3.139 \(\int \frac{\tan (a+i \log (x))}{x} \, dx\)

Optimal. Leaf size=14 \[ i \log (\cos (a+i \log (x))) \]

[Out]

I*Log[Cos[a + I*Log[x]]]

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Rubi [A]  time = 0.0127726, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3475} \[ i \log (\cos (a+i \log (x))) \]

Antiderivative was successfully verified.

[In]

Int[Tan[a + I*Log[x]]/x,x]

[Out]

I*Log[Cos[a + I*Log[x]]]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.021204, size = 14, normalized size = 1. \[ i \log (\cos (a+i \log (x))) \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[a + I*Log[x]]/x,x]

[Out]

I*Log[Cos[a + I*Log[x]]]

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Maple [A]  time = 0.013, size = 17, normalized size = 1.2 \begin{align*} -{\frac{i}{2}}\ln \left ( 1+ \left ( \tan \left ( a+i\ln \left ( x \right ) \right ) \right ) ^{2} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.997935, size = 14, normalized size = 1. \begin{align*} -i \, \log \left (\sec \left (a + i \, \log \left (x\right )\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*log(sec(a + I*log(x)))

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Fricas [A]  time = 0.485468, size = 59, normalized size = 4.21 \begin{align*} i \, \log \left (x\right ) + i \, \log \left (e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: PolynomialError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: PolynomialError

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Giac [A]  time = 1.14129, size = 14, normalized size = 1. \begin{align*} i \, \log \left (\cos \left (a + i \, \log \left (x\right )\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*log(cos(a + I*log(x)))