Integrand size = 28, antiderivative size = 90 \[ \int \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2 \, dx=\frac {4 (-1)^{3/4} a^2 \sqrt {d} \arctan \left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}+\frac {4 i a^2 \sqrt {d \tan (e+f x)}}{f}-\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f} \] Output:
4*(-1)^(3/4)*a^2*d^(1/2)*arctan((-1)^(3/4)*(d*tan(f*x+e))^(1/2)/d^(1/2))/f +4*I*a^2*(d*tan(f*x+e))^(1/2)/f-2/3*a^2*(d*tan(f*x+e))^(3/2)/d/f
Time = 0.39 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.83 \[ \int \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2 \, dx=-\frac {2 a^2 \left ((3+3 i) \sqrt {2} \sqrt {d} \text {arctanh}\left (\frac {(1+i) \sqrt {d \tan (e+f x)}}{\sqrt {2} \sqrt {d}}\right )+\sqrt {d \tan (e+f x)} (-6 i+\tan (e+f x))\right )}{3 f} \] Input:
Integrate[Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x])^2,x]
Output:
(-2*a^2*((3 + 3*I)*Sqrt[2]*Sqrt[d]*ArcTanh[((1 + I)*Sqrt[d*Tan[e + f*x]])/ (Sqrt[2]*Sqrt[d])] + Sqrt[d*Tan[e + f*x]]*(-6*I + Tan[e + f*x])))/(3*f)
Time = 0.46 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 4026, 3042, 4011, 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.
| \(\displaystyle \int (a+i a \tan (e+f x))^2 \sqrt {d \tan (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+i a \tan (e+f x))^2 \sqrt {d \tan (e+f x)}dx\) |
| \(\Big \downarrow \) 4026 |
| \(\displaystyle -\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\int \sqrt {d \tan (e+f x)} \left (2 i \tan (e+f x) a^2+2 a^2\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\int \sqrt {d \tan (e+f x)} \left (2 i \tan (e+f x) a^2+2 a^2\right )dx\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \frac {2 a^2 d \tan (e+f x)-2 i a^2 d}{\sqrt {d \tan (e+f x)}}dx-\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\frac {4 i a^2 \sqrt {d \tan (e+f x)}}{f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {2 a^2 d \tan (e+f x)-2 i a^2 d}{\sqrt {d \tan (e+f x)}}dx-\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\frac {4 i a^2 \sqrt {d \tan (e+f x)}}{f}\) |
| \(\Big \downarrow \) 4016 |
| \(\displaystyle -\frac {8 a^4 d^2 \int \frac {1}{-2 i a^2 d^2-2 a^2 \tan (e+f x) d^2}d\sqrt {d \tan (e+f x)}}{f}-\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\frac {4 i a^2 \sqrt {d \tan (e+f x)}}{f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {4 (-1)^{3/4} a^2 \sqrt {d} \arctan \left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}-\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\frac {4 i a^2 \sqrt {d \tan (e+f x)}}{f}\) |
Input:
Int[Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x])^2,x]
Output:
(4*(-1)^(3/4)*a^2*Sqrt[d]*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d] ])/f + ((4*I)*a^2*Sqrt[d*Tan[e + f*x]])/f - (2*a^2*(d*Tan[e + f*x])^(3/2)) /(3*d*f)
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_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
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_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[d^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*( m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e + f* x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && !LeQ [m, -1] && !(EqQ[m, 2] && EqQ[a, 0])
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (73 ) = 146\).
