3.193 \(\int \frac {\cot (a+i \log (x))}{x^4} \, dx\)

Optimal. Leaf size=45 \[ -\frac {2 i e^{-2 i a}}{x}+2 i e^{-3 i a} \tanh ^{-1}\left (e^{-i a} x\right )-\frac {i}{3 x^3} \]

[Out]

-1/3*I/x^3-2*I/exp(2*I*a)/x+2*I*arctanh(x/exp(I*a))/exp(3*I*a)

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Rubi [F]  time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cot (a+i \log (x))}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[a + I*Log[x]]/x^4,x]

[Out]

Defer[Int][Cot[a + I*Log[x]]/x^4, x]

Rubi steps

\begin {align*} \int \frac {\cot (a+i \log (x))}{x^4} \, dx &=\int \frac {\cot (a+i \log (x))}{x^4} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 70, normalized size = 1.56 \[ -\frac {2 \sin (2 a)}{x}-\frac {2 i \cos (2 a)}{x}+2 i \cos (3 a) \tanh ^{-1}(x \cos (a)-i x \sin (a))+2 \sin (3 a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-\frac {i}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + I*Log[x]]/x^4,x]

[Out]

(-1/3*I)/x^3 - ((2*I)*Cos[2*a])/x + (2*I)*ArcTanh[x*Cos[a] - I*x*Sin[a]]*Cos[3*a] - (2*Sin[2*a])/x + 2*ArcTanh
[x*Cos[a] - I*x*Sin[a]]*Sin[3*a]

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fricas [A]  time = 0.59, size = 55, normalized size = 1.22 \[ \frac {{\left (3 i \, x^{3} e^{\left (-i \, a\right )} \log \left (x + e^{\left (i \, a\right )}\right ) - 3 i \, x^{3} e^{\left (-i \, a\right )} \log \left (x - e^{\left (i \, a\right )}\right ) - 6 i \, x^{2} - i \, e^{\left (2 i \, a\right )}\right )} e^{\left (-2 i \, a\right )}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x))/x^4,x, algorithm="fricas")

[Out]

1/3*(3*I*x^3*e^(-I*a)*log(x + e^(I*a)) - 3*I*x^3*e^(-I*a)*log(x - e^(I*a)) - 6*I*x^2 - I*e^(2*I*a))*e^(-2*I*a)
/x^3

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giac [A]  time = 0.26, size = 49, normalized size = 1.09 \[ i \, e^{\left (-3 i \, a\right )} \log \left (i \, x + i \, e^{\left (i \, a\right )}\right ) - i \, e^{\left (-3 i \, a\right )} \log \left (-i \, x + i \, e^{\left (i \, a\right )}\right ) - \frac {2 i \, e^{\left (-2 i \, a\right )}}{x} - \frac {i}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x))/x^4,x, algorithm="giac")

[Out]

I*e^(-3*I*a)*log(I*x + I*e^(I*a)) - I*e^(-3*I*a)*log(-I*x + I*e^(I*a)) - 2*I*e^(-2*I*a)/x - 1/3*I/x^3

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maple [A]  time = 0.06, size = 35, normalized size = 0.78 \[ -\frac {i}{3 x^{3}}-\frac {2 i {\mathrm e}^{-2 i a}}{x}+2 i \arctanh \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{-3 i a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a+I*ln(x))/x^4,x)

[Out]

-1/3*I/x^3-2*I*exp(-2*I*a)/x+2*I*arctanh(x*exp(-I*a))*exp(-3*I*a)

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maxima [B]  time = 0.34, size = 142, normalized size = 3.16 \[ -\frac {3 \, x^{3} {\left (-i \, \cos \left (3 \, a\right ) - \sin \left (3 \, a\right )\right )} \log \left (x^{2} + 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) + 3 \, x^{3} {\left (i \, \cos \left (3 \, a\right ) + \sin \left (3 \, a\right )\right )} \log \left (x^{2} - 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) + {\left ({\left (6 \, \cos \left (3 \, a\right ) - 6 i \, \sin \left (3 \, a\right )\right )} \arctan \left (\sin \relax (a), x + \cos \relax (a)\right ) + {\left (6 \, \cos \left (3 \, a\right ) - 6 i \, \sin \left (3 \, a\right )\right )} \arctan \left (\sin \relax (a), x - \cos \relax (a)\right )\right )} x^{3} + 12 \, x^{2} {\left (i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} + 2 i}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x))/x^4,x, algorithm="maxima")

[Out]

-1/6*(3*x^3*(-I*cos(3*a) - sin(3*a))*log(x^2 + 2*x*cos(a) + cos(a)^2 + sin(a)^2) + 3*x^3*(I*cos(3*a) + sin(3*a
))*log(x^2 - 2*x*cos(a) + cos(a)^2 + sin(a)^2) + ((6*cos(3*a) - 6*I*sin(3*a))*arctan2(sin(a), x + cos(a)) + (6
*cos(3*a) - 6*I*sin(3*a))*arctan2(sin(a), x - cos(a)))*x^3 + 12*x^2*(I*cos(2*a) + sin(2*a)) + 2*I)/x^3

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mupad [B]  time = 2.21, size = 44, normalized size = 0.98 \[ \frac {\mathrm {atan}\left (\frac {x}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )\,2{}\mathrm {i}}{{\left (-{\mathrm {e}}^{a\,2{}\mathrm {i}}\right )}^{3/2}}-\frac {2{}\mathrm {i}\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,x^2+\frac {1}{3}{}\mathrm {i}}{x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a + log(x)*1i)/x^4,x)

[Out]

(atan(x/(-exp(a*2i))^(1/2))*2i)/(-exp(a*2i))^(3/2) - (x^2*exp(-a*2i)*2i + 1i/3)/x^3

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sympy [A]  time = 0.30, size = 54, normalized size = 1.20 \[ - \left (i \log {\left (x - e^{i a} \right )} - i \log {\left (x + e^{i a} \right )}\right ) e^{- 3 i a} - \frac {\left (6 i x^{2} + i e^{2 i a}\right ) e^{- 2 i a}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*ln(x))/x**4,x)

[Out]

-(I*log(x - exp(I*a)) - I*log(x + exp(I*a)))*exp(-3*I*a) - (6*I*x**2 + I*exp(2*I*a))*exp(-2*I*a)/(3*x**3)

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