\(\int \frac {(a+i a \tan (e+f x))^{3/2}}{(c+d \tan (e+f x))^{3/2}} \, dx\) [1162]

Optimal result

Integrand size = 32, antiderivative size = 129 \[ \int \frac {(a+i a \tan (e+f x))^{3/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=-\frac {2 i \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{(c-i d)^{3/2} f}-\frac {2 a \sqrt {a+i a \tan (e+f x)}}{(i c+d) f \sqrt {c+d \tan (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {3626, 3625, 214} \[ \int \frac {(a+i a \tan (e+f x))^{3/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=-\frac {2 i \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f (c-i d)^{3/2}}-\frac {2 a \sqrt {a+i a \tan (e+f x)}}{f (d+i c) \sqrt {c+d \tan (e+f x)}} \]

[In]

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

[Out]

((-2*I)*Sqrt[2]*a^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e +
f*x]])])/((c - I*d)^(3/2)*f) - (2*a*Sqrt[a + I*a*Tan[e + f*x]])/((I*c + d)*f*Sqrt[c + d*Tan[e + f*x]])

Rule 214

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]

Rule 3625

Int[Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
-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] && NeQ[c^2 + d^2, 0]

Rule 3626

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[a*b*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m - 1)*(a*c - b*d))), x] + Dist[2*(a^2/(a
*c - b*d)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1), 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] && EqQ[m + n, 0] && GtQ[m, 1/2]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 a \sqrt {a+i a \tan (e+f x)}}{(i c+d) f \sqrt {c+d \tan (e+f x)}}+\frac {(2 a) \int \frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}} \, dx}{c-i d} \\ & = -\frac {2 a \sqrt {a+i a \tan (e+f x)}}{(i c+d) f \sqrt {c+d \tan (e+f x)}}+\frac {\left (4 a^3\right ) \text {Subst}\left (\int \frac {1}{a c-i a d-2 a^2 x^2} \, dx,x,\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}}\right )}{(i c+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 )}{(c-i d)^{3/2} f}-\frac {2 a \sqrt {a+i a \tan (e+f x)}}{(i c+d) f \sqrt {c+d \tan (e+f x)}} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(981\) vs. \(2(129)=258\).

Time = 6.69 (sec) , antiderivative size = 981, normalized size of antiderivative = 7.60 \[ \int \frac {(a+i a \tan (e+f x))^{3/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=-\frac {2 d (a+i a \tan (e+f x))^{3/2}}{\left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 \left (\frac {a \left (\frac {1}{2} i a^2 (c+3 i d)+a^2 d\right ) \left (-\frac {2 \sqrt {2} \arctan \left (\frac {\sqrt {-a c+i a d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {2} a \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {-a c+i a d}}-\frac {2 (-1)^{3/4} \sqrt {c+i d} \sqrt {\frac {1}{\frac {c}{c+i d}+\frac {i d}{c+i d}}} \sqrt {\frac {c}{c+i d}+\frac {i d}{c+i d}} \arcsin \left (\frac {\sqrt [4]{-1} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+i d} \sqrt {\frac {c}{c+i d}+\frac {i d}{c+i d}}}\right ) \sqrt {\frac {c+d \tan (e+f x)}{c+i d}}}{\sqrt {a} \sqrt {d} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {(i a c-a d)^2 \sqrt {\frac {i a}{-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}}} \left (-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}\right )^2 \sqrt {\frac {i a (c+d \tan (e+f x))}{i a c-a d}} \sqrt {1+\frac {i a d (a+i a \tan (e+f x))}{(i a c-a d) \left (-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}\right )}} \left (\frac {2 i a d (a+i a \tan (e+f x))}{(i a c-a d) \left (-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}\right )}-\frac {2 \sqrt [4]{-1} \sqrt {a} \sqrt {d} \text {arcsinh}\left (\frac {\sqrt [4]{-1} \sqrt {a} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {i a c-a d} \sqrt {-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}}}\right ) \sqrt {a+i a \tan (e+f x)}}{\sqrt {i a c-a d} \sqrt {-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}} \sqrt {1+\frac {i a d (a+i a \tan (e+f x))}{(i a c-a d) \left (-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}\right )}}}\right )}{2 a d f \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}\right )}{a \left (c^2+d^2\right )} \]

[In]

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

[Out]

(-2*d*(a + I*a*Tan[e + f*x])^(3/2))/((c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) + (2*((a*((I/2)*a^2*(c + (3*I)*d)
+ a^2*d)*((-2*Sqrt[2]*ArcTan[(Sqrt[-(a*c) + I*a*d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[2]*a*Sqrt[c + d*Tan[e +
f*x]])])/Sqrt[-(a*c) + I*a*d] - (2*(-1)^(3/4)*Sqrt[c + I*d]*Sqrt[(c/(c + I*d) + (I*d)/(c + I*d))^(-1)]*Sqrt[c/
(c + I*d) + (I*d)/(c + I*d)]*ArcSin[((-1)^(1/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + I*d]*Sqr
t[c/(c + I*d) + (I*d)/(c + I*d)])]*Sqrt[(c + d*Tan[e + f*x])/(c + I*d)])/(Sqrt[a]*Sqrt[d]*Sqrt[c + d*Tan[e + f
*x]])))/f - ((I*a*c - a*d)^2*Sqrt[(I*a)/(-((a^2*c)/(I*a*c - a*d)) - (I*a^2*d)/(I*a*c - a*d))]*(-((a^2*c)/(I*a*
c - a*d)) - (I*a^2*d)/(I*a*c - a*d))^2*Sqrt[(I*a*(c + d*Tan[e + f*x]))/(I*a*c - a*d)]*Sqrt[1 + (I*a*d*(a + I*a
*Tan[e + f*x]))/((I*a*c - a*d)*(-((a^2*c)/(I*a*c - a*d)) - (I*a^2*d)/(I*a*c - a*d)))]*(((2*I)*a*d*(a + I*a*Tan
[e + f*x]))/((I*a*c - a*d)*(-((a^2*c)/(I*a*c - a*d)) - (I*a^2*d)/(I*a*c - a*d))) - (2*(-1)^(1/4)*Sqrt[a]*Sqrt[
d]*ArcSinh[((-1)^(1/4)*Sqrt[a]*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[I*a*c - a*d]*Sqrt[-((a^2*c)/(I*a*c -
a*d)) - (I*a^2*d)/(I*a*c - a*d)])]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[I*a*c - a*d]*Sqrt[-((a^2*c)/(I*a*c - a*d)
) - (I*a^2*d)/(I*a*c - a*d)]*Sqrt[1 + (I*a*d*(a + I*a*Tan[e + f*x]))/((I*a*c - a*d)*(-((a^2*c)/(I*a*c - a*d))
- (I*a^2*d)/(I*a*c - a*d)))])))/(2*a*d*f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])))/(a*(c^2 + d^2)
)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2563 vs. \(2 (103 ) = 206\).

