Integrand size = 30, antiderivative size = 267 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {i \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{2 a (c-i d)^{5/2} f}+\frac {(i c-6 d) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{2 a (c+i d)^{7/2} f}+\frac {d (3 i c+7 d)}{6 a (i c-d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {1}{2 (i c-d) f (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}+\frac {d \left (c^2-14 i c d-5 d^2\right )}{2 a (c-i d)^2 (c+i d)^3 f \sqrt {c+d \tan (e+f x)}} \]
-1/2*I*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/a/(c-I*d)^(5/2)/f+1/2 *(I*c-6*d)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/a/(c+I*d)^(7/2)/f +1/2*d*(c^2-14*I*c*d-5*d^2)/a/(c-I*d)^2/(c+I*d)^3/f/(c+d*tan(f*x+e))^(1/2) +1/6*d*(3*I*c+7*d)/a/(I*c-d)/(c^2+d^2)/f/(c+d*tan(f*x+e))^(3/2)-1/2/(I*c-d )/f/(a+I*a*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 = 1.21 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.54 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {i \left (-\frac {(c+i d) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )}{c-i d}+\frac {(c+6 i d) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c+i d}\right )}{c+i d}+\frac {3 i}{-i+\tan (e+f x)}\right )}{6 a (c+i d) f (c+d \tan (e+f x))^{3/2}} \]
((-1/6*I)*(-(((c + I*d)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f* x])/(c - I*d)])/(c - I*d)) + ((c + (6*I)*d)*Hypergeometric2F1[-3/2, 1, -1/ 2, (c + d*Tan[e + f*x])/(c + I*d)])/(c + I*d) + (3*I)/(-I + Tan[e + f*x])) )/(a*(c + I*d)*f*(c + d*Tan[e + f*x])^(3/2))
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.
| \(\displaystyle \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4035 |
| \(\displaystyle \frac {\int \frac {a (2 i c-7 d)+5 i a d \tan (e+f x)}{2 (c+d \tan (e+f x))^{5/2}}dx}{2 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a (2 i c-7 d)+5 i a d \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}}dx}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (2 i c-7 d)+5 i a d \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}}dx}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {\int -\frac {a \left (7 c d-i \left (2 c^2+5 d^2\right )\right )-a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {\int -\frac {a \left (2 i c^2-7 d c+5 i d^2\right )+a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a \left (7 c d-i \left (2 c^2+5 d^2\right )\right )-a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {\int -\frac {a \left (2 i c^2-7 d c+5 i d^2\right )+a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a \left (7 c d-i \left (2 c^2+5 d^2\right )\right )-a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {\int -\frac {a \left (2 i c^2-7 d c+5 i d^2\right )+a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a \left (7 c d-i \left (2 c^2+5 d^2\right )\right )-a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {\int -\frac {a \left (2 i c^2-7 d c+5 i d^2\right )+a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a \left (7 c d-i \left (2 c^2+5 d^2\right )\right )-a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {\int -\frac {a \left (2 i c^2-7 d c+5 i d^2\right )+a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a \left (7 c d-i \left (2 c^2+5 d^2\right )\right )-a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {\int -\frac {a \left (2 i c^2-7 d c+5 i d^2\right )+a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a \left (7 c d-i \left (2 c^2+5 d^2\right )\right )-a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {\int -\frac {a \left (2 i c^2-7 d c+5 i d^2\right )+a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a \left (7 c d-i \left (2 c^2+5 d^2\right )\right )-a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {\int -\frac {a \left (2 i c^2-7 d c+5 i d^2\right )+a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a \left (7 c d-i \left (2 c^2+5 d^2\right )\right )-a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {\int -\frac {a \left (2 i c^2-7 d c+5 i