3.3.78 \(\int \cot ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx\) [278]

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

[Out]

1/4*I*a^(1/3)*x*2^(1/3)+1/4*a^(1/3)*ln(cos(d*x+c))*2^(1/3)/d+4/9*a^(1/3)*ln(tan(d*x+c))/d-4/3*a^(1/3)*ln(a^(1/
3)-(a+I*a*tan(d*x+c))^(1/3))/d+3/4*a^(1/3)*ln(2^(1/3)*a^(1/3)-(a+I*a*tan(d*x+c))^(1/3))*2^(1/3)/d+8/9*a^(1/3)*
arctan(1/3*(a^(1/3)+2*(a+I*a*tan(d*x+c))^(1/3))/a^(1/3)*3^(1/2))*3^(1/2)/d-1/2*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-1/6*I*cot(d*x+c)*(a+I*a*tan(d*x+c))^(1/3)/d-
1/2*cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(1/3)/d

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Rubi [A]
time = 0.39, antiderivative size = 327, normalized size of antiderivative = 1.00, 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 = {3642, 3679, 3681, 3562, 59, 631, 210, 31, 3680} \begin {gather*} \frac {8 \sqrt [3]{a} \text {ArcTan}\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} \text {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 {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}} \end {gather*}

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 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 3642

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 3679

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 3680

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]

Rule 3681

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]

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) \text {Subst}\left (\int \frac {1}{x (a+i a x)^{2/3}} \, dx,x,\tan (c+d x)\right )}{9 d}-\frac {a \text {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 ) \text {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 ) \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 (4 a^{2/3}\right ) \text {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 ) \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 {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 ) \text {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 ) \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 {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*}

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Mathematica [F]
time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

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 = 0.40, size = 0, normalized size = 0.00 \[\int \left (\cot ^{3}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}\, dx\]

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 [A]
time = 0.53, size = 306, normalized size = 0.94 \begin {gather*} -\frac {a^{2} {\left (\frac {18 \, \sqrt {3} 2^{\frac {1}{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 )}{a^{\frac {5}{3}}} - \frac {6 \, {\left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{2} + a^{3}} - \frac {32 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {5}{3}}} + \frac {9 \cdot 2^{\frac {1}{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 )}{a^{\frac {5}{3}}} - \frac {18 \cdot 2^{\frac {1}{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 )}{a^{\frac {5}{3}}} - \frac {16 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {5}{3}}} + \frac {32 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {5}{3}}}\right )}}{36 \, d} \end {gather*}

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]

-1/36*a^2*(18*sqrt(3)*2^(1/3)*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))/a^(5/3) - 6*((I*a*tan(d*x + c) + a)^(4/3) + 2*(I*a*tan(d*x + c) + a)^(1/3)*a)/((I*a*tan(d*x + c) + a)^2
*a - 2*(I*a*tan(d*x + c) + a)*a^2 + a^3) - 32*sqrt(3)*arctan(1/3*sqrt(3)*(2*(I*a*tan(d*x + c) + a)^(1/3) + a^(
1/3))/a^(1/3))/a^(5/3) + 9*2^(1/3)*log(2^(2/3)*a^(2/3) + 2^(1/3)*(I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + (I*a*t
an(d*x + c) + a)^(2/3))/a^(5/3) - 18*2^(1/3)*log(-2^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^(1/3))/a^(5/3) - 16
*log((I*a*tan(d*x + c) + a)^(2/3) + (I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + a^(2/3))/a^(5/3) + 32*log((I*a*tan(
d*x + c) + a)^(1/3) - a^(1/3))/a^(5/3))/d

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 678 vs. \(2 (239) = 478\).
time = 0.60, size = 678, normalized size = 2.07 \begin {gather*} \frac {6 \cdot 2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} {\left (2 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} - 9 \, \left (\frac {1}{4}\right )^{\frac {1}{3}} {\left ({\left (i \, \sqrt {3} d + d\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\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 {a}{d^{3}}\right )^{\frac {1}{3}} \log \left (\left (\frac {1}{4}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} d + d\right )} \left (\frac {a}{d^{3}}\right )^{\frac {1}{3}} + 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 )}\right ) - 9 \, \left (\frac {1}{4}\right )^{\frac {1}{3}} {\left ({\left (-i \, \sqrt {3} d + d\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\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 {a}{d^{3}}\right )^{\frac {1}{3}} \log \left (\left (\frac {1}{4}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} d + d\right )} \left (\frac {a}{d^{3}}\right )^{\frac {1}{3}} + 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 )}\right ) + 18 \, \left (\frac {1}{4}\right )^{\frac {1}{3}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \left (\frac {a}{d^{3}}\right )^{\frac {1}{3}} \log \left (-2 \, \left (\frac {1}{4}\right )^{\frac {1}{3}} d \left (\frac {a}{d^{3}}\right )^{\frac {1}{3}} + 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 )}\right ) - 8 \, {\left ({\left (i \, \sqrt {3} d + d\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\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 {a}{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 )} - \frac {1}{2} \, {\left (i \, \sqrt {3} d + d\right )} \left (-\frac {a}{d^{3}}\right )^{\frac {1}{3}}\right ) - 8 \, {\left ({\left (-i \, \sqrt {3} d + d\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\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 {a}{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 )} - \frac {1}{2} \, {\left (-i \, \sqrt {3} d + d\right )} \left (-\frac {a}{d^{3}}\right )^{\frac {1}{3}}\right ) + 16 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \left (-\frac {a}{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 )} + d \left (-\frac {a}{d^{3}}\right )^{\frac {1}{3}}\right )}{18 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}

