3.290 \(\int \frac{\tan ^4(c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx\)

Optimal. Leaf size=282 \[ -\frac{39 i (a+i a \tan (c+d x))^{5/3}}{20 a^2 d}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{15 i \tan ^2(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{i \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{45 i (a+i a \tan (c+d x))^{2/3}}{8 a d}+\frac{3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{i \log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{x}{4 \sqrt [3]{2} \sqrt [3]{a}} \]

[Out]

-x/(4*2^(1/3)*a^(1/3)) + ((I/2)*Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/
3))])/(2^(1/3)*a^(1/3)*d) + ((I/4)*Log[Cos[c + d*x]])/(2^(1/3)*a^(1/3)*d) + (((3*I)/4)*Log[2^(1/3)*a^(1/3) - (
a + I*a*Tan[c + d*x])^(1/3)])/(2^(1/3)*a^(1/3)*d) - (((15*I)/8)*Tan[c + d*x]^2)/(d*(a + I*a*Tan[c + d*x])^(1/3
)) + (3*Tan[c + d*x]^3)/(8*d*(a + I*a*Tan[c + d*x])^(1/3)) + (((45*I)/8)*(a + I*a*Tan[c + d*x])^(2/3))/(a*d) -
(((39*I)/20)*(a + I*a*Tan[c + d*x])^(5/3))/(a^2*d)

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Rubi [A]  time = 0.442492, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {3560, 3595, 3592, 3527, 3481, 55, 617, 204, 31} \[ -\frac{39 i (a+i a \tan (c+d x))^{5/3}}{20 a^2 d}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{15 i \tan ^2(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{i \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{45 i (a+i a \tan (c+d x))^{2/3}}{8 a d}+\frac{3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{i \log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{x}{4 \sqrt [3]{2} \sqrt [3]{a}} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^4/(a + I*a*Tan[c + d*x])^(1/3),x]

[Out]

-x/(4*2^(1/3)*a^(1/3)) + ((I/2)*Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/
3))])/(2^(1/3)*a^(1/3)*d) + ((I/4)*Log[Cos[c + d*x]])/(2^(1/3)*a^(1/3)*d) + (((3*I)/4)*Log[2^(1/3)*a^(1/3) - (
a + I*a*Tan[c + d*x])^(1/3)])/(2^(1/3)*a^(1/3)*d) - (((15*I)/8)*Tan[c + d*x]^2)/(d*(a + I*a*Tan[c + d*x])^(1/3
)) + (3*Tan[c + d*x]^3)/(8*d*(a + I*a*Tan[c + d*x])^(1/3)) + (((45*I)/8)*(a + I*a*Tan[c + d*x])^(2/3))/(a*d) -
(((39*I)/20)*(a + I*a*Tan[c + d*x])^(5/3))/(a^2*d)

Rule 3560

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

Rule 3595

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*b - a*B)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n)/(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 - 1)*Simp[A*(a*c*m + b*d*n
) - B*(b*c*m + a*d*n) - d*(b*B*(m - n) - a*A*(m + n))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A,
B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && GtQ[n, 0]

Rule 3592

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]

Rule 3527

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] + Dist[(b*c + a*d)/b, Int[(a + b*Tan[e + f*x])^m, x], x] /; FreeQ[{a, b, c,
d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] &&  !LtQ[m, 0]

Rule 3481

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 55

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), 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 617

