Integrand size = 35, antiderivative size = 153 \[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{\sqrt {c-i c \tan (e+f x)}} \, dx=\frac {6 i a^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{\sqrt {c} f}-\frac {2 i a (a+i a \tan (e+f x))^{3/2}}{f \sqrt {c-i c \tan (e+f x)}}-\frac {3 i a^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c f} \]
6*I*a^(5/2)*arctan(c^(1/2)*(a+I*a*tan(f*x+e))^(1/2)/a^(1/2)/(c-I*c*tan(f*x +e))^(1/2))/f/c^(1/2)-3*I*a^2*(a+I*a*tan(f*x+e))^(1/2)*(c-I*c*tan(f*x+e))^ (1/2)/c/f-2*I*a*(a+I*a*tan(f*x+e))^(3/2)/f/(c-I*c*tan(f*x+e))^(1/2)
Time = 2.98 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.25 \[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{\sqrt {c-i c \tan (e+f x)}} \, dx=\frac {i a^{5/2} \left (\frac {4 \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{\sqrt {c}}+\frac {2 \arcsin \left (\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \sqrt {1-i \tan (e+f x)}}{\sqrt {c-i c \tan (e+f x)}}-\frac {\sqrt {a} \left (5+4 i \tan (e+f x)+\tan ^2(e+f x)\right )}{\sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}\right )}{f} \]
(I*a^(5/2)*((4*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/Sqrt[c] + (2*ArcSin[Sqrt[a + I*a*Tan[e + f*x]]/(Sq rt[2]*Sqrt[a])]*Sqrt[1 - I*Tan[e + f*x]])/Sqrt[c - I*c*Tan[e + f*x]] - (Sq rt[a]*(5 + (4*I)*Tan[e + f*x] + Tan[e + f*x]^2))/(Sqrt[a + I*a*Tan[e + f*x ]]*Sqrt[c - I*c*Tan[e + f*x]])))/f
Time = 0.30 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3042, 4006, 57, 60, 45, 218}
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.
| \(\displaystyle \int \frac {(a+i a \tan (e+f x))^{5/2}}{\sqrt {c-i c \tan (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (e+f x))^{5/2}}{\sqrt {c-i c \tan (e+f x)}}dx\) |
| \(\Big \downarrow \) 4006 |
| \(\displaystyle \frac {a c \int \frac {(i \tan (e+f x) a+a)^{3/2}}{(c-i c \tan (e+f x))^{3/2}}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {a c \left (-\frac {3 a \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c-i c \tan (e+f x)}}d\tan (e+f x)}{c}-\frac {2 i (a+i a \tan (e+f x))^{3/2}}{c \sqrt {c-i c \tan (e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {a c \left (-\frac {3 a \left (a \int \frac {1}{\sqrt {i \tan (e+f x) a+a} \sqrt {c-i c \tan (e+f x)}}d\tan (e+f x)+\frac {i \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c}\right )}{c}-\frac {2 i (a+i a \tan (e+f x))^{3/2}}{c \sqrt {c-i c \tan (e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 45 |
| \(\displaystyle \frac {a c \left (-\frac {3 a \left (2 a \int \frac {1}{i a+\frac {i c (i \tan (e+f x) a+a)}{c-i c \tan (e+f x)}}d\frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c-i c \tan (e+f x)}}+\frac {i \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c}\right )}{c}-\frac {2 i (a+i a \tan (e+f x))^{3/2}}{c \sqrt {c-i c \tan (e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a c \left (-\frac {3 a \left (\frac {i \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c}-\frac {2 i \sqrt {a} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{\sqrt {c}}\right )}{c}-\frac {2 i (a+i a \tan (e+f x))^{3/2}}{c \sqrt {c-i c \tan (e+f x)}}\right )}{f}\) |
(a*c*(((-2*I)*(a + I*a*Tan[e + f*x])^(3/2))/(c*Sqrt[c - I*c*Tan[e + f*x]]) - (3*a*(((-2*I)*Sqrt[a]*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt [a]*Sqrt[c - I*c*Tan[e + f*x]])])/Sqrt[c] + (I*Sqrt[a + I*a*Tan[e + f*x]]* Sqrt[c - I*c*Tan[e + f*x]])/c))/c))/f
3.11.17.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && !GtQ[c, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c , d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
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_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(c/f) Subst[Int[(a + b*x)^(m - 1)*( c + d*x)^(n - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (124 ) = 248\).
Time = 0.82 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.95
| method | result | size |
| derivativedivides | \(\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (3 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \left (\tan ^{2}\left (f x +e \right )\right )-3 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c -6 i \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \tan \left (f x +e \right )-6 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )-\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan ^{2}\left (f x +e \right )\right )+5 \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{f c \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan \left (f x +e \right )+i\right )^{2} \sqrt {a c}}\) | \(299\) |
| default | \(\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (3 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \left (\tan ^{2}\left (f x +e \right )\right )-3 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c -6 i \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \tan \left (f x +e \right )-6 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )-\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan ^{2}\left (f x +e \right )\right )+5 \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{f c \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan \left (f x +e \right )+i\right )^{2} \sqrt {a c}}\) | \(299\) |
I/f*(a*(1+I*tan(f*x+e)))^(1/2)*(-c*(I*tan(f*x+e)-1))^(1/2)*a^2/c*(3*I*ln(( c*a*tan(f*x+e)+(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c)^(1/2))*a*c* tan(f*x+e)^2-3*I*ln((c*a*tan(f*x+e)+(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/ 2))/(a*c)^(1/2))*a*c-6*I*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+ e)-6*ln((c*a*tan(f*x+e)+(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c)^(1 /2))*a*c*tan(f*x+e)-(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^2+ 5*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c*(1+tan(f*x+e)^2))^(1/2)/( tan(f*x+e)+I)^2/(a*c)^(1/2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (115) = 230\).
