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

Optimal. Leaf size=8 \[ -\frac{1}{\cosh (x)+1} \]

[Out]

-(1 + Cosh[x])^(-1)

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Rubi [A]  time = 0.0209781, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2667, 32} \[ -\frac{1}{\cosh (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]/(1 + Cosh[x])^2,x]

[Out]

-(1 + Cosh[x])^(-1)

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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0105182, size = 12, normalized size = 1.5 \[ -\frac{1}{2} \text{sech}^2\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]/(1 + Cosh[x])^2,x]

[Out]

-Sech[x/2]^2/2

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Maple [A]  time = 0.004, size = 9, normalized size = 1.1 \begin{align*} - \left ( 1+\cosh \left ( x \right ) \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(1+cosh(x))

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Maxima [A]  time = 1.06784, size = 11, normalized size = 1.38 \begin{align*} -\frac{1}{\cosh \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(cosh(x) + 1)

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Fricas [B]  time = 2.10478, size = 122, normalized size = 15.25 \begin{align*} -\frac{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{\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)/(1+cosh(x))^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.431506, size = 7, normalized size = 0.88 \begin{align*} - \frac{1}{\cosh{\left (x \right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(cosh(x) + 1)

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Giac [A]  time = 1.15624, size = 14, normalized size = 1.75 \begin{align*} -\frac{2 \, e^{x}}{{\left (e^{x} + 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*e^x/(e^x + 1)^2