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 204

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

Rule 31

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

Rubi steps

\begin{align*} \int \frac{\tan ^4(c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx &=\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{3 \int \frac{\tan ^2(c+d x) \left (3 a-\frac{1}{3} i a \tan (c+d x)\right )}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx}{8 a}\\ &=-\frac{15 i \tan ^2(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{9 \int \tan (c+d x) (a+i a \tan (c+d x))^{2/3} \left (\frac{20 i a^2}{3}+\frac{52}{9} a^2 \tan (c+d x)\right ) \, dx}{16 a^3}\\ &=-\frac{15 i \tan ^2(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{39 i (a+i a \tan (c+d x))^{5/3}}{20 a^2 d}+\frac{9 \int (a+i a \tan (c+d x))^{2/3} \left (-\frac{52 a^2}{9}+\frac{20}{3} i a^2 \tan (c+d x)\right ) \, dx}{16 a^3}\\ &=-\frac{15 i \tan ^2(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{45 i (a+i a \tan (c+d x))^{2/3}}{8 a d}-\frac{39 i (a+i a \tan (c+d x))^{5/3}}{20 a^2 d}+\frac{\int (a+i a \tan (c+d x))^{2/3} \, dx}{2 a}\\ &=-\frac{15 i \tan ^2(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{45 i (a+i a \tan (c+d x))^{2/3}}{8 a d}-\frac{39 i (a+i a \tan (c+d x))^{5/3}}{20 a^2 d}-\frac{i \operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt [3]{a+x}} \, dx,x,i a \tan (c+d x)\right )}{2 d}\\ &=-\frac{x}{4 \sqrt [3]{2} \sqrt [3]{a}}+\frac{i \log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{15 i \tan ^2(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{45 i (a+i a \tan (c+d x))^{2/3}}{8 a d}-\frac{39 i (a+i a \tan (c+d x))^{5/3}}{20 a^2 d}+\frac{(3 i) \operatorname{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 )}{4 d}-\frac{(3 i) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}\\ &=-\frac{x}{4 \sqrt [3]{2} \sqrt [3]{a}}+\frac{i \log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{15 i \tan ^2(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{45 i (a+i a \tan (c+d x))^{2/3}}{8 a d}-\frac{39 i (a+i a \tan (c+d x))^{5/3}}{20 a^2 d}-\frac{(3 i) \operatorname{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 \sqrt [3]{2} \sqrt [3]{a} d}\\ &=-\frac{x}{4 \sqrt [3]{2} \sqrt [3]{a}}+\frac{i \sqrt{3} \tan ^{-1}\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} \sqrt [3]{a} d}+\frac{i \log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{15 i \tan ^2(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{45 i (a+i a \tan (c+d x))^{2/3}}{8 a d}-\frac{39 i (a+i a \tan (c+d x))^{5/3}}{20 a^2 d}\\ \end{align*}

Mathematica [C]  time = 1.23264, size = 125, normalized size = 0.44 \[ \frac{15 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right ) (\tan (c+d x)-i)+3 i \sec ^3(c+d x) (2 i \sin (c+d x)+7 i \sin (3 (c+d x))+37 \cos (c+d x)+12 \cos (3 (c+d x)))}{40 d \sqrt [3]{a+i a \tan (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^4/(a + I*a*Tan[c + d*x])^(1/3),x]

[Out]

((3*I)*Sec[c + d*x]^3*(37*Cos[c + d*x] + 12*Cos[3*(c + d*x)] + (2*I)*Sin[c + d*x] + (7*I)*Sin[3*(c + d*x)]) +
15*Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(-I + Tan[c + d*x]))/(40*d*(a
+ I*a*Tan[c + d*x])^(1/3))

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Maple [A]  time = 0.021, size = 227, normalized size = 0.8 \begin{align*}{\frac{{\frac{3\,i}{8}}}{d{a}^{3}} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{8}{3}}}}-{\frac{{\frac{6\,i}{5}}}{{a}^{2}d} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{5}{3}}}}+{\frac{3\,i}{ad} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}}+{\frac{{\frac{3\,i}{2}}}{d}{\frac{1}{\sqrt [3]{a+ia\tan \left ( dx+c \right ) }}}}+{\frac{{\frac{i}{4}}{2}^{{\frac{2}{3}}}}{d}\ln \left ( \sqrt [3]{a+ia\tan \left ( dx+c \right ) }-\sqrt [3]{2}\sqrt [3]{a} \right ){\frac{1}{\sqrt [3]{a}}}}-{\frac{{\frac{i}{8}}{2}^{{\frac{2}{3}}}}{d}\ln \left ( \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}+\sqrt [3]{2}\sqrt [3]{a}\sqrt [3]{a+ia\tan \left ( dx+c \right ) }+{2}^{{\frac{2}{3}}}{a}^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{a}}}}+{\frac{{\frac{i}{4}}\sqrt{3}{2}^{{\frac{2}{3}}}}{d}\arctan \left ({\frac{\sqrt{3}}{3} \left ({{2}^{{\frac{2}{3}}}\sqrt [3]{a+ia\tan \left ( dx+c \right ) }{\frac{1}{\sqrt [3]{a}}}}+1 \right ) } \right ){\frac{1}{\sqrt [3]{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^4/(a+I*a*tan(d*x+c))^(1/3),x)

