Integrand size = 30, antiderivative size = 446 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {i \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 (c-i d)^{5/2} f}+\frac {\left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 (c+i d)^{11/2} f}+\frac {d \left (6 c^3+33 i c^2 d-83 c d^2+154 i d^3\right )}{48 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac {i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac {2 c^2+11 i c d-30 d^2}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}+\frac {d \left (2 c^4+11 i c^3 d-26 c^2 d^2+253 i c d^3+150 d^4\right )}{16 a^3 (c-i d)^2 (c+i d)^5 f \sqrt {c+d \tan (e+f x)}} \]
-1/8*I*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/a^3/(c-I*d)^(5/2)/f+1 /16*(2*I*c^3-16*c^2*d-61*I*c*d^2+152*d^3)*arctanh((c+d*tan(f*x+e))^(1/2)/( c+I*d)^(1/2))/a^3/(c+I*d)^(11/2)/f+1/16*d*(2*c^4+11*I*c^3*d-26*c^2*d^2+253 *I*c*d^3+150*d^4)/a^3/(c-I*d)^2/(c+I*d)^5/f/(c+d*tan(f*x+e))^(1/2)+1/48*d* (6*c^3+33*I*c^2*d-83*c*d^2+154*I*d^3)/a^3/(c-I*d)/(c+I*d)^4/f/(c+d*tan(f*x +e))^(3/2)-1/6/(I*c-d)/f/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(3/2)+1/8*( I*c-4*d)/a/(c+I*d)^2/f/(a+I*a*tan(f*x+e))^2/(c+d*tan(f*x+e))^(3/2)+1/16*(2 *c^2+11*I*c*d-30*d^2)/(I*c-d)^3/f/(a^3+I*a^3*tan(f*x+e))/(c+d*tan(f*x+e))^ (3/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 2.68 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.54 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {-\frac {i \left (2 (c+i d)^4 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )+\left (-2 c^4-14 i c^3 d+45 c^2 d^2+91 i c d^3+152 d^4\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c+i d}\right )\right )}{(c-i d) (c+i d)^2}+\frac {8 (c+i d)}{(-i+\tan (e+f x))^3}+\frac {6 i (c+4 i d)}{(-i+\tan (e+f x))^2}+\frac {-6 c^2-33 i c d+90 d^2}{(c+i d) (-i+\tan (e+f x))}}{48 a^3 (c+i d)^2 f (c+d \tan (e+f x))^{3/2}} \]
-1/48*(((-I)*(2*(c + I*d)^4*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c - I*d)] + (-2*c^4 - (14*I)*c^3*d + 45*c^2*d^2 + (91*I)*c*d^3 + 152*d^4)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c + I*d)]) )/((c - I*d)*(c + I*d)^2) + (8*(c + I*d))/(-I + Tan[e + f*x])^3 + ((6*I)*( c + (4*I)*d))/(-I + Tan[e + f*x])^2 + (-6*c^2 - (33*I)*c*d + 90*d^2)/((c + I*d)*(-I + Tan[e + f*x])))/(a^3*(c + I*d)^2*f*(c + d*Tan[e + f*x])^(3/2))
Time = 2.68 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.11, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.633, Rules used = {3042, 4042, 27, 3042, 4079, 3042, 4079, 25, 3042, 4012, 3042, 4012, 3042, 4022, 3042, 4020, 25, 73, 221}
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.
