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

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

[Out]

-1/(1+cosh(x))

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Rubi [A]  time = 0.02, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 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]

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])

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*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.50 \[ -\frac {1}{2} \text {sech}^2\left (\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]/(1 + Cosh[x])^2,x]

[Out]

-1/2*Sech[x/2]^2

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fricas [B]  time = 1.54, size = 31, normalized size = 3.88 \[ -\frac {2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}}{\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) + 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 2 \, \cosh \relax (x) + 1} \]

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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giac [A]  time = 0.14, size = 10, normalized size = 1.25 