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

Optimal. Leaf size=260 \[ -\frac {\sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{d}+\frac {\sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{2^{2/3} d}-\frac {\sqrt [3]{a} \log (\tan (c+d x))}{2 d}+\frac {3 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac {3 \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac {\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac {i \sqrt [3]{a} x}{2\ 2^{2/3}} \]

[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-1/2*a^(1/3)*ln(tan(d*x+c))/d+3/2*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-a^(1/3)*arc
tan(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

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Rubi [A]  time = 0.28, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {3562, 3481, 57, 617, 204, 31, 3599} \[ -\frac {\sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{d}+\frac {\sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{2^{2/3} d}-\frac {\sqrt [3]{a} \log (\tan (c+d x))}{2 d}+\frac {3 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac {3 \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac {\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac {i \sqrt [3]{a} x}{2\ 2^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 3562

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[a/(
a*c - b*d), Int[(a + b*Tan[e + f*x])^m, x], x] - Dist[d/(a*c - b*d), Int[((a + b*Tan[e + f*x])^m*(b + a*Tan[e
+ f*x]))/(c + d*Tan[e + f*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]

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

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Mathematica [F]  time = 0.66, size = 0, normalized size = 0.00 \[ \int \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.70, size = 387, normalized size = 1.49 \[ \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\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 ) + \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\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 ) + \frac {1}{2} \, {\left (-i \, \sqrt {3} - 1\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 ) + \frac {1}{2} \, {\left (i \, \sqrt {3} - 1\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 ) + \left (\frac {1}{4}\right )^{\frac {1}{3}} \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 ) + \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 ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(1/4)^(1/3)*(I*sqrt(3) - 1)*(-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)) + 1/2*(1/4)^(1/3)*(-I*sqrt(3) - 1)*(-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)) + 1/2*(-I*sqrt(3) - 1)*(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)) + 1/2*(I*sqrt(3) - 1)*(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)) + (1/4)^(1/3)*(-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)) +
(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))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(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), x)

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maple [F]  time = 0.83, size = 0, normalized size = 0.00 \[ \int \cot \left (d x +c \right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.95, size = 233, normalized size = 0.90 \[ \frac {2 \, \sqrt {3} 2^{\frac {1}{3}} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) - 4 \, \sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) + 2^{\frac {1}{3}} a^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) - 2 \cdot 2^{\frac {1}{3}} a^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right ) - 2 \, a^{\frac {1}{3}} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 4 \, a^{\frac {1}{3}} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.31, size = 328, normalized size = 1.26 \[ \ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}-d\,{\left (\frac {a}{d^3}\right )}^{1/3}\right )\,{\left (\frac {a}{d^3}\right )}^{1/3}+\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+2^{1/3}\,d\,{\left (-\frac {a}{d^3}\right )}^{1/3}\right )\,{\left (-\frac {a}{4\,d^3}\right )}^{1/3}-\ln \left (\frac {2^{1/3}\,d\,{\left (-\frac {a}{d^3}\right )}^{1/3}}{2}-{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+\frac {2^{1/3}\,\sqrt {3}\,d\,{\left (-\frac {a}{d^3}\right )}^{1/3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a}{4\,d^3}\right )}^{1/3}+\frac {\ln \left (2\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+d\,{\left (\frac {a}{d^3}\right )}^{1/3}-\sqrt {3}\,d\,{\left (\frac {a}{d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {a}{d^3}\right )}^{1/3}}{2}-\frac {\ln \left (2\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+d\,{\left (\frac {a}{d^3}\right )}^{1/3}+\sqrt {3}\,d\,{\left (\frac {a}{d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {a}{d^3}\right )}^{1/3}}{2}+\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}-\frac {2^{1/3}\,d\,{\left (-\frac {a}{d^3}\right )}^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,d\,{\left (-\frac {a}{d^3}\right )}^{1/3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a}{4\,d^3}\right )}^{1/3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(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), x)

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