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

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

[Out]

(I/3)/x^3 - (2*I)/(E^((2*I)*a)*x) - ((2*I)*ArcTan[x/E^(I*a)])/E^((3*I)*a)

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Rubi [F]  time = 0.0270119, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\tan (a+i \log (x))}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tan[a + I*Log[x]]/x^4,x]

[Out]

Defer[Int][Tan[a + I*Log[x]]/x^4, x]

Rubi steps

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

Mathematica [A]  time = 0.0249357, size = 70, normalized size = 1.56 \[ -\frac{2 \sin (2 a)}{x}-\frac{2 i \cos (2 a)}{x}-2 i \cos (3 a) \tan ^{-1}(x \cos (a)-i x \sin (a))-2 \sin (3 a) \tan ^{-1}(x \cos (a)-i x \sin (a))+\frac{i}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[a + I*Log[x]]/x^4,x]

[Out]

(I/3)/x^3 - ((2*I)*Cos[2*a])/x - (2*I)*ArcTan[x*Cos[a] - I*x*Sin[a]]*Cos[3*a] - (2*Sin[2*a])/x - 2*ArcTan[x*Co
s[a] - I*x*Sin[a]]*Sin[3*a]

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \begin{align*} \int{\frac{\tan \left ( a+i\ln \left ( x \right ) \right ) }{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.67508, size = 212, normalized size = 4.71 \begin{align*} -\frac{6 \, x^{3}{\left (-i \, \cos \left (3 \, a\right ) - \sin \left (3 \, a\right )\right )} \arctan \left (\frac{2 \, x \cos \left (a\right )}{x^{2} + \cos \left (a\right )^{2} - 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}, \frac{x^{2} - \cos \left (a\right )^{2} - \sin \left (a\right )^{2}}{x^{2} + \cos \left (a\right )^{2} - 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}\right ) + x^{3}{\left (3 \, \cos \left (3 \, a\right ) - 3 i \, \sin \left (3 \, a\right )\right )} \log \left (\frac{x^{2} + \cos \left (a\right )^{2} + 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}{x^{2} + \cos \left (a\right )^{2} - 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}\right ) + 12 \, x^{2}{\left (i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} - 2 i}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{-i \, e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i}{x^{4} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-I*e^(2*I*a - 2*log(x)) + I)/(x^4*e^(2*I*a - 2*log(x)) + x^4), x)

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Sympy [A]  time = 1.25665, size = 53, normalized size = 1.18 \begin{align*} \left (- \log{\left (x - i e^{i a} \right )} + \log{\left (x + i 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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-log(x - I*exp(I*a)) + log(x + I*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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Giac [A]  time = 1.14238, size = 38, normalized size = 0.84 \begin{align*} -2 i \, \arctan \left (x e^{\left (-i \, a\right )}\right ) e^{\left (-3 i \, a\right )} - \frac{2 i \, e^{\left (-2 i \, a\right )}}{x} + \frac{i}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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