\(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}dx\) |
\(\Big \downarrow \) 4042 |
\(\displaystyle -\frac {\int -\frac {3 (a (2 i c-5 d)+3 i a d \tan (e+f x))}{2 (i \tan (e+f x) a+a)^2 (c+d \tan (e+f x))^{5/2}}dx}{6 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {a (2 i c-5 d)+3 i a d \tan (e+f x)}{(i \tan (e+f x) a+a)^2 (c+d \tan (e+f x))^{5/2}}dx}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {a (2 i c-5 d)+3 i a d \tan (e+f x)}{(i \tan (e+f x) a+a)^2 (c+d \tan (e+f x))^{5/2}}dx}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 4079 |
\(\displaystyle \frac {-\frac {\int \frac {\left (4 c^2+15 i d c-32 d^2\right ) a^2+7 (c+4 i d) d \tan (e+f x) a^2}{(i \tan (e+f x) a+a) (c+d \tan (e+f x))^{5/2}}dx}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {\int \frac {\left (4 c^2+15 i d c-32 d^2\right ) a^2+7 (c+4 i d) d \tan (e+f x) a^2}{(i \tan (e+f x) a+a) (c+d \tan (e+f x))^{5/2}}dx}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 4079 |
\(\displaystyle \frac {-\frac {\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {\int -\frac {\left (4 i c^3-22 d c^2-67 i d^2 c+154 d^3\right ) a^3+5 d \left (2 i c^2-11 d c-30 i d^2\right ) \tan (e+f x) a^3}{(c+d \tan (e+f x))^{5/2}}dx}{2 a^2 (-d+i c)}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {-\frac {\frac {\int \frac {\left (4 i c^3-22 d c^2-67 i d^2 c+154 d^3\right ) a^3+5 d \left (2 i c^2-11 d c-30 i d^2\right ) \tan (e+f x) a^3}{(c+d \tan (e+f x))^{5/2}}dx}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {\frac {\int \frac {\left (4 i c^3-22 d c^2-67 i d^2 c+154 d^3\right ) a^3+5 d \left (2 i c^2-11 d c-30 i d^2\right ) \tan (e+f x) a^3}{(c+d \tan (e+f x))^{5/2}}dx}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \frac {-\frac {\frac {\frac {\int \frac {\left (4 i c^4-22 d c^3-57 i d^2 c^2+99 d^3 c-150 i d^4\right ) a^3+d \left (6 i c^3-33 d c^2-83 i d^2 c-154 d^3\right ) \tan (e+f x) a^3}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {\frac {\frac {\int \frac {\left (4 i c^4-22 d c^3-57 i d^2 c^2+99 d^3 c-150 i d^4\right ) a^3+d \left (6 i c^3-33 d c^2-83 i d^2 c-154 d^3\right ) \tan (e+f x) a^3}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \frac {-\frac {\frac {\frac {\frac {\int \frac {\left (4 i c^5-22 d c^4-51 i d^2 c^3+66 d^3 c^2-233 i d^4 c-154 d^5\right ) a^3+d \left (2 i c^4-11 d c^3-26 i d^2 c^2-253 d^3 c+150 i d^4\right ) \tan (e+f x) a^3}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}+\frac {2 a^3 d \left (2 i c^4-11 c^3 d-26 i c^2 d^2-253 c d^3+150 i d^4\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {\frac {\frac {\frac {\int \frac {\left (4 i c^5-22 d c^4-51 i d^2 c^3+66 d^3 c^2-233 i d^4 c-154 d^5\right ) a^3+d \left (2 i c^4-11 d c^3-26 i d^2 c^2-253 d^3 c+150 i d^4\right ) \tan (e+f x) a^3}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}+\frac {2 a^3 d \left (2 i c^4-11 c^3 d-26 i c^2 d^2-253 c d^3+150 i d^4\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 4022 |
\(\displaystyle \frac {-\frac {\frac {\frac {\frac {a^3 (c-i d)^2 \left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+2 a^3 (-d+i c)^5 \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}+\frac {2 a^3 d \left (2 i c^4-11 c^3 d-26 i c^2 d^2-253 c d^3+150 i d^4\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {\frac {\frac {\frac {a^3 (c-i d)^2 \left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+2 a^3 (-d+i c)^5 \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}+\frac {2 a^3 d \left (2 i c^4-11 c^3 d-26 i c^2 d^2-253 c d^3+150 i d^4\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 4020 |