Time = 0.26 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.24 \[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{\sqrt {c-i c \tan (e+f x)}} \, dx=-\frac {3 \, \sqrt {\frac {a^{5}}{c f^{2}}} c f \log \left (\frac {4 \, {\left (2 \, {\left (a^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + a^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - {\left (i \, c f e^{\left (2 i \, f x + 2 i \, e\right )} - i \, c f\right )} \sqrt {\frac {a^{5}}{c f^{2}}}\right )}}{a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2}}\right ) - 3 \, \sqrt {\frac {a^{5}}{c f^{2}}} c f \log \left (\frac {4 \, {\left (2 \, {\left (a^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + a^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - {\left (-i \, c f e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c f\right )} \sqrt {\frac {a^{5}}{c f^{2}}}\right )}}{a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2}}\right ) + 4 \, {\left (2 i \, a^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + 3 i \, a^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{2 \, c f} \]
-1/2*(3*sqrt(a^5/(c*f^2))*c*f*log(4*(2*(a^2*e^(3*I*f*x + 3*I*e) + a^2*e^(I *f*x + I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1)) - (I*c*f*e^(2*I*f*x + 2*I*e) - I*c*f)*sqrt(a^5/(c*f^2)))/(a^2*e^(2* I*f*x + 2*I*e) + a^2)) - 3*sqrt(a^5/(c*f^2))*c*f*log(4*(2*(a^2*e^(3*I*f*x + 3*I*e) + a^2*e^(I*f*x + I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/( e^(2*I*f*x + 2*I*e) + 1)) - (-I*c*f*e^(2*I*f*x + 2*I*e) + I*c*f)*sqrt(a^5/ (c*f^2)))/(a^2*e^(2*I*f*x + 2*I*e) + a^2)) + 4*(2*I*a^2*e^(3*I*f*x + 3*I*e ) + 3*I*a^2*e^(I*f*x + I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^( 2*I*f*x + 2*I*e) + 1)))/(c*f)
\[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{\sqrt {c-i c \tan (e+f x)}} \, dx=\int \frac {\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {5}{2}}}{\sqrt {- i c \left (\tan {\left (e + f x \right )} + i\right )}}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (115) = 230\).
Time = 0.54 (sec) , antiderivative size = 540, normalized size of antiderivative = 3.53 \[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{\sqrt {c-i c \tan (e+f x)}} \, dx=\frac {{\left (6 \, {\left (a^{2} \cos \left (2 \, f x + 2 \, e\right ) + i \, a^{2} \sin \left (2 \, f x + 2 \, e\right ) + a^{2}\right )} \arctan \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ), \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) + 6 \, {\left (a^{2} \cos \left (2 \, f x + 2 \, e\right ) + i \, a^{2} \sin \left (2 \, f x + 2 \, e\right ) + a^{2}\right )} \arctan \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ), -\sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) - 4 \, {\left (2 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + 2 i \, a^{2} \sin \left (2 \, f x + 2 \, e\right ) + 3 \, a^{2}\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 3 \, {\left (i \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) - a^{2} \sin \left (2 \, f x + 2 \, e\right ) + i \, a^{2}\right )} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) + 3 \, {\left (-i \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2} \sin \left (2 \, f x + 2 \, e\right ) - i \, a^{2}\right )} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} - 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) + 4 \, {\left (-2 i \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + 2 \, a^{2} \sin \left (2 \, f x + 2 \, e\right ) - 3 i \, a^{2}\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )\right )} \sqrt {a} \sqrt {c}}{-2 \, {\left (i \, c \cos \left (2 \, f x + 2 \, e\right ) - c \sin \left (2 \, f x + 2 \, e\right ) + i \, c\right )} f} \]
(6*(a^2*cos(2*f*x + 2*e) + I*a^2*sin(2*f*x + 2*e) + a^2)*arctan2(cos(1/2*a rctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))), sin(1/2*arctan2(sin(2*f*x + 2 *e), cos(2*f*x + 2*e))) + 1) + 6*(a^2*cos(2*f*x + 2*e) + I*a^2*sin(2*f*x + 2*e) + a^2)*arctan2(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))), -sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) - 4*(2*a^2*cos (2*f*x + 2*e) + 2*I*a^2*sin(2*f*x + 2*e) + 3*a^2)*cos(1/2*arctan2(sin(2*f* x + 2*e), cos(2*f*x + 2*e))) + 3*(I*a^2*cos(2*f*x + 2*e) - a^2*sin(2*f*x + 2*e) + I*a^2)*log(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*sin(1/2*arcta n2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) + 3*(-I*a^2*cos(2*f*x + 2*e) + a^2*sin(2*f*x + 2*e) - I*a^2)*log(cos(1/2*arctan2(sin(2*f*x + 2*e), cos( 2*f*x + 2*e)))^2 + sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) + 4*(-2*I*a^ 2*cos(2*f*x + 2*e) + 2*a^2*sin(2*f*x + 2*e) - 3*I*a^2)*sin(1/2*arctan2(sin (2*f*x + 2*e), cos(2*f*x + 2*e))))*sqrt(a)*sqrt(c)/((-2*I*c*cos(2*f*x + 2* e) + 2*c*sin(2*f*x + 2*e) - 2*I*c)*f)
\[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{\sqrt {c-i c \tan (e+f x)}} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {-i \, c \tan \left (f x + e\right ) + c}} \,d x } \]
Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{\sqrt {c-i c \tan (e+f x)}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}} \,d x \]