3.635 \(\int \log (\cosh (x)) \sinh (x) \, dx\)

Optimal. Leaf size=11 \[ \cosh (x) \log (\cosh (x))-\cosh (x) \]

[Out]

-Cosh[x] + Cosh[x]*Log[Cosh[x]]

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Rubi [A]  time = 0.0112569, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2638, 2554} \[ \cosh (x) \log (\cosh (x))-\cosh (x) \]

Antiderivative was successfully verified.

[In]

Int[Log[Cosh[x]]*Sinh[x],x]

[Out]

-Cosh[x] + Cosh[x]*Log[Cosh[x]]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rubi steps

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

Mathematica [A]  time = 0.0065869, size = 11, normalized size = 1. \[ \cosh (x) \log (\cosh (x))-\cosh (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[Cosh[x]]*Sinh[x],x]

[Out]

-Cosh[x] + Cosh[x]*Log[Cosh[x]]

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Maple [A]  time = 0.004, size = 12, normalized size = 1.1 \begin{align*} -\cosh \left ( x \right ) +\cosh \left ( x \right ) \ln \left ( \cosh \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(cosh(x))*sinh(x),x)

[Out]

-cosh(x)+cosh(x)*ln(cosh(x))

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Maxima [A]  time = 0.947536, size = 15, normalized size = 1.36 \begin{align*} \cosh \left (x\right ) \log \left (\cosh \left (x\right )\right ) - \cosh \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(cosh(x))*sinh(x),x, algorithm="maxima")

[Out]

cosh(x)*log(cosh(x)) - cosh(x)

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Fricas [B]  time = 2.20455, size = 185, normalized size = 16.82 \begin{align*} -\frac{\cosh \left (x\right )^{2} -{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \log \left (\cosh \left (x\right )\right ) + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(cosh(x))*sinh(x),x, algorithm="fricas")

[Out]

-1/2*(cosh(x)^2 - (cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*log(cosh(x)) + 2*cosh(x)*sinh(x) + sinh(x)^2
+ 1)/(cosh(x) + sinh(x))

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Sympy [A]  time = 0.939075, size = 10, normalized size = 0.91 \begin{align*} \log{\left (\cosh{\left (x \right )} \right )} \cosh{\left (x \right )} - \cosh{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(cosh(x))*sinh(x),x)

[Out]

log(cosh(x))*cosh(x) - cosh(x)

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Giac [B]  time = 1.06502, size = 43, normalized size = 3.91 \begin{align*} \frac{1}{2} \,{\left (e^{\left (-x\right )} + e^{x}\right )} \log \left (\frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x}\right ) - \frac{1}{2} \, e^{\left (-x\right )} - \frac{1}{2} \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(cosh(x))*sinh(x),x, algorithm="giac")

[Out]

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