\(\int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx\) [1108]

Optimal result

Integrand size = 30, antiderivative size = 131 \[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx=-\frac {4 i a^2 (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {4 a^2 (i c+d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 i a^2 (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f} \]

[Out]

-4*I*a^2*(c-I*d)^(3/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/f+4*a^2*(I*c+d)*(c+d*tan(f*x+e))^(1/2)/f+
4/3*I*a^2*(c+d*tan(f*x+e))^(3/2)/f-2/5*a^2*(c+d*tan(f*x+e))^(5/2)/d/f

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3624, 3609, 3618, 65, 214} \[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx=-\frac {4 i a^2 (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}-\frac {2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f}+\frac {4 i a^2 (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {4 a^2 (d+i c) \sqrt {c+d \tan (e+f x)}}{f} \]

[In]

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

[Out]

((-4*I)*a^2*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + (4*a^2*(I*c + d)*Sqrt[c + d*T
an[e + f*x]])/f + (((4*I)/3)*a^2*(c + d*Tan[e + f*x])^(3/2))/f - (2*a^2*(c + d*Tan[e + f*x])^(5/2))/(5*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 3609

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]

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 3624

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])

Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f}+\int \left (2 a^2+2 i a^2 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2} \, dx \\ & = \frac {4 i a^2 (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f}+\int \sqrt {c+d \tan (e+f x)} \left (2 a^2 (c-i d)+2 a^2 (i c+d) \tan (e+f x)\right ) \, dx \\ & = \frac {4 a^2 (i c+d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 i a^2 (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f}+\int \frac {2 a^2 (c-i d)^2+2 i a^2 (c-i d)^2 \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx \\ & = \frac {4 a^2 (i c+d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 i a^2 (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f}+\frac {\left (4 i a^4 (c-i d)^4\right ) \text {Subst}\left (\int \frac {1}{\left (-4 a^4 (c-i d)^4+2 a^2 (c-i d)^2 x\right ) \sqrt {c-\frac {i d x}{2 a^2 (c-i d)^2}}} \, dx,x,2 i a^2 (c-i d)^2 \tan (e+f x)\right )}{f} \\ & = \frac {4 a^2 (i c+d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 i a^2 (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f}-\frac {\left (16 a^6 (c-i d)^6\right ) \text {Subst}\left (\int \frac {1}{-4 a^4 (c-i d)^4-\frac {4 i a^4 c (c-i d)^4}{d}+\frac {4 i a^4 (c-i d)^4 x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f} \\ & = -\frac {4 i a^2 (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {4 a^2 (i c+d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 i a^2 (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.54 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.95 \[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx=-\frac {2 a^2 \left (30 \sqrt {c-i d} d (i c+d) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+\sqrt {c+d \tan (e+f x)} \left (3 c^2-40 i c d-30 d^2+2 (3 c-5 i d) d \tan (e+f x)+3 d^2 \tan ^2(e+f x)\right )\right )}{15 d f} \]

[In]

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

[Out]

(-2*a^2*(30*Sqrt[c - I*d]*d*(I*c + d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + Sqrt[c + d*Tan[e + f*x
]]*(3*c^2 - (40*I)*c*d - 30*d^2 + 2*(3*c - (5*I)*d)*d*Tan[e + f*x] + 3*d^2*Tan[e + f*x]^2)))/(15*d*f)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 869 vs. \(2 (110 ) = 220\).

Time = 0.69 (sec) , antiderivative size = 870, normalized size of antiderivative = 6.64

method	result	size
derivativedivides	\(\frac {2 a^{2} \left (-\frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 i d \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 i d c \sqrt {c +d \tan \left (f x +e \right )}+2 d^{2} \sqrt {c +d \tan \left (f x +e \right )}-2 d \left (\frac {-\frac {\left (i c^{2} \sqrt {c^{2}+d^{2}}-i d^{2} \sqrt {c^{2}+d^{2}}+i c^{3}+i c \,d^{2}+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}\, c^{3}-i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}-\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 (i c^{2} \sqrt {c^{2}+d^{2}}-i d^{2} \sqrt {c^{2}+d^{2}}+i c^{3}+i c \,d^{2}+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}}}+\frac {\frac {\left (i c^{2} \sqrt {c^{2}+d^{2}}-i d^{2} \sqrt {c^{2}+d^{2}}+i c^{3}+i c \,d^{2}+2 c d \sqrt {c^{2}+d^{2}}+c^{2} d +d^{3}\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^{3}+i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}+\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 (i c^{2} \sqrt {c^{2}+d^{2}}-i d^{2} \sqrt {c^{2}+d^{2}}+i c^{3}+i c \,d^{2}+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 {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}}}\right )\right )}{f d}\)	\(870\)
default	\(\frac {2 a^{2} \left (-\frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 i d \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 i d c \sqrt {c +d \tan \left (f x +e \right )}+2 d^{2} \sqrt {c +d \tan \left (f x +e \right )}-2 d \left (\frac {-\frac {\left (i c^{2} \sqrt {c^{2}+d^{2}}-i d^{2} \sqrt {c^{2}+d^{2}}+i c^{3}+i c \,d^{2}+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}\, c^{3}-i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}-\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 (i c^{2} \sqrt {c^{2}+d^{2}}-i d^{2} \sqrt {c^{2}+d^{2}}+i c^{3}+i c \,d^{2}+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}}}+\frac {\frac {\left (i c^{2} \sqrt {c^{2}+d^{2}}-i d^{2} \sqrt {c^{2}+d^{2}}+i c^{3}+i c \,d^{2}+2 c d \sqrt {c^{2}+d^{2}}+c^{2} d +d^{3}\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^{3}+i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}+\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 (i c^{2} \sqrt {c^{2}+d^{2}}-i d^{2} \sqrt {c^{2}+d^{2}}+i c^{3}+i c \,d^{2}+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 {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}}}\right )\right )}{f d}\)	\(870\)
parts	\(\text {Expression too large to display}\)	\(2493\)

