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

Optimal. Leaf size=30 \[ \frac{1}{2} e^{-2 a} \log \left (1-e^{2 a} x^4\right )+\frac{x^4}{4} \]

[Out]

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

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Rubi [F]  time = 0.0285749, 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 = {} \[ \int x^3 \coth (a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^3*Coth[a + 2*Log[x]],x]

[Out]

Defer[Int][x^3*Coth[a + 2*Log[x]], x]

Rubi steps

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

Mathematica [B]  time = 0.0241018, size = 64, normalized size = 2.13 \[ \frac{1}{2} \cosh (2 a) \log \left (x^4 \sinh (a)+x^4 \cosh (a)+\sinh (a)-\cosh (a)\right )-\frac{1}{2} \sinh (2 a) \log \left (x^4 \sinh (a)+x^4 \cosh (a)+\sinh (a)-\cosh (a)\right )+\frac{x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Coth[a + 2*Log[x]],x]

[Out]

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

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Maple [A]  time = 0.026, size = 24, normalized size = 0.8 \begin{align*}{\frac{{x}^{4}}{4}}+{\frac{{{\rm e}^{-2\,a}}\ln \left ({{\rm e}^{2\,a}}{x}^{4}-1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.21452, size = 49, normalized size = 1.63 \begin{align*} \frac{1}{4} \, x^{4} + \frac{1}{2} \, e^{\left (-2 \, a\right )} \log \left (x^{2} e^{a} + 1\right ) + \frac{1}{2} \, e^{\left (-2 \, 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^3*coth(a+2*log(x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.53483, size = 72, normalized size = 2.4 \begin{align*} \frac{1}{4} \,{\left (x^{4} e^{\left (2 \, a\right )} + 2 \, \log \left (x^{4} e^{\left (2 \, a\right )} - 1\right )\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)),x, algorithm="fricas")

[Out]

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

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

[Out]

Integral(x**3*coth(a + 2*log(x)), x)

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Giac [A]  time = 1.14286, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{4} \, x^{4} + \frac{1}{2} \, 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)),x, algorithm="giac")

[Out]

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