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]) + Log[1 + Cosh[x]]

________________________________________________________________________________________

Rubi [A]  time = 0.0402647, antiderivative size = 14, normalized size of antiderivative = 1., 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 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])

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])

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 = 0.0092576, 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

________________________________________________________________________________________

Maple [A]  time = 0.014, size = 15, normalized size = 1.1 \begin{align*} 2\, \left ( 1+\cosh \left ( x \right ) \right ) ^{-1}+\ln \left ( 1+\cosh \left ( x \right ) \right ) \end{align*}

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 = 1.15641, size = 42, normalized size = 3. \begin{align*} 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 ) \end{align*}

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)

________________________________________________________________________________________

Fricas [B]  time = 1.86647, size = 336, normalized size = 24. \begin{align*} -\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} \end{align*}

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)

________________________________________________________________________________________

Sympy [B]  time = 0.914517, size = 194, normalized size = 13.86 \begin{align*} \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 )} \cosh ^{2}{\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} + 4 \cosh{\left (x \right )} + 2} + \frac{2 \sinh ^{2}{\left (x \right )} \cosh{\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} + 4 \cosh{\left (x \right )} + 2} - \frac{\cosh ^{4}{\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} + 4 \cosh{\left (x \right )} + 2} - \frac{2 \cosh ^{3}{\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} + 4 \cosh{\left (x \right )} + 2} + \frac{4 \cosh{\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} + 4 \cosh{\left (x \right )} + 2} + \frac{3}{2 \cosh ^{2}{\left (x \right )} + 4 \cosh{\left (x \right )} + 2} \end{align*}

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*cosh(x)**2/(2*cosh(x)**2 + 4*cosh(
x) + 2) + 2*sinh(x)**2*cosh(x)/(2*cosh(x)**2 + 4*cosh(x) + 2) - cosh(x)**4/(2*cosh(x)**2 + 4*cosh(x) + 2) - 2*
cosh(x)**3/(2*cosh(x)**2 + 4*cosh(x) + 2) + 4*cosh(x)/(2*cosh(x)**2 + 4*cosh(x) + 2) + 3/(2*cosh(x)**2 + 4*cos
h(x) + 2)

________________________________________________________________________________________

Giac [A]  time = 1.14942, size = 28, normalized size = 2. \begin{align*} -x + \frac{4 \, e^{x}}{{\left (e^{x} + 1\right )}^{2}} + 2 \, \log \left (e^{x} + 1\right ) \end{align*}

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)