3.3.95 \(\int \frac {\cot (c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx\) [295]

Optimal. Leaf size=286 \[ -\frac {i x}{4 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d}-\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a} d}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a} d}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac {3}{2 d \sqrt [3]{a+i a \tan (c+d x)}} \]

[Out]

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

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Rubi [A]
time = 0.30, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3643, 3560, 3562, 57, 631, 210, 31, 3677, 3680} \begin {gather*} \frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d}-\frac {\sqrt {3} \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 \sqrt [3]{2} \sqrt [3]{a} d}+\frac {3}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a} d}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a} d}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {i x}{4 \sqrt [3]{2} \sqrt [3]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), 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 3560

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a*((a + b*Tan[c + d*x])^n/(2*b*d*n)), x] +
Dist[1/(2*a), Int[(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 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 3643

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 3677

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*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*
f*m*(b*c - a*d))), x] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Si
mp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x
] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] &&  !GtQ[n,
0]

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]

Rubi steps

\begin {align*} \int \frac {\cot (c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx &=i \int \frac {1}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx-\frac {i \int \frac {\cot (c+d x) (i a+a \tan (c+d x))}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx}{a}\\ &=\frac {3}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {(3 i) \int \cot (c+d x) (a+i a \tan (c+d x))^{2/3} \left (\frac {2 i a^2}{3}+\frac {2}{3} a^2 \tan (c+d x)\right ) \, dx}{2 a^3}+\frac {i \int (a+i a \tan (c+d x))^{2/3} \, dx}{2 a}\\ &=\frac {3}{2 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{(a-x) \sqrt [3]{a+x}} \, dx,x,i a \tan (c+d x)\right )}{2 d}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt [3]{a+i a x}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {i x}{4 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a} d}+\frac {3}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {3 \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 )}{4 d}+\frac {3 \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 )}{2 d}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a} d}+\frac {3 \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 )}{4 \sqrt [3]{2} \sqrt [3]{a} d}\\ &=-\frac {i x}{4 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a} d}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a} d}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac {3}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {3 \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 )}{\sqrt [3]{a} d}+\frac {3 \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 \sqrt [3]{2} \sqrt [3]{a} d}\\ &=-\frac {i x}{4 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a} d}-\frac {\sqrt {3} \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 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a} d}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a} d}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac {3}{2 d \sqrt [3]{a+i a \tan (c+d x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 1.29, size = 195, normalized size = 0.68 \begin {gather*} \frac {3 \left (2 \left (1+e^{2 i d x}\right ) \cos (c)+e^{2 i d x} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right ) (\cos (c)+i \sin (c))-4 e^{2 i d x} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right ) (\cos (c)+i \sin (c))+2 i \left (-1+e^{2 i d x}\right ) \sin (c)\right )}{4 d \left (\left (1+e^{2 i d x}\right ) \cos (c)+i \left (-1+e^{2 i d x}\right ) \sin (c)\right ) \sqrt [3]{a+i a \tan (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(2*(1 + E^((2*I)*d*x))*Cos[c] + E^((2*I)*d*x)*Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2
*I)*(c + d*x)))]*(Cos[c] + I*Sin[c]) - 4*E^((2*I)*d*x)*Hypergeometric2F1[2/3, 1, 5/3, (2*E^((2*I)*(c + d*x)))/
(1 + E^((2*I)*(c + d*x)))]*(Cos[c] + I*Sin[c]) + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c]))/(4*d*((1 + E^((2*I)*d*x))
*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c])*(a + I*a*Tan[c + d*x])^(1/3))

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

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 = 0.54, size = 249, normalized size = 0.87 \begin {gather*} -\frac {\frac {2 \, \sqrt {3} 2^{\frac {2}{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 {1}{3}}} - \frac {2^{\frac {2}{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 {1}{3}}} + \frac {2 \cdot 2^{\frac {2}{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 {1}{3}}} - \frac {8 \, \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 {1}{3}}} + \frac {4 \, \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 {1}{3}}} - \frac {8 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}} - \frac {12}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}}{8 \, d} \end {gather*}

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]