d^2\right )+a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a \left (7 c d-i \left (2 c^2+5 d^2\right )\right )-a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {\int -\frac {a \left (2 i c^2-7 d c+5 i d^2\right )+a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a \left (7 c d-i \left (2 c^2+5 d^2\right )\right )-a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {\int -\frac {a \left (2 i c^2-7 d c+5 i d^2\right )+a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a \left (7 c d-i \left (2 c^2+5 d^2\right )\right )-a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {\int -\frac {a \left (2 i c^2-7 d c+5 i d^2\right )+a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a \left (7 c d-i \left (2 c^2+5 d^2\right )\right )-a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 a d (7 d+3 i c)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {\int -\frac {a \left (2 i c^2-7 d c+5 i d^2\right )+a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}}{4 a^2 (-d+i c)}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
3.12.34.3.1 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_.)*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[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-a)*((c + d*Tan[e + f*x])^(n + 1)/(2*f*(b* c - a*d)*(a + b*Tan[e + f*x]))), x] + Simp[1/(2*a*(b*c - a*d)) Int[(c + d *Tan[e + f*x])^n*Simp[b*c + a*d*(n - 1) - b*d*n*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] && !GtQ[n, 0]
Time = 0.56 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.36
| method | result | size |
| derivativedivides | \(\frac {2 d^{2} \left (\frac {\left (i c^{4}-6 i c^{2} d^{2}+i d^{4}-4 c^{3} d +4 c \,d^{3}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{4 \left (i d -c \right )^{\frac {5}{2}} \left (i d +c \right )^{4} d^{2}}-\frac {3 i c^{2}+i d^{2}-2 c d}{\left (i d -c \right )^{2} \left (i d +c \right )^{4} \sqrt {c +d \tan \left (f x +e \right )}}-\frac {i c^{3}+i c \,d^{2}-c^{2} d -d^{3}}{3 \left (i d -c \right )^{2} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {-\frac {\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) d \sqrt {c +d \tan \left (f x +e \right )}}{\left (i d +c \right ) \left (-d \tan \left (f x +e \right )+i d \right )}-\frac {\left (i c^{5}+2 i d^{2} c^{3}+i c \,d^{4}-6 c^{4} d -12 c^{2} d^{3}-6 d^{5}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{\left (i d +c \right ) \sqrt {-i d -c}}}{4 d^{2} \left (i d +c \right )^{4} \left (i d -c \right )^{2}}\right )}{f a}\) | \(364\) |
| default | \(\frac {2 d^{2} \left (\frac {\left (i c^{4}-6 i c^{2} d^{2}+i d^{4}-4 c^{3} d +4 c \,d^{3}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{4 \left (i d -c \right )^{\frac {5}{2}} \left (i d +c \right )^{4} d^{2}}-\frac {3 i c^{2}+i d^{2}-2 c d}{\left (i d -c \right )^{2} \left (i d +c \right )^{4} \sqrt {c +d \tan \left (f x +e \right )}}-\frac {i c^{3}+i c \,d^{2}-c^{2} d -d^{3}}{3 \left (i d -c \right )^{2} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {-\frac {\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) d \sqrt {c +d \tan \left (f x +e \right )}}{\left (i d +c \right ) \left (-d \tan \left (f x +e \right )+i d \right )}-\frac {\left (i c^{5}+2 i d^{2} c^{3}+i c \,d^{4}-6 c^{4} d -12 c^{2} d^{3}-6 d^{5}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{\left (i d +c \right ) \sqrt {-i d -c}}}{4 d^{2} \left (i d +c \right )^{4} \left (i d -c \right )^{2}}\right )}{f a}\) | \(364\) |
2/f/a*d^2*(1/4/(I*d-c)^(5/2)/(c+I*d)^4*(-6*I*c^2*d^2+I*d^4-4*c^3*d+4*c*d^3 +I*c^4)/d^2*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))-1/(I*d-c)^2/(c+I* d)^4*(3*I*c^2+I*d^2-2*c*d)/(c+d*tan(f*x+e))^(1/2)-1/3/(I*d-c)^2/(c+I*d)^4* (I*c^3+I*c*d^2-c^2*d-d^3)/(c+d*tan(f*x+e))^(3/2)+1/4/d^2/(c+I*d)^4/(I*d-c) ^2*(-(c^4+2*c^2*d^2+d^4)*d/(c+I*d)*(c+d*tan(f*x+e))^(1/2)/(-d*tan(f*x+e)+I *d)-(-6*c^4*d-12*c^2*d^3-6*d^5+I*c^5+2*I*c^3*d^2+I*c*d^4)/(c+I*d)/(-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. 2659 vs. \(2 (213) = 426\).
