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]

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

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Rubi [F]  time = 0.03, 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 = {} \[ \int x^3 \coth (a+2 \log (x)) \, dx \]

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*}

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Mathematica [B]  time = 0.03, 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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fricas [A]  time = 0.39, size = 28, normalized size = 0.93 \[ \frac {1}{4} \, {\left (x^{4} e^{\left (2 \, a\right )} + 2 \, \log \left (x^{4} e^{\left (2 \, a\right )} - 1\right )\right )} e^{\left (-2 \, a\right )} \]

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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giac [A]  time = 0.13, size = 24, normalized size = 0.80 \[ \frac {1}{4} \, x^{4} + \frac {1}{2} \, e^{\left (-2 \, a\right )} \log \left ({\left | x^{4} e^{\left (2 \, a\right )} - 1 \right |}\right ) \]

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))

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maple [A]  time = 0.10, size = 24, normalized size = 0.80 \[ \frac {x^{4}}{4}+\frac {{\mathrm e}^{-2 a} \ln \left (-1+{\mathrm e}^{2 a} x^{4}\right )}{2} \]

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

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maxima [A]  time = 0.38, size = 36, normalized size = 1.20 \[ \frac {1}{4} \, x^{4} + \frac {1}{2} \, e^{\left (-2 \, a\right )} \log \left (x^{2} e^{a} + 1\right ) + \frac {1}{2} \, e^{\left (-2 \, a\right )} \log \left (x^{2} e^{a} - 1\right ) \]

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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mupad [B]  time = 1.25, size = 23, normalized size = 0.77 \[ \frac {\ln \left (x^4-{\mathrm {e}}^{-2\,a}\right )\,{\mathrm {e}}^{-2\,a}}{2}+\frac {x^4}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*coth(a + 2*log(x)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*coth(a + 2*log(x)), x)

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