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

Optimal. Leaf size=14 \[ \frac{\sinh ^3(x)}{3 (\cosh (x)+1)^3} \]

[Out]

Sinh[x]^3/(3*(1 + Cosh[x])^3)

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Rubi [A]  time = 0.0322497, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2671} \[ \frac{\sinh ^3(x)}{3 (\cosh (x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^2/(1 + Cosh[x])^3,x]

[Out]

Sinh[x]^3/(3*(1 + Cosh[x])^3)

Rule 2671

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[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*} \int \frac{\sinh ^2(x)}{(1+\cosh (x))^3} \, dx &=\frac{\sinh ^3(x)}{3 (1+\cosh (x))^3}\\ \end{align*}

Mathematica [A]  time = 0.0264682, size = 12, normalized size = 0.86 \[ \frac{1}{3} \tanh ^3\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Tanh[x/2]^3/3

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Maple [A]  time = 0.011, size = 9, normalized size = 0.6 \begin{align*}{\frac{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.06615, size = 66, normalized size = 4.71 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 2.11234, size = 127, normalized size = 9.07 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 1.50567, size = 7, normalized size = 0.5 \begin{align*} \frac{\tanh ^{3}{\left (\frac{x}{2} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tanh(x/2)**3/3

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Giac [A]  time = 1.16736, size = 22, normalized size = 1.57 \begin{align*} -\frac{2 \,{\left (3 \, e^{\left (2 \, x\right )} + 1\right )}}{3 \,{\left (e^{x} + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*e^(2*x) + 1)/(e^x + 1)^3