3.153 \(\int x \coth (a+2 \log (x)) \, dx\)

Optimal. Leaf size=23 \[ \frac{x^2}{2}-e^{-a} \tanh ^{-1}\left (e^a x^2\right ) \]

[Out]

x^2/2 - ArcTanh[E^a*x^2]/E^a

________________________________________________________________________________________

Rubi [F]  time = 0.0156127, 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 = {} \[ \int x \coth (a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x*Coth[a + 2*Log[x]],x]

[Out]

Defer[Int][x*Coth[a + 2*Log[x]], x]

Rubi steps

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

Mathematica [A]  time = 0.181986, size = 26, normalized size = 1.13 \[ (\sinh (a)-\cosh (a)) \tanh ^{-1}\left (x^2 (\sinh (a)+\cosh (a))\right )+\frac{x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Coth[a + 2*Log[x]],x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.024, size = 37, normalized size = 1.6 \begin{align*}{\frac{{x}^{2}}{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(x*coth(a+2*ln(x)),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)

________________________________________________________________________________________

Maxima [A]  time = 1.04383, size = 49, normalized size = 2.13 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} + 1\right ) + \frac{1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.59376, size = 81, normalized size = 3.52 \begin{align*} \frac{1}{2} \,{\left (x^{2} e^{a} - \log \left (x^{2} e^{a} + 1\right ) + \log \left (x^{2} e^{a} - 1\right )\right )} e^{\left (-a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*coth(a + 2*log(x)), x)

________________________________________________________________________________________

Giac [A]  time = 1.1168, size = 50, normalized size = 2.17 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} + 1\right ) + \frac{1}{2} \, e^{\left (-a\right )} \log \left ({\left | x^{2} e^{a} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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