3.153 \(\int x^3 \tanh ^2(a+2 \log (x)) \, dx\)

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

[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 = {} \[ \int x^3 \tanh ^2(a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^3*Tanh[a + 2*Log[x]]^2,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.12, 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*Tanh[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 = 0.51, size = 58, normalized size = 1.23 \[ \frac {x^{8} e^{\left (4 \, a\right )} + x^{4} e^{\left (2 \, a\right )} - 4 \, {\left (x^{4} e^{\left (2 \, a\right )} + 1\right )} \log \left (x^{4} e^{\left (2 \, a\right )} + 1\right ) - 4}{4 \, {\left (x^{4} e^{\left (4 \, a\right )} + e^{\left (2 \, a\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*tanh(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 = 0.14, size = 39, normalized size = 0.83 \[ \frac {1}{4} \, x^{4} + \frac {x^{4}}{x^{4} e^{\left (2 \, a\right )} + 1} - e^{\left (-2 \, a\right )} \log \left (x^{4} e^{\left (2 \, a\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*tanh(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(x^4*e^(2*a) + 1)

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maple [A]  time = 0.08, size = 42, normalized size = 0.89 \[ \frac {x^{4}}{4}-\frac {{\mathrm e}^{-2 a}}{1+{\mathrm e}^{2 a} x^{4}}-{\mathrm e}^{-2 a} \ln \left (1+{\mathrm e}^{2 a} x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*tanh(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 = 0.34, size = 40, normalized size = 0.85 \[ \frac {1}{4} \, x^{4} - e^{\left (-2 \, a\right )} \log \left (x^{4} e^{\left (2 \, a\right )} + 1\right ) - \frac {1}{x^{4} e^{\left (4 \, a\right )} + e^{\left (2 \, a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.12, size = 39, normalized size = 0.83 \[ \frac {x^4}{4}-\frac {{\mathrm {e}}^{-2\,a}}{{\mathrm {e}}^{2\,a}\,x^4+1}-{\mathrm {e}}^{-2\,a}\,\ln \left (x^4+{\mathrm {e}}^{-2\,a}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*tanh(a + 2*log(x))^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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