3.285 \(\int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx\)

Optimal. Leaf size=315 \[ -\frac{4 i a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} d}+\frac{i \sqrt [3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{d}-\frac{2 i a^{4/3} \log (\tan (c+d x))}{3 d}+\frac{2 i a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{d}-\frac{3 i a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac{i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}+\frac{a^{4/3} x}{2^{2/3}}+\frac{i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d} \]

[Out]

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

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Rubi [A]  time = 0.529468, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {3561, 3594, 3600, 3481, 57, 617, 204, 31, 3599} \[ -\frac{4 i a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} d}+\frac{i \sqrt [3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{d}-\frac{2 i a^{4/3} \log (\tan (c+d x))}{3 d}+\frac{2 i a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{d}-\frac{3 i a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac{i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}+\frac{a^{4/3} x}{2^{2/3}}+\frac{i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(4/3),x]

[Out]

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

Rule 3561

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*(n + 1)*(c^2 + d^2)), x] - Dist[1/(a*(c^2 + d^2)*
(n + 1)), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)*Simp[b*d*m - a*c*(n + 1) + a*d*(m + n + 1)*T
an[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] && LtQ[n, -1] && (IntegerQ[n] || IntegersQ[2*m, 2*n])

Rule 3594

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

Rule 3600

Int[(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*tan[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(A*b + a*B)/(b*c + a*d), Int[(a + b*Tan[e + f*x])^m, x], x] - Dist[(B*c
- A*d)/(b*c + a*d), Int[((a + b*Tan[e + f*x])^m*(a - b*Tan[e + f*x]))/(c + d*Tan[e + f*x]), x], x] /; FreeQ[{
a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 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 57

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 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]

Rule 3599

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

Rubi steps

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

Mathematica [A]  time = 7.54937, size = 587, normalized size = 1.86 \[ \frac{i \sqrt [3]{e^{i d x}} e^{-\frac{1}{3} i (5 c+2 d x)} \sqrt [3]{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt [3]{1+e^{2 i (c+d x)}} (a+i a \tan (c+d x))^{4/3} \left (-6 \log \left (1-\frac{e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}\right )+4\ 2^{2/3} \log \left (1-\frac{\sqrt [3]{2} e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}\right )+3 \log \left (\frac{\left (1+e^{2 i (c+d x)}\right )^{2/3}+e^{\frac{2}{3} i (c+d x)} \sqrt [3]{1+e^{2 i (c+d x)}}+e^{\frac{4}{3} i (c+d x)}}{\left (1+e^{2 i (c+d x)}\right )^{2/3}}\right )-2\ 2^{2/3} \log \left (\frac{\left (1+e^{2 i (c+d x)}\right )^{2/3}+\sqrt [3]{2} e^{\frac{2}{3} i (c+d x)} \sqrt [3]{1+e^{2 i (c+d x)}}+2^{2/3} e^{\frac{4}{3} i (c+d x)}}{\left (1+e^{2 i (c+d x)}\right )^{2/3}}\right )+6 \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}}{\sqrt{3}}\right )-4\ 2^{2/3} \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{2} e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}}{\sqrt{3}}\right )\right )}{3\ 2^{2/3} d \sec ^{\frac{4}{3}}(c+d x) (\cos (d x)+i \sin (d x))^{4/3}}+\frac{\cos (c+d x) (a+i a \tan (c+d x))^{4/3} (\csc (c) (\cos (c)-i \sin (c)) \sin (d x) \csc (c+d x)+\cot (c) (-\cos (c)+i \sin (c)))}{d (\cos (d x)+i \sin (d x))} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(4/3),x]

[Out]

