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

Optimal. Leaf size=299 \[ -\frac{i \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} d}+\frac{i \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2^{2/3} d}-\frac{i \sqrt [3]{a} \log (\tan (c+d x))}{6 d}+\frac{i \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac{3 i \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac{i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{\cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}+\frac{\sqrt [3]{a} x}{2\ 2^{2/3}} \]

[Out]

(a^(1/3)*x)/(2*2^(2/3)) - (I*a^(1/3)*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(Sq
rt[3]*d) + (I*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) - ((I/6)*a^(1/3)*Log[Tan[c + d*x]])/d + ((I/2)*a^(1/3)
*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/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) - (Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(1/3))/d

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Rubi [A]  time = 0.424409, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3561, 3600, 3481, 57, 617, 204, 31, 3599} \[ -\frac{i \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} d}+\frac{i \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2^{2/3} d}-\frac{i \sqrt [3]{a} \log (\tan (c+d x))}{6 d}+\frac{i \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac{3 i \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac{i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{\cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}+\frac{\sqrt [3]{a} x}{2\ 2^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^(1/3)*x)/(2*2^(2/3)) - (I*a^(1/3)*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(Sq
rt[3]*d) + (I*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) - ((I/6)*a^(1/3)*Log[Tan[c + d*x]])/d + ((I/2)*a^(1/3)
*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/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) - (Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(1/3))/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 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) \sqrt [3]{a+i a \tan (c+d x)} \, dx &=-\frac{\cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}+\frac{\int \cot (c+d x) \left (\frac{i a}{3}-\frac{2}{3} a \tan (c+d x)\right ) \sqrt [3]{a+i a \tan (c+d x)} \, dx}{a}\\ &=-\frac{\cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}+\frac{i \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt [3]{a+i a \tan (c+d x)} \, dx}{3 a}-\int \sqrt [3]{a+i a \tan (c+d x)} \, dx\\ &=-\frac{\cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}+\frac{(i a) \operatorname{Subst}\left (\int \frac{1}{x (a+i a x)^{2/3}} \, dx,x,\tan (c+d x)\right )}{3 d}+\frac{(i a) \operatorname{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{i \sqrt [3]{a} \log (\tan (c+d x))}{6 d}-\frac{\cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{\left (i \sqrt [3]{a}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}+\frac{\left (3 i \sqrt [3]{a}\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^{2/3} d}-\frac{\left (i a^{2/3}\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 )}{2 d}+\frac{\left (3 i a^{2/3}\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 )}{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{i \sqrt [3]{a} \log (\tan (c+d x))}{6 d}+\frac{i \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 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{\cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}+\frac{\left (i \sqrt [3]{a}\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]{a}\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 )}{2^{2/3} d}\\ &=\frac{\sqrt [3]{a} x}{2\ 2^{2/3}}-\frac{i \sqrt [3]{a} \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} \sqrt [3]{a} \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^{2/3} d}-\frac{i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{i \sqrt [3]{a} \log (\tan (c+d x))}{6 d}+\frac{i \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 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{\cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}\\ \end{align*}

Mathematica [F]  time = 180.008, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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Maple [F]  time = 0.112, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( dx+c \right ) \right ) ^{2}\sqrt [3]{a+ia\tan \left ( dx+c \right ) }\, 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))^(1/3),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.35767, size = 1679, normalized size = 5.62 \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))^(1/3),x, algorithm="fricas")

[Out]

1/2*(2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*(-2*I*e^(2*I*d*x + 2*I*c) - 2*I)*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)*(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)) + ((I*sqrt(3)*
d - d)*e^(2*I*d*x + 2*I*c) - I*sqrt(3)*d + d)*(-1/27*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) - 1/2*(3*sqrt(3)*d + 3*I*d)*(-1/27*I*a/d^3)^(1/3)) + ((-I*sqrt(3)*d - d)*e^(2*I*
d*x + 2*I*c) + I*sqrt(3)*d + d)*(-1/27*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) + 1/2*(3*sqrt(3)*d - 3*I*d)*(-1/27*I*a/d^3)^(1/3)) + 2*(d*e^(2*I*d*x + 2*I*c) - d)*(-1/27*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) + 3*I*d*(-1/27*I*a/d^3)^(1/
3)))/(d*e^(2*I*d*x + 2*I*c) - d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{a \left (i \tan{\left (c + d x \right )} + 1\right )} \cot ^{2}{\left (c + d x \right )}\, dx \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))**(1/3),x)

[Out]

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

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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{1}{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))^(1/3),x, algorithm="giac")

[Out]

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