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

Optimal. Leaf size=29 \[ \frac{x^4}{4}-\frac{1}{2} e^{-2 a} \log \left (e^{2 a} x^4+1\right ) \]

[Out]

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

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Rubi [F]  time = 0.0291397, 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 = {} \[ \int x^3 \tanh (a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^3*Tanh[a + 2*Log[x]],x]

[Out]

Defer[Int][x^3*Tanh[a + 2*Log[x]], x]

Rubi steps

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

Mathematica [B]  time = 0.0251789, size = 64, normalized size = 2.21 \[ -\frac{1}{2} \cosh (2 a) \log \left (x^4 \sinh (a)+x^4 \cosh (a)-\sinh (a)+\cosh (a)\right )+\frac{1}{2} \sinh (2 a) \log \left (x^4 \sinh (a)+x^4 \cosh (a)-\sinh (a)+\cosh (a)\right )+\frac{x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Tanh[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 = 0.021, size = 24, normalized size = 0.8 \begin{align*}{\frac{{x}^{4}}{4}}-{\frac{{{\rm e}^{-2\,a}}\ln \left ( 1+{{\rm e}^{2\,a}}{x}^{4} \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*tanh(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 = 1.23674, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{4} \, x^{4} - \frac{1}{2} \, 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)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.44727, size = 72, normalized size = 2.48 \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*tanh(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} \tanh{\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)),x)

[Out]

Integral(x**3*tanh(a + 2*log(x)), x)

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Giac [A]  time = 1.16495, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{4} \, x^{4} - \frac{1}{2} \, 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)),x, algorithm="giac")

[Out]

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