-1/8*(2*sqrt(3)*2^(2/3)*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^(1/3) - 2^(2/3)*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^(1/3) + 2*2^(2/3)*log(-2^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^(1/3))/a^(1/3) - 8*sqrt(3)*arctan(1/
3*sqrt(3)*(2*(I*a*tan(d*x + c) + a)^(1/3) + a^(1/3))/a^(1/3))/a^(1/3) + 4*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^(1/3) - 8*log((I*a*tan(d*x + c) + a)^(1/3) - a^(1/3))/a^(1/3)
- 12/(I*a*tan(d*x + c) + a)^(1/3))/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. 580 vs. \(2 (206) = 412\).
time = 0.96, size = 580, normalized size = 2.03 \begin {gather*} \frac {{\left (2 \, \left (\frac {1}{2}\right )^{\frac {1}{3}} a d \left (-\frac {1}{a d^{3}}\right )^{\frac {1}{3}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-2 \, \left (\frac {1}{2}\right )^{\frac {2}{3}} a d^{2} \left (-\frac {1}{a d^{3}}\right )^{\frac {2}{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 ) + 4 \, a d \left (\frac {1}{a d^{3}}\right )^{\frac {1}{3}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-a d^{2} \left (\frac {1}{a d^{3}}\right )^{\frac {2}{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 {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} a d + a d\right )} \left (-\frac {1}{a d^{3}}\right )^{\frac {1}{3}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-\left (\frac {1}{2}\right )^{\frac {2}{3}} {\left (i \, \sqrt {3} a d^{2} - a d^{2}\right )} \left (-\frac {1}{a d^{3}}\right )^{\frac {2}{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 {1}{2}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} a d + a d\right )} \left (-\frac {1}{a d^{3}}\right )^{\frac {1}{3}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-\left (\frac {1}{2}\right )^{\frac {2}{3}} {\left (-i \, \sqrt {3} a d^{2} - a d^{2}\right )} \left (-\frac {1}{a d^{3}}\right )^{\frac {2}{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 ) + 3 \cdot 2^{\frac {2}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {2}{3}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (\frac {4}{3} i \, d x + \frac {4}{3} i \, c\right )} - 2 \, {\left (-i \, \sqrt {3} a d + a d\right )} \left (\frac {1}{a d^{3}}\right )^{\frac {1}{3}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (\frac {1}{2} \, {\left (i \, \sqrt {3} a d^{2} + a d^{2}\right )} \left (\frac {1}{a d^{3}}\right )^{\frac {2}{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 ) - 2 \, {\left (i \, \sqrt {3} a d + a d\right )} \left (\frac {1}{a d^{3}}\right )^{\frac {1}{3}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (\frac {1}{2} \, {\left (-i \, \sqrt {3} a d^{2} + a d^{2}\right )} \left (\frac {1}{a d^{3}}\right )^{\frac {2}{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 )\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a d} \end {gather*}

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]

1/4*(2*(1/2)^(1/3)*a*d*(-1/(a*d^3))^(1/3)*e^(2*I*d*x + 2*I*c)*log(-2*(1/2)^(2/3)*a*d^2*(-1/(a*d^3))^(2/3) + 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)) + 4*a*d*(1/(a*d^3))^(1/3)*e^(2*I*d*x + 2*I*
c)*log(-a*d^2*(1/(a*d^3))^(2/3) + 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)
^(1/3)*(I*sqrt(3)*a*d + a*d)*(-1/(a*d^3))^(1/3)*e^(2*I*d*x + 2*I*c)*log(-(1/2)^(2/3)*(I*sqrt(3)*a*d^2 - a*d^2)
*(-1/(a*d^3))^(2/3) + 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)^(1/3)*(-I*s
qrt(3)*a*d + a*d)*(-1/(a*d^3))^(1/3)*e^(2*I*d*x + 2*I*c)*log(-(1/2)^(2/3)*(-I*sqrt(3)*a*d^2 - a*d^2)*(-1/(a*d^
3))^(2/3) + 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)) + 3*2^(2/3)*(a/(e^(2*I*d*x +
2*I*c) + 1))^(2/3)*(e^(2*I*d*x + 2*I*c) + 1)*e^(4/3*I*d*x + 4/3*I*c) - 2*(-I*sqrt(3)*a*d + a*d)*(1/(a*d^3))^(1
/3)*e^(2*I*d*x + 2*I*c)*log(1/2*(I*sqrt(3)*a*d^2 + a*d^2)*(1/(a*d^3))^(2/3) + 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)) - 2*(I*sqrt(3)*a*d + a*d)*(1/(a*d^3))^(1/3)*e^(2*I*d*x + 2*I*c)*log(1/2*(
-I*sqrt(3)*a*d^2 + a*d^2)*(1/(a*d^3))^(2/3) + 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)))*e^(-2*I*d*x - 2*I*c)/(a*d)