Time = 1.39 (sec) , antiderivative size = 2564, normalized size of antiderivative = 19.88

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2564\)
default	\(\text {Expression too large to display}\)	\(2564\)

[In]

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

[Out]

1/2/f*2^(1/2)*(a*(1+I*tan(f*x+e)))^(1/2)*(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))*a*c^2+I*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))*
a*c*d*(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))*a*c*d*(I*a*d)^(1/2)*tan(f*x+e)+2*2^(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))*a*c^4*(-a*(I*d-c
))^(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))*a*d^2*(I*a*d)^(1/2)*tan(f*x+e)+ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*t
an(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))*(I*a*d)^(1
/2)*a*c*d+I*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))*a*c^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))*a*c^2*(I*a*d)^(1/2
)-2*(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(1/2)*d^3+I*2^(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))*a*c*
d*(-a*(I*d-c))^(1/2)*tan(f*x+e)+2*I*2^(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))*a*c^2*d^2*(-a*(I*d-c))^(1/2)*tan(f*x+e)+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))*2^(1/2)*(-a*(I*d-c))
^(1/2)*a*c*d+I*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))*2^(1/2)*(-a*(I*d-c))^(1/2)*a*c^2+I*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))*a*d^2*(I*a*d)^(1/2)*tan(f*x
+e)-I*2^(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))*a*d^2*(-a*(I*d-c))^(1/2)*tan(f*x+e)+2*I*2^(1/2)*c*d^2*(I*a*d)^(1/2)*(-a*(I*d-c))^(1/2)*(a*(1+
I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2)-I*2^(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*ta
n(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*a*c*d*(-a*(I*d-c))^(1/2)+2*I*2^(1/2)*c^3*(-a*(I*d-c))^(1/2)
*(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+2^(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))*a*d^2*(-a*(I*d-c))^(1/2)*tan(f*x+e)-2*(a
*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(1/2)*c^2*d+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))*2^(1/2)*(-a*(I*d
-c))^(1/2)*a*c^2*d^2+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))*2^(1/2)*(-a*(I*d-c))^(1/2)*a*c^3*d*tan(f*x+e)+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))*2^(1/2)*(-a*(I*d-c))^(1/2)*a*c*d^
3*tan(f*x+e)+I*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))*a*c*d*(I*a*d)^(1/2)*tan(f*x+e)+2^(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))*a*c*d*(-a*(I*d-c))^(1/2
)*tan(f*x+e)+2*I*2^(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))*a*d^4*(-a*(I*d-c))^(1/2)*tan(f*x+e)+2*I*2^(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))*a*c^3*d*(-a*(I*d-c))^(1/2)+2*I*
2^(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))*a*c*d^3*(-a*(I*d-c))^(1/2))*a/(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2)/(c^2+d^2)^2/(I*a*d)^(1/2)
/(-a*(I*d-c))^(1/2)/(c+d*tan(f*x+e))^(1/2)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 551 vs. \(2 (97) = 194\).

Time = 0.25 (sec) , antiderivative size = 551, normalized size of antiderivative = 4.27 \[ \int \frac {(a+i a \tan (e+f x))^{3/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=\frac {4 \, \sqrt {2} {\left (i \, a e^{\left (3 i \, f x + 3 i \, e\right )} + i \, a e^{\left (i \, f x + i \, e\right )}\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}} + {\left ({\left (c^{2} - 2 i \, c d - d^{2}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (c^{2} + d^{2}\right )} f\right )} \sqrt {\frac {8 i \, a^{3}}{{\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\right )} f^{2}}} \log \left (\frac {{\left ({\left (i \, c^{2} + 2 \, c d - i \, d^{2}\right )} f \sqrt {\frac {8 i \, a^{3}}{{\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\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 ) - {\left ({\left (c^{2} - 2 i \, c d - d^{2}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (c^{2} + d^{2}\right )} f\right )} \sqrt {\frac {8 i \, a^{3}}{{\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\right )} f^{2}}} \log \left (\frac {{\left ({\left (-i \, c^{2} - 2 \, c d + i \, d^{2}\right )} f \sqrt {\frac {8 i \, a^{3}}{{\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\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 )}{2 \, {\left ({\left (c^{2} - 2 i \, c d - d^{2}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (c^{2} + d^{2}\right )} f\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-1)]

Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{3/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^{3/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Non regular value [0,0] was discarded and replaced randomly by 0=[-59,-77]Warning, replacing -59 by -5, a s
ubstitution

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{3/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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