\(\displaystyle \frac {-\frac {\frac {\frac {\frac {\frac {2 i a^3 (-d+i c)^5 \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{f}-\frac {i a^3 (c-i d)^2 \left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{f}}{c^2+d^2}+\frac {2 a^3 d \left (2 i c^4-11 c^3 d-26 i c^2 d^2-253 c d^3+150 i d^4\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {-\frac {\frac {\frac {\frac {\frac {i a^3 (c-i d)^2 \left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{f}-\frac {2 i a^3 (-d+i c)^5 \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{f}}{c^2+d^2}+\frac {2 a^3 d \left (2 i c^4-11 c^3 d-26 i c^2 d^2-253 c d^3+150 i d^4\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {-\frac {\frac {\frac {\frac {\frac {2 a^3 (c-i d)^2 \left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}+\frac {4 a^3 (-d+i c)^5 \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}}{c^2+d^2}+\frac {2 a^3 d \left (2 i c^4-11 c^3 d-26 i c^2 d^2-253 c d^3+150 i d^4\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {-\frac {\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}+\frac {\frac {\frac {\frac {2 a^3 (c-i d)^2 \left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}+\frac {4 a^3 (-d+i c)^5 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}}{c^2+d^2}+\frac {2 a^3 d \left (2 i c^4-11 c^3 d-26 i c^2 d^2-253 c d^3+150 i d^4\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\) |
-1/6*1/((I*c - d)*f*(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2)) + (-1/2*(a*(c + (4*I)*d))/((c + I*d)*f*(a + I*a*Tan[e + f*x])^2*(c + d*Tan[ e + f*x])^(3/2)) - ((a^2*((2*I)*c^2 - 11*c*d - (30*I)*d^2))/((c + I*d)*f*( a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)) + ((2*a^3*d*((6*I)*c^3 - 33*c^2*d - (83*I)*c*d^2 - 154*d^3))/(3*(c^2 + d^2)*f*(c + d*Tan[e + f*x]) ^(3/2)) + (((4*a^3*(I*c - d)^5*ArcTan[Tan[e + f*x]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f) + (2*a^3*(c - I*d)^2*((2*I)*c^3 - 16*c^2*d - (61*I)*c*d^2 + 152 *d^3)*ArcTan[Tan[e + f*x]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f))/(c^2 + d^2) + (2*a^3*d*((2*I)*c^4 - 11*c^3*d - (26*I)*c^2*d^2 - 253*c*d^3 + (150*I)*d^4 ))/((c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]))/(c^2 + d^2))/(2*a^2*(I*c - d) ))/(4*a^2*(I*c - d)))/(4*a^2*(I*c - d))
3.12.36.3.1 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)^(-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_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d)) In t[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(a*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*( b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Free Q[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && !GtQ[n, 0]
Time = 0.79 (sec) , antiderivative size = 743, normalized size of antiderivative = 1.67
method | result | size |
derivativedivides | \(\frac {2 d^{4} \left (\frac {\left (i c^{6}-15 i c^{4} d^{2}+15 i c^{2} d^{4}-i d^{6}-6 c^{5} d +20 c^{3} d^{3}-6 c \,d^{5}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{16 \left (i d -c \right )^{\frac {5}{2}} \left (i d +c \right )^{6} d^{4}}-\frac {-5 i c^{2}-3 i d^{2}+2 c d}{\left (i d -c \right )^{2} \left (i d +c \right )^{6} \sqrt {c +d \tan \left (f x +e \right )}}-\frac {-i c^{3}-i c \,d^{2}+c^{2} d +d^{3}}{3 \left (i d -c \right )^{2} \left (i d +c \right )^{6} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {i \left (\frac {\frac {d \left (2 i c^{8}-82 i c^{6} d^{2}-116 i c^{4} d^{4}+22 i c^{2} d^{6}+54 i d^{8}-19 c^{7} d +85 c^{5} d^{3}+227 c^{3} d^{5}+123 c \,d^{7}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{6 i c^{2} d -2 i d^{3}+2 c^{3}-6 c \,d^{2}}-\frac {2 d \left (3 i c^{9}-166 i c^{7} d^{2}-44 i c^{5} d^{4}+422 i c^{3} d^{6}+297 i c \,d^{8}-33 c^{8} d +282 c^{6} d^{3}+572 