3.150 \(\int \frac {\sinh ^3(x)}{(1+\cosh (x))^3} \, dx\)

Optimal. Leaf size=14 \[ \frac {2}{\cosh (x)+1}+\log (\cosh (x)+1) \]

[Out]

2/(1+cosh(x))+ln(1+cosh(x))

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2667, 43} \[ \frac {2}{\cosh (x)+1}+\log (\cosh (x)+1) \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^3/(1 + Cosh[x])^3,x]

[Out]

2/(1 + Cosh[x]) + Log[1 + Cosh[x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps

\begin {align*} \int \frac {\sinh ^3(x)}{(1+\cosh (x))^3} \, dx &=-\operatorname {Subst}\left (\int \frac {1-x}{(1+x)^2} \, dx,x,\cosh (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {1}{-1-x}+\frac {2}{(1+x)^2}\right ) \, dx,x,\cosh (x)\right )\\ &=\frac {2}{1+\cosh (x)}+\log (1+\cosh (x))\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 20, normalized size = 1.43 \[ 2 \log \left (\cosh \left (\frac {x}{2}\right )\right )-\tanh ^2\left (\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]^3/(1 + Cosh[x])^3,x]

[Out]

2*Log[Cosh[x/2]] - Tanh[x/2]^2

________________________________________________________________________________________

fricas [B]  time = 0.55, size = 89, normalized size = 6.36 \[ -\frac {x \cosh \relax (x)^{2} + x \sinh \relax (x)^{2} + 2 \, {\left (x - 2\right )} \cosh \relax (x) - 2 \, {\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) + 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 2 \, \cosh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + 2 \, {\left (x \cosh \relax (x) + x - 2\right )} \sinh \relax (x) + x}{\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) + 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 2 \, \cosh \relax (x) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^3/(1+cosh(x))^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.12, size = 21, normalized size = 1.50 \[ -x + \frac {4 \, e^{x}}{{\left (e^{x} + 1\right )}^{2}} + 2 \, \log \left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^3/(1+cosh(x))^3,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 15, normalized size = 1.07 \[ \frac {2}{1+\cosh \relax (x )}+\ln \left (1+\cosh \relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^3/(1+cosh(x))^3,x)

[Out]

2/(1+cosh(x))+ln(1+cosh(x))

________________________________________________________________________________________

maxima [B]  time = 0.30, size = 31, normalized size = 2.21 \[ x + \frac {4 \, e^{\left (-x\right )}}{2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1} + 2 \, \log \left (e^{\left (-x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^3/(1+cosh(x))^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.98, size = 14, normalized size = 1.00 \[ \ln \left (\mathrm {cosh}\relax (x)+1\right )+\frac {2}{\mathrm {cosh}\relax (x)+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^3/(cosh(x) + 1)^3,x)

[Out]

log(cosh(x) + 1) + 2/(cosh(x) + 1)

________________________________________________________________________________________

sympy [B]  time = 0.59, size = 126, normalized size = 9.00 \[ \frac {2 \log {\left (\cosh {\relax (x )} + 1 \right )} \cosh ^{2}{\relax (x )}}{2 \cosh ^{2}{\relax (x )} + 4 \cosh {\relax (x )} + 2} + \frac {4 \log {\left (\cosh {\relax (x )} + 1 \right )} \cosh {\relax (x )}}{2 \cosh ^{2}{\relax (x )} + 4 \cosh {\relax (x )} + 2} + \frac {2 \log {\left (\cosh {\relax (x )} + 1 \right )}}{2 \cosh ^{2}{\relax (x )} + 4 \cosh {\relax (x )} + 2} - \frac {\sinh ^{2}{\relax (x )}}{2 \cosh ^{2}{\relax (x )} + 4 \cosh {\relax (x )} + 2} + \frac {2 \cosh {\relax (x )}}{2 \cosh ^{2}{\relax (x )} + 4 \cosh {\relax (x )} + 2} + \frac {2}{2 \cosh ^{2}{\relax (x )} + 4 \cosh {\relax (x )} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)**3/(1+cosh(x))**3,x)

[Out]

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

________________________________________________________________________________________