((I/3)*(E^(I*d*x))^(1/3)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/3)*(1 + E^((2*I)*(c + d*x)))^(1/3)*(6*
Sqrt[3]*ArcTan[(1 + (2*E^(((2*I)/3)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))^(1/3))/Sqrt[3]] - 4*2^(2/3)*Sqrt[3]*
ArcTan[(1 + (2*2^(1/3)*E^(((2*I)/3)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))^(1/3))/Sqrt[3]] - 6*Log[1 - E^(((2*I
)/3)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))^(1/3)] + 4*2^(2/3)*Log[1 - (2^(1/3)*E^(((2*I)/3)*(c + d*x)))/(1 + E^
((2*I)*(c + d*x)))^(1/3)] + 3*Log[(E^(((4*I)/3)*(c + d*x)) + E^(((2*I)/3)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))
^(1/3) + (1 + E^((2*I)*(c + d*x)))^(2/3))/(1 + E^((2*I)*(c + d*x)))^(2/3)] - 2*2^(2/3)*Log[(2^(2/3)*E^(((4*I)/
3)*(c + d*x)) + 2^(1/3)*E^(((2*I)/3)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(1/3) + (1 + E^((2*I)*(c + d*x)))^(2
/3))/(1 + E^((2*I)*(c + d*x)))^(2/3)])*(a + I*a*Tan[c + d*x])^(4/3))/(2^(2/3)*d*E^((I/3)*(5*c + 2*d*x))*Sec[c
+ d*x]^(4/3)*(Cos[d*x] + I*Sin[d*x])^(4/3)) + (Cos[c + d*x]*(Cot[c]*(-Cos[c] + I*Sin[c]) + Csc[c]*Csc[c + d*x]
*(Cos[c] - I*Sin[c])*Sin[d*x])*(a + I*a*Tan[c + d*x])^(4/3))/(d*(Cos[d*x] + I*Sin[d*x]))

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Maple [F]  time = 0.123, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( dx+c \right ) \right ) ^{2} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{4}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(4/3),x)

[Out]

int(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(4/3),x)

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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(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(4/3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.90179, size = 1785, normalized size = 5.67 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(4/3),x, algorithm="fricas")

[Out]

1/2*(2^(1/3)*(-2*I*a*e^(2*I*d*x + 2*I*c) - 2*I*a)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)
+ ((I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) - I*sqrt(3)*d + d)*(-64/27*I*a^4/d^3)^(1/3)*log(1/8*(8*2^(1/3)*a*(a/(
e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) - (3*sqrt(3)*d + 3*I*d)*(-64/27*I*a^4/d^3)^(1/3))/a) +
((-I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) + I*sqrt(3)*d + d)*(-64/27*I*a^4/d^3)^(1/3)*log(1/8*(8*2^(1/3)*a*(a/(
e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) + (3*sqrt(3)*d - 3*I*d)*(-64/27*I*a^4/d^3)^(1/3))/a) +
2*(d*e^(2*I*d*x + 2*I*c) - d)*(-64/27*I*a^4/d^3)^(1/3)*log(1/4*(4*2^(1/3)*a*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/
3)*e^(2/3*I*d*x + 2/3*I*c) + 3*I*(-64/27*I*a^4/d^3)^(1/3)*d)/a) + ((I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) - I*s
qrt(3)*d + d)*(2*I*a^4/d^3)^(1/3)*log(1/2*(2*2^(1/3)*a*(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)*(2*I*a^4/d^3)^(1/3))/a) + ((-I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) + I*sqrt(3)*d + d)*
(2*I*a^4/d^3)^(1/3)*log(1/2*(2*2^(1/3)*a*(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)*(2*I*a^4/d^3)^(1/3))/a) + 2*(d*e^(2*I*d*x + 2*I*c) - d)*(2*I*a^4/d^3)^(1/3)*log((2^(1/3)*a*(a/(e^(2
*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) - I*(2*I*a^4/d^3)^(1/3)*d)/a))/(d*e^(2*I*d*x + 2*I*c) - d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**2*(a+I*a*tan(d*x+c))**(4/3),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{4}{3}} \cot \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(4/3),x, algorithm="giac")

[Out]

integrate((I*a*tan(d*x + c) + a)^(4/3)*cot(d*x + c)^2, x)