[In]

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

[Out]

2/f*a^2/d*(-1/5*(c+d*tan(f*x+e))^(5/2)+2/3*I*d*(c+d*tan(f*x+e))^(3/2)+2*I*d*c*(c+d*tan(f*x+e))^(1/2)+2*d^2*(c+
d*tan(f*x+e))^(1/2)-2*d*(1/2/(2*(c^2+d^2)^(1/2)+2*c)^(1/2)/(c^2+d^2)^(1/2)*(-1/2*(I*c^2*(c^2+d^2)^(1/2)-I*d^2*
(c^2+d^2)^(1/2)+I*c^3+I*c*d^2+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^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*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^3+1/2*(I*c^2*(c^2+d^2)^(1/2)-I*d^
2*(c^2+d^2)^(1/2)+I*c^3+I*c*d^2+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)*arctan(((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))
)+1/2/(2*(c^2+d^2)^(1/2)+2*c)^(1/2)/(c^2+d^2)^(1/2)*(1/2*(I*c^2*(c^2+d^2)^(1/2)-I*d^2*(c^2+d^2)^(1/2)+I*c^3+I*
c*d^2+2*c*d*(c^2+d^2)^(1/2)+c^2*d+d^3)*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^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*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^3-1/2*(I*c^2*(c^2+d^2)^(1/2)-I*d^2*(c^2+d^2)^(1/2)+I*c^3+I
*c*d^2+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)*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)))))

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. 646 vs. \(2 (105) = 210\).

Time = 0.30 (sec) , antiderivative size = 646, normalized size of antiderivative = 4.93 \[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx=\frac {15 \, {\left (d f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, d f e^{\left (2 i \, f x + 2 i \, e\right )} + d f\right )} \sqrt {-\frac {a^{4} c^{3} - 3 i \, a^{4} c^{2} d - 3 \, a^{4} c d^{2} + i \, a^{4} d^{3}}{f^{2}}} \log \left (\frac {2 \, {\left (-i \, a^{2} c^{2} - a^{2} c d + {\left (f e^{\left (2 i \, f x + 2 i \, e\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}} \sqrt {-\frac {a^{4} c^{3} - 3 i \, a^{4} c^{2} d - 3 \, a^{4} c d^{2} + i \, a^{4} d^{3}}{f^{2}}} + {\left (-i \, a^{2} c^{2} - 2 \, a^{2} c d + i \, a^{2} d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{-i \, a^{2} c - a^{2} d}\right ) - 15 \, {\left (d f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, d f e^{\left (2 i \, f x + 2 i \, e\right )} + d f\right )} \sqrt {-\frac {a^{4} c^{3} - 3 i \, a^{4} c^{2} d - 3 \, a^{4} c d^{2} + i \, a^{4} d^{3}}{f^{2}}} \log \left (\frac {2 \, {\left (-i \, a^{2} c^{2} - a^{2} c d - {\left (f e^{\left (2 i \, f x + 2 i \, e\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}} \sqrt {-\frac {a^{4} c^{3} - 3 i \, a^{4} c^{2} d - 3 \, a^{4} c d^{2} + i \, a^{4} d^{3}}{f^{2}}} + {\left (-i \, a^{2} c^{2} - 2 \, a^{2} c d + i \, a^{2} d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{-i \, a^{2} c - a^{2} d}\right ) - 2 \, {\left (3 \, a^{2} c^{2} - 34 i \, a^{2} c d - 23 \, a^{2} d^{2} + {\left (3 \, a^{2} c^{2} - 46 i \, a^{2} c d - 43 \, a^{2} d^{2}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, {\left (3 \, a^{2} c^{2} - 40 i \, a^{2} c d - 27 \, a^{2} d^{2}\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}}}{15 \, {\left (d f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, d f e^{\left (2 i \, f x + 2 i \, e\right )} + d f\right )}} \]

[In]

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

[Out]

