\(\int \frac {(d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^2} \, dx\) [174]

Optimal result

Integrand size = 28, antiderivative size = 297 \[ \int \frac {(d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^2} \, dx=\frac {\left (\frac {1}{16}+\frac {3 i}{16}\right ) d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 f}-\frac {\left (\frac {1}{16}+\frac {3 i}{16}\right ) d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 f}+\frac {\left (\frac {1}{32}-\frac {3 i}{32}\right ) d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 f}-\frac {\left (\frac {1}{32}-\frac {3 i}{32}\right ) d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 f}+\frac {3 d \sqrt {d \tan (e+f x)}}{8 a^2 f (1+i \tan (e+f x))}-\frac {d \sqrt {d \tan (e+f x)}}{4 f (a+i a \tan (e+f x))^2} \]

[Out]

(1/32+3/32*I)*d^(3/2)*arctan(1-2^(1/2)*(d*tan(f*x+e))^(1/2)/d^(1/2))/a^2/f*2^(1/2)-(1/32+3/32*I)*d^(3/2)*arcta
n(1+2^(1/2)*(d*tan(f*x+e))^(1/2)/d^(1/2))/a^2/f*2^(1/2)+(1/64-3/64*I)*d^(3/2)*ln(d^(1/2)-2^(1/2)*(d*tan(f*x+e)
)^(1/2)+d^(1/2)*tan(f*x+e))/a^2/f*2^(1/2)+(-1/64+3/64*I)*d^(3/2)*ln(d^(1/2)+2^(1/2)*(d*tan(f*x+e))^(1/2)+d^(1/
2)*tan(f*x+e))/a^2/f*2^(1/2)+3/8*d*(d*tan(f*x+e))^(1/2)/a^2/f/(1+I*tan(f*x+e))-1/4*d*(d*tan(f*x+e))^(1/2)/f/(a
+I*a*tan(f*x+e))^2

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {3639, 3677, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {(d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^2} \, dx=\frac {\left (\frac {1}{16}+\frac {3 i}{16}\right ) d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 f}-\frac {\left (\frac {1}{16}+\frac {3 i}{16}\right ) d^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} a^2 f}+\frac {\left (\frac {1}{32}-\frac {3 i}{32}\right ) d^{3/2} \log \left (\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a^2 f}-\frac {\left (\frac {1}{32}-\frac {3 i}{32}\right ) d^{3/2} \log \left (\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a^2 f}+\frac {3 d \sqrt {d \tan (e+f x)}}{8 a^2 f (1+i \tan (e+f x))}-\frac {d \sqrt {d \tan (e+f x)}}{4 f (a+i a \tan (e+f x))^2} \]

[In]

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

[Out]

((1/16 + (3*I)/16)*d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^2*f) - ((1/16 + (3*I
)/16)*d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^2*f) + ((1/32 - (3*I)/32)*d^(3/2)
*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^2*f) - ((1/32 - (3*I)/32)*d^(3
/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^2*f) + (3*d*Sqrt[d*Tan[e +
f*x]])/(8*a^2*f*(1 + I*Tan[e + f*x])) - (d*Sqrt[d*Tan[e + f*x]])/(4*f*(a + I*a*Tan[e + f*x])^2)

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

Rule 3615

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 3639

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

Rule 3677

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*
f*m*(b*c - a*d))), x] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Si
mp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x
] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] &&  !GtQ[n,
0]

