Integrand size = 36, antiderivative size = 98 \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {4 \sqrt [4]{-1} a^2 (i A+B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}+\frac {2 a^2 (i A-B) \sqrt {\tan (c+d x)}}{d}-\frac {2 A \left (a^2+i a^2 \tan (c+d x)\right )}{d \sqrt {\tan (c+d x)}} \]
-4*(-1)^(1/4)*a^2*(I*A+B)*arctan((-1)^(3/4)*tan(d*x+c)^(1/2))/d+2*a^2*(I*A -B)*tan(d*x+c)^(1/2)/d-2*A*(a^2+I*a^2*tan(d*x+c))/d/tan(d*x+c)^(1/2)
Time = 1.37 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.76 \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 a^2 \left (A-(1+i) \sqrt {2} (A-i B) \text {arctanh}\left (\frac {(1+i) \sqrt {\tan (c+d x)}}{\sqrt {2}}\right ) \sqrt {\tan (c+d x)}+B \tan (c+d x)\right )}{d \sqrt {\tan (c+d x)}} \]
(-2*a^2*(A - (1 + I)*Sqrt[2]*(A - I*B)*ArcTanh[((1 + I)*Sqrt[Tan[c + d*x]] )/Sqrt[2]]*Sqrt[Tan[c + d*x]] + B*Tan[c + d*x]))/(d*Sqrt[Tan[c + d*x]])
Time = 0.55 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3042, 4076, 27, 3042, 4075, 3042, 4016, 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.
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\tan (c+d x)^{3/2}}dx\) |
| \(\Big \downarrow \) 4076 |
| \(\displaystyle 2 \int \frac {(i \tan (c+d x) a+a) (a (3 i A+B)+a (A+i B) \tan (c+d x))}{2 \sqrt {\tan (c+d x)}}dx-\frac {2 A \left (a^2+i a^2 \tan (c+d x)\right )}{d \sqrt {\tan (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(i \tan (c+d x) a+a) (a (3 i A+B)+a (A+i B) \tan (c+d x))}{\sqrt {\tan (c+d x)}}dx-\frac {2 A \left (a^2+i a^2 \tan (c+d x)\right )}{d \sqrt {\tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(i \tan (c+d x) a+a) (a (3 i A+B)+a (A+i B) \tan (c+d x))}{\sqrt {\tan (c+d x)}}dx-\frac {2 A \left (a^2+i a^2 \tan (c+d x)\right )}{d \sqrt {\tan (c+d x)}}\) |
| \(\Big \downarrow \) 4075 |
| \(\displaystyle \int \frac {2 a^2 (i A+B)-2 a^2 (A-i B) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx+\frac {2 a^2 (-B+i A) \sqrt {\tan (c+d x)}}{d}-\frac {2 A \left (a^2+i a^2 \tan (c+d x)\right )}{d \sqrt {\tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {2 a^2 (i A+B)-2 a^2 (A-i B) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx+\frac {2 a^2 (-B+i A) \sqrt {\tan (c+d x)}}{d}-\frac {2 A \left (a^2+i a^2 \tan (c+d x)\right )}{d \sqrt {\tan (c+d x)}}\) |
| \(\Big \downarrow \) 4016 |
| \(\displaystyle \frac {8 a^4 (B+i A)^2 \int \frac {1}{2 (i A+B) a^2+2 (A-i B) \tan (c+d x) a^2}d\sqrt {\tan (c+d x)}}{d}+\frac {2 a^2 (-B+i A) \sqrt {\tan (c+d x)}}{d}-\frac {2 A \left (a^2+i a^2 \tan (c+d x)\right )}{d \sqrt {\tan (c+d x)}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {4 \sqrt [4]{-1} a^2 (B+i A) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}+\frac {2 a^2 (-B+i A) \sqrt {\tan (c+d x)}}{d}-\frac {2 A \left (a^2+i a^2 \tan (c+d x)\right )}{d \sqrt {\tan (c+d x)}}\) |
(-4*(-1)^(1/4)*a^2*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d + (2 *a^2*(I*A - B)*Sqrt[Tan[c + d*x]])/d - (2*A*(a^2 + I*a^2*Tan[c + d*x]))/(d *Sqrt[Tan[c + d*x]])
3.2.23.3.1 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_)^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[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2*(c^2/f) Subst[Int[1/(b*c - d*x^2), x], x, Sqrt[b *Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && EqQ[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_)]), x_Symbol] :> Simp[B *d*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f* x])^m*Simp[A*c - B*d + (B*c + A*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && !LeQ[m, -1]
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^2)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(b*c + a*d)*(n + 1))), x] - Simp[a/(d*(b*c + a*d)*(n + 1)) Int[ (a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*b*d*(m - n - 2) - B*(b*c*(m - 1) + a*d*(n + 1)) + (a*A*d*(m + n) - B*(a*c*(m - 1) + b *d*(n + 1)))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (85 ) = 170\).