1/15*(15*(d*f*e^(4*I*f*x + 4*I*e) + 2*d*f*e^(2*I*f*x + 2*I*e) + d*f)*sqrt(-(a^4*c^3 - 3*I*a^4*c^2*d - 3*a^4*c*
d^2 + I*a^4*d^3)/f^2)*log(2*(-I*a^2*c^2 - a^2*c*d + (f*e^(2*I*f*x + 2*I*e) + 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))*sqrt(-(a^4*c^3 - 3*I*a^4*c^2*d - 3*a^4*c*d^2 + I*a^4*d^3)/f^2) + (
-I*a^2*c^2 - 2*a^2*c*d + I*a^2*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/(-I*a^2*c - a^2*d)) - 15*(d*f*e^
(4*I*f*x + 4*I*e) + 2*d*f*e^(2*I*f*x + 2*I*e) + d*f)*sqrt(-(a^4*c^3 - 3*I*a^4*c^2*d - 3*a^4*c*d^2 + I*a^4*d^3)
/f^2)*log(2*(-I*a^2*c^2 - a^2*c*d - (f*e^(2*I*f*x + 2*I*e) + 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))*sqrt(-(a^4*c^3 - 3*I*a^4*c^2*d - 3*a^4*c*d^2 + I*a^4*d^3)/f^2) + (-I*a^2*c^2 - 2*a
^2*c*d + I*a^2*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/(-I*a^2*c - a^2*d)) - 2*(3*a^2*c^2 - 34*I*a^2*c*
d - 23*a^2*d^2 + (3*a^2*c^2 - 46*I*a^2*c*d - 43*a^2*d^2)*e^(4*I*f*x + 4*I*e) + 2*(3*a^2*c^2 - 40*I*a^2*c*d - 2
7*a^2*d^2)*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*
f*e^(4*I*f*x + 4*I*e) + 2*d*f*e^(2*I*f*x + 2*I*e) + d*f)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate((I*a*tan(f*x + e) + a)^2*(d*tan(f*x + e) + c)^(3/2), 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. 282 vs. \(2 (105) = 210\).

Time = 0.86 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.15 \[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx=-\frac {8 \, {\left (-i \, a^{2} c^{2} - 2 \, a^{2} c d + i \, a^{2} d^{2}\right )} \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{\sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} f {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} - \frac {2 \, {\left (3 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} a^{2} d^{4} f^{4} - 10 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} a^{2} d^{5} f^{4} - 30 i \, \sqrt {d \tan \left (f x + e\right ) + c} a^{2} c d^{5} f^{4} - 30 \, \sqrt {d \tan \left (f x + e\right ) + c} a^{2} d^{6} f^{4}\right )}}{15 \, d^{5} f^{5}} \]

[In]

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

[Out]

-8*(-I*a^2*c^2 - 2*a^2*c*d + I*a^2*d^2)*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*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))))/(sqrt(-2*c + 2*sqrt(c^2 + d^2))*f*(-I*d/(c - sqrt(c^2 + d^2)) + 1)) - 2/15*(3*(d*tan(f*
x + e) + c)^(5/2)*a^2*d^4*f^4 - 10*I*(d*tan(f*x + e) + c)^(3/2)*a^2*d^5*f^4 - 30*I*sqrt(d*tan(f*x + e) + c)*a^
2*c*d^5*f^4 - 30*sqrt(d*tan(f*x + e) + c)*a^2*d^6*f^4)/(d^5*f^5)

Mupad [B] (verification not implemented)

Time = 11.76 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.50 \[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx=-\left (\frac {2\,a^2\,\left (c-d\,1{}\mathrm {i}\right )}{3\,d\,f}-\frac {2\,a^2\,\left (c+d\,1{}\mathrm {i}\right )}{3\,d\,f}\right )\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}-\left (c-d\,1{}\mathrm {i}\right )\,\left (\frac {2\,a^2\,\left (c-d\,1{}\mathrm {i}\right )}{d\,f}-\frac {2\,a^2\,\left (c+d\,1{}\mathrm {i}\right )}{d\,f}\right )\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}-\frac {2\,a^2\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}}{5\,d\,f}+\frac {\sqrt {4{}\mathrm {i}}\,a^2\,\mathrm {atan}\left (\frac {\sqrt {4{}\mathrm {i}}\,{\left (-d-c\,1{}\mathrm {i}\right )}^{3/2}\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}\,1{}\mathrm {i}}{2\,\left (c^2\,1{}\mathrm {i}+2\,c\,d-d^2\,1{}\mathrm {i}\right )}\right )\,{\left (-d-c\,1{}\mathrm {i}\right )}^{3/2}\,2{}\mathrm {i}}{f} \]

[In]

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

[Out]

(4i^(1/2)*a^2*atan((4i^(1/2)*(- c*1i - d)^(3/2)*(c + d*tan(e + f*x))^(1/2)*1i)/(2*(2*c*d + c^2*1i - d^2*1i)))*
(- c*1i - d)^(3/2)*2i)/f - (c - d*1i)*((2*a^2*(c - d*1i))/(d*f) - (2*a^2*(c + d*1i))/(d*f))*(c + d*tan(e + f*x
))^(1/2) - (2*a^2*(c + d*tan(e + f*x))^(5/2))/(5*d*f) - ((2*a^2*(c - d*1i))/(3*d*f) - (2*a^2*(c + d*1i))/(3*d*
f))*(c + d*tan(e + f*x))^(3/2)