Integrand size = 32, antiderivative size = 151 \[ \int \frac {(a+i a \tan (e+f x))^{3/2}}{\sqrt {c+d \tan (e+f x)}} \, dx=-\frac {2 (-1)^{3/4} a^{3/2} \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {d} f}-\frac {2 i \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{\sqrt {c-i d} f} \]
-2*I*a^(3/2)*arctanh(2^(1/2)*a^(1/2)*(c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2)/ (a+I*a*tan(f*x+e))^(1/2))*2^(1/2)/f/(c-I*d)^(1/2)-2*(-1)^(3/4)*a^(3/2)*arc tanh((-1)^(3/4)*d^(1/2)*(a+I*a*tan(f*x+e))^(1/2)/a^(1/2)/(c+d*tan(f*x+e))^ (1/2))/f/d^(1/2)
Time = 1.87 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.34 \[ \int \frac {(a+i a \tan (e+f x))^{3/2}}{\sqrt {c+d \tan (e+f x)}} \, dx=\frac {2 a \left (-\frac {\sqrt {2} \sqrt {-a (c-i d)} \arctan \left (\frac {\sqrt {-a (c-i d)} \sqrt {a+i a \tan (e+f x)}}{\sqrt {2} a \sqrt {c+d \tan (e+f x)}}\right )}{i c+d}+\frac {\sqrt [4]{-1} \sqrt {a} \sqrt {c+i d} \arcsin \left (\frac {\sqrt [4]{-1} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+i d}}\right ) \sqrt {\frac {c+d \tan (e+f x)}{c+i d}}}{\sqrt {d} \sqrt {c+d \tan (e+f x)}}\right )}{f} \]
(2*a*(-((Sqrt[2]*Sqrt[-(a*(c - I*d))]*ArcTan[(Sqrt[-(a*(c - I*d))]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[2]*a*Sqrt[c + d*Tan[e + f*x]])])/(I*c + d)) + ( (-1)^(1/4)*Sqrt[a]*Sqrt[c + I*d]*ArcSin[((-1)^(1/4)*Sqrt[d]*Sqrt[a + I*a*T an[e + f*x]])/(Sqrt[a]*Sqrt[c + I*d])]*Sqrt[(c + d*Tan[e + f*x])/(c + I*d) ])/(Sqrt[d]*Sqrt[c + d*Tan[e + f*x]])))/f
Time = 0.74 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 4038, 3042, 4027, 221, 4082, 66, 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.
\(\displaystyle \int \frac {(a+i a \tan (e+f x))^{3/2}}{\sqrt {c+d \tan (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+i a \tan (e+f x))^{3/2}}{\sqrt {c+d \tan (e+f x)}}dx\) |
\(\Big \downarrow \) 4038 |
\(\displaystyle 2 a \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx+i \int \frac {\sqrt {i \tan (e+f x) a+a} (\tan (e+f x) a+i a)}{\sqrt {c+d \tan (e+f x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 a \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx+i \int \frac {\sqrt {i \tan (e+f x) a+a} (\tan (e+f x) a+i a)}{\sqrt {c+d \tan (e+f x)}}dx\) |
\(\Big \downarrow \) 4027 |
\(\displaystyle i \int \frac {\sqrt {i \tan (e+f x) a+a} (\tan (e+f x) a+i a)}{\sqrt {c+d \tan (e+f x)}}dx-\frac {4 i a^3 \int \frac {1}{a (c-i d)-\frac {2 a^2 (c+d \tan (e+f x))}{i \tan (e+f x) a+a}}d\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {i \tan (e+f x) a+a}}}{f}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle i \int \frac {\sqrt {i \tan (e+f x) a+a} (\tan (e+f x) a+i a)}{\sqrt {c+d \tan (e+f x)}}dx-\frac {2 i \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f \sqrt {c-i d}}\) |
\(\Big \downarrow \) 4082 |
\(\displaystyle -\frac {a^2 \int \frac {1}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{f}-\frac {2 i \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f \sqrt {c-i d}}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle -\frac {2 a^2 \int \frac {1}{i a-\frac {d (i \tan (e+f x) a+a)}{c+d \tan (e+f x)}}d\frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}}{f}-\frac {2 i \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f \sqrt {c-i d}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {2 (-1)^{3/4} a^{3/2} \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {d} f}-\frac {2 i \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f \sqrt {c-i d}}\) |
(-2*(-1)^(3/4)*a^(3/2)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f* x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[d]*f) - ((2*I)*Sqrt[2]*a^( 3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqr t[a + I*a*Tan[e + f*x]])])/(Sqrt[c - I*d]*f)
3.12.56.3.1 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] && !GtQ[c - a*(d/b), 0]
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[Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[-2*a*(b/f) Subst[Int[1/(a*c - b*d - 2* a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && N eQ[c^2 + d^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(3/2)/Sqrt[(c_.) + (d_.)*tan[(e_ .) + (f_.)*(x_)]], x_Symbol] :> Simp[2*a Int[Sqrt[a + b*Tan[e + f*x]]/Sqr t[c + d*Tan[e + f*x]], x], x] + Simp[b/a Int[(b + a*Tan[e + f*x])*(Sqrt[a + b*Tan[e + f*x]]/Sqrt[c + d*Tan[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