Time = 1.46 (sec) , antiderivative size = 311, normalized size of antiderivative = 3.46
| method | result | size |
| derivativedivides | \(\frac {2 a^{2} \left (-\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 i d \sqrt {d \tan \left (f x +e \right )}-2 d^{2} \left (\frac {i \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f d}\) | \(311\) |
| default | \(\frac {2 a^{2} \left (-\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 i d \sqrt {d \tan \left (f x +e \right )}-2 d^{2} \left (\frac {i \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f d}\) | \(311\) |
| parts | \(\frac {a^{2} d \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 f \left (d^{2}\right )^{\frac {1}{4}}}+\frac {2 i a^{2} \left (2 \sqrt {d \tan \left (f x +e \right )}-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4}\right )}{f}-\frac {2 a^{2} \left (\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-\frac {d^{2} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f d}\) | \(451\) |
Input:
int((d*tan(f*x+e))^(1/2)*(a+I*a*tan(f*x+e))^2,x,method=_RETURNVERBOSE)
Output:
2/f*a^2/d*(-1/3*(d*tan(f*x+e))^(3/2)+2*I*d*(d*tan(f*x+e))^(1/2)-2*d^2*(1/8 *I/d*(d^2)^(1/4)*2^(1/2)*(ln((d*tan(f*x+e)+(d^2)^(1/4)*(d*tan(f*x+e))^(1/2 )*2^(1/2)+(d^2)^(1/2))/(d*tan(f*x+e)-(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1 /2)+(d^2)^(1/2)))+2*arctan(2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1)-2*a rctan(-2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1))-1/8/(d^2)^(1/4)*2^(1/2 )*(ln((d*tan(f*x+e)-(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2))/ (d*tan(f*x+e)+(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2)))+2*arc tan(2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1)-2*arctan(-2^(1/2)/(d^2)^(1 /4)*(d*tan(f*x+e))^(1/2)+1))))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (72) = 144\).
Time = 0.09 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.40 \[ \int \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2 \, dx=\frac {3 \, \sqrt {\frac {16 i \, a^{4} d}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {{\left (-4 i \, a^{2} d e^{\left (2 i \, f x + 2 i \, e\right )} + \sqrt {\frac {16 i \, a^{4} d}{f^{2}}} {\left (i \, f e^{\left (2 i \, f x + 2 i \, e\right )} + i \, f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{2 \, a^{2}}\right ) - 3 \, \sqrt {\frac {16 i \, a^{4} d}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {{\left (-4 i \, a^{2} d e^{\left (2 i \, f x + 2 i \, e\right )} + \sqrt {\frac {16 i \, a^{4} d}{f^{2}}} {\left (-i \, f e^{\left (2 i \, f x + 2 i \, e\right )} - i \, f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{2 \, a^{2}}\right ) - 8 \, {\left (-7 i \, a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 5 i \, a^{2}\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{12 \, {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \] Input:
integrate((d*tan(f*x+e))^(1/2)*(a+I*a*tan(f*x+e))^2,x, algorithm="fricas")
Output:
1/12*(3*sqrt(16*I*a^4*d/f^2)*(f*e^(2*I*f*x + 2*I*e) + f)*log(1/2*(-4*I*a^2 *d*e^(2*I*f*x + 2*I*e) + sqrt(16*I*a^4*d/f^2)*(I*f*e^(2*I*f*x + 2*I*e) + I *f)*sqrt((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) + 1)))*e^(- 2*I*f*x - 2*I*e)/a^2) - 3*sqrt(16*I*a^4*d/f^2)*(f*e^(2*I*f*x + 2*I*e) + f) *log(1/2*(-4*I*a^2*d*e^(2*I*f*x + 2*I*e) + sqrt(16*I*a^4*d/f^2)*(-I*f*e^(2 *I*f*x + 2*I*e) - I*f)*sqrt((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-2*I*f*x - 2*I*e)/a^2) - 8*(-7*I*a^2*e^(2*I*f*x + 2*I*e) - 5*I*a^2)*sqrt((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) + 1 )))/(f*e^(2*I*f*x + 2*I*e) + f)
\[ \int \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2 \, dx=- a^{2} \left (\int \left (- \sqrt {d \tan {\left (e + f x \right )}}\right )\, dx + \int \sqrt {d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}\, dx + \int \left (- 2 i \sqrt {d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}\right )\, dx\right ) \] Input:
integrate((d*tan(f*x+e))**(1/2)*(a+I*a*tan(f*x+e))**2,x)
Output:
-a**2*(Integral(-sqrt(d*tan(e + f*x)), x) + Integral(sqrt(d*tan(e + f*x))* tan(e + f*x)**2, x) + Integral(-2*I*sqrt(d*tan(e + f*x))*tan(e + f*x), x))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (72) = 144\).
