\(\int \frac {(a+i a \tan (e+f x))^3}{\sqrt {c+d \tan (e+f x)}} \, dx\) [1119]

Optimal result

Integrand size = 30, antiderivative size = 126 \[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {c+d \tan (e+f x)}} \, dx=-\frac {8 i a^3 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d} f}+\frac {4 a^3 (i c-4 d) \sqrt {c+d \tan (e+f x)}}{3 d^2 f}-\frac {2 \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}{3 d f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3637, 3673, 3618, 65, 214} \[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {c+d \tan (e+f x)}} \, dx=-\frac {8 i a^3 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {4 a^3 (-4 d+i c) \sqrt {c+d \tan (e+f x)}}{3 d^2 f}-\frac {2 \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}{3 d f} \]

[In]

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

[Out]

((-8*I)*a^3*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f) + (4*a^3*(I*c - 4*d)*Sqrt[c + d
*Tan[e + f*x]])/(3*d^2*f) - (2*(a^3 + I*a^3*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/(3*d*f)

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, 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 3618

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

Rule 3637

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Dist[a/(d*(m + n - 1
)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n*Simp[b*c*(m - 2) + a*d*(m + 2*n) + (a*c*(m - 2) +
b*d*(3*m + 2*n - 4))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a
^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 1] && NeQ[m + n - 1, 0] && (IntegerQ[m] || Intege
rsQ[2*m, 2*n])

Rule 3673

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}{3 d f}+\frac {(2 a) \int \frac {(a+i a \tan (e+f x)) (a (i c+2 d)+a (c+4 i d) \tan (e+f x))}{\sqrt {c+d \tan (e+f x)}} \, dx}{3 d} \\ & = \frac {4 a^3 (i c-4 d) \sqrt {c+d \tan (e+f x)}}{3 d^2 f}-\frac {2 \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}{3 d f}+\frac {(2 a) \int \frac {6 a^2 d+6 i a^2 d \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{3 d} \\ & = \frac {4 a^3 (i c-4 d) \sqrt {c+d \tan (e+f x)}}{3 d^2 f}-\frac {2 \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}{3 d f}+\frac {\left (24 i a^5 d\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {i x}{6 a^2}} \left (-36 a^4 d^2+6 a^2 d x\right )} \, dx,x,6 i a^2 d \tan (e+f x)\right )}{f} \\ & = \frac {4 a^3 (i c-4 d) \sqrt {c+d \tan (e+f x)}}{3 d^2 f}-\frac {2 \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}{3 d f}-\frac {\left (288 a^7 d\right ) \text {Subst}\left (\int \frac {1}{-36 i a^4 c d-36 a^4 d^2+36 i a^4 d x^2} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{f} \\ & = -\frac {8 i a^3 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d} f}+\frac {4 a^3 (i c-4 d) \sqrt {c+d \tan (e+f x)}}{3 d^2 f}-\frac {2 \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}{3 d f} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.35 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.73 \[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {c+d \tan (e+f x)}} \, dx=\frac {i a^2 \left (-\frac {8 a \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}+\frac {2 a (2 c+9 i d-d \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}{3 d^2}\right )}{f} \]

[In]

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

[Out]

(I*a^2*((-8*a*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + (2*a*(2*c + (9*I)*d - d*Tan[e +
f*x])*Sqrt[c + d*Tan[e + f*x]])/(3*d^2)))/f

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 799 vs. \(2 (107 ) = 214\).

Time = 0.90 (sec) , antiderivative size = 800, normalized size of antiderivative = 6.35