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[b*(B/f) Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[ a^2 + b^2, 0] && EqQ[A*b + a*B, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 982 vs. \(2 (114 ) = 228\).
Time = 1.39 (sec) , antiderivative size = 983, normalized size of antiderivative = 6.51
method | result | size |
derivativedivides | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {c +d \tan \left (f x +e \right )}\, a^{2} \left (i \sqrt {i a d}\, \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) c +i \sqrt {i a d}\, \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) d +i \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) c -i \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) d +\sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) c^{2}+\sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) d^{2}-\sqrt {i a d}\, \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) c +\sqrt {i a d}\, \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) d +\sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) c +\sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) d \right ) \sqrt {2}}{2 f \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, \left (c^{2}+d^{2}\right ) \sqrt {i a d}\, \sqrt {-a \left (i d -c \right )}}\) | \(983\) |
default | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {c +d \tan \left (f x +e \right )}\, a^{2} \left (i \sqrt {i a d}\, \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) c +i \sqrt {i a d}\, \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) d +i \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) c -i \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) d +\sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) c^{2}+\sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) d^{2}-\sqrt {i a d}\, \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) c +\sqrt {i a d}\, \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) d +\sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) c +\sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) d \right ) \sqrt {2}}{2 f \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, \left (c^{2}+d^{2}\right ) \sqrt {i a d}\, \sqrt {-a \left (i d -c \right )}}\) | \(983\) |
1/2/f*(a*(1+I*tan(f*x+e)))^(1/2)*(c+d*tan(f*x+e))^(1/2)*a^2*(I*(I*a*d)^(1/ 2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c) )^(1/2)*(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*c+I*( I*a*d)^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*( -a*(I*d-c))^(1/2)*(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2))/(tan(f*x+e) +I))*d+I*2^(1/2)*(-a*(I*d-c))^(1/2)*ln(1/2*(2*I*a*d*tan(f*x+e)+I*a*c+2*(a* (1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2)) *c-I*2^(1/2)*(-a*(I*d-c))^(1/2)*ln(1/2*(2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(1+I *tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*d+2 ^(1/2)*(-a*(I*d-c))^(1/2)*ln(1/2*(2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(1+I*tan(f *x+e))*(c+d*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*c^2+2^(1/ 2)*(-a*(I*d-c))^(1/2)*ln(1/2*(2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(1+I*tan(f*x+e ))*(c+d*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*d^2-(I*a*d)^( 1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d- c))^(1/2)*(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*c+( I*a*d)^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*( -a*(I*d-c))^(1/2)*(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2))/(tan(f*x+e) +I))*d+2^(1/2)*(-a*(I*d-c))^(1/2)*ln(1/2*(2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(1 +I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*c +2^(1/2)*(-a*(I*d-c))^(1/2)*ln(1/2*(2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(1+I*...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (109) = 218\).
