3.141 \(\int \frac{\tan (a+i \log (x))}{x^3} \, dx\)

Optimal. Leaf size=35 \[ \frac{i}{2 x^2}-i e^{-2 i a} \log \left (1+\frac{e^{2 i a}}{x^2}\right ) \]

[Out]

(I/2)/x^2 - (I*Log[1 + E^((2*I)*a)/x^2])/E^((2*I)*a)

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Rubi [F]  time = 0.0267757, 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 = {} \[ \int \frac{\tan (a+i \log (x))}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tan[a + I*Log[x]]/x^3,x]

[Out]

Defer[Int][Tan[a + I*Log[x]]/x^3, x]

Rubi steps

\begin{align*} \int \frac{\tan (a+i \log (x))}{x^3} \, dx &=\int \frac{\tan (a+i \log (x))}{x^3} \, dx\\ \end{align*}

Mathematica [B]  time = 0.0318199, size = 132, normalized size = 3.77 \[ -\frac{1}{2} i \cos (2 a) \log \left (2 x^2 \cos (2 a)+x^4+1\right )-\frac{1}{2} \sin (2 a) \log \left (2 x^2 \cos (2 a)+x^4+1\right )+\cos (2 a) \left (-\tan ^{-1}\left (\frac{\left (x^2+1\right ) \cos (a)}{\sin (a)-x^2 \sin (a)}\right )\right )+i \sin (2 a) \tan ^{-1}\left (\frac{\left (x^2+1\right ) \cos (a)}{\sin (a)-x^2 \sin (a)}\right )+2 \sin (2 a) \log (x)+2 i \cos (2 a) \log (x)+\frac{i}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[a + I*Log[x]]/x^3,x]

[Out]

(I/2)/x^2 - ArcTan[((1 + x^2)*Cos[a])/(Sin[a] - x^2*Sin[a])]*Cos[2*a] + (2*I)*Cos[2*a]*Log[x] - (I/2)*Cos[2*a]
*Log[1 + x^4 + 2*x^2*Cos[2*a]] + I*ArcTan[((1 + x^2)*Cos[a])/(Sin[a] - x^2*Sin[a])]*Sin[2*a] + 2*Log[x]*Sin[2*
a] - (Log[1 + x^4 + 2*x^2*Cos[2*a]]*Sin[2*a])/2

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Maple [A]  time = 0.057, size = 43, normalized size = 1.2 \begin{align*}{\frac{{\frac{i}{2}}}{{x}^{2}}}-i \left ({\frac{\ln \left ( \left ({{\rm e}^{ia}} \right ) ^{2}+{x}^{2} \right ) }{ \left ({{\rm e}^{ia}} \right ) ^{2}}}-2\,{\frac{\ln \left ( x \right ) }{ \left ({{\rm e}^{ia}} \right ) ^{2}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(a+I*ln(x))/x^3,x)

[Out]

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

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Maxima [B]  time = 1.08791, size = 130, normalized size = 3.71 \begin{align*} -\frac{x^{2}{\left (i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \log \left (x^{4} + 2 \, x^{2} \cos \left (2 \, a\right ) + \cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right ) -{\left ({\left (2 \, \cos \left (2 \, a\right ) - 2 i \, \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (2 \, a\right ), x^{2} + \cos \left (2 \, a\right )\right ) + 4 \,{\left (i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \log \left (x\right )\right )} x^{2} - i}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+I*log(x))/x^3,x, algorithm="maxima")

[Out]

-1/2*(x^2*(I*cos(2*a) + sin(2*a))*log(x^4 + 2*x^2*cos(2*a) + cos(2*a)^2 + sin(2*a)^2) - ((2*cos(2*a) - 2*I*sin
(2*a))*arctan2(sin(2*a), x^2 + cos(2*a)) + 4*(I*cos(2*a) + sin(2*a))*log(x))*x^2 - I)/x^2

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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^{3} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+I*log(x))/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.696203, size = 39, normalized size = 1.11 \begin{align*} 2 i e^{- 2 i a} \log{\left (x \right )} - i e^{- 2 i a} \log{\left (x^{2} + e^{2 i a} \right )} + \frac{i}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+I*ln(x))/x**3,x)

[Out]

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

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Giac [A]  time = 1.13705, size = 45, normalized size = 1.29 \begin{align*} -i \, e^{\left (-2 i \, a\right )} \log \left (-i \, x^{2} - i \, e^{\left (2 i \, a\right )}\right ) + 2 i \, e^{\left (-2 i \, a\right )} \log \left (x\right ) + \frac{i}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+I*log(x))/x^3,x, algorithm="giac")

[Out]

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