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]

1/18*(6*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)*((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)) - 9*(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)) + 18*(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*((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)) - 8*((-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)) + 16*(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{i a \left (\tan {\left (c + d x \right )} - i\right )} \cot ^{3}{\left (c + d x \right )}\, dx \end {gather*}

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Not invertible Error: Bad Argument Value

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Mupad [B]
time = 4.41, size = 417, normalized size = 1.28 \begin {gather*} \frac {8\,\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+d\,{\left (-\frac {a}{d^3}\right )}^{1/3}\right )\,{\left (-\frac {a}{d^3}\right )}^{1/3}}{9}+\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}-2^{1/3}\,d\,{\left (\frac {a}{d^3}\right )}^{1/3}\right )\,{\left (\frac {a}{4\,d^3}\right )}^{1/3}+\frac {\frac {a\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{4/3}}{6}+\frac {a^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{3}}{d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2+a^2\,d-2\,a\,d\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}-\frac {4\,\ln \left (d\,{\left (-\frac {a}{d^3}\right )}^{1/3}-2\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+\sqrt {3}\,d\,{\left (-\frac {a}{d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {a}{d^3}\right )}^{1/3}}{9}+\frac {4\,\ln \left (2\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}-d\,{\left (-\frac {a}{d^3}\right )}^{1/3}+\sqrt {3}\,d\,{\left (-\frac {a}{d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {a}{d^3}\right )}^{1/3}}{9}+\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+\frac {2^{1/3}\,d\,{\left (\frac {a}{d^3}\right )}^{1/3}}{2}-\frac {2^{1/3}\,\sqrt {3}\,d\,{\left (\frac {a}{d^3}\right )}^{1/3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {a}{4\,d^3}\right )}^{1/3}-\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+\frac {2^{1/3}\,d\,{\left (\frac {a}{d^3}\right )}^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,d\,{\left (\frac {a}{d^3}\right )}^{1/3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {a}{4\,d^3}\right )}^{1/3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*log((a*(tan(c + d*x)*1i + 1))^(1/3) + d*(-a/d^3)^(1/3))*(-a/d^3)^(1/3))/9 + log((a*(tan(c + d*x)*1i + 1))^(
1/3) - 2^(1/3)*d*(a/d^3)^(1/3))*(a/(4*d^3))^(1/3) + ((a*(a + a*tan(c + d*x)*1i)^(4/3))/6 + (a^2*(a + a*tan(c +
d*x)*1i)^(1/3))/3)/(d*(a + a*tan(c + d*x)*1i)^2 + a^2*d - 2*a*d*(a + a*tan(c + d*x)*1i)) - (4*log(d*(-a/d^3)^
(1/3) - 2*(a*(tan(c + d*x)*1i + 1))^(1/3) + 3^(1/2)*d*(-a/d^3)^(1/3)*1i)*(3^(1/2)*1i + 1)*(-a/d^3)^(1/3))/9 +
(4*log(2*(a*(tan(c + d*x)*1i + 1))^(1/3) - d*(-a/d^3)^(1/3) + 3^(1/2)*d*(-a/d^3)^(1/3)*1i)*(3^(1/2)*1i - 1)*(-
a/d^3)^(1/3))/9 + log((a*(tan(c + d*x)*1i + 1))^(1/3) + (2^(1/3)*d*(a/d^3)^(1/3))/2 - (2^(1/3)*3^(1/2)*d*(a/d^
3)^(1/3)*1i)/2)*((3^(1/2)*1i)/2 - 1/2)*(a/(4*d^3))^(1/3) - log((a*(tan(c + d*x)*1i + 1))^(1/3) + (2^(1/3)*d*(a
/d^3)^(1/3))/2 + (2^(1/3)*3^(1/2)*d*(a/d^3)^(1/3)*1i)/2)*((3^(1/2)*1i)/2 + 1/2)*(a/(4*d^3))^(1/3)

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