[Out]

3/8*I/d/a^3*(a+I*a*tan(d*x+c))^(8/3)-6/5*I/d/a^2*(a+I*a*tan(d*x+c))^(5/3)+3*I/d/a*(a+I*a*tan(d*x+c))^(2/3)+3/2
*I/d/(a+I*a*tan(d*x+c))^(1/3)+1/4*I/d/a^(1/3)*2^(2/3)*ln((a+I*a*tan(d*x+c))^(1/3)-2^(1/3)*a^(1/3))-1/8*I/d/a^(
1/3)*2^(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))+1/4*I/d/a^(
1/3)*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)*(2^(2/3)/a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+1))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^4/(a+I*a*tan(d*x+c))^(1/3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.74475, size = 1453, normalized size = 5.15 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^4/(a+I*a*tan(d*x+c))^(1/3),x, algorithm="fricas")

[Out]

1/20*(2^(2/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(2/3)*(57*I*e^(6*I*d*x + 6*I*c) + 117*I*e^(4*I*d*x + 4*I*c) + 105*
I*e^(2*I*d*x + 2*I*c) + 15*I)*e^(4/3*I*d*x + 4/3*I*c) + 20*(a*d*e^(6*I*d*x + 6*I*c) + 2*a*d*e^(4*I*d*x + 4*I*c
) + a*d*e^(2*I*d*x + 2*I*c))*(-1/16*I/(a*d^3))^(1/3)*log(8*a*d^2*(-1/16*I/(a*d^3))^(2/3) + 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)) + ((-10*I*sqrt(3)*a*d - 10*a*d)*e^(6*I*d*x + 6*I*c) + (-20*I
*sqrt(3)*a*d - 20*a*d)*e^(4*I*d*x + 4*I*c) + (-10*I*sqrt(3)*a*d - 10*a*d)*e^(2*I*d*x + 2*I*c))*(-1/16*I/(a*d^3
))^(1/3)*log((4*I*sqrt(3)*a*d^2 - 4*a*d^2)*(-1/16*I/(a*d^3))^(2/3) + 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)) + ((10*I*sqrt(3)*a*d - 10*a*d)*e^(6*I*d*x + 6*I*c) + (20*I*sqrt(3)*a*d - 20*a*d)*e
^(4*I*d*x + 4*I*c) + (10*I*sqrt(3)*a*d - 10*a*d)*e^(2*I*d*x + 2*I*c))*(-1/16*I/(a*d^3))^(1/3)*log((-4*I*sqrt(3
)*a*d^2 - 4*a*d^2)*(-1/16*I/(a*d^3))^(2/3) + 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)))/(a*d*e^(6*I*d*x + 6*I*c) + 2*a*d*e^(4*I*d*x + 4*I*c) + a*d*e^(2*I*d*x + 2*I*c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{4}{\left (c + d x \right )}}{\sqrt [3]{a \left (i \tan{\left (c + d x \right )} + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**4/(a+I*a*tan(d*x+c))**(1/3),x)

[Out]

Integral(tan(c + d*x)**4/(a*(I*tan(c + d*x) + 1))**(1/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (d x + c\right )^{4}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^4/(a+I*a*tan(d*x+c))^(1/3),x, algorithm="giac")

[Out]

integrate(tan(d*x + c)^4/(I*a*tan(d*x + c) + a)^(1/3), x)