3.140 \(\int \frac {\tan (a+i \log (x))}{x^2} \, dx\)

Optimal. Leaf size=29 \[ 2 i e^{-i a} \tan ^{-1}\left (e^{-i a} x\right )+\frac {i}{x} \]

[Out]

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

Verification is Not applicable to the result.

[In]

Int[Tan[a + I*Log[x]]/x^2,x]

[Out]

Defer[Int][Tan[a + I*Log[x]]/x^2, x]

Rubi steps

\begin {align*} \int \frac {\tan (a+i \log (x))}{x^2} \, dx &=\int \frac {\tan (a+i \log (x))}{x^2} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 44, normalized size = 1.52 \[ 2 i \cos (a) \tan ^{-1}(x \cos (a)-i x \sin (a))+2 \sin (a) \tan ^{-1}(x \cos (a)-i x \sin (a))+\frac {i}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[a + I*Log[x]]/x^2,x]

[Out]

I/x + (2*I)*ArcTan[x*Cos[a] - I*x*Sin[a]]*Cos[a] + 2*ArcTan[x*Cos[a] - I*x*Sin[a]]*Sin[a]

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fricas [B]  time = 0.51, size = 39, normalized size = 1.34 \[ -\frac {{\left (x \log \left (x + i \, e^{\left (i \, a\right )}\right ) - x \log \left (x - i \, e^{\left (i \, a\right )}\right ) - i \, e^{\left (i \, a\right )}\right )} e^{\left (-i \, a\right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+I*log(x))/x^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.44, size = 28, normalized size = 0.97 \[ -\frac {2 \, \arctan \left (\frac {i \, x}{\sqrt {-e^{\left (2 i \, a\right )}}}\right )}{\sqrt {-e^{\left (2 i \, a\right )}}} + \frac {i}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+I*log(x))/x^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 24, normalized size = 0.83 \[ \frac {i}{x}+2 i \arctan \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{-i a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(a+I*ln(x))/x^2,x)

[Out]

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

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maxima [B]  time = 0.46, size = 127, normalized size = 4.38 \[ \frac {2 \, x {\left (-i \, \cos \relax (a) - \sin \relax (a)\right )} \arctan \left (\frac {2 \, x \cos \relax (a)}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}, \frac {x^{2} - \cos \relax (a)^{2} - \sin \relax (a)^{2}}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}\right ) + x {\left (\cos \relax (a) - i \, \sin \relax (a)\right )} \log \left (\frac {x^{2} + \cos \relax (a)^{2} + 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}\right ) + 2 i}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+I*log(x))/x^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.27, size = 27, normalized size = 0.93 \[ \frac {\mathrm {atan}\left (\frac {x}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )\,2{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}+\frac {1{}\mathrm {i}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(a + log(x)*1i)/x^2,x)

[Out]

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

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sympy [A]  time = 0.23, size = 27, normalized size = 0.93 \[ \left (\log {\left (x - i e^{i a} \right )} - \log {\left (x + i e^{i a} \right )}\right ) e^{- i a} + \frac {i}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+I*ln(x))/x**2,x)

[Out]

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

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