c^{4} d^{5}+166 c^{2} d^{7}-91 d^{9}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {d \left (2 i c^{10}-146 i c^{8} d^{2}+192 i c^{6} d^{4}+760 i c^{4} d^{6}+350 i c^{2} d^{8}-70 i d^{10}-25 c^{9} d +340 c^{7} d^{3}+458 c^{5} d^{5}-204 c^{3} d^{7}-297 c \,d^{9}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{6 i c^{2} d -2 i d^{3}+2 c^{3}-6 c \,d^{2}}}{\left (-d \tan \left (f x +e \right )+i d \right )^{3}}-\frac {\left (20 i c^{8} d -250 i c^{6} d^{3}-408 i c^{4} d^{5}+14 i c^{2} d^{7}+152 i d^{9}+2 c^{9}-91 c^{7} d^{2}+177 c^{5} d^{4}+635 c^{3} d^{6}+365 c \,d^{8}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right ) \sqrt {-i d -c}}\right )}{16 d^{4} \left (i d +c \right )^{6} \left (i d -c \right )^{2}}\right )}{f \,a^{3}}\) | \(743\) |
default | \(\frac {2 d^{4} \left (\frac {\left (i c^{6}-15 i c^{4} d^{2}+15 i c^{2} d^{4}-i d^{6}-6 c^{5} d +20 c^{3} d^{3}-6 c \,d^{5}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{16 \left (i d -c \right )^{\frac {5}{2}} \left (i d +c \right )^{6} d^{4}}-\frac {-5 i c^{2}-3 i d^{2}+2 c d}{\left (i d -c \right )^{2} \left (i d +c \right )^{6} \sqrt {c +d \tan \left (f x +e \right )}}-\frac {-i c^{3}-i c \,d^{2}+c^{2} d +d^{3}}{3 \left (i d -c \right )^{2} \left (i d +c \right )^{6} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {i \left (\frac {\frac {d \left (2 i c^{8}-82 i c^{6} d^{2}-116 i c^{4} d^{4}+22 i c^{2} d^{6}+54 i d^{8}-19 c^{7} d +85 c^{5} d^{3}+227 c^{3} d^{5}+123 c \,d^{7}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{6 i c^{2} d -2 i d^{3}+2 c^{3}-6 c \,d^{2}}-\frac {2 d \left (3 i c^{9}-166 i c^{7} d^{2}-44 i c^{5} d^{4}+422 i c^{3} d^{6}+297 i c \,d^{8}-33 c^{8} d +282 c^{6} d^{3}+572 c^{4} d^{5}+166 c^{2} d^{7}-91 d^{9}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {d \left (2 i c^{10}-146 i c^{8} d^{2}+192 i c^{6} d^{4}+760 i c^{4} d^{6}+350 i c^{2} d^{8}-70 i d^{10}-25 c^{9} d +340 c^{7} d^{3}+458 c^{5} d^{5}-204 c^{3} d^{7}-297 c \,d^{9}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{6 i c^{2} d -2 i d^{3}+2 c^{3}-6 c \,d^{2}}}{\left (-d \tan \left (f x +e \right )+i d \right )^{3}}-\frac {\left (20 i c^{8} d -250 i c^{6} d^{3}-408 i c^{4} d^{5}+14 i c^{2} d^{7}+152 i d^{9}+2 c^{9}-91 c^{7} d^{2}+177 c^{5} d^{4}+635 c^{3} d^{6}+365 c \,d^{8}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right ) \sqrt {-i d -c}}\right )}{16 d^{4} \left (i d +c \right )^{6} \left (i d -c \right )^{2}}\right )}{f \,a^{3}}\) | \(743\) |
2/f/a^3*d^4*(1/16/(I*d-c)^(5/2)/(c+I*d)^6*(-I*d^6+15*I*c^2*d^4+20*c^3*d^3- 6*c*d^5-15*I*c^4*d^2+I*c^6-6*c^5*d)/d^4*arctan((c+d*tan(f*x+e))^(1/2)/(I*d -c)^(1/2))-1/(I*d-c)^2/(c+I*d)^6*(-5*I*c^2-3*I*d^2+2*c*d)/(c+d*tan(f*x+e)) ^(1/2)-1/3/(I*d-c)^2/(c+I*d)^6*(-I*c^3-I*c*d^2+c^2*d+d^3)/(c+d*tan(f*x+e)) ^(3/2)+1/16*I/d^4/(c+I*d)^6/(I*d-c)^2*((1/2*d*(2*I*c^8-82*I*c^6*d^2-116*I* c^4*d^4+22*I*c^2*d^6+54*I*d^8-19*c^7*d+85*c^5*d^3+227*c^3*d^5+123*c*d^7)/( 3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(5/2)-2/3*d*(-33*c^8*d+282*c ^6*d^3+572*c^4*d^5+166*c^2*d^7-91*d^9+3*I*c^9-166*I*c^7*d^2-44*I*c^5*d^4+4 22*I*c^3*d^6+297*I*c*d^8)/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^( 3/2)+1/2*d*(2*I*c^10-146*I*c^8*d^2+192*I*c^6*d^4+760*I*c^4*d^6+350*I*c^2*d ^8-70*I*d^10-25*c^9*d+340*c^7*d^3+458*c^5*d^5-204*c^3*d^7-297*c*d^9)/(3*I* c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(1/2))/(-d*tan(f*x+e)+I*d)^3-1/2 *(-91*c^7*d^2+177*c^5*d^4+635*c^3*d^6+365*c*d^8+20*I*c^8*d-250*I*c^6*d^3-4 08*I*c^4*d^5+14*I*c^2*d^7+152*I*d^9+2*c^9)/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)/( -I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3803 vs. \(2 (364) = 728\).