Time = 0.12 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.18 \[ \int \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2 \, dx=\frac {3 \, a^{2} d^{2} {\left (-\frac {\left (2 i - 2\right ) \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\left (2 i - 2\right ) \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\left (i + 1\right ) \, \sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {\left (i + 1\right ) \, \sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} - 4 \, \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} a^{2} + 24 i \, \sqrt {d \tan \left (f x + e\right )} a^{2} d}{6 \, d f} \] Input:
integrate((d*tan(f*x+e))^(1/2)*(a+I*a*tan(f*x+e))^2,x, algorithm="maxima")
Output:
1/6*(3*a^2*d^2*(-(2*I - 2)*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2 *sqrt(d*tan(f*x + e)))/sqrt(d))/sqrt(d) - (2*I - 2)*sqrt(2)*arctan(-1/2*sq rt(2)*(sqrt(2)*sqrt(d) - 2*sqrt(d*tan(f*x + e)))/sqrt(d))/sqrt(d) - (I + 1 )*sqrt(2)*log(d*tan(f*x + e) + sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(d) + d)/s qrt(d) + (I + 1)*sqrt(2)*log(d*tan(f*x + e) - sqrt(2)*sqrt(d*tan(f*x + e)) *sqrt(d) + d)/sqrt(d)) - 4*(d*tan(f*x + e))^(3/2)*a^2 + 24*I*sqrt(d*tan(f* x + e))*a^2*d)/(d*f)
Time = 0.18 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.16 \[ \int \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2 \, dx=\frac {2 \, a^{2} {\left (\frac {6 \, \sqrt {2} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {d \tan \left (f x + e\right )} {\left | d \right |}}{i \, \sqrt {2} d^{\frac {3}{2}} + \sqrt {2} \sqrt {d} {\left | d \right |}}\right )}{\frac {i \, d}{{\left | d \right |}} + 1} - \frac {\sqrt {d \tan \left (f x + e\right )} d^{3} \tan \left (f x + e\right ) - 6 i \, \sqrt {d \tan \left (f x + e\right )} d^{3}}{d^{3}}\right )}}{3 \, f} \] Input:
integrate((d*tan(f*x+e))^(1/2)*(a+I*a*tan(f*x+e))^2,x, algorithm="giac")
Output:
2/3*a^2*(6*sqrt(2)*sqrt(d)*arctan(2*sqrt(d*tan(f*x + e))*abs(d)/(I*sqrt(2) *d^(3/2) + sqrt(2)*sqrt(d)*abs(d)))/(I*d/abs(d) + 1) - (sqrt(d*tan(f*x + e ))*d^3*tan(f*x + e) - 6*I*sqrt(d*tan(f*x + e))*d^3)/d^3)/f
Time = 1.14 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.82 \[ \int \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2 \, dx=\frac {a^2\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,4{}\mathrm {i}}{f}-\frac {2\,a^2\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{3\,d\,f}-\frac {2\,\sqrt {4{}\mathrm {i}}\,a^2\,\sqrt {d}\,\mathrm {atanh}\left (\frac {\sqrt {4{}\mathrm {i}}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {d}}\right )}{f} \] Input:
int((d*tan(e + f*x))^(1/2)*(a + a*tan(e + f*x)*1i)^2,x)
Output:
(a^2*(d*tan(e + f*x))^(1/2)*4i)/f - (2*a^2*(d*tan(e + f*x))^(3/2))/(3*d*f) - (2*4i^(1/2)*a^2*d^(1/2)*atanh((4i^(1/2)*(d*tan(e + f*x))^(1/2))/(2*d^(1 /2))))/f
\[ \int \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2 \, dx=\frac {\sqrt {d}\, a^{2} \left (4 \sqrt {\tan \left (f x +e \right )}\, i -2 \left (\int \frac {\sqrt {\tan \left (f x +e \right )}}{\tan \left (f x +e \right )}d x \right ) f i +\left (\int \sqrt {\tan \left (f x +e \right )}d x \right ) f -\left (\int \sqrt {\tan \left (f x +e \right )}\, \tan \left (f x +e \right )^{2}d x \right ) f \right )}{f} \] Input:
int((d*tan(f*x+e))^(1/2)*(a+I*a*tan(f*x+e))^2,x)
Output:
(sqrt(d)*a**2*(4*sqrt(tan(e + f*x))*i - 2*int(sqrt(tan(e + f*x))/tan(e + f *x),x)*f*i + int(sqrt(tan(e + f*x)),x)*f - int(sqrt(tan(e + f*x))*tan(e + f*x)**2,x)*f))/f