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

Optimal. Leaf size=327 \[ \frac{8 \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} d}-\frac{\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2^{2/3} d}+\frac{4 \sqrt [3]{a} \log (\tan (c+d x))}{9 d}-\frac{4 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{3 d}+\frac{3 \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}+\frac{\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}-\frac{i \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{6 d}+\frac{i \sqrt [3]{a} x}{2\ 2^{2/3}} \]

[Out]

((I/2)*a^(1/3)*x)/2^(2/3) + (8*a^(1/3)*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(
3*Sqrt[3]*d) - (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) + (a^(1/3)*Log[Cos[c + d*x]])/(2*2^(2/3)*d) + (4*a^(1/3)*Log[Tan[c + d*x]])/(9*d) - (4*a^(1/3)*Log[a
^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(3*d) + (3*a^(1/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)]
)/(2*2^(2/3)*d) - ((I/6)*Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(1/3))/d - (Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])
^(1/3))/(2*d)

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Rubi [A]  time = 0.565707, antiderivative size = 327, 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, 3598, 3600, 3481, 57, 617, 204, 31, 3599} \[ \frac{8 \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} d}-\frac{\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2^{2/3} d}+\frac{4 \sqrt [3]{a} \log (\tan (c+d x))}{9 d}-\frac{4 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{3 d}+\frac{3 \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}+\frac{\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}-\frac{i \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{6 d}+\frac{i \sqrt [3]{a} x}{2\ 2^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/2)*a^(1/3)*x)/2^(2/3) + (8*a^(1/3)*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(
3*Sqrt[3]*d) - (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) + (a^(1/3)*Log[Cos[c + d*x]])/(2*2^(2/3)*d) + (4*a^(1/3)*Log[Tan[c + d*x]])/(9*d) - (4*a^(1/3)*Log[a
^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(3*d) + (3*a^(1/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)]
)/(2*2^(2/3)*d) - ((I/6)*Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(1/3))/d - (Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])
^(1/3))/(2*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 3598

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*d - B*c)*(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*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n
+ 1)*Simp[A*(b*d*m - a*c*(n + 1)) - B*(b*c*m + a*d*(n + 1)) - a*(B*c - A*d)*(m + n + 1)*Tan[e + f*x], 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] && 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 ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx &=-\frac{\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}+\frac{\int \cot ^2(c+d x) \left (\frac{i a}{3}-\frac{5}{3} a \tan (c+d x)\right ) \sqrt [3]{a+i a \tan (c+d x)} \, dx}{2 a}\\ &=-\frac{i \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{6 d}-\frac{\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}+\frac{\int \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)} \left (-\frac{16 a^2}{9}-\frac{2}{9} i a^2 \tan (c+d x)\right ) \, dx}{2 a^2}\\ &=-\frac{i \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{6 d}-\frac{\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}-i \int \sqrt [3]{a+i a \tan (c+d x)} \, dx-\frac{8 \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt [3]{a+i a \tan (c+d x)} \, dx}{9 a}\\ &=-\frac{i \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{6 d}-\frac{\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}-\frac{(8 a) \operatorname{Subst}\left (\int \frac{1}{x (a+i a x)^{2/3}} \, dx,x,\tan (c+d x)\right )}{9 d}-\frac{a \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{2/3}} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=\frac{i \sqrt [3]{a} x}{2\ 2^{2/3}}+\frac{\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}+\frac{4 \sqrt [3]{a} \log (\tan (c+d x))}{9 d}-\frac{i \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{6 d}-\frac{\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}+\frac{\left (4 \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 )}{3 d}-\frac{\left (3 \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 (4 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 )}{3 d}-\frac{\left (3 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{i \sqrt [3]{a} x}{2\ 2^{2/3}}+\frac{\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}+\frac{4 \sqrt [3]{a} \log (\tan (c+d x))}{9 d}-\frac{4 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{3 d}+\frac{3 \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 \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{6 d}-\frac{\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}-\frac{\left (8 \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 )}{3 d}+\frac{\left (3 \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{i \sqrt [3]{a} x}{2\ 2^{2/3}}+\frac{8 \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 )}{3 \sqrt{3} d}-\frac{\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{\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}+\frac{4 \sqrt [3]{a} \log (\tan (c+d x))}{9 d}-\frac{4 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{3 d}+\frac{3 \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 \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{6 d}-\frac{\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\\ \end{align*}

Mathematica [F]  time = 180.004, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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Maple [F]  time = 0.123, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( dx+c \right ) \right ) ^{3}\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)^3*(a+I*a*tan(d*x+c))^(1/3),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.28138, size = 2055, normalized size = 6.28 \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)^3*(a+I*a*tan(d*x+c))^(1/3),x, algorithm="fricas")

[Out]

1/54*(18*2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*(2*e^(4*I*d*x + 4*I*c) + 3*e^(2*I*d*x + 2*I*c) + 1)*e^(2/
3*I*d*x + 2/3*I*c) - 9*(1/4)^(1/3)*(3*(I*sqrt(3)*d + d)*e^(4*I*d*x + 4*I*c) + 6*(-I*sqrt(3)*d - d)*e^(2*I*d*x
+ 2*I*c) + 3*I*sqrt(3)*d + 3*d)*(a/d^3)^(1/3)*log((1/4)^(1/3)*(I*sqrt(3)*d + d)*(a/d^3)^(1/3) + 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)) - 27*(1/4)^(1/3)*((-I*sqrt(3)*d + d)*e^(4*I*d*x + 4*I*c
) + 2*(I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) - I*sqrt(3)*d + d)*(a/d^3)^(1/3)*log((1/4)^(1/3)*(-I*sqrt(3)*d + d
)*(a/d^3)^(1/3) + 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)) + 54*(1/4)^(1/3)*(d*e^(
4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*(a/d^3)^(1/3)*log(-2*(1/4)^(1/3)*d*(a/d^3)^(1/3) + 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)) - 8*(3*(I*sqrt(3)*d + d)*e^(4*I*d*x + 4*I*c) + 6*(-I
*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) + 3*I*sqrt(3)*d + 3*d)*(-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*(I*sqrt(3)*d + d)*(-a/d^3)^(1/3)) - 24*((-I*sqrt(3)*d + d)*e^(4*I*d*
x + 4*I*c) + 2*(I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) - I*sqrt(3)*d + d)*(-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*(-I*sqrt(3)*d + d)*(-a/d^3)^(1/3)) + 48*(d*e^(4*I*d*x +
4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*(-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) + d*(-a/d^3)^(1/3)))/(d*e^(4*I*d*x + 4*I*c) - 2*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 ^{3}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**3*(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)**3, 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 )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(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)^3, x)