3.3.22 \(\int \log (\cosh ^2(x)) \sinh (x) \, dx\) [222]

Optimal. Leaf size=13 \[ -2 \cosh (x)+\cosh (x) \log \left (\cosh ^2(x)\right ) \]

[Out]

-2*cosh(x)+cosh(x)*ln(cosh(x)^2)

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Rubi [A]
time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2718, 2634, 12} \begin {gather*} \cosh (x) \log \left (\cosh ^2(x)\right )-2 \cosh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[Cosh[x]^2]*Sinh[x],x]

[Out]

-2*Cosh[x] + Cosh[x]*Log[Cosh[x]^2]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 \left (\cosh ^2(x)\right ) \sinh (x) \, dx &=\cosh (x) \log \left (\cosh ^2(x)\right )-\int 2 \sinh (x) \, dx\\ &=\cosh (x) \log \left (\cosh ^2(x)\right )-2 \int \sinh (x) \, dx\\ &=-2 \cosh (x)+\cosh (x) \log \left (\cosh ^2(x)\right )\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} -2 \cosh (x)+\cosh (x) \log \left (\cosh ^2(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[Cosh[x]^2]*Sinh[x],x]

[Out]

-2*Cosh[x] + Cosh[x]*Log[Cosh[x]^2]

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Maple [A]
time = 0.12, size = 14, normalized size = 1.08

method	result	size
derivativedivides	\(-2 \cosh \left (x \right )+\cosh \left (x \right ) \ln \left (\cosh ^{2}\left (x \right )\right )\)	\(14\)
default	\(-2 \cosh \left (x \right )+\cosh \left (x \right ) \ln \left (\cosh ^{2}\left (x \right )\right )\)	\(14\)
risch	\(-\left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x} \ln \left ({\mathrm e}^{x}\right )+\frac {\left (-4-4 \,{\mathrm e}^{2 x}-i \pi \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2}\right )^{3}+i \pi \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}-i \pi \mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right )^{3}-4 \ln \left (2\right ) {\mathrm e}^{2 x}+4 \,{\mathrm e}^{2 x} \ln \left (1+{\mathrm e}^{2 x}\right )-i \mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right ) \mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right )^{2} \pi +2 i \mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right )^{2} \mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \pi +i \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-2 x}\right ) \pi +i \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \pi +i \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2}\right )^{2} \mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right ) \pi +i \pi \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{3} {\mathrm e}^{2 x}-i \pi \mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right )^{3} {\mathrm e}^{2 x}-i \pi \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2}\right )^{3} {\mathrm e}^{2 x}-2 i \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \pi -i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-2 x}\right ) \mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2}\right ) {\mathrm e}^{2 x}+i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-2 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2}\right )^{2} {\mathrm e}^{2 x}+i \pi \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) {\mathrm e}^{2 x}-i \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2}\right ) \mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x}\right ) \pi +i \pi \,\mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2}\right )^{2} {\mathrm e}^{2 x}-2 i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{2 x}-i \pi \mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right )^{2} \mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right ) {\mathrm e}^{2 x}+2 i \pi \,\mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right )^{2} {\mathrm e}^{2 x}+4 \ln \left (1+{\mathrm e}^{2 x}\right )-4 \ln \left (2\right )\right ) {\mathrm e}^{-x}}{4}\)	\(613\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(cosh(x)^2)*sinh(x),x,method=_RETURNVERBOSE)

[Out]

-2*cosh(x)+cosh(x)*ln(cosh(x)^2)

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Maxima [A]
time = 0.27, size = 12, normalized size = 0.92 \begin {gather*} 2 \, \cosh \left (x\right ) \log \left (\cosh \left (x\right )\right ) - 2 \, \cosh \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(cosh(x)^2)*sinh(x),x, algorithm="maxima")

[Out]

2*cosh(x)*log(cosh(x)) - 2*cosh(x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (13) = 26\).
time = 0.37, size = 62, normalized size = 4.77 \begin {gather*} -\frac {2 \, \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 (\frac {1}{2} \, \cosh \left (x\right )^{2} + \frac {1}{2} \, \sinh \left (x\right )^{2} + \frac {1}{2}\right ) + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} + 2}{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)^2)*sinh(x),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.48, size = 14, normalized size = 1.08 \begin {gather*} \log {\left (\cosh ^{2}{\left (x \right )} \right )} \cosh {\left (x \right )} - 2 \cosh {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(cosh(x)**2)*sinh(x),x)

[Out]

log(cosh(x)**2)*cosh(x) - 2*cosh(x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (13) = 26\).
time = 4.67, size = 37, normalized size = 2.85 \begin {gather*} {\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 ) - {\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)^2)*sinh(x),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.38, size = 9, normalized size = 0.69 \begin {gather*} 2\,\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)^2)*sinh(x),x)

[Out]

2*cosh(x)*(log(cosh(x)) - 1)

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