Time = 0.04 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.21
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (-2 B \left (\sqrt {\tan }\left (d x +c \right )\right )-\frac {2 A}{\sqrt {\tan \left (d x +c \right )}}+\frac {\left (2 i A +2 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (2 i B -2 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\) | \(217\) |
| default | \(\frac {a^{2} \left (-2 B \left (\sqrt {\tan }\left (d x +c \right )\right )-\frac {2 A}{\sqrt {\tan \left (d x +c \right )}}+\frac {\left (2 i A +2 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (2 i B -2 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\) | \(217\) |
| parts | \(\frac {\left (2 i A \,a^{2}+B \,a^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4 d}+\frac {\left (2 i B \,a^{2}-A \,a^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4 d}+\frac {A \,a^{2} \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}-\frac {2}{\sqrt {\tan \left (d x +c \right )}}\right )}{d}-\frac {B \,a^{2} \left (2 \left (\sqrt {\tan }\left (d x +c \right )\right )-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\) | \(414\) |
1/d*a^2*(-2*B*tan(d*x+c)^(1/2)-2*A/tan(d*x+c)^(1/2)+1/4*(2*B+2*I*A)*2^(1/2 )*(ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+ tan(d*x+c)))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan( d*x+c)^(1/2)))+1/4*(-2*A+2*I*B)*2^(1/2)*(ln((1-2^(1/2)*tan(d*x+c)^(1/2)+ta n(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))+2*arctan(1+2^(1/2)*tan( d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (82) = 164\).
Time = 0.26 (sec) , antiderivative size = 386, normalized size of antiderivative = 3.94 \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (-\frac {2 \, {\left ({\left (A - i \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{2}}\right ) - \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (-\frac {2 \, {\left ({\left (A - i \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{2}}\right ) - 2 \, {\left ({\left (i \, A + B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (i \, A - B\right )} a^{2}\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{d e^{\left (2 i \, d x + 2 i \, c\right )} - d} \]
(sqrt(-(-I*A^2 - 2*A*B + I*B^2)*a^4/d^2)*(d*e^(2*I*d*x + 2*I*c) - d)*log(- 2*((A - I*B)*a^2*e^(2*I*d*x + 2*I*c) + sqrt(-(-I*A^2 - 2*A*B + I*B^2)*a^4/ d^2)*(d*e^(2*I*d*x + 2*I*c) + d)*sqrt((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I *d*x + 2*I*c) + 1)))*e^(-2*I*d*x - 2*I*c)/((-I*A - B)*a^2)) - sqrt(-(-I*A^ 2 - 2*A*B + I*B^2)*a^4/d^2)*(d*e^(2*I*d*x + 2*I*c) - d)*log(-2*((A - I*B)* a^2*e^(2*I*d*x + 2*I*c) - sqrt(-(-I*A^2 - 2*A*B + I*B^2)*a^4/d^2)*(d*e^(2* I*d*x + 2*I*c) + d)*sqrt((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-2*I*d*x - 2*I*c)/((-I*A - B)*a^2)) - 2*((I*A + B)*a^2*e^(2*I*d *x + 2*I*c) + (I*A - B)*a^2)*sqrt((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1)))/(d*e^(2*I*d*x + 2*I*c) - d)
\[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=- a^{2} \left (\int \left (- \frac {A}{\tan ^{\frac {3}{2}}{\left (c + d x \right )}}\right )\, dx + \int A \sqrt {\tan {\left (c + d x \right )}}\, dx + \int \left (- \frac {B}{\sqrt {\tan {\left (c + d x \right )}}}\right )\, dx + \int B \tan ^{\frac {3}{2}}{\left (c + d x \right )}\, dx + \int \left (- \frac {2 i A}{\sqrt {\tan {\left (c + d x \right )}}}\right )\, dx + \int \left (- 2 i B \sqrt {\tan {\left (c + d x \right )}}\right )\, dx\right ) \]
-a**2*(Integral(-A/tan(c + d*x)**(3/2), x) + Integral(A*sqrt(tan(c + d*x)) , x) + Integral(-B/sqrt(tan(c + d*x)), x) + Integral(B*tan(c + d*x)**(3/2) , x) + Integral(-2*I*A/sqrt(tan(c + d*x)), x) + Integral(-2*I*B*sqrt(tan(c + d*x)), x))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (82) = 164\).