Time = 1.74 (sec) , antiderivative size = 2659, normalized size of antiderivative = 9.96 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \]
1/24*(6*((I*a*c^7 + a*c^6*d + 3*I*a*c^5*d^2 + 3*a*c^4*d^3 + 3*I*a*c^3*d^4 + 3*a*c^2*d^5 + I*a*c*d^6 + a*d^7)*f*e^(6*I*f*x + 6*I*e) + 2*(I*a*c^7 - a* c^6*d + 3*I*a*c^5*d^2 - 3*a*c^4*d^3 + 3*I*a*c^3*d^4 - 3*a*c^2*d^5 + I*a*c* d^6 - a*d^7)*f*e^(4*I*f*x + 4*I*e) + (I*a*c^7 - 3*a*c^6*d - I*a*c^5*d^2 - 5*a*c^4*d^3 - 5*I*a*c^3*d^4 - a*c^2*d^5 - 3*I*a*c*d^6 + a*d^7)*f*e^(2*I*f* x + 2*I*e))*sqrt(1/4*I/((-I*a^2*c^5 - 5*a^2*c^4*d + 10*I*a^2*c^3*d^2 + 10* a^2*c^2*d^3 - 5*I*a^2*c*d^4 - a^2*d^5)*f^2))*log(-2*(2*((I*a*c^3 + 3*a*c^2 *d - 3*I*a*c*d^2 - a*d^3)*f*e^(2*I*f*x + 2*I*e) + (I*a*c^3 + 3*a*c^2*d - 3 *I*a*c*d^2 - a*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(1/4*I/((-I*a^2*c^5 - 5*a^2*c^4*d + 10*I*a^2*c^ 3*d^2 + 10*a^2*c^2*d^3 - 5*I*a^2*c*d^4 - a^2*d^5)*f^2)) - (c - I*d)*e^(2*I *f*x + 2*I*e) - c)*e^(-2*I*f*x - 2*I*e)) + 6*((-I*a*c^7 - a*c^6*d - 3*I*a* c^5*d^2 - 3*a*c^4*d^3 - 3*I*a*c^3*d^4 - 3*a*c^2*d^5 - I*a*c*d^6 - a*d^7)*f *e^(6*I*f*x + 6*I*e) + 2*(-I*a*c^7 + a*c^6*d - 3*I*a*c^5*d^2 + 3*a*c^4*d^3 - 3*I*a*c^3*d^4 + 3*a*c^2*d^5 - I*a*c*d^6 + a*d^7)*f*e^(4*I*f*x + 4*I*e) + (-I*a*c^7 + 3*a*c^6*d + I*a*c^5*d^2 + 5*a*c^4*d^3 + 5*I*a*c^3*d^4 + a*c^ 2*d^5 + 3*I*a*c*d^6 - a*d^7)*f*e^(2*I*f*x + 2*I*e))*sqrt(1/4*I/((-I*a^2*c^ 5 - 5*a^2*c^4*d + 10*I*a^2*c^3*d^2 + 10*a^2*c^2*d^3 - 5*I*a^2*c*d^4 - a^2* d^5)*f^2))*log(-2*(2*((-I*a*c^3 - 3*a*c^2*d + 3*I*a*c*d^2 + a*d^3)*f*e^(2* I*f*x + 2*I*e) + (-I*a*c^3 - 3*a*c^2*d + 3*I*a*c*d^2 + a*d^3)*f)*sqrt((...
\[ \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=- \frac {i \int \frac {1}{c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} - i c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} + 2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )} - 2 i c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} + d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{3}{\left (e + f x \right )} - i d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}}\, dx}{a} \]
-I*Integral(1/(c**2*sqrt(c + d*tan(e + f*x))*tan(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)**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))*t an(e + f*x)**3 - I*d**2*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**2), x)/a
Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (213) = 426\).
Time = 1.32 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.04 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=d^{2} {\left (\frac {{\left (-i \, c + 6 \, d\right )} \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} + i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{{\left (a c^{3} d^{2} f + 3 i \, a c^{2} d^{3} f - 3 \, a c d^{4} f - i \, a d^{5} f\right )} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} {\left (\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {2 \, {\left (9 \, {\left (d \tan \left (f x + e\right ) + c\right )} c + c^{2} - 3 \, {\left (i \, d \tan \left (f x + e\right ) + i \, c\right )} d + d^{2}\right )}}{-3 \, {\left (-i \, a c^{5} f + a c^{4} d f - 2 i \, a c^{3} d^{2} f + 2 \, a c^{2} d^{3} f - i \, a c d^{4} f + a d^{5} f\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} + \frac {2 \, \sqrt {d \tan \left (f x + e\right ) + c}}{-4 \, {\left (i \, a c^{3} d f - 3 \, a c^{2} d^{2} f - 3 i \, a c d^{3} f + a d^{4} f\right )} {\left (i \, d \tan \left (f x + e\right ) + d\right )}} + \frac {i \, \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{{\left (a c^{2} d^{2} f - 2 i \, a c d^{3} f - a d^{4} f\right )} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}}\right )} \]
d^2*((-I*c + 6*d)*arctan(2*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*s qrt(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*c^3*d^2*f + 3*I*a*c^2*d^3*f - 3*a*c*d^4*f - I*a*d^5*f)*sqrt(-2*c + 2*sq rt(c^2 + d^2))*(I*d/(c - sqrt(c^2 + d^2)) + 1)) + 2*(9*(d*tan(f*x + e) + c )*c + c^2 - 3*(I*d*tan(f*x + e) + I*c)*d + d^2)/((3*I*a*c^5*f - 3*a*c^4*d* f + 6*I*a*c^3*d^2*f - 6*a*c^2*d^3*f + 3*I*a*c*d^4*f - 3*a*d^5*f)*(d*tan(f* x + e) + c)^(3/2)) + 2*sqrt(d*tan(f*x + e) + c)/((-4*I*a*c^3*d*f + 12*a*c^ 2*d^2*f + 12*I*a*c*d^3*f - 4*a*d^4*f)*(I*d*tan(f*x + e) + d)) + I*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 - sq rt(c^2 + d^2)*sqrt(-2*c + 2*sqrt(c^2 + d^2))))/((a*c^2*d^2*f - 2*I*a*c*d^3 *f - a*d^4*f)*sqrt(-2*c + 2*sqrt(c^2 + d^2))*(-I*d/(c - sqrt(c^2 + d^2)) + 1)))
Time = 100.54 (sec) , antiderivative size = 69981, normalized size of antiderivative = 262.10 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \]
log((a*f*(139*c*d^7 - d^8*30i + c^2*d^6*180i - 62*c^3*d^5 + c^4*d^4*10i - c^5*d^3))/2 - (((a*f*(208*a^2*c^2*d^11*f^2 - a^2*c*d^12*f^2*320i - 112*a^2 *d^13*f^2 - a^2*c^3*d^10*f^2*640i + 1312*a^2*c^4*d^9*f^2 + 1568*a^2*c^6*d^ 7*f^2 + a^2*c^7*d^6*f^2*640i + 592*a^2*c^8*d^5*f^2 + a^2*c^9*d^4*f^2*320i + 16*a^2*c^10*d^3*f^2))/2 - 2*(c + d*tan(e + f*x))^(1/2)*(a^2*d^2*f^2 - a^ 2*c^2*f^2 + a^2*c*d*f^2*2i)*((4480*c^2*d^9 - 560*d^11 - c*d^10*2800i + c^3 *d^8*4480i - 560*c^4*d^7 - c^5*d^6*112i - 224*c^6*d^5 + c^7*d^4*32i - a^2* c^10*f^2*(((3920*c*d^12 - 16240*c^3*d^10 + 5712*c^5*d^8 + 304*c^7*d^6 + 32 *c^9*d^4)/(a^2*c^12*f^2 + a^2*d^12*f^2 + 6*a^2*c^2*d^10*f^2 + 15*a^2*c^4*d ^8*f^2 + 20*a^2*c^6*d^6*f^2 + 15*a^2*c^8*d^4*f^2 + 6*a^2*c^10*d^2*f^2) + ( (10640*c^2*d^11 - 560*d^13 - 14000*c^4*d^9 + 560*c^6*d^7 + 160*c^8*d^5)*1i )/(a^2*c^12*f^2 + a^2*d^12*f^2 + 6*a^2*c^2*d^10*f^2 + 15*a^2*c^4*d^8*f^2 + 20*a^2*c^6*d^6*f^2 + 15*a^2*c^8*d^4*f^2 + 6*a^2*c^10*d^2*f^2))^2 - 4*(256 *d^6 + 256*c^2*d^4)*(((60*c*d^7 + 10*c^3*d^5)*1i)/(a^4*c^12*f^4 + a^4*d^12 *f^4 + 6*a^4*c^2*d^10*f^4 + 15*a^4*c^4*d^8*f^4 + 20*a^4*c^6*d^6*f^4 + 15*a ^4*c^8*d^4*f^4 + 6*a^4*c^10*d^2*f^4) + (36*d^8 - 13*c^2*d^6 + c^4*d^4)/(a^ 4*c^12*f^4 + a^4*d^12*f^4 + 6*a^4*c^2*d^10*f^4 + 15*a^4*c^4*d^8*f^4 + 20*a ^4*c^6*d^6*f^4 + 15*a^4*c^8*d^4*f^4 + 6*a^4*c^10*d^2*f^4)))^(1/2)*1i + a^2 *d^10*f^2*(((3920*c*d^12 - 16240*c^3*d^10 + 5712*c^5*d^8 + 304*c^7*d^6 + 3 2*c^9*d^4)/(a^2*c^12*f^2 + a^2*d^12*f^2 + 6*a^2*c^2*d^10*f^2 + 15*a^2*c...