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]

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

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Rubi [F]  time = 0.0678357, 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 = {} \[ \int x^3 \tanh ^2(a+2 \log (x)) \, dx \]

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*}

Mathematica [A]  time = 0.10442, 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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Maple [A]  time = 0.019, size = 42, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}}{4}}-{\frac{{{\rm e}^{-2\,a}}}{1+{{\rm e}^{2\,a}}{x}^{4}}}-{{\rm e}^{-2\,a}}\ln \left ( 1+{{\rm e}^{2\,a}}{x}^{4} \right ) \end{align*}

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 = 1.54095, size = 54, normalized size = 1.15 \begin{align*} \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 )}} \end{align*}

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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Fricas [A]  time = 1.88471, 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*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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \tanh ^{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*tanh(a+2*ln(x))**2,x)

[Out]

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

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Giac [A]  time = 1.1802, size = 53, normalized size = 1.13 \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 (x^{4} e^{\left (2 \, a\right )} + 1\right ) \end{align*}

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)