method	result	size
derivativedivides	\(\frac {2 a^{3} \left (-\frac {i \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+i c \sqrt {c +d \tan \left (f x +e \right )}-3 d \sqrt {c +d \tan \left (f x +e \right )}-4 d^{2} \left (\frac {\frac {\left (i \sqrt {c^{2}+d^{2}}+i c -d \right ) \ln \left (d \tan \left (f x +e \right )+c +\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d -\frac {\left (i \sqrt {c^{2}+d^{2}}+i c -d \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}+\frac {-\frac {\left (2 i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}+2 i c^{3}+2 i c \,d^{2}-c d \sqrt {c^{2}+d^{2}}-c^{2} d -d^{3}\right ) \ln \left (\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-d \tan \left (f x +e \right )-c -\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (-i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}\, c^{2}-i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{3}-i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}\, c d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2} d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{3}+\frac {\left (2 i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}+2 i c^{3}+2 i c \,d^{2}-c d \sqrt {c^{2}+d^{2}}-c^{2} d -d^{3}\right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-2 \sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}\, \left (\sqrt {c^{2}+d^{2}}\, c +c^{2}+d^{2}\right )}\right )\right )}{f \,d^{2}}\)	\(800\)
default	\(\frac {2 a^{3} \left (-\frac {i \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+i c \sqrt {c +d \tan \left (f x +e \right )}-3 d \sqrt {c +d \tan \left (f x +e \right )}-4 d^{2} \left (\frac {\frac {\left (i \sqrt {c^{2}+d^{2}}+i c -d \right ) \ln \left (d \tan \left (f x +e \right )+c +\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d -\frac {\left (i \sqrt {c^{2}+d^{2}}+i c -d \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}+\frac {-\frac {\left (2 i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}+2 i c^{3}+2 i c \,d^{2}-c d \sqrt {c^{2}+d^{2}}-c^{2} d -d^{3}\right ) \ln \left (\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-d \tan \left (f x +e \right )-c -\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (-i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}\, c^{2}-i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{3}-i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}\, c d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2} d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{3}+\frac {\left (2 i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}+2 i c^{3}+2 i c \,d^{2}-c d \sqrt {c^{2}+d^{2}}-c^{2} d -d^{3}\right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-2 \sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}\, \left (\sqrt {c^{2}+d^{2}}\, c +c^{2}+d^{2}\right )}\right )\right )}{f \,d^{2}}\)	\(800\)
parts	\(\text {Expression too large to display}\)	\(3867\)

[In]

int((a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/f*a^3/d^2*(-1/3*I*(c+d*tan(f*x+e))^(3/2)+I*c*(c+d*tan(f*x+e))^(1/2)-3*d*(c+d*tan(f*x+e))^(1/2)-4*d^2*(1/2/(2
*(c^2+d^2)^(1/2)+2*c)^(1/2)/(c^2+d^2)^(1/2)*(1/2*(I*(c^2+d^2)^(1/2)+I*c-d)*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^
(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))+2*(I*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c-(2*(c^2+d^2)^(1/2)+2
*c)^(1/2)*d-1/2*(I*(c^2+d^2)^(1/2)+I*c-d)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(
(2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)))+1/2/(2*(c^2+d^2)^(1/2
)+2*c)^(1/2)/(c^2+d^2)^(1/2)/((c^2+d^2)^(1/2)*c+c^2+d^2)*(-1/2*(2*I*(c^2+d^2)^(1/2)*c^2+I*d^2*(c^2+d^2)^(1/2)+
2*I*c^3+2*I*c*d^2-c*d*(c^2+d^2)^(1/2)-c^2*d-d^3)*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan
(f*x+e)-c-(c^2+d^2)^(1/2))+2*(-I*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*c^2-I*(2*(c^2+d^2)^(1/2)+2*c)^(
1/2)*c^3-I*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*c*d+(2*(c^2+d^2)^
(1/2)+2*c)^(1/2)*c^2*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^3+1/2*(2*I*(c^2+d^2)^(1/2)*c^2+I*d^2*(c^2+d^2)^(1/2)+2*
I*c^3+2*I*c*d^2-c*d*(c^2+d^2)^(1/2)-c^2*d-d^3)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*ar
ctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(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. 427 vs. \(2 (102) = 204\).

