3.157 \(\int \frac{\coth (a+2 \log (x))}{x^3} \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{2 x^2}-e^a \tanh ^{-1}\left (e^a x^2\right ) \]

[Out]

1/(2*x^2) - E^a*ArcTanh[E^a*x^2]

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Rubi [F]  time = 0.0213417, 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 = {} \[ \int \frac{\coth (a+2 \log (x))}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Coth[a + 2*Log[x]]/x^3,x]

[Out]

Defer[Int][Coth[a + 2*Log[x]]/x^3, x]

Rubi steps

\begin{align*} \int \frac{\coth (a+2 \log (x))}{x^3} \, dx &=\int \frac{\coth (a+2 \log (x))}{x^3} \, dx\\ \end{align*}

Mathematica [A]  time = 0.156472, size = 27, normalized size = 1.29 \[ \frac{1}{2 x^2}-(\sinh (a)+\cosh (a)) \tanh ^{-1}\left (\frac{\cosh (a)-\sinh (a)}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[a + 2*Log[x]]/x^3,x]

[Out]

1/(2*x^2) - ArcTanh[(Cosh[a] - Sinh[a])/x^2]*(Cosh[a] + Sinh[a])

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Maple [A]  time = 0.026, size = 35, normalized size = 1.7 \begin{align*}{\frac{1}{2\,{x}^{2}}}-{\frac{{{\rm e}^{a}}\ln \left ( -{{\rm e}^{a}}{x}^{2}-1 \right ) }{2}}+{\frac{{{\rm e}^{a}}\ln \left ( -{{\rm e}^{a}}{x}^{2}+1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a+2*ln(x))/x^3,x)

[Out]

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

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Maxima [A]  time = 1.02956, size = 41, normalized size = 1.95 \begin{align*} -\frac{1}{2} \, e^{a} \log \left (\frac{1}{x^{2}} + e^{a}\right ) + \frac{1}{2} \, e^{a} \log \left (\frac{1}{x^{2}} - e^{a}\right ) + \frac{1}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+2*log(x))/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.5844, size = 92, normalized size = 4.38 \begin{align*} -\frac{x^{2} e^{a} \log \left (x^{2} e^{a} + 1\right ) - x^{2} e^{a} \log \left (x^{2} e^{a} - 1\right ) - 1}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+2*log(x))/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth{\left (a + 2 \log{\left (x \right )} \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+2*ln(x))/x**3,x)

[Out]

Integral(coth(a + 2*log(x))/x**3, x)

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Giac [A]  time = 1.10794, size = 45, normalized size = 2.14 \begin{align*} -\frac{1}{2} \, e^{a} \log \left (x^{2} e^{a} + 1\right ) + \frac{1}{2} \, e^{a} \log \left ({\left | x^{2} e^{a} - 1 \right |}\right ) + \frac{1}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+2*log(x))/x^3,x, algorithm="giac")

[Out]

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