Rubi steps \begin{align*} \text {integral}& = -\frac {d \sqrt {d \tan (e+f x)}}{4 f (a+i a \tan (e+f x))^2}-\frac {\int \frac {-\frac {a d^2}{2}+\frac {5}{2} i a d^2 \tan (e+f x)}{\sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))} \, dx}{4 a^2} \\ & = \frac {3 d \sqrt {d \tan (e+f x)}}{8 a^2 f (1+i \tan (e+f x))}-\frac {d \sqrt {d \tan (e+f x)}}{4 f (a+i a \tan (e+f x))^2}-\frac {\int \frac {\frac {a^2 d^3}{2}+\frac {3}{2} i a^2 d^3 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{8 a^4 d} \\ & = \frac {3 d \sqrt {d \tan (e+f x)}}{8 a^2 f (1+i \tan (e+f x))}-\frac {d \sqrt {d \tan (e+f x)}}{4 f (a+i a \tan (e+f x))^2}-\frac {\text {Subst}\left (\int \frac {\frac {a^2 d^4}{2}+\frac {3}{2} i a^2 d^3 x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{4 a^4 d f} \\ & = \frac {3 d \sqrt {d \tan (e+f x)}}{8 a^2 f (1+i \tan (e+f x))}-\frac {d \sqrt {d \tan (e+f x)}}{4 f (a+i a \tan (e+f x))^2}-\frac {\left (\left (\frac {1}{16}-\frac {3 i}{16}\right ) d^2\right ) \text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a^2 f}-\frac {\left (\left (\frac {1}{16}+\frac {3 i}{16}\right ) d^2\right ) \text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a^2 f} \\ & = \frac {3 d \sqrt {d \tan (e+f x)}}{8 a^2 f (1+i \tan (e+f x))}-\frac {d \sqrt {d \tan (e+f x)}}{4 f (a+i a \tan (e+f x))^2}--\frac {\left (\left (\frac {1}{32}-\frac {3 i}{32}\right ) d^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}+2 x}{-d-\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 f}--\frac {\left (\left (\frac {1}{32}-\frac {3 i}{32}\right ) d^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}-2 x}{-d+\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 f}-\frac {\left (\left (\frac {1}{32}+\frac {3 i}{32}\right ) d^2\right ) \text {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a^2 f}-\frac {\left (\left (\frac {1}{32}+\frac {3 i}{32}\right ) d^2\right ) \text {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a^2 f} \\ & = \frac {\left (\frac {1}{32}-\frac {3 i}{32}\right ) d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 f}-\frac {\left (\frac {1}{32}-\frac {3 i}{32}\right ) d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 f}+\frac {3 d \sqrt {d \tan (e+f x)}}{8 a^2 f (1+i \tan (e+f x))}-\frac {d \sqrt {d \tan (e+f x)}}{4 f (a+i a \tan (e+f x))^2}--\frac {\left (\left (\frac {1}{16}+\frac {3 i}{16}\right ) d^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 f}-\frac {\left (\left (\frac {1}{16}+\frac {3 i}{16}\right ) d^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 f} \\ & = \frac {\left (\frac {1}{16}+\frac {3 i}{16}\right ) d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 f}-\frac {\left (\frac {1}{16}+\frac {3 i}{16}\right ) d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 f}+\frac {\left (\frac {1}{32}-\frac {3 i}{32}\right ) d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 f}-\frac {\left (\frac {1}{32}-\frac {3 i}{32}\right ) d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 f}+\frac {3 d \sqrt {d \tan (e+f x)}}{8 a^2 f (1+i \tan (e+f x))}-\frac {d \sqrt {d \tan (e+f x)}}{4 f (a+i a \tan (e+f x))^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.10 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.61 \[ \int \frac {(d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^2} \, dx=-\frac {d \sec ^2(e+f x) \left (4 \sqrt [4]{-1} \sqrt {d} \arctan \left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right ) (\cos (2 (e+f x))+i \sin (2 (e+f x)))-2 \sqrt [4]{-1} \sqrt {d} \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right ) (\cos (2 (e+f x))+i \sin (2 (e+f x)))+(1+\cos (2 (e+f x))+3 i \sin (2 (e+f x))) \sqrt {d \tan (e+f x)}\right )}{16 a^2 f (-i+\tan (e+f x))^2} \]

[In]

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

[Out]

-1/16*(d*Sec[e + f*x]^2*(4*(-1)^(1/4)*Sqrt[d]*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]]*(Cos[2*(e + f*
x)] + I*Sin[2*(e + f*x)]) - 2*(-1)^(1/4)*Sqrt[d]*ArcTanh[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]]*(Cos[2*(e
+ f*x)] + I*Sin[2*(e + f*x)]) + (1 + Cos[2*(e + f*x)] + (3*I)*Sin[2*(e + f*x)])*Sqrt[d*Tan[e + f*x]]))/(a^2*f*
(-I + Tan[e + f*x])^2)