Time = 0.25 (sec) , antiderivative size = 427, normalized size of antiderivative = 3.39 \[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {c+d \tan (e+f x)}} \, dx=\frac {3 \, {\left (d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + d^{2} f\right )} \sqrt {-\frac {64 i \, a^{6}}{{\left (i \, c + d\right )} f^{2}}} \log \left (\frac {{\left (8 \, a^{3} c + \sqrt {-\frac {64 i \, a^{6}}{{\left (i \, c + d\right )} f^{2}}} {\left ({\left (i \, c + d\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, c + d\right )} f\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} + 8 \, {\left (a^{3} c - i \, a^{3} d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{4 \, a^{3}}\right ) - 3 \, {\left (d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + d^{2} f\right )} \sqrt {-\frac {64 i \, a^{6}}{{\left (i \, c + d\right )} f^{2}}} \log \left (\frac {{\left (8 \, a^{3} c + \sqrt {-\frac {64 i \, a^{6}}{{\left (i \, c + d\right )} f^{2}}} {\left ({\left (-i \, c - d\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-i \, c - d\right )} f\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} + 8 \, {\left (a^{3} c - i \, a^{3} d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{4 \, a^{3}}\right ) - 16 \, {\left (-i \, a^{3} c + 4 \, a^{3} d + {\left (-i \, a^{3} c + 5 \, a^{3} d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{12 \, {\left (d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + d^{2} f\right )}} \]

[In]

integrate((a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

1/12*(3*(d^2*f*e^(2*I*f*x + 2*I*e) + d^2*f)*sqrt(-64*I*a^6/((I*c + d)*f^2))*log(1/4*(8*a^3*c + sqrt(-64*I*a^6/
((I*c + d)*f^2))*((I*c + d)*f*e^(2*I*f*x + 2*I*e) + (I*c + d)*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d
)/(e^(2*I*f*x + 2*I*e) + 1)) + 8*(a^3*c - I*a^3*d)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/a^3) - 3*(d^2*f*e
^(2*I*f*x + 2*I*e) + d^2*f)*sqrt(-64*I*a^6/((I*c + d)*f^2))*log(1/4*(8*a^3*c + sqrt(-64*I*a^6/((I*c + d)*f^2))
*((-I*c - d)*f*e^(2*I*f*x + 2*I*e) + (-I*c - d)*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x
+ 2*I*e) + 1)) + 8*(a^3*c - I*a^3*d)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/a^3) - 16*(-I*a^3*c + 4*a^3*d +
(-I*a^3*c + 5*a^3*d)*e^(2*I*f*x + 2*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)))/(d^2*f*e^(2*I*f*x + 2*I*e) + d^2*f)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (102) = 204\).

Time = 0.64 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.89 \[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {c+d \tan (e+f x)}} \, dx=\frac {16 i \, a^{3} \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{\sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} f {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} - \frac {2 \, {\left (i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} a^{3} d^{4} f^{2} - 3 i \, \sqrt {d \tan \left (f x + e\right ) + c} a^{3} c d^{4} f^{2} + 9 \, \sqrt {d \tan \left (f x + e\right ) + c} a^{3} d^{5} f^{2}\right )}}{3 \, d^{6} f^{3}} \]

[In]

integrate((a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(1/2),x, algorithm="giac")

[Out]

16*I*a^3*arctan(2*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(-2*c + 2*sqr
t(c^2 + d^2)) - I*sqrt(-2*c + 2*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-2*c + 2*sqrt(c^2 + d^2))))/(sqrt(-2
*c + 2*sqrt(c^2 + d^2))*f*(-I*d/(c - sqrt(c^2 + d^2)) + 1)) - 2/3*(I*(d*tan(f*x + e) + c)^(3/2)*a^3*d^4*f^2 -
3*I*sqrt(d*tan(f*x + e) + c)*a^3*c*d^4*f^2 + 9*sqrt(d*tan(f*x + e) + c)*a^3*d^5*f^2)/(d^6*f^3)

Mupad [B] (verification not implemented)

Time = 7.46 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.94 \[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {c+d \tan (e+f x)}} \, dx=-\left (\frac {a^3\,\left (c-d\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d^2\,f}-\frac {a^3\,\left (c+d\,1{}\mathrm {i}\right )\,4{}\mathrm {i}}{d^2\,f}\right )\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}-\frac {a^3\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,2{}\mathrm {i}}{3\,d^2\,f}+\frac {a^3\,\mathrm {atan}\left (\frac {\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {-c+d\,1{}\mathrm {i}}}\right )\,8{}\mathrm {i}}{f\,\sqrt {-c+d\,1{}\mathrm {i}}} \]

[In]

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

[Out]

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