3.1049 \(\int \cosh ^x(x) (\log (\cosh (x))+x \tanh (x)) \, dx\)

Optimal. Leaf size=4 \[ \cosh ^x(x) \]

[Out]

cosh(x)^x

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Rubi [A]  time = 0.14, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6742, 2553} \[ \cosh ^x(x) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^x*(Log[Cosh[x]] + x*Tanh[x]),x]

[Out]

Cosh[x]^x

Rule 2553

Int[Log[u_]*(u_)^((a_.)*(x_)), x_Symbol] :> Simp[u^(a*x)/a, x] - Int[SimplifyIntegrand[x*u^(a*x - 1)*D[u, x],
x], x] /; FreeQ[a, x] && InverseFunctionFreeQ[u, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \cosh ^x(x) (\log (\cosh (x))+x \tanh (x)) \, dx &=\int \left (\cosh ^x(x) \log (\cosh (x))+x \cosh ^{-1+x}(x) \sinh (x)\right ) \, dx\\ &=\int \cosh ^x(x) \log (\cosh (x)) \, dx+\int x \cosh ^{-1+x}(x) \sinh (x) \, dx\\ &=\cosh ^x(x)\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 4, normalized size = 1.00 \[ \cosh ^x(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^x*(Log[Cosh[x]] + x*Tanh[x]),x]

[Out]

Cosh[x]^x

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fricas [B]  time = 0.41, size = 13, normalized size = 3.25 \[ \cosh \left (x \log \left (\cosh \relax (x)\right )\right ) + \sinh \left (x \log \left (\cosh \relax (x)\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^x*(log(cosh(x))+x*tanh(x)),x, algorithm="fricas")

[Out]

cosh(x*log(cosh(x))) + sinh(x*log(cosh(x)))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x \tanh \relax (x) + \log \left (\cosh \relax (x)\right )\right )} \cosh \relax (x)^{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^x*(log(cosh(x))+x*tanh(x)),x, algorithm="giac")

[Out]

integrate((x*tanh(x) + log(cosh(x)))*cosh(x)^x, x)

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maple [A]  time = 0.14, size = 5, normalized size = 1.25 \[ \cosh ^{x}\relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^x*(ln(cosh(x))+x*tanh(x)),x)

[Out]

cosh(x)^x

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maxima [B]  time = 0.62, size = 21, normalized size = 5.25 \[ e^{\left (-x^{2} - x \log \relax (2) + x \log \left (e^{\left (2 \, x\right )} + 1\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^x*(log(cosh(x))+x*tanh(x)),x, algorithm="maxima")

[Out]

e^(-x^2 - x*log(2) + x*log(e^(2*x) + 1))

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mupad [B]  time = 1.95, size = 4, normalized size = 1.00 \[ {\mathrm {cosh}\relax (x)}^x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^x*(log(cosh(x)) + x*tanh(x)),x)

[Out]

cosh(x)^x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)**x*(ln(cosh(x))+x*tanh(x)),x)

[Out]

Integral((x*tanh(x) + log(cosh(x)))*cosh(x)**x, x)

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