\(\int \frac {\sinh ^3(x)}{(1+\cosh (x))^3} \, dx\) [150]

Optimal result

Integrand size = 11, antiderivative size = 14 \[ \int \frac {\sinh ^3(x)}{(1+\cosh (x))^3} \, dx=\frac {2}{1+\cosh (x)}+\log (1+\cosh (x)) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2746, 45} \[ \int \frac {\sinh ^3(x)}{(1+\cosh (x))^3} \, dx=\frac {2}{\cosh (x)+1}+\log (\cosh (x)+1) \]

[In]

Int[Sinh[x]^3/(1 + Cosh[x])^3,x]

[Out]

2/(1 + Cosh[x]) + Log[1 + Cosh[x]]

Rule 45

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 2746

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*} \text {integral}& = -\text {Subst}\left (\int \frac {1-x}{(1+x)^2} \, dx,x,\cosh (x)\right ) \\ & = -\text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {\sinh ^3(x)}{(1+\cosh (x))^3} \, dx=2 \log \left (\cosh \left (\frac {x}{2}\right )\right )-\tanh ^2\left (\frac {x}{2}\right ) \]

[In]

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

[Out]

2*Log[Cosh[x/2]] - Tanh[x/2]^2

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07

method	result	size
derivativedivides	\(\frac {2}{\cosh \left (x \right )+1}+\ln \left (\cosh \left (x \right )+1\right )\)	\(15\)
default	\(\frac {2}{\cosh \left (x \right )+1}+\ln \left (\cosh \left (x \right )+1\right )\)	\(15\)
risch	\(-x +\frac {4 \,{\mathrm e}^{x}}{\left ({\mathrm e}^{x}+1\right )^{2}}+2 \ln \left ({\mathrm e}^{x}+1\right )\)	\(22\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (14) = 28\).

Time = 0.26 (sec) , antiderivative size = 89, normalized size of antiderivative = 6.36 \[ \int \frac {\sinh ^3(x)}{(1+\cosh (x))^3} \, dx=-\frac {x \cosh \left (x\right )^{2} + x \sinh \left (x\right )^{2} + 2 \, {\left (x - 2\right )} \cosh \left (x\right ) - 2 \, {\left (\cosh \left (x\right )^{2} + 2 \, {\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 2 \, {\left (x \cosh \left (x\right ) + x - 2\right )} \sinh \left (x\right ) + x}{\cosh \left (x\right )^{2} + 2 \, {\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (12) = 24\).

Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 9.00 \[ \int \frac {\sinh ^3(x)}{(1+\cosh (x))^3} \, dx=\frac {2 \log {\left (\cosh {\left (x \right )} + 1 \right )} \cosh ^{2}{\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} + 4 \cosh {\left (x \right )} + 2} + \frac {4 \log {\left (\cosh {\left (x \right )} + 1 \right )} \cosh {\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} + 4 \cosh {\left (x \right )} + 2} + \frac {2 \log {\left (\cosh {\left (x \right )} + 1 \right )}}{2 \cosh ^{2}{\left (x \right )} + 4 \cosh {\left (x \right )} + 2} - \frac {\sinh ^{2}{\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} + 4 \cosh {\left (x \right )} + 2} + \frac {2 \cosh {\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} + 4 \cosh {\left (x \right )} + 2} + \frac {2}{2 \cosh ^{2}{\left (x \right )} + 4 \cosh {\left (x \right )} + 2} \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (14) = 28\).

Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.21 \[ \int \frac {\sinh ^3(x)}{(1+\cosh (x))^3} \, dx=x + \frac {4 \, e^{\left (-x\right )}}{2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1} + 2 \, \log \left (e^{\left (-x\right )} + 1\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.50 \[ \int \frac {\sinh ^3(x)}{(1+\cosh (x))^3} \, dx=-x + \frac {4 \, e^{x}}{{\left (e^{x} + 1\right )}^{2}} + 2 \, \log \left (e^{x} + 1\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 1.84 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh ^3(x)}{(1+\cosh (x))^3} \, dx=\ln \left (\mathrm {cosh}\left (x\right )+1\right )+\frac {2}{\mathrm {cosh}\left (x\right )+1} \]

[In]

int(sinh(x)^3/(cosh(x) + 1)^3,x)

[Out]

log(cosh(x) + 1) + 2/(cosh(x) + 1)