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]

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

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*}

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Mathematica [B]  time = 0.03, 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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fricas [A]  time = 0.78, size = 28, normalized size = 0.97 \[ \frac {1}{4} \, {\left (x^{4} e^{\left (2 \, a\right )} - 2 \, \log \left (x^{4} e^{\left (2 \, a\right )} + 1\right )\right )} e^{\left (-2 \, a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*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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giac [A]  time = 0.13, size = 23, normalized size = 0.79 \[ \frac {1}{4} \, x^{4} - \frac {1}{2} \, e^{\left (-2 \, a\right )} \log \left (x^{4} e^{\left (2 \, a\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.09, size = 24, normalized size = 0.83 \[ \frac {x^{4}}{4}-\frac {{\mathrm e}^{-2 a} \ln \left (1+{\mathrm e}^{2 a} x^{4}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*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 = 0.32, size = 23, normalized size = 0.79 \[ \frac {1}{4} \, x^{4} - \frac {1}{2} \, e^{\left (-2 \, a\right )} \log \left (x^{4} e^{\left (2 \, a\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.08, size = 21, normalized size = 0.72 \[ \frac {x^4}{4}-\frac {{\mathrm {e}}^{-2\,a}\,\ln \left (x^4+{\mathrm {e}}^{-2\,a}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*tanh(a + 2*log(x)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*tanh(a + 2*log(x)), x)

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