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.40

method	result	size
derivativedivides	\(\frac {2 d^{3} \left (-\frac {\frac {-\frac {3 i \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{2}-\frac {d \sqrt {d \tan \left (f x +e \right )}}{2}}{\left (i d \tan \left (f x +e \right )+d \right )^{2}}+\frac {i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {-i d}}\right )}{2 \sqrt {-i d}}}{8 d}-\frac {i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {i d}}\right )}{8 d \sqrt {i d}}\right )}{f \,a^{2}}\)	\(118\)
default	\(\frac {2 d^{3} \left (-\frac {\frac {-\frac {3 i \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{2}-\frac {d \sqrt {d \tan \left (f x +e \right )}}{2}}{\left (i d \tan \left (f x +e \right )+d \right )^{2}}+\frac {i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {-i d}}\right )}{2 \sqrt {-i d}}}{8 d}-\frac {i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {i d}}\right )}{8 d \sqrt {i d}}\right )}{f \,a^{2}}\)	\(118\)

[In]

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

[Out]

2/f/a^2*d^3*(-1/8/d*((-3/2*I*(d*tan(f*x+e))^(3/2)-1/2*d*(d*tan(f*x+e))^(1/2))/(I*d*tan(f*x+e)+d)^2+1/2*I/(-I*d
)^(1/2)*arctan((d*tan(f*x+e))^(1/2)/(-I*d)^(1/2)))-1/8*I/d/(I*d)^(1/2)*arctan((d*tan(f*x+e))^(1/2)/(I*d)^(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. 566 vs. \(2 (217) = 434\).

Time = 0.25 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.91 \[ \int \frac {(d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^2} \, dx=\frac {{\left (4 \, a^{2} f \sqrt {-\frac {i \, d^{3}}{16 \, a^{4} f^{2}}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (-\frac {2 \, {\left (i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 4 \, {\left (a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{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 {i \, d^{3}}{16 \, a^{4} f^{2}}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{d}\right ) - 4 \, a^{2} f \sqrt {-\frac {i \, d^{3}}{16 \, a^{4} f^{2}}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (-\frac {2 \, {\left (i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 4 \, {\left (a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{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 {i \, d^{3}}{16 \, a^{4} f^{2}}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{d}\right ) + 4 \, a^{2} f \sqrt {\frac {i \, d^{3}}{64 \, a^{4} f^{2}}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (\frac {{\left (i \, d^{2} + 8 \, {\left (a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{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 {i \, d^{3}}{64 \, a^{4} f^{2}}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{8 \, a^{2} f}\right ) - 4 \, a^{2} f \sqrt {\frac {i \, d^{3}}{64 \, a^{4} f^{2}}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (\frac {{\left (i \, d^{2} - 8 \, {\left (a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{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 {i \, d^{3}}{64 \, a^{4} f^{2}}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{8 \, a^{2} f}\right ) + {\left (2 \, d e^{\left (4 i \, f x + 4 i \, e\right )} + d e^{\left (2 i \, f x + 2 i \, e\right )} - d\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 (-4 i \, f x - 4 i \, e\right )}}{16 \, a^{2} f} \]

[In]

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

[Out]

