3.222 \(\int \log (\cosh ^2(x)) \sinh (x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.02, 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 = {2638, 2554, 12} \[ \cosh (x) \log \left (\cosh ^2(x)\right )-2 \cosh (x) \]

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 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 \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 \[ \cosh (x) \log \left (\cosh ^2(x)\right )-2 \cosh (x) \]

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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fricas [B]  time = 0.46, size = 62, normalized size = 4.77 \[ -\frac {2 \, \cosh \relax (x)^{2} - {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \log \left (\frac {1}{2} \, \cosh \relax (x)^{2} + \frac {1}{2} \, \sinh \relax (x)^{2} + \frac {1}{2}\right ) + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} + 2}{2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]

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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giac [B]  time = 0.18, size = 31, normalized size = 2.38 \[ {\left (e^{\left (-x\right )} + e^{x}\right )} \log \left (\frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x}\right ) - e^{\left (-x\right )} - e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.29, size = 14, normalized size = 1.08 \[ \cosh \relax (x ) \ln \left (\cosh ^{2}\relax (x )\right )-2 \cosh \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(cosh(x)^2)*sinh(x),x)

[Out]

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

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maxima [A]  time = 0.44, size = 12, normalized size = 0.92 \[ 2 \, \cosh \relax (x) \log \left (\cosh \relax (x)\right ) - 2 \, \cosh \relax (x) \]

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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mupad [B]  time = 0.38, size = 9, normalized size = 0.69 \[ 2\,\mathrm {cosh}\relax (x)\,\left (\ln \left (\mathrm {cosh}\relax (x)\right )-1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(cosh(x)^2)*sinh(x),x)

[Out]

2*cosh(x)*(log(cosh(x)) - 1)

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sympy [A]  time = 0.91, size = 14, normalized size = 1.08 \[ 2 \log {\left (\cosh {\relax (x )} \right )} \cosh {\relax (x )} - 2 \cosh {\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(cosh(x))*cosh(x) - 2*cosh(x)

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