\(\int \frac {(a+i a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx\) [153]

Optimal result

Integrand size = 28, antiderivative size = 66 \[ \int \frac {(a+i a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx=-\frac {4 (-1)^{3/4} a^2 \arctan \left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{3/2} f}-\frac {2 a^2}{d f \sqrt {d \tan (e+f x)}} \] Output:

-4*(-1)^(3/4)*a^2*arctan((-1)^(3/4)*(d*tan(f*x+e))^(1/2)/d^(1/2))/d^(3/2)/ 
f-2*a^2/d/f/(d*tan(f*x+e))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03 \[ \int \frac {(a+i a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx=\frac {2 a^2 \left ((1+i) \sqrt {2} \sqrt {d} \text {arctanh}\left (\frac {(1+i) \sqrt {d \tan (e+f x)}}{\sqrt {2} \sqrt {d}}\right )-\frac {d}{\sqrt {d \tan (e+f x)}}\right )}{d^2 f} \] Input:

Integrate[(a + I*a*Tan[e + f*x])^2/(d*Tan[e + f*x])^(3/2),x]
 

Output:

(2*a^2*((1 + I)*Sqrt[2]*Sqrt[d]*ArcTanh[((1 + I)*Sqrt[d*Tan[e + f*x]])/(Sq 
rt[2]*Sqrt[d])] - d/Sqrt[d*Tan[e + f*x]]))/(d^2*f)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3042, 4025, 27, 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.

\(\displaystyle \int \frac {(a+i a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+i a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}}dx\)

\(\Big \downarrow \) 4025

\(\displaystyle -\frac {2 a^2}{d f \sqrt {d \tan (e+f x)}}+\frac {\int \frac {2 \left (i a^2 d-a^2 d \tan (e+f x)\right )}{\sqrt {d \tan (e+f x)}}dx}{d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 a^2}{d f \sqrt {d \tan (e+f x)}}+\frac {2 \int \frac {i a^2 d-a^2 d \tan (e+f x)}{\sqrt {d \tan (e+f x)}}dx}{d^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 a^2}{d f \sqrt {d \tan (e+f x)}}+\frac {2 \int \frac {i a^2 d-a^2 d \tan (e+f x)}{\sqrt {d \tan (e+f x)}}dx}{d^2}\)

\(\Big \downarrow \) 4016

\(\displaystyle -\frac {2 a^2}{d f \sqrt {d \tan (e+f x)}}-\frac {4 a^4 \int \frac {1}{i a^2 d^2+a^2 \tan (e+f x) d^2}d\sqrt {d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 218

\(\displaystyle -\frac {4 (-1)^{3/4} a^2 \arctan \left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{3/2} f}-\frac {2 a^2}{d f \sqrt {d \tan (e+f x)}}\)

Input:

Int[(a + I*a*Tan[e + f*x])^2/(d*Tan[e + f*x])^(3/2),x]
 

Output:

(-4*(-1)^(3/4)*a^2*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(d^( 
3/2)*f) - (2*a^2)/(d*f*Sqrt[d*Tan[e + f*x]])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4016

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]
 

rule 4025

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 
 1)/(b*f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2)   Int[(a + b*Tan[e 
+ f*x])^(m + 1)*Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta 
n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] 
 && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.55 (sec) , antiderivative size = 291, normalized size of antiderivative = 4.41

method	result	size
derivativedivides	\(\frac {2 a^{2} \left (-\frac {1}{\sqrt {d \tan \left (f x +e \right )}}+\frac {i \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f d}\)	\(291\)
default	\(\frac {2 a^{2} \left (-\frac {1}{\sqrt {d \tan \left (f x +e \right )}}+\frac {i \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f d}\)	\(291\)
parts	\(\frac {2 a^{2} d \left (-\frac {1}{d^{2} \sqrt {d \tan \left (f x +e \right )}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d^{2} \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f}+\frac {i a^{2} \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{2 f \,d^{2}}-\frac {a^{2} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 f d \left (d^{2}\right )^{\frac {1}{4}}}\)	\(442\)

Input:

int((a+I*a*tan(f*x+e))^2/(d*tan(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

2/f*a^2/d*(-1/(d*tan(f*x+e))^(1/2)+1/4*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*arctan(-2^(1/2)/(d^2)^(1/4)*(d*tan(f* 
x+e))^(1/2)+1))-1/4/(d^2)^(1/4)*2^(1/2)*(ln((d*tan(f*x+e)-(d^2)^(1/4)*(d*t 
an(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*arctan(-2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1)))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 349, normalized size of antiderivative = 5.29 \[ \int \frac {(a+i a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx=-\frac {{\left (d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - d^{2} f\right )} \sqrt {\frac {16 i \, a^{4}}{d^{3} f^{2}}} \log \left (\frac {{\left (-4 i \, a^{2} d e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d^{2} f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {16 i \, a^{4}}{d^{3} f^{2}}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{2 \, a^{2}}\right ) - {\left (d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - d^{2} f\right )} \sqrt {\frac {16 i \, a^{4}}{d^{3} f^{2}}} \log \left (\frac {{\left (-4 i \, a^{2} d e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-i \, d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - i \, d^{2} f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {16 i \, a^{4}}{d^{3} f^{2}}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{2 \, a^{2}}\right ) + 8 \, {\left (i \, a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, a^{2}\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{4 \, {\left (d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - d^{2} f\right )}} \] Input:

integrate((a+I*a*tan(f*x+e))^2/(d*tan(f*x+e))^(3/2),x, algorithm="fricas")
 

