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

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

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 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]

Rule 2638

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

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Mathematica [A]  time = 0.01, size = 11, normalized size = 1.00 \[ \cosh (x) \log (\cosh (x))-\cosh (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cosh[x] + Cosh[x]*Log[Cosh[x]]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \log (\cosh (x)) \sinh (x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [B]  time = 1.12, size = 46, normalized size = 4.18 \[ -\frac {\cosh \relax (x)^{2} - {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \log \left (\cosh \relax (x)\right ) + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1}{2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]

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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giac [B]  time = 1.12, size = 32, normalized size = 2.91 \[ \frac {1}{2} \, {\left (e^{\left (-x\right )} + e^{x}\right )} \log \left (\frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x}\right ) - \frac {1}{2} \, e^{\left (-x\right )} - \frac {1}{2} \, e^{x} \]

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

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maple [A]  time = 0.16, size = 12, normalized size = 1.09

method	result	size
derivativedivides	\(-\cosh \relax (x )+\cosh \relax (x ) \ln \left (\cosh \relax (x )\right )\)	\(12\)
default	\(-\cosh \relax (x )+\cosh \relax (x ) \ln \left (\cosh \relax (x )\right )\)	\(12\)
risch	\(-\frac {\left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x} \ln \left ({\mathrm e}^{x}\right )}{2}-\frac {\left (-i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{2}+i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right ) {\mathrm e}^{2 x}+i \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right ) \mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \pi -i \pi \,\mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{2}+i \pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{3}+i \pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{3} {\mathrm e}^{2 x}-i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{2} {\mathrm e}^{2 x}-i \pi \,\mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{2} {\mathrm e}^{2 x}+2+2 \ln \relax (2) {\mathrm e}^{2 x}-2 \,{\mathrm e}^{2 x} \ln \left (1+{\mathrm e}^{2 x}\right )+2 \,{\mathrm e}^{2 x}+2 \ln \relax (2)-2 \ln \left (1+{\mathrm e}^{2 x}\right )\right ) {\mathrm e}^{-x}}{4}\)	\(309\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(cosh(x))*sinh(x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.43, size = 11, normalized size = 1.00 \[ \cosh \relax (x) \log \left (\cosh \relax (x)\right ) - \cosh \relax (x) \]

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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mupad [B]  time = 0.36, size = 8, normalized size = 0.73 \[ \mathrm {cosh}\relax (x)\,\left (\ln \left (\mathrm {cosh}\relax (x)\right )-1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cosh(x)*(log(cosh(x)) - 1)

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sympy [A]  time = 0.95, size = 10, normalized size = 0.91 \[ \log {\left (\cosh {\relax (x )} \right )} \cosh {\relax (x )} - \cosh {\relax (x )} \]

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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