3.7.35 \(\int \log (\cosh (x)) \sinh (x) \, dx\) [635]

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

[Out]

-cosh(x)+cosh(x)*ln(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 = {2718, 2634} \begin {gather*} \cosh (x) \log (\cosh (x))-\cosh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Cosh[x] + Cosh[x]*Log[Cosh[x]]

Rule 2634

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 2718

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 \begin {gather*} -\cosh (x)+\cosh (x) \log (\cosh (x)) \end {gather*}

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 = 0.08, size = 12, normalized size = 1.09

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

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 = 1.25, size = 11, normalized size = 1.00 \begin {gather*} \cosh \left (x\right ) \log \left (\cosh \left (x\right )\right ) - \cosh \left (x\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (11) = 22\).
time = 1.03, size = 46, normalized size = 4.18 \begin {gather*} -\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 {gather*}

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 = 0.19, size = 10, normalized size = 0.91 \begin {gather*} \log {\left (\cosh {\left (x \right )} \right )} \cosh {\left (x \right )} - \cosh {\left (x \right )} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (11) = 22\).
time = 1.40, size = 38, normalized size = 3.45 \begin {gather*} \frac {1}{2} \, {\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} \log \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )}\right ) - \frac {1}{2} \, {\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.36, size = 8, normalized size = 0.73 \begin {gather*} \mathrm {cosh}\left (x\right )\,\left (\ln \left (\mathrm {cosh}\left (x\right )\right )-1\right ) \end {gather*}

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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