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]

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

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Rubi [F]  time = 0.07, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.11, 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])

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fricas [A]  time = 0.39, size = 61, normalized size = 1.30 \[ \frac {x^{8} e^{\left (4 \, a\right )} - x^{4} e^{\left (2 \, a\right )} + 4 \, {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \log \left (x^{4} e^{\left (2 \, a\right )} - 1\right ) - 4}{4 \, {\left (x^{4} e^{\left (4 \, a\right )} - e^{\left (2 \, a\right )}\right )}} \]

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))

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giac [A]  time = 0.11, size = 40, normalized size = 0.85 \[ \frac {1}{4} \, x^{4} - \frac {x^{4}}{x^{4} e^{\left (2 \, a\right )} - 1} + e^{\left (-2 \, a\right )} \log \left ({\left | x^{4} e^{\left (2 \, a\right )} - 1 \right |}\right ) \]

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))

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maple [A]  time = 0.10, size = 41, normalized size = 0.87 \[ \frac {x^{4}}{4}-\frac {{\mathrm e}^{-2 a}}{-1+{\mathrm e}^{2 a} x^{4}}+{\mathrm e}^{-2 a} \ln \left (-1+{\mathrm e}^{2 a} x^{4}\right ) \]

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

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maxima [A]  time = 0.33, size = 53, normalized size = 1.13 \[ \frac {1}{4} \, x^{4} + e^{\left (-2 \, a\right )} \log \left (x^{2} e^{a} + 1\right ) + e^{\left (-2 \, a\right )} \log \left (x^{2} e^{a} - 1\right ) - \frac {1}{x^{4} e^{\left (4 \, a\right )} - e^{\left (2 \, a\right )}} \]

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))

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mupad [B]  time = 1.25, size = 40, normalized size = 0.85 \[ \ln \left (x^4-{\mathrm {e}}^{-2\,a}\right )\,{\mathrm {e}}^{-2\,a}-\frac {{\mathrm {e}}^{-2\,a}}{x^4\,{\mathrm {e}}^{2\,a}-1}+\frac {x^4}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*coth(a + 2*log(x))^2,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \coth ^{2}{\left (a + 2 \log {\relax (x )} \right )}\, dx \]

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)

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