3.158 \(\int x^3 \coth ^2(a+2 \log (x)) \, dx\)

Optimal. Leaf size=47 \[ \frac{e^{-2 a}}{1-e^{2 a} x^4}+e^{-2 a} \log \left (1-e^{2 a} x^4\right )+\frac{x^4}{4} \]

[Out]

x^4/4 + 1/(E^(2*a)*(1 - E^(2*a)*x^4)) + Log[1 - E^(2*a)*x^4]/E^(2*a)

________________________________________________________________________________________

Rubi [F]  time = 0.0662787, 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 = {} \[ \int x^3 \coth ^2(a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^3*Coth[a + 2*Log[x]]^2,x]

[Out]

Defer[Int][x^3*Coth[a + 2*Log[x]]^2, x]

Rubi steps

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

Mathematica [A]  time = 0.102571, size = 86, normalized size = 1.83 \[ \frac{\sinh (3 a)-\cosh (3 a)}{\left (x^4+1\right ) \sinh (a)+\left (x^4-1\right ) \cosh (a)}+\cosh (2 a) \log \left (\left (x^4+1\right ) \sinh (a)+\left (x^4-1\right ) \cosh (a)\right )-\sinh (2 a) \log \left (\left (x^4+1\right ) \sinh (a)+\left (x^4-1\right ) \cosh (a)\right )+\frac{x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Coth[a + 2*Log[x]]^2,x]

[Out]

x^4/4 + Cosh[2*a]*Log[(-1 + x^4)*Cosh[a] + (1 + x^4)*Sinh[a]] - Log[(-1 + x^4)*Cosh[a] + (1 + x^4)*Sinh[a]]*Si
nh[2*a] + (-Cosh[3*a] + Sinh[3*a])/((-1 + x^4)*Cosh[a] + (1 + x^4)*Sinh[a])

________________________________________________________________________________________

Maple [A]  time = 0.018, size = 41, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}}{4}}-{\frac{{{\rm e}^{-2\,a}}}{{{\rm e}^{2\,a}}{x}^{4}-1}}+{{\rm e}^{-2\,a}}\ln \left ({{\rm e}^{2\,a}}{x}^{4}-1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4-exp(-2*a)/(exp(2*a)*x^4-1)+exp(-2*a)*ln(exp(2*a)*x^4-1)

________________________________________________________________________________________

Maxima [A]  time = 1.05107, size = 72, normalized size = 1.53 \begin{align*} \frac{1}{4} \, x^{4} + e^{\left (-2 \, a\right )} \log \left (x^{2} e^{a} + 1\right ) + e^{\left (-2 \, a\right )} \log \left (x^{2} e^{a} - 1\right ) - \frac{1}{x^{4} e^{\left (4 \, a\right )} - e^{\left (2 \, a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4 + e^(-2*a)*log(x^2*e^a + 1) + e^(-2*a)*log(x^2*e^a - 1) - 1/(x^4*e^(4*a) - e^(2*a))

________________________________________________________________________________________

Fricas [A]  time = 2.46811, size = 140, normalized size = 2.98 \begin{align*} \frac{x^{8} e^{\left (4 \, a\right )} - x^{4} e^{\left (2 \, a\right )} + 4 \,{\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \log \left (x^{4} e^{\left (2 \, a\right )} - 1\right ) - 4}{4 \,{\left (x^{4} e^{\left (4 \, a\right )} - e^{\left (2 \, a\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(x^8*e^(4*a) - x^4*e^(2*a) + 4*(x^4*e^(2*a) - 1)*log(x^4*e^(2*a) - 1) - 4)/(x^4*e^(4*a) - e^(2*a))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*coth(a + 2*log(x))**2, x)

________________________________________________________________________________________

Giac [A]  time = 1.12165, size = 54, normalized size = 1.15 \begin{align*} \frac{1}{4} \, x^{4} - \frac{x^{4}}{x^{4} e^{\left (2 \, a\right )} - 1} + e^{\left (-2 \, a\right )} \log \left ({\left | x^{4} e^{\left (2 \, a\right )} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4 - x^4/(x^4*e^(2*a) - 1) + e^(-2*a)*log(abs(x^4*e^(2*a) - 1))