*b*Elliptic E[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))))/(a^2 + b^2))/(5*(a^2 + b^2))))/(b* d^2*f*Sqrt[d*Sec[e + f*x]])
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[ (-(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[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -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[(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. 36193 vs. \(2 (730 ) = 1460\).
Time = 27.50 (sec) , antiderivative size = 36194, normalized size of antiderivative = 44.14
Input:
int(1/(d*sec(f*x+e))^(5/2)/(a+b*tan(f*x+e))^3,x,method=_RETURNVERBOSE)
Output:
result too large to display
Timed out. \[ \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^3} \, dx=\text {Timed out} \] Input:
integrate(1/(d*sec(f*x+e))^(5/2)/(a+b*tan(f*x+e))^3,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^3} \, dx=\int \frac {1}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{2}} \left (a + b \tan {\left (e + f x \right )}\right )^{3}}\, dx \] Input:
integrate(1/(d*sec(f*x+e))**(5/2)/(a+b*tan(f*x+e))**3,x)
Output:
Integral(1/((d*sec(e + f*x))**(5/2)*(a + b*tan(e + f*x))**3), x)
Exception generated. \[ \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(d*sec(f*x+e))^(5/2)/(a+b*tan(f*x+e))^3,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^3} \, dx=\int { \frac {1}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{2}} {\left (b \tan \left (f x + e\right ) + a\right )}^{3}} \,d x } \] Input:
integrate(1/(d*sec(f*x+e))^(5/2)/(a+b*tan(f*x+e))^3,x, algorithm="giac")
Output:
integrate(1/((d*sec(f*x + e))^(5/2)*(b*tan(f*x + e) + a)^3), x)
Timed out. \[ \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^3} \, dx=\int \frac {1}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/2}\,{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3} \,d x \] Input:
int(1/((d/cos(e + f*x))^(5/2)*(a + b*tan(e + f*x))^3),x)
Output:
int(1/((d/cos(e + f*x))^(5/2)*(a + b*tan(e + f*x))^3), x)
\[ \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^3} \, dx=\frac {\sqrt {d}\, \left (\int \frac {\sqrt {\sec \left (f x +e \right )}}{\sec \left (f x +e \right )^{3} \tan \left (f x +e \right )^{3} b^{3}+3 \sec \left (f x +e \right )^{3} \tan \left (f x +e \right )^{2} a \,b^{2}+3 \sec \left (f x +e \right )^{3} \tan \left (f x +e \right ) a^{2} b +\sec \left (f x +e \right )^{3} a^{3}}d x \right )}{d^{3}} \] Input:
int(1/(d*sec(f*x+e))^(5/2)/(a+b*tan(f*x+e))^3,x)
Output:
(sqrt(d)*int(sqrt(sec(e + f*x))/(sec(e + f*x)**3*tan(e + f*x)**3*b**3 + 3* sec(e + f*x)**3*tan(e + f*x)**2*a*b**2 + 3*sec(e + f*x)**3*tan(e + f*x)*a* *2*b + sec(e + f*x)**3*a**3),x))/d**3