Time = 0.25 (sec) , antiderivative size = 533, normalized size of antiderivative = 3.53 \[ \int \frac {(a+i a \tan (e+f x))^{3/2}}{\sqrt {c+d \tan (e+f x)}} \, dx=\frac {1}{2} \, \sqrt {-\frac {8 i \, a^{3}}{{\left (i \, c + d\right )} f^{2}}} \log \left (\frac {{\left ({\left (i \, c + d\right )} f \sqrt {-\frac {8 i \, a^{3}}{{\left (i \, c + d\right )} f^{2}}} e^{\left (i \, f x + i \, e\right )} + 2 \, \sqrt {2} {\left (a e^{\left (2 i \, f x + 2 i \, e\right )} + a\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{2 \, a}\right ) - \frac {1}{2} \, \sqrt {-\frac {8 i \, a^{3}}{{\left (i \, c + d\right )} f^{2}}} \log \left (\frac {{\left ({\left (-i \, c - d\right )} f \sqrt {-\frac {8 i \, a^{3}}{{\left (i \, c + d\right )} f^{2}}} e^{\left (i \, f x + i \, e\right )} + 2 \, \sqrt {2} {\left (a e^{\left (2 i \, f x + 2 i \, e\right )} + a\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{2 \, a}\right ) - \frac {1}{2} \, \sqrt {-\frac {4 i \, a^{3}}{d f^{2}}} \log \left (\frac {{\left (d f \sqrt {-\frac {4 i \, a^{3}}{d f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} {\left (a e^{\left (2 i \, f x + 2 i \, e\right )} + a\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a}\right ) + \frac {1}{2} \, \sqrt {-\frac {4 i \, a^{3}}{d f^{2}}} \log \left (-\frac {{\left (d f \sqrt {-\frac {4 i \, a^{3}}{d f^{2}}} e^{\left (i \, f x + i \, e\right )} - \sqrt {2} {\left (a e^{\left (2 i \, f x + 2 i \, e\right )} + a\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a}\right ) \]
1/2*sqrt(-8*I*a^3/((I*c + d)*f^2))*log(1/2*((I*c + d)*f*sqrt(-8*I*a^3/((I* c + d)*f^2))*e^(I*f*x + I*e) + 2*sqrt(2)*(a*e^(2*I*f*x + 2*I*e) + a)*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( a/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-I*f*x - I*e)/a) - 1/2*sqrt(-8*I*a^3/((I* c + d)*f^2))*log(1/2*((-I*c - d)*f*sqrt(-8*I*a^3/((I*c + d)*f^2))*e^(I*f*x + I*e) + 2*sqrt(2)*(a*e^(2*I*f*x + 2*I*e) + a)*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(a/(e^(2*I*f*x + 2*I*e ) + 1)))*e^(-I*f*x - I*e)/a) - 1/2*sqrt(-4*I*a^3/(d*f^2))*log((d*f*sqrt(-4 *I*a^3/(d*f^2))*e^(I*f*x + I*e) + sqrt(2)*(a*e^(2*I*f*x + 2*I*e) + a)*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 (a/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-I*f*x - I*e)/a) + 1/2*sqrt(-4*I*a^3/(d* f^2))*log(-(d*f*sqrt(-4*I*a^3/(d*f^2))*e^(I*f*x + I*e) - sqrt(2)*(a*e^(2*I *f*x + 2*I*e) + a)*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(a/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-I*f*x - I*e)/a)
\[ \int \frac {(a+i a \tan (e+f x))^{3/2}}{\sqrt {c+d \tan (e+f x)}} \, dx=\int \frac {\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}}}{\sqrt {c + d \tan {\left (e + f x \right )}}}\, dx \]
Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^{3/2}}{\sqrt {c+d \tan (e+f x)}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((d^2-2*c*d-c^2)>0)', see `assume ?` for mor
Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^{3/2}}{\sqrt {c+d \tan (e+f x)}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Non regular value [0,0] was discard ed and replaced randomly by 0=[59,-68]Warning, replacing 59 by 96, a subst itution v
Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{3/2}}{\sqrt {c+d \tan (e+f x)}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \]