Integrand size = 45, antiderivative size = 169 \[ \int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\frac {2 a^{3/2} (i A+2 B) \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{\sqrt {c} f}-\frac {(i A+B) (a+i a \tan (e+f x))^{3/2}}{f \sqrt {c-i c \tan (e+f x)}}-\frac {a (i A+2 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c f} \]
2*a^(3/2)*(I*A+2*B)*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)-a*(I*A+2*B)*(a+I*a*tan(f*x+e))^(1/2)*(c-I*c* tan(f*x+e))^(1/2)/c/f-(I*A+B)*(a+I*a*tan(f*x+e))^(3/2)/f/(c-I*c*tan(f*x+e) )^(1/2)
Time = 5.59 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.88 \[ \int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\frac {2 a^{3/2} (i A+2 B) \arcsin \left (\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \sqrt {1-i \tan (e+f x)} \sqrt {a+i a \tan (e+f x)}-a^2 (-i+\tan (e+f x)) (-2 A+3 i B+B \tan (e+f x))}{f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \]
(2*a^(3/2)*(I*A + 2*B)*ArcSin[Sqrt[a + I*a*Tan[e + f*x]]/(Sqrt[2]*Sqrt[a]) ]*Sqrt[1 - I*Tan[e + f*x]]*Sqrt[a + I*a*Tan[e + f*x]] - a^2*(-I + Tan[e + f*x])*(-2*A + (3*I)*B + B*Tan[e + f*x]))/(f*Sqrt[a + I*a*Tan[e + f*x]]*Sqr t[c - I*c*Tan[e + f*x]])
Time = 0.39 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3042, 4071, 87, 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.
| \(\displaystyle \int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}}dx\) |
| \(\Big \downarrow \) 4071 |
| \(\displaystyle \frac {a c \int \frac {\sqrt {i \tan (e+f x) a+a} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {a c \left (-\frac {(A-2 i B) \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c-i c \tan (e+f x)}}d\tan (e+f x)}{c}-\frac {(B+i A) (a+i a \tan (e+f x))^{3/2}}{a c \sqrt {c-i c \tan (e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {a c \left (-\frac {(A-2 i B) \left (a \int \frac {1}{\sqrt {i \tan (e+f x) a+a} \sqrt {c-i c \tan (e+f x)}}d\tan (e+f x)+\frac {i \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c}\right )}{c}-\frac {(B+i A) (a+i a \tan (e+f x))^{3/2}}{a c \sqrt {c-i c \tan (e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 45 |
| \(\displaystyle \frac {a c \left (-\frac {(A-2 i B) \left (2 a \int \frac {1}{i a+\frac {i c (i \tan (e+f x) a+a)}{c-i c \tan (e+f x)}}d\frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c-i c \tan (e+f x)}}+\frac {i \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c}\right )}{c}-\frac {(B+i A) (a+i a \tan (e+f x))^{3/2}}{a c \sqrt {c-i c \tan (e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a c \left (-\frac {(A-2 i B) \left (\frac {i \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c}-\frac {2 i \sqrt {a} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{\sqrt {c}}\right )}{c}-\frac {(B+i A) (a+i a \tan (e+f x))^{3/2}}{a c \sqrt {c-i c \tan (e+f x)}}\right )}{f}\) |
(a*c*(-(((I*A + B)*(a + I*a*Tan[e + f*x])^(3/2))/(a*c*Sqrt[c - I*c*Tan[e + f*x]])) - ((A - (2*I)*B)*(((-2*I)*Sqrt[a]*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Ta n[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
3.8.99.3.1 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 + 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si mp[a*(c/f) Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1)*(A + B*x), x], x , Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, 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. 496 vs. \(2 (140 ) = 280\).
Time = 0.35 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.94
| method | result | size |
| derivativedivides | \(\frac {\left (2 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{2}-2 i A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )-A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{2}-2 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -4 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-4 B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )-B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+2 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}+A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +2 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+3 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right ) \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a}{f c \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \left (i+\tan \left (f x +e \right )\right )^{2} \sqrt {a c}}\) | \(497\) |
| default | \(\frac {\left (2 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{2}-2 i A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )-A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{2}-2 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -4 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-4 B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )-B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+2 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}+A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +2 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+3 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right ) \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a}{f c \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \left (i+\tan \left (f x +e \right )\right )^{2} \sqrt {a c}}\) | \(497\) |
| parts | \(\frac {i A \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 \left (i \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) \tan \left (f x +e \right )^{2} a c -i \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -2 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-2 \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) \tan \left (f x +e \right ) a c +2 \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{f c \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \left (i+\tan \left (f x +e \right )\right )^{2} \sqrt {a c}}+\frac {B \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 \left (2 i \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) \tan \left (f x +e \right )^{2} a c -2 i \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -4 \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) \tan \left (f x +e \right ) a c -4 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-\tan \left (f x +e \right )^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+3 \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{f c \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \left (i+\tan \left (f x +e \right )\right )^{2} \sqrt {a c}}\) | \(564\) |
int((a+I*a*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(1/2),x,m ethod=_RETURNVERBOSE)
1/f*(2*I*B*ln((a*c*tan(f*x+e)+(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a *c)^(1/2))*a*c*tan(f*x+e)^2-2*I*A*ln((a*c*tan(f*x+e)+(a*c)^(1/2)*(a*c*(1+t an(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a*c*tan(f*x+e)-A*ln((a*c*tan(f*x+e)+(a*c )^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a*c*tan(f*x+e)^2-2*I*B* ln((a*c*tan(f*x+e)+(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))* a*c-4*I*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)-4*B*ln((a*c* tan(f*x+e)+(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a*c*tan( f*x+e)-B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^2+2*I*A*(a*c) ^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)+A*ln((a*c*tan(f*x+e)+(a*c)^(1/2)*(a*c* (1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a*c+2*A*(a*c)^(1/2)*(a*c*(1+tan(f*x+ e)^2))^(1/2)*tan(f*x+e)+3*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))*(a*( 1+I*tan(f*x+e)))^(1/2)*(-c*(I*tan(f*x+e)-1))^(1/2)*a/c/(a*c*(1+tan(f*x+e)^ 2))^(1/2)/(I+tan(f*x+e))^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. 438 vs. \(2 (131) = 262\).
Time = 0.28 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.59 \[ \int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\frac {c \sqrt {\frac {{\left (A^{2} - 4 i \, A B - 4 \, B^{2}\right )} a^{3}}{c f^{2}}} f \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (-i \, A - 2 \, B\right )} a e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-i \, A - 2 \, B\right )} a e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} + {\left (c f e^{\left (2 i \, f x + 2 i \, e\right )} - c f\right )} \sqrt {\frac {{\left (A^{2} - 4 i \, A B - 4 \, B^{2}\right )} a^{3}}{c f^{2}}}\right )}}{{\left (i \, A + 2 \, B\right )} a e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, A + 2 \, B\right )} a}\right ) - c \sqrt {\frac {{\left (A^{2} - 4 i \, A B - 4 \, B^{2}\right )} a^{3}}{c f^{2}}} f \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (-i \, A - 2 \, B\right )} a e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-i \, A - 2 \, B\right )} a e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - {\left (c f e^{\left (2 i \, f x + 2 i \, e\right )} - c f\right )} \sqrt {\frac {{\left (A^{2} - 4 i \, A B - 4 \, B^{2}\right )} a^{3}}{c f^{2}}}\right )}}{{\left (i \, A + 2 \, B\right )} a e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, A + 2 \, B\right )} a}\right ) - 4 \, {\left ({\left (i \, A + B\right )} a e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (i \, A + 2 \, B\right )} a e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{2 \, c f} \]
integrate((a+I*a*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(1/ 2),x, algorithm="fricas")
1/2*(c*sqrt((A^2 - 4*I*A*B - 4*B^2)*a^3/(c*f^2))*f*log(-4*(2*((-I*A - 2*B) *a*e^(3*I*f*x + 3*I*e) + (-I*A - 2*B)*a*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*e^(2*I*f*x + 2*I *e) - c*f)*sqrt((A^2 - 4*I*A*B - 4*B^2)*a^3/(c*f^2)))/((I*A + 2*B)*a*e^(2* I*f*x + 2*I*e) + (I*A + 2*B)*a)) - c*sqrt((A^2 - 4*I*A*B - 4*B^2)*a^3/(c*f ^2))*f*log(-4*(2*((-I*A - 2*B)*a*e^(3*I*f*x + 3*I*e) + (-I*A - 2*B)*a*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*e^(2*I*f*x + 2*I*e) - c*f)*sqrt((A^2 - 4*I*A*B - 4*B^2)*a^3/ (c*f^2)))/((I*A + 2*B)*a*e^(2*I*f*x + 2*I*e) + (I*A + 2*B)*a)) - 4*((I*A + B)*a*e^(3*I*f*x + 3*I*e) + (I*A + 2*B)*a*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)
\[ \int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\int \frac {\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}} \left (A + B \tan {\left (e + f x \right )}\right )}{\sqrt {- i c \left (\tan {\left (e + f x \right )} + i\right )}}\, dx \]
Integral((I*a*(tan(e + f*x) - I))**(3/2)*(A + B*tan(e + f*x))/sqrt(-I*c*(t an(e + f*x) + I)), x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 599 vs. \(2 (131) = 262\).
Time = 0.47 (sec) , antiderivative size = 599, normalized size of antiderivative = 3.54 \[ \int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\frac {{\left (2 \, {\left ({\left (A - 2 i \, B\right )} a \cos \left (2 \, f x + 2 \, e\right ) - {\left (-i \, A - 2 \, B\right )} a \sin \left (2 \, f x + 2 \, e\right ) + {\left (A - 2 i \, B\right )} a\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 ) + 2 \, {\left ({\left (A - 2 i \, B\right )} a \cos \left (2 \, f x + 2 \, e\right ) - {\left (-i \, A - 2 \, B\right )} a \sin \left (2 \, f x + 2 \, e\right ) + {\left (A - 2 i \, B\right )} a\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 ({\left (A - i \, B\right )} a \cos \left (2 \, f x + 2 \, e\right ) + {\left (i \, A + B\right )} a \sin \left (2 \, f x + 2 \, e\right ) + {\left (A - 2 i \, B\right )} a\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 ) + {\left ({\left (i \, A + 2 \, B\right )} a \cos \left (2 \, f x + 2 \, e\right ) - {\left (A - 2 i \, B\right )} a \sin \left (2 \, f x + 2 \, e\right ) + {\left (i \, A + 2 \, B\right )} a\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 ) + {\left ({\left (-i \, A - 2 \, B\right )} a \cos \left (2 \, f x + 2 \, e\right ) + {\left (A - 2 i \, B\right )} a \sin \left (2 \, f x + 2 \, e\right ) + {\left (-i \, A - 2 \, B\right )} a\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 ({\left (i \, A + B\right )} a \cos \left (2 \, f x + 2 \, e\right ) - {\left (A - i \, B\right )} a \sin \left (2 \, f x + 2 \, e\right ) + {\left (i \, A + 2 \, B\right )} a\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} \]
integrate((a+I*a*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(1/ 2),x, algorithm="maxima")
(2*((A - 2*I*B)*a*cos(2*f*x + 2*e) - (-I*A - 2*B)*a*sin(2*f*x + 2*e) + (A - 2*I*B)*a)*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) + 2*((A - 2*I*B) *a*cos(2*f*x + 2*e) - (-I*A - 2*B)*a*sin(2*f*x + 2*e) + (A - 2*I*B)*a)*arc tan2(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))), -sin(1/2*arctan 2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) - 4*((A - I*B)*a*cos(2*f*x + 2 *e) + (I*A + B)*a*sin(2*f*x + 2*e) + (A - 2*I*B)*a)*cos(1/2*arctan2(sin(2* f*x + 2*e), cos(2*f*x + 2*e))) + ((I*A + 2*B)*a*cos(2*f*x + 2*e) - (A - 2* I*B)*a*sin(2*f*x + 2*e) + (I*A + 2*B)*a)*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) + ( (-I*A - 2*B)*a*cos(2*f*x + 2*e) + (A - 2*I*B)*a*sin(2*f*x + 2*e) + (-I*A - 2*B)*a)*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*((I*A + B)*a*cos(2*f*x + 2*e) - (A - I*B)*a*sin(2*f*x + 2*e) + (I*A + 2*B)*a)*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)
\[ \int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{\sqrt {-i \, c \tan \left (f x + e\right ) + c}} \,d x } \]
integrate((a+I*a*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(1/ 2),x, algorithm="giac")
Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\int \frac {\left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}} \,d x \]