### 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 = 0.139696, antiderivative size = 4, normalized size of antiderivative = 1., 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 6742

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

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]

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*}

Mathematica [A]  time = 0.0707959, size = 4, normalized size = 1. $\cosh ^x(x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Cosh[x]^x

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Maple [A]  time = 0.02, size = 5, normalized size = 1.3 \begin{align*} \left ( \cosh \left ( x \right ) \right ) ^{x} \end{align*}

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 = 2.23277, size = 28, normalized size = 7. \begin{align*} e^{\left (-x^{2} - x \log \left (2\right ) + x \log \left (e^{\left (2 \, x\right )} + 1\right )\right )} \end{align*}

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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Fricas [B]  time = 2.07264, size = 61, normalized size = 15.25 \begin{align*} \cosh \left (x \log \left (\cosh \left (x\right )\right )\right ) + \sinh \left (x \log \left (\cosh \left (x\right )\right )\right ) \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x \tanh \left (x\right ) + \log \left (\cosh \left (x\right )\right )\right )} \cosh \left (x\right )^{x}\,{d x} \end{align*}

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)