1/16*(4*a^2*f*sqrt(-1/16*I*d^3/(a^4*f^2))*e^(4*I*f*x + 4*I*e)*log(-2*(I*d^2*e^(2*I*f*x + 2*I*e) + 4*(a^2*f*e^(
2*I*f*x + 2*I*e) + a^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(-1/16*I*d^3/(a
^4*f^2)))*e^(-2*I*f*x - 2*I*e)/d) - 4*a^2*f*sqrt(-1/16*I*d^3/(a^4*f^2))*e^(4*I*f*x + 4*I*e)*log(-2*(I*d^2*e^(2
*I*f*x + 2*I*e) - 4*(a^2*f*e^(2*I*f*x + 2*I*e) + a^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(-1/16*I*d^3/(a^4*f^2)))*e^(-2*I*f*x - 2*I*e)/d) + 4*a^2*f*sqrt(1/64*I*d^3/(a^4*f^2))*e^(4*I*f*
x + 4*I*e)*log(1/8*(I*d^2 + 8*(a^2*f*e^(2*I*f*x + 2*I*e) + a^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(1/64*I*d^3/(a^4*f^2)))*e^(-2*I*f*x - 2*I*e)/(a^2*f)) - 4*a^2*f*sqrt(1/64*I*d^3/(a^4*
f^2))*e^(4*I*f*x + 4*I*e)*log(1/8*(I*d^2 - 8*(a^2*f*e^(2*I*f*x + 2*I*e) + a^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(1/64*I*d^3/(a^4*f^2)))*e^(-2*I*f*x - 2*I*e)/(a^2*f)) + (2*d*e^(4*I*f*
x + 4*I*e) + d*e^(2*I*f*x + 2*I*e) - d)*sqrt((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-
4*I*f*x - 4*I*e)/(a^2*f)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [A] (verification not implemented)

none

Time = 0.61 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.67 \[ \int \frac {(d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^2} \, dx=-\frac {1}{8} \, d^{3} {\left (\frac {2 i \, \sqrt {2} \arctan \left (\frac {8 \, \sqrt {d^{2}} \sqrt {d \tan \left (f x + e\right )}}{4 i \, \sqrt {2} d^{\frac {3}{2}} + 4 \, \sqrt {2} \sqrt {d^{2}} \sqrt {d}}\right )}{a^{2} d^{\frac {3}{2}} f {\left (\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} + \frac {i \, \sqrt {2} \arctan \left (\frac {8 \, \sqrt {d^{2}} \sqrt {d \tan \left (f x + e\right )}}{-4 i \, \sqrt {2} d^{\frac {3}{2}} + 4 \, \sqrt {2} \sqrt {d^{2}} \sqrt {d}}\right )}{a^{2} d^{\frac {3}{2}} f {\left (-\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} + \frac {3 i \, \sqrt {d \tan \left (f x + e\right )} d \tan \left (f x + e\right ) + \sqrt {d \tan \left (f x + e\right )} d}{{\left (d \tan \left (f x + e\right ) - i \, d\right )}^{2} a^{2} d f}\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.46 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.60 \[ \int \frac {(d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^2} \, dx=\frac {\frac {d^3\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{8\,a^2\,f}+\frac {d^2\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,3{}\mathrm {i}}{8\,a^2\,f}}{-d^2\,{\mathrm {tan}\left (e+f\,x\right )}^2+d^2\,\mathrm {tan}\left (e+f\,x\right )\,2{}\mathrm {i}+d^2}-\mathrm {atan}\left (\frac {8\,a^2\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {-\frac {d^3\,1{}\mathrm {i}}{64\,a^4\,f^2}}}{d^2}\right )\,\sqrt {-\frac {d^3\,1{}\mathrm {i}}{64\,a^4\,f^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {16\,a^2\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {\frac {d^3\,1{}\mathrm {i}}{256\,a^4\,f^2}}}{d^2}\right )\,\sqrt {\frac {d^3\,1{}\mathrm {i}}{256\,a^4\,f^2}}\,2{}\mathrm {i} \]

[In]

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

[Out]

((d^3*(d*tan(e + f*x))^(1/2))/(8*a^2*f) + (d^2*(d*tan(e + f*x))^(3/2)*3i)/(8*a^2*f))/(d^2*tan(e + f*x)*2i + d^
2 - d^2*tan(e + f*x)^2) - atan((8*a^2*f*(d*tan(e + f*x))^(1/2)*(-(d^3*1i)/(64*a^4*f^2))^(1/2))/d^2)*(-(d^3*1i)
/(64*a^4*f^2))^(1/2)*2i - atan((16*a^2*f*(d*tan(e + f*x))^(1/2)*((d^3*1i)/(256*a^4*f^2))^(1/2))/d^2)*((d^3*1i)
/(256*a^4*f^2))^(1/2)*2i