Time = 16.97 (sec) , antiderivative size = 3803, normalized size of antiderivative = 8.53 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \]
1/192*(48*((-I*a^3*c^9 + a^3*c^8*d - 4*I*a^3*c^7*d^2 + 4*a^3*c^6*d^3 - 6*I *a^3*c^5*d^4 + 6*a^3*c^4*d^5 - 4*I*a^3*c^3*d^6 + 4*a^3*c^2*d^7 - I*a^3*c*d ^8 + a^3*d^9)*f*e^(10*I*f*x + 10*I*e) + 2*(-I*a^3*c^9 + 3*a^3*c^8*d + 8*a^ 3*c^6*d^3 + 6*I*a^3*c^5*d^4 + 6*a^3*c^4*d^5 + 8*I*a^3*c^3*d^6 + 3*I*a^3*c* d^8 - a^3*d^9)*f*e^(8*I*f*x + 8*I*e) + (-I*a^3*c^9 + 5*a^3*c^8*d + 8*I*a^3 *c^7*d^2 + 14*I*a^3*c^5*d^4 - 14*a^3*c^4*d^5 - 8*a^3*c^2*d^7 - 5*I*a^3*c*d ^8 + a^3*d^9)*f*e^(6*I*f*x + 6*I*e))*sqrt(1/64*I/((-I*a^6*c^5 - 5*a^6*c^4* d + 10*I*a^6*c^3*d^2 + 10*a^6*c^2*d^3 - 5*I*a^6*c*d^4 - a^6*d^5)*f^2))*log (-2*(8*((I*a^3*c^3 + 3*a^3*c^2*d - 3*I*a^3*c*d^2 - a^3*d^3)*f*e^(2*I*f*x + 2*I*e) + (I*a^3*c^3 + 3*a^3*c^2*d - 3*I*a^3*c*d^2 - a^3*d^3)*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(1/64 *I/((-I*a^6*c^5 - 5*a^6*c^4*d + 10*I*a^6*c^3*d^2 + 10*a^6*c^2*d^3 - 5*I*a^ 6*c*d^4 - a^6*d^5)*f^2)) - (c - I*d)*e^(2*I*f*x + 2*I*e) - c)*e^(-2*I*f*x - 2*I*e)) + 48*((I*a^3*c^9 - a^3*c^8*d + 4*I*a^3*c^7*d^2 - 4*a^3*c^6*d^3 + 6*I*a^3*c^5*d^4 - 6*a^3*c^4*d^5 + 4*I*a^3*c^3*d^6 - 4*a^3*c^2*d^7 + I*a^3 *c*d^8 - a^3*d^9)*f*e^(10*I*f*x + 10*I*e) + 2*(I*a^3*c^9 - 3*a^3*c^8*d - 8 *a^3*c^6*d^3 - 6*I*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 8*I*a^3*c^3*d^6 - 3*I*a^3 *c*d^8 + a^3*d^9)*f*e^(8*I*f*x + 8*I*e) + (I*a^3*c^9 - 5*a^3*c^8*d - 8*I*a ^3*c^7*d^2 - 14*I*a^3*c^5*d^4 + 14*a^3*c^4*d^5 + 8*a^3*c^2*d^7 + 5*I*a^3*c *d^8 - a^3*d^9)*f*e^(6*I*f*x + 6*I*e))*sqrt(1/64*I/((-I*a^6*c^5 - 5*a^6...
\[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx=\frac {i \int \frac {1}{c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{3}{\left (e + f x \right )} - 3 i c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )} - 3 c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} + i c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} + 2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{4}{\left (e + f x \right )} - 6 i c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{3}{\left (e + f x \right )} - 6 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )} + 2 i c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} + d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{5}{\left (e + f x \right )} - 3 i d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{4}{\left (e + f x \right )} - 3 d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{3}{\left (e + f x \right )} + i d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}}\, dx}{a^{3}} \]
I*Integral(1/(c**2*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**3 - 3*I*c**2*sqr t(c + d*tan(e + f*x))*tan(e + f*x)**2 - 3*c**2*sqrt(c + d*tan(e + f*x))*ta n(e + f*x) + I*c**2*sqrt(c + d*tan(e + f*x)) + 2*c*d*sqrt(c + d*tan(e + f* x))*tan(e + f*x)**4 - 6*I*c*d*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**3 - 6 *c*d*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**2 + 2*I*c*d*sqrt(c + d*tan(e + f*x))*tan(e + f*x) + d**2*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**5 - 3*I* d**2*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**4 - 3*d**2*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**3 + I*d**2*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**2), x)/a**3
Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1042 vs. \(2 (364) = 728\).
Time = 1.50 (sec) , antiderivative size = 1042, normalized size of antiderivative = 2.34 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \]
-1/48*d^4*(6*(-2*I*c^3 + 16*c^2*d + 61*I*c*d^2 - 152*d^3)*arctan(-2*(sqrt( d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt( -2*c + 2*sqrt(c^2 + d^2)) + I*sqrt(-2*c + 2*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-2*c + 2*sqrt(c^2 + d^2))))/((a^3*c^5*d^4*f + 5*I*a^3*c^4*d^5* f - 10*a^3*c^3*d^6*f - 10*I*a^3*c^2*d^7*f + 5*a^3*c*d^8*f + I*a^3*d^9*f)*s qrt(-2*c + 2*sqrt(c^2 + d^2))*(I*d/(c - sqrt(c^2 + d^2)) + 1)) - 12*I*arct an(2*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c ))/(c*sqrt(-2*c + 2*sqrt(c^2 + d^2)) - I*sqrt(-2*c + 2*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-2*c + 2*sqrt(c^2 + d^2))))/((a^3*c^2*d^4*f - 2*I*a ^3*c*d^5*f - a^3*d^6*f)*sqrt(-2*c + 2*sqrt(c^2 + d^2))*(-I*d/(c - sqrt(c^2 + d^2)) + 1)) - (6*I*(d*tan(f*x + e) + c)^4*c^4 - 12*I*(d*tan(f*x + e) + c)^3*c^5 + 6*I*(d*tan(f*x + e) + c)^2*c^6 - 33*(d*tan(f*x + e) + c)^4*c^3* d + 84*(d*tan(f*x + e) + c)^3*c^4*d - 51*(d*tan(f*x + e) + c)^2*c^5*d - 78 *I*(d*tan(f*x + e) + c)^4*c^2*d^2 + 256*I*(d*tan(f*x + e) + c)^3*c^3*d^2 - 198*I*(d*tan(f*x + e) + c)^2*c^4*d^2 - 759*(d*tan(f*x + e) + c)^4*c*d^3 + 1856*(d*tan(f*x + e) + c)^3*c^2*d^3 - 1446*(d*tan(f*x + e) + c)^2*c^3*d^3 + 384*(d*tan(f*x + e) + c)*c^4*d^3 + 32*c^5*d^3 + 450*I*(d*tan(f*x + e) + c)^4*d^4 + 844*I*(d*tan(f*x + e) + c)^3*c*d^4 - 2334*I*(d*tan(f*x + e) + c)^2*c^2*d^4 - 960*(-I*d*tan(f*x + e) - I*c)*c^3*d^4 + 96*I*c^4*d^4 + 1196 *(d*tan(f*x + e) + c)^3*d^5 - 243*(d*tan(f*x + e) + c)^2*c*d^5 - 576*(d...
Timed out. \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx=\text {Hanged} \]