\(\int \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx\) [273]

Optimal result

Integrand size = 26, antiderivative size = 185 \[ \int \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\frac {\sqrt [3]{a} x}{2\ 2^{2/3}}+\frac {i \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{2^{2/3} d}-\frac {i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac {3 i \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac {3 i (a+i a \tan (c+d x))^{4/3}}{4 a d} \]

[Out]

1/4*a^(1/3)*x*2^(1/3)-1/4*I*a^(1/3)*ln(cos(d*x+c))*2^(1/3)/d-3/4*I*a^(1/3)*ln(2^(1/3)*a^(1/3)-(a+I*a*tan(d*x+c
))^(1/3))*2^(1/3)/d+1/2*I*a^(1/3)*arctan(1/3*(a^(1/3)+2^(2/3)*(a+I*a*tan(d*x+c))^(1/3))/a^(1/3)*3^(1/2))*3^(1/
2)*2^(1/3)/d-3/4*I*(a+I*a*tan(d*x+c))^(4/3)/a/d

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3624, 3562, 59, 631, 210, 31} \[ \int \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\frac {i \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{2^{2/3} d}-\frac {3 i (a+i a \tan (c+d x))^{4/3}}{4 a d}-\frac {3 i \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac {i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}+\frac {\sqrt [3]{a} x}{2\ 2^{2/3}} \]

[In]

Int[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(1/3),x]

[Out]

(a^(1/3)*x)/(2*2^(2/3)) + (I*Sqrt[3]*a^(1/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*
a^(1/3))])/(2^(2/3)*d) - ((I/2)*a^(1/3)*Log[Cos[c + d*x]])/(2^(2/3)*d) - (((3*I)/2)*a^(1/3)*Log[2^(1/3)*a^(1/3
) - (a + I*a*Tan[c + d*x])^(1/3)])/(2^(2/3)*d) - (((3*I)/4)*(a + I*a*Tan[c + d*x])^(4/3))/(a*d)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 59

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]
&& PosQ[(b*c - a*d)/b]

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 3562

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[-b/d, Subst[Int[(a + x)^(n - 1)/(a - x), x]
, x, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 + b^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 {3 i (a+i a \tan (c+d x))^{4/3}}{4 a d}-\int \sqrt [3]{a+i a \tan (c+d x)} \, dx \\ & = -\frac {3 i (a+i a \tan (c+d x))^{4/3}}{4 a d}+\frac {(i a) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{2/3}} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = \frac {\sqrt [3]{a} x}{2\ 2^{2/3}}-\frac {i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac {3 i (a+i a \tan (c+d x))^{4/3}}{4 a d}+\frac {\left (3 i \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}+\frac {\left (3 i a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d} \\ & = \frac {\sqrt [3]{a} x}{2\ 2^{2/3}}-\frac {i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac {3 i \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac {3 i (a+i a \tan (c+d x))^{4/3}}{4 a d}-\frac {\left (3 i \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}\right )}{2^{2/3} d} \\ & = \frac {\sqrt [3]{a} x}{2\ 2^{2/3}}+\frac {i \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{2^{2/3} d}-\frac {i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac {3 i \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac {3 i (a+i a \tan (c+d x))^{4/3}}{4 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.08 \[ \int \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\frac {i \left (2 \sqrt [3]{2} \sqrt {3} a^{4/3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \sqrt [3]{2} a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )+\sqrt [3]{2} a^{4/3} \log \left (2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+i a \tan (c+d x)}+(a+i a \tan (c+d x))^{2/3}\right )-3 (a+i a \tan (c+d x))^{4/3}\right )}{4 a d} \]

[In]

Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(1/3),x]

[Out]

((I/4)*(2*2^(1/3)*Sqrt[3]*a^(4/3)*ArcTan[(1 + (2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/a^(1/3))/Sqrt[3]] - 2*2^(
1/3)*a^(4/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)] + 2^(1/3)*a^(4/3)*Log[2^(2/3)*a^(2/3) + 2^(1/
3)*a^(1/3)*(a + I*a*Tan[c + d*x])^(1/3) + (a + I*a*Tan[c + d*x])^(2/3)] - 3*(a + I*a*Tan[c + d*x])^(4/3)))/(a*
d)

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.85

method	result	size
derivativedivides	\(\frac {3 i \left (-\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}{4}-\left (\frac {2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {2}{3}}}-\frac {2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {2}{3}}}\right ) a^{2}\right )}{d a}\)	\(157\)
default	\(\frac {3 i \left (-\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}{4}-\left (\frac {2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {2}{3}}}-\frac {2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {2}{3}}}\right ) a^{2}\right )}{d a}\)	\(157\)

