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]

1/3*I/x^3-2*I/exp(2*I*a)/x-2*I*arctan(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 = {} \[ \int \frac {\tan (a+i \log (x))}{x^4} \, dx \]

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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.48, size = 53, normalized size = 1.18 \[ \frac {{\left (3 \, x^{3} \log \left (x + i \, e^{\left (i \, a\right )}\right ) - 3 \, x^{3} \log \left (x - i \, e^{\left (i \, a\right )}\right ) - 6 i \, x^{2} e^{\left (i \, a\right )} + i \, e^{\left (3 i \, a\right )}\right )} e^{\left (-3 i \, a\right )}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.29, size = 28, normalized size = 0.62 \[ -2 i \, \arctan \left (x e^{\left (-i \, a\right )}\right ) e^{\left (-3 i \, a\right )} - \frac {2 i \, e^{\left (-2 i \, a\right )}}{x} + \frac {i}{3 \, x^{3}} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.46, size = 157, normalized size = 3.49 \[ -\frac {6 \, x^{3} {\left (-i \, \cos \left (3 \, a\right ) - \sin \left (3 \, a\right )\right )} \arctan \left (\frac {2 \, x \cos \relax (a)}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}, \frac {x^{2} - \cos \relax (a)^{2} - \sin \relax (a)^{2}}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}\right ) + x^{3} {\left (3 \, \cos \left (3 \, a\right ) - 3 i \, \sin \left (3 \, a\right )\right )} \log \left (\frac {x^{2} + \cos \relax (a)^{2} + 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}\right ) + 12 \, x^{2} {\left (i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} - 2 i}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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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mupad [B]  time = 2.30, size = 40, normalized size = 0.89 \[ -\frac {\mathrm {atan}\left (\frac {x}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )\,2{}\mathrm {i}}{{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\right )}^{3/2}}-\frac {x^2\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,2{}\mathrm {i}-\frac {1}{3}{}\mathrm {i}}{x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(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 = 0.29, size = 53, normalized size = 1.18 \[ \left (- \log {\left (x - i e^{i a} \right )} + \log {\left (x + i e^{i a} \right )}\right ) e^{- 3 i a} + \frac {\left (- 6 i x^{2} + i e^{2 i a}\right ) e^{- 2 i a}}{3 x^{3}} \]

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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