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

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Mupad [B]
time = 0.25, size = 559, normalized size = 1.95 \begin {gather*} \frac {3}{2\,d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}+\ln \left (\left (746496\,a^7\,d^9\,{\left (\frac {1}{a\,d^3}\right )}^{2/3}-528768\,a^6\,d^7\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\right )\,{\left (\frac {1}{a\,d^3}\right )}^{1/3}-217728\,a^6\,d^6\right )\,{\left (\frac {1}{a\,d^3}\right )}^{1/3}+\ln \left (\left (746496\,a^7\,d^9\,{\left (-\frac {1}{16\,a\,d^3}\right )}^{2/3}-528768\,a^6\,d^7\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\right )\,{\left (-\frac {1}{16\,a\,d^3}\right )}^{1/3}-217728\,a^6\,d^6\right )\,{\left (-\frac {1}{16\,a\,d^3}\right )}^{1/3}+\frac {\ln \left (217728\,a^6\,d^6+\frac {\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (528768\,a^6\,d^7\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}-186624\,a^7\,d^9\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (\frac {1}{a\,d^3}\right )}^{2/3}\right )\,{\left (\frac {1}{a\,d^3}\right )}^{1/3}}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {1}{a\,d^3}\right )}^{1/3}}{2}-\frac {\ln \left (217728\,a^6\,d^6-\frac {\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (528768\,a^6\,d^7\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}-186624\,a^7\,d^9\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (\frac {1}{a\,d^3}\right )}^{2/3}\right )\,{\left (\frac {1}{a\,d^3}\right )}^{1/3}}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {1}{a\,d^3}\right )}^{1/3}}{2}+\ln \left (217728\,a^6\,d^6+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (528768\,a^6\,d^7\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}-746496\,a^7\,d^9\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,{\left (-\frac {1}{16\,a\,d^3}\right )}^{2/3}\right )\,{\left (-\frac {1}{16\,a\,d^3}\right )}^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {1}{16\,a\,d^3}\right )}^{1/3}-\ln \left (217728\,a^6\,d^6-\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (528768\,a^6\,d^7\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}-746496\,a^7\,d^9\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,{\left (-\frac {1}{16\,a\,d^3}\right )}^{2/3}\right )\,{\left (-\frac {1}{16\,a\,d^3}\right )}^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {1}{16\,a\,d^3}\right )}^{1/3} \end {gather*}

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]

3/(2*d*(a + a*tan(c + d*x)*1i)^(1/3)) + log((746496*a^7*d^9*(1/(a*d^3))^(2/3) - 528768*a^6*d^7*(a + a*tan(c +
d*x)*1i)^(1/3))*(1/(a*d^3))^(1/3) - 217728*a^6*d^6)*(1/(a*d^3))^(1/3) + log((746496*a^7*d^9*(-1/(16*a*d^3))^(2
/3) - 528768*a^6*d^7*(a + a*tan(c + d*x)*1i)^(1/3))*(-1/(16*a*d^3))^(1/3) - 217728*a^6*d^6)*(-1/(16*a*d^3))^(1
/3) + (log(217728*a^6*d^6 + ((3^(1/2)*1i - 1)*(528768*a^6*d^7*(a + a*tan(c + d*x)*1i)^(1/3) - 186624*a^7*d^9*(
3^(1/2)*1i - 1)^2*(1/(a*d^3))^(2/3))*(1/(a*d^3))^(1/3))/2)*(3^(1/2)*1i - 1)*(1/(a*d^3))^(1/3))/2 - (log(217728
*a^6*d^6 - ((3^(1/2)*1i + 1)*(528768*a^6*d^7*(a + a*tan(c + d*x)*1i)^(1/3) - 186624*a^7*d^9*(3^(1/2)*1i + 1)^2
*(1/(a*d^3))^(2/3))*(1/(a*d^3))^(1/3))/2)*(3^(1/2)*1i + 1)*(1/(a*d^3))^(1/3))/2 + log(217728*a^6*d^6 + ((3^(1/
2)*1i)/2 - 1/2)*(528768*a^6*d^7*(a + a*tan(c + d*x)*1i)^(1/3) - 746496*a^7*d^9*((3^(1/2)*1i)/2 - 1/2)^2*(-1/(1
6*a*d^3))^(2/3))*(-1/(16*a*d^3))^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(-1/(16*a*d^3))^(1/3) - log(217728*a^6*d^6 - ((
3^(1/2)*1i)/2 + 1/2)*(528768*a^6*d^7*(a + a*tan(c + d*x)*1i)^(1/3) - 746496*a^7*d^9*((3^(1/2)*1i)/2 + 1/2)^2*(
-1/(16*a*d^3))^(2/3))*(-1/(16*a*d^3))^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(-1/(16*a*d^3))^(1/3)

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