\(\int \frac {\sinh ^2(x)}{(1+\cosh (x))^3} \, dx\) [148]

Optimal result

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

[Out]

1/3*sinh(x)^3/(1+cosh(x))^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2750} \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^3} \, dx=\frac {\sinh ^3(x)}{3 (\cosh (x)+1)^3} \]

[In]

Int[Sinh[x]^2/(1 + Cosh[x])^3,x]

[Out]

Sinh[x]^3/(3*(1 + Cosh[x])^3)

Rule 2750

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*m)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^
2, 0] && EqQ[Simplify[m + p + 1], 0] &&  !ILtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sinh ^3(x)}{3 (1+\cosh (x))^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^3} \, dx=\frac {1}{3} \tanh ^3\left (\frac {x}{2}\right ) \]

[In]

Integrate[Sinh[x]^2/(1 + Cosh[x])^3,x]

[Out]

Tanh[x/2]^3/3

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64

method	result	size
default	\(\frac {\tanh \left (\frac {x}{2}\right )^{3}}{3}\)	\(9\)
risch	\(-\frac {2 \left (3 \,{\mathrm e}^{2 x}+1\right )}{3 \left ({\mathrm e}^{x}+1\right )^{3}}\)	\(17\)

[In]

int(sinh(x)^2/(cosh(x)+1)^3,x,method=_RETURNVERBOSE)

[Out]

1/3*tanh(1/2*x)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (12) = 24\).

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.36 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^3} \, dx=-\frac {4 \, {\left (2 \, \cosh \left (x\right ) + \sinh \left (x\right )\right )}}{3 \, {\left (\cosh \left (x\right )^{2} + 2 \, {\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) + 3\right )}} \]

[In]

integrate(sinh(x)^2/(1+cosh(x))^3,x, algorithm="fricas")

[Out]

-4/3*(2*cosh(x) + sinh(x))/(cosh(x)^2 + 2*(cosh(x) + 1)*sinh(x) + sinh(x)^2 + 4*cosh(x) + 3)

Sympy [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.50 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^3} \, dx=\frac {\tanh ^{3}{\left (\frac {x}{2} \right )}}{3} \]

[In]

integrate(sinh(x)**2/(1+cosh(x))**3,x)

[Out]

tanh(x/2)**3/3

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (12) = 24\).

Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 3.50 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^3} \, dx=\frac {2 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1} + \frac {2}{3 \, {\left (3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1\right )}} \]

[In]

integrate(sinh(x)^2/(1+cosh(x))^3,x, algorithm="maxima")

[Out]

2*e^(-2*x)/(3*e^(-x) + 3*e^(-2*x) + e^(-3*x) + 1) + 2/3/(3*e^(-x) + 3*e^(-2*x) + e^(-3*x) + 1)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^3} \, dx=-\frac {2 \, {\left (3 \, e^{\left (2 \, x\right )} + 1\right )}}{3 \, {\left (e^{x} + 1\right )}^{3}} \]

[In]

integrate(sinh(x)^2/(1+cosh(x))^3,x, algorithm="giac")

[Out]

-2/3*(3*e^(2*x) + 1)/(e^x + 1)^3

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^3} \, dx=-\frac {2\,\left (3\,{\mathrm {e}}^{2\,x}+1\right )}{3\,{\left ({\mathrm {e}}^x+1\right )}^3} \]

[In]

int(sinh(x)^2/(cosh(x) + 1)^3,x)

[Out]

-(2*(3*exp(2*x) + 1))/(3*(exp(x) + 1)^3)