Output:

-1/4*((d^2*f*e^(2*I*f*x + 2*I*e) - d^2*f)*sqrt(16*I*a^4/(d^3*f^2))*log(1/2 
*(-4*I*a^2*d*e^(2*I*f*x + 2*I*e) + (I*d^2*f*e^(2*I*f*x + 2*I*e) + I*d^2*f) 
*sqrt((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(16* 
I*a^4/(d^3*f^2)))*e^(-2*I*f*x - 2*I*e)/a^2) - (d^2*f*e^(2*I*f*x + 2*I*e) - 
 d^2*f)*sqrt(16*I*a^4/(d^3*f^2))*log(1/2*(-4*I*a^2*d*e^(2*I*f*x + 2*I*e) + 
 (-I*d^2*f*e^(2*I*f*x + 2*I*e) - I*d^2*f)*sqrt((-I*d*e^(2*I*f*x + 2*I*e) + 
 I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(16*I*a^4/(d^3*f^2)))*e^(-2*I*f*x - 2 
*I*e)/a^2) + 8*(I*a^2*e^(2*I*f*x + 2*I*e) + 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)))/(d^2*f*e^(2*I*f*x + 2*I*e) - d^ 
2*f)
 

Sympy [F]

\[ \int \frac {(a+i a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx=- a^{2} \left (\int \left (- \frac {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\right )\, dx + \int \frac {\tan ^{2}{\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx + \int \left (- \frac {2 i \tan {\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\right )\, dx\right ) \] Input:

integrate((a+I*a*tan(f*x+e))**2/(d*tan(f*x+e))**(3/2),x)
 

Output:

-a**2*(Integral(-1/(d*tan(e + f*x))**(3/2), x) + Integral(tan(e + f*x)**2/ 
(d*tan(e + f*x))**(3/2), x) + Integral(-2*I*tan(e + f*x)/(d*tan(e + f*x))* 
*(3/2), x))
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.67 \[ \int \frac {(a+i a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx=-\frac {a^{2} {\left (-\frac {\left (2 i - 2\right ) \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\left (2 i - 2\right ) \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\left (i + 1\right ) \, \sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {\left (i + 1\right ) \, \sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} + \frac {4 \, a^{2}}{\sqrt {d \tan \left (f x + e\right )}}}{2 \, d f} \] Input:

integrate((a+I*a*tan(f*x+e))^2/(d*tan(f*x+e))^(3/2),x, algorithm="maxima")
 

Output:

-1/2*(a^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*sqrt(2) 
*(sqrt(2)*sqrt(d) - 2*sqrt(d*tan(f*x + e)))/sqrt(d))/sqrt(d) - (I + 1)*sqr 
t(2)*log(d*tan(f*x + e) + sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(d) + d)/sqrt(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*a^2/sqrt(d*tan(f*x + e)))/(d*f)
                                                                                    
                                                                                    
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.15 \[ \int \frac {(a+i a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx=-\frac {2 \, a^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {d \tan \left (f x + e\right )} {\left | d \right |}}{i \, \sqrt {2} d^{\frac {3}{2}} + \sqrt {2} \sqrt {d} {\left | d \right |}}\right )}{\sqrt {d} {\left (\frac {i \, d}{{\left | d \right |}} + 1\right )}} + \frac {1}{\sqrt {d \tan \left (f x + e\right )}}\right )}}{d f} \] Input:

integrate((a+I*a*tan(f*x+e))^2/(d*tan(f*x+e))^(3/2),x, algorithm="giac")
 

Output:

-2*a^2*(2*sqrt(2)*arctan(2*sqrt(d*tan(f*x + e))*abs(d)/(I*sqrt(2)*d^(3/2) 
+ sqrt(2)*sqrt(d)*abs(d)))/(sqrt(d)*(I*d/abs(d) + 1)) + 1/sqrt(d*tan(f*x + 
 e)))/(d*f)
 

Mupad [B] (verification not implemented)

Time = 1.14 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.83 \[ \int \frac {(a+i a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx=-\frac {2\,a^2}{d\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}+\frac {2\,\sqrt {4{}\mathrm {i}}\,a^2\,\mathrm {atanh}\left (\frac {\sqrt {4{}\mathrm {i}}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {d}}\right )}{d^{3/2}\,f} \] Input:

int((a + a*tan(e + f*x)*1i)^2/(d*tan(e + f*x))^(3/2),x)
 

Output:

(2*4i^(1/2)*a^2*atanh((4i^(1/2)*(d*tan(e + f*x))^(1/2))/(2*d^(1/2))))/(d^( 
3/2)*f) - (2*a^2)/(d*f*(d*tan(e + f*x))^(1/2))
 

Reduce [F]

\[ \int \frac {(a+i a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx=\frac {\sqrt {d}\, a^{2} \left (\int \frac {\sqrt {\tan \left (f x +e \right )}}{\tan \left (f x +e \right )^{2}}d x +2 \left (\int \frac {\sqrt {\tan \left (f x +e \right )}}{\tan \left (f x +e \right )}d x \right ) i -\left (\int \sqrt {\tan \left (f x +e \right )}d x \right )\right )}{d^{2}} \] Input:

int((a+I*a*tan(f*x+e))^2/(d*tan(f*x+e))^(3/2),x)
 

Output:

(sqrt(d)*a**2*(int(sqrt(tan(e + f*x))/tan(e + f*x)**2,x) + 2*int(sqrt(tan( 
e + f*x))/tan(e + f*x),x)*i - int(sqrt(tan(e + f*x)),x)))/d**2