[In]

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

[Out]

3*I/d/a*(-1/4*(a+I*a*tan(d*x+c))^(4/3)-(1/6*2^(1/3)/a^(2/3)*ln((a+I*a*tan(d*x+c))^(1/3)-2^(1/3)*a^(1/3))-1/12*
2^(1/3)/a^(2/3)*ln((a+I*a*tan(d*x+c))^(2/3)+2^(1/3)*a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+2^(2/3)*a^(2/3))-1/6*2^(1
/3)/a^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2^(2/3)/a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+1)))*a^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. 294 vs. \(2 (128) = 256\).

Time = 0.27 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.59 \[ \int \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\frac {{\left ({\left (i \, \sqrt {3} d - d\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, \sqrt {3} d - d\right )} \left (\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}} \log \left (2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} + {\left (\sqrt {3} d + i \, d\right )} \left (\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}}\right ) + {\left ({\left (-i \, \sqrt {3} d - d\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, \sqrt {3} d - d\right )} \left (\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}} \log \left (2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} - {\left (\sqrt {3} d - i \, d\right )} \left (\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}}\right ) + 2 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \left (\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}} \log \left (2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} - 2 i \, d \left (\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}}\right ) - 3 i \cdot 2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {8}{3} i \, d x + \frac {8}{3} i \, c\right )}}{2 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.83 \[ \int \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\frac {i \, {\left (2 \, \sqrt {3} 2^{\frac {1}{3}} a^{\frac {10}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) + 2^{\frac {1}{3}} a^{\frac {10}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) - 2 \cdot 2^{\frac {1}{3}} a^{\frac {10}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right ) - 3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} a^{2}\right )}}{4 \, a^{3} d} \]

[In]

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

[Out]

1/4*I*(2*sqrt(3)*2^(1/3)*a^(10/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^(1/3) + 2*(I*a*tan(d*x + c) + a)^(1/3)
)/a^(1/3)) + 2^(1/3)*a^(10/3)*log(2^(2/3)*a^(2/3) + 2^(1/3)*(I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + (I*a*tan(d*
x + c) + a)^(2/3)) - 2*2^(1/3)*a^(10/3)*log(-2^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^(1/3)) - 3*(I*a*tan(d*x
+ c) + a)^(4/3)*a^2)/(a^3*d)

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.85 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.05 \[ \int \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=-\frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{4/3}\,3{}\mathrm {i}}{4\,a\,d}+\frac {{\left (\frac {1}{4}{}\mathrm {i}\right )}^{1/3}\,a^{1/3}\,\ln \left (18\,{\left (\frac {1}{4}{}\mathrm {i}\right )}^{1/3}\,a^{4/3}\,d^2+a\,d^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\,9{}\mathrm {i}\right )}{d}+\frac {{\left (\frac {1}{4}{}\mathrm {i}\right )}^{1/3}\,a^{1/3}\,\ln \left (a\,d^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\,9{}\mathrm {i}+18\,{\left (\frac {1}{4}{}\mathrm {i}\right )}^{1/3}\,a^{4/3}\,d^2\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{d}-\frac {{\left (\frac {1}{4}{}\mathrm {i}\right )}^{1/3}\,a^{1/3}\,\ln \left (a\,d^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\,9{}\mathrm {i}-18\,{\left (\frac {1}{4}{}\mathrm {i}\right )}^{1/3}\,a^{4/3}\,d^2\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{d} \]

[In]

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

[Out]

((1i/4)^(1/3)*a^(1/3)*log(18*(1i/4)^(1/3)*a^(4/3)*d^2 + a*d^2*(a + a*tan(c + d*x)*1i)^(1/3)*9i))/d - ((a + a*t
an(c + d*x)*1i)^(4/3)*3i)/(4*a*d) + ((1i/4)^(1/3)*a^(1/3)*log(a*d^2*(a + a*tan(c + d*x)*1i)^(1/3)*9i + 18*(1i/
4)^(1/3)*a^(4/3)*d^2*((3^(1/2)*1i)/2 - 1/2))*((3^(1/2)*1i)/2 - 1/2))/d - ((1i/4)^(1/3)*a^(1/3)*log(a*d^2*(a +
a*tan(c + d*x)*1i)^(1/3)*9i - 18*(1i/4)^(1/3)*a^(4/3)*d^2*((3^(1/2)*1i)/2 + 1/2))*((3^(1/2)*1i)/2 + 1/2))/d