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]

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

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Rubi [A]  time = 0.42, antiderivative size = 299, normalized size of antiderivative = 1.00, 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 31

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

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 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 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 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 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 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]

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]

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*}

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Mathematica [F]  time = 180.01, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]

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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fricas [B]  time = 1.67, size = 550, normalized size = 1.84 \[ \frac {2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} {\left (-2 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 2 i\right )} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} + {\left ({\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 {i \, a}{4 \, 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 )} + {\left (\sqrt {3} d + i \, d\right )} \left (\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}}\right ) + {\left ({\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 {i \, a}{4 \, 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 )} - {\left (\sqrt {3} d - i \, d\right )} \left (\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}}\right ) + 2 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \left (\frac {i \, a}{4 \, 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 )} - 2 i \, d \left (\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}}\right ) + {\left ({\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 {i \, a}{27 \, 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 {3}{2} \, {\left (\sqrt {3} d + i \, d\right )} \left (-\frac {i \, a}{27 \, d^{3}}\right )^{\frac {1}{3}}\right ) + {\left ({\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 {i \, a}{27 \, 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 {3}{2} \, {\left (\sqrt {3} d - i \, d\right )} \left (-\frac {i \, a}{27 \, d^{3}}\right )^{\frac {1}{3}}\right ) + 2 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \left (-\frac {i \, a}{27 \, 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 )} + 3 i \, d \left (-\frac {i \, a}{27 \, d^{3}}\right )^{\frac {1}{3}}\right )}{2 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]

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) - 3/2*(sqrt(3)*d + 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) + 3/2*(sqrt(3)*d - 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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} \cot \left (d x + c\right )^{2}\,{d x} \]

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)

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maple [F]  time = 0.75, size = 0, normalized size = 0.00 \[ \int \left (\cot ^{2}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^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 [A]  time = 0.88, size = 261, normalized size = 0.87 \[ \frac {i \, {\left (\frac {6 \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{a^{\frac {2}{3}}} - \frac {4 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {2}{3}}} + \frac {3 \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right )}{a^{\frac {2}{3}}} - \frac {6 \cdot 2^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}{a^{\frac {2}{3}}} - \frac {2 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {2}{3}}} + \frac {4 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {2}{3}}} + \frac {12 i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}{a \tan \left (d x + c\right )}\right )} a}{12 \, d} \]

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]

1/12*I*(6*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^(2/3) - 4*sqrt(3)*arctan(1/3*sqrt(3)*(2*(I*a*tan(d*x + c) + a)^(1/3) + a^(1/3))/a^(1/3))/a^(2/3) + 3*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*tan(d*x + c) + a)^(2/3))/a^(2/3)
- 6*2^(1/3)*log(-2^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^(1/3))/a^(2/3) - 2*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^(2/3) + 4*log((I*a*tan(d*x + c) + a)^(1/3) - a^(1/3))/a^(
2/3) + 12*I*(I*a*tan(d*x + c) + a)^(1/3)/(a*tan(d*x + c)))*a/d

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mupad [B]  time = 4.20, size = 806, normalized size = 2.70 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(((a^7*d^5*(a + a*tan(c + d*x)*1i)^(1/3)*81i - 1458*a^7*d^6*((a*1i)/(4*d^3))^(1/3))*((a*1i)/(4*d^3))^(2/3)
+ a^8*d^3*225i)*((a*1i)/(4*d^3))^(1/3) + 90*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/3))*((a*1i)/(4*d^3))^(1/3) + lo
g(((a^7*d^5*(a + a*tan(c + d*x)*1i)^(1/3)*81i - 1458*a^7*d^6*(-(a*1i)/(27*d^3))^(1/3))*(-(a*1i)/(27*d^3))^(2/3
) + a^8*d^3*225i)*(-(a*1i)/(27*d^3))^(1/3) + 90*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/3))*(-(a*1i)/(27*d^3))^(1/3
) + (log(90*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/3) + ((3^(1/2)*1i - 1)*(a^8*d^3*225i + ((3^(1/2)*1i - 1)^2*(a^7
*d^5*(a + a*tan(c + d*x)*1i)^(1/3)*81i - 729*a^7*d^6*(3^(1/2)*1i - 1)*((a*1i)/(4*d^3))^(1/3))*((a*1i)/(4*d^3))
^(2/3))/4)*((a*1i)/(4*d^3))^(1/3))/2)*(3^(1/2)*1i - 1)*((a*1i)/(4*d^3))^(1/3))/2 - (log(90*a^8*d^2*(a + a*tan(
c + d*x)*1i)^(1/3) - ((3^(1/2)*1i + 1)*(a^8*d^3*225i + ((3^(1/2)*1i + 1)^2*(a^7*d^5*(a + a*tan(c + d*x)*1i)^(1
/3)*81i + 729*a^7*d^6*(3^(1/2)*1i + 1)*((a*1i)/(4*d^3))^(1/3))*((a*1i)/(4*d^3))^(2/3))/4)*((a*1i)/(4*d^3))^(1/
3))/2)*(3^(1/2)*1i + 1)*((a*1i)/(4*d^3))^(1/3))/2 + (log(90*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/3) + ((3^(1/2)*
1i - 1)*(a^8*d^3*225i + ((3^(1/2)*1i - 1)^2*(a^7*d^5*(a + a*tan(c + d*x)*1i)^(1/3)*81i - 729*a^7*d^6*(3^(1/2)*
1i - 1)*(-(a*1i)/(27*d^3))^(1/3))*(-(a*1i)/(27*d^3))^(2/3))/4)*(-(a*1i)/(27*d^3))^(1/3))/2)*(3^(1/2)*1i - 1)*(
-(a*1i)/(27*d^3))^(1/3))/2 - (log(90*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/3) - ((3^(1/2)*1i + 1)*(a^8*d^3*225i +
((3^(1/2)*1i + 1)^2*(a^7*d^5*(a + a*tan(c + d*x)*1i)^(1/3)*81i + 729*a^7*d^6*(3^(1/2)*1i + 1)*(-(a*1i)/(27*d^
3))^(1/3))*(-(a*1i)/(27*d^3))^(2/3))/4)*(-(a*1i)/(27*d^3))^(1/3))/2)*(3^(1/2)*1i + 1)*(-(a*1i)/(27*d^3))^(1/3)
)/2 - (a + a*tan(c + d*x)*1i)^(1/3)/(d*tan(c + d*x))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt [3]{i a \left (\tan {\left (c + d x \right )} - i\right )} \cot ^{2}{\left (c + d x \right )}\, dx \]

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

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