3.190 \(\int \frac{\cot (a+i \log (x))}{x} \, dx\)

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

[Out]

(-I)*Log[Sin[a + I*Log[x]]]

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Rubi [A]  time = 0.012911, 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} \[ -i \log (\sin (a+i \log (x))) \]

Antiderivative was successfully verified.

[In]

Int[Cot[a + I*Log[x]]/x,x]

[Out]

(-I)*Log[Sin[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{\cot (a+i \log (x))}{x} \, dx &=\operatorname{Subst}(\int \cot (a+i x) \, dx,x,\log (x))\\ &=-i \log (\sin (a+i \log (x)))\\ \end{align*}

Mathematica [A]  time = 0.0254794, size = 25, normalized size = 1.79 \[ -i (\log (\tan (a+i \log (x)))+\log (\cos (a+i \log (x)))) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + I*Log[x]]/x,x]

[Out]

(-I)*(Log[Cos[a + I*Log[x]]] + Log[Tan[a + I*Log[x]]])

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Maple [A]  time = 0.017, size = 17, normalized size = 1.2 \begin{align*}{\frac{i}{2}}\ln \left ( \left ( \cot \left ( a+i\ln \left ( x \right ) \right ) \right ) ^{2}+1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.12153, size = 14, normalized size = 1. \begin{align*} -i \, \log \left (\sin \left (a + i \, \log \left (x\right )\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*log(sin(a + I*log(x)))

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Fricas [A]  time = 0.482536, size = 61, normalized size = 4.36 \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(cot(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(cot(a+I*ln(x))/x,x)

[Out]

Exception raised: PolynomialError

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Giac [B]  time = 1.22991, size = 78, normalized size = 5.57 \begin{align*} -\frac{1}{2} i \, \log \left (-\frac{1}{8} \,{\left (\frac{{\left (x^{2} + 1\right )}^{2}}{x^{2}} - \frac{{\left (x^{2} - 1\right )}^{2}}{x^{2}}\right )} \cos \left (2 \, a\right ) + \frac{{\left (x^{2} + 1\right )}^{2}}{8 \, x^{2}} + \frac{{\left (x^{2} - 1\right )}^{2}}{8 \, x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*log(-1/8*((x^2 + 1)^2/x^2 - (x^2 - 1)^2/x^2)*cos(2*a) + 1/8*(x^2 + 1)^2/x^2 + 1/8*(x^2 - 1)^2/x^2)