Time = 0.43 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.73 \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {4 \, B a^{2} \sqrt {\tan \left (d x + c\right )} - {\left (2 \, \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )} a^{2} + \frac {4 \, A a^{2}}{\sqrt {\tan \left (d x + c\right )}}}{2 \, d} \]
-1/2*(4*B*a^2*sqrt(tan(d*x + c)) - (2*sqrt(2)*((I - 1)*A + (I + 1)*B)*arct an(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*((I - 1)*A + (I + 1)*B)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) - sqrt(2) *(-(I + 1)*A + (I - 1)*B)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + sqrt(2)*(-(I + 1)*A + (I - 1)*B)*log(-sqrt(2)*sqrt(tan(d*x + c)) + ta n(d*x + c) + 1))*a^2 + 4*A*a^2/sqrt(tan(d*x + c)))/d
Time = 0.84 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.70 \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 \, B a^{2} \sqrt {\tan \left (d x + c\right )}}{d} + \frac {\left (2 i - 2\right ) \, \sqrt {2} {\left (A a^{2} - i \, B a^{2}\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{d} - \frac {2 \, A a^{2}}{d \sqrt {\tan \left (d x + c\right )}} \]
-2*B*a^2*sqrt(tan(d*x + c))/d + (2*I - 2)*sqrt(2)*(A*a^2 - I*B*a^2)*arctan (-(1/2*I - 1/2)*sqrt(2)*sqrt(tan(d*x + c)))/d - 2*A*a^2/(d*sqrt(tan(d*x + c)))
Time = 7.71 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.07 \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2\,A\,a^2}{d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}}-\frac {2\,B\,a^2\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}}{d}+\frac {\sqrt {2}\,A\,a^2\,\ln \left (-4\,A\,a^2\,d+\sqrt {2}\,A\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\left (-2-2{}\mathrm {i}\right )\right )\,\left (1+1{}\mathrm {i}\right )}{d}-\frac {\sqrt {4{}\mathrm {i}}\,A\,a^2\,\ln \left (-4\,A\,a^2\,d+2\,\sqrt {4{}\mathrm {i}}\,A\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}{d}+\frac {\sqrt {2}\,B\,a^2\,\ln \left (B\,a^2\,d\,4{}\mathrm {i}+\sqrt {2}\,B\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\left (-2+2{}\mathrm {i}\right )\right )\,\left (1-\mathrm {i}\right )}{d}-\frac {\sqrt {-4{}\mathrm {i}}\,B\,a^2\,\ln \left (B\,a^2\,d\,4{}\mathrm {i}+2\,\sqrt {-4{}\mathrm {i}}\,B\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}{d} \]
(2^(1/2)*A*a^2*log(- 4*A*a^2*d - 2^(1/2)*A*a^2*d*tan(c + d*x)^(1/2)*(2 + 2 i))*(1 + 1i))/d - (2*B*a^2*tan(c + d*x)^(1/2))/d - (2*A*a^2)/(d*tan(c + d* x)^(1/2)) - (4i^(1/2)*A*a^2*log(2*4i^(1/2)*A*a^2*d*tan(c + d*x)^(1/2) - 4* A*a^2*d))/d + (2^(1/2)*B*a^2*log(B*a^2*d*4i - 2^(1/2)*B*a^2*d*tan(c + d*x) ^(1/2)*(2 - 2i))*(1 - 1i))/d - ((-4i)^(1/2)*B*a^2*log(B*a^2*d*4i + 2*(-4i) ^(1/2)*B*a^2*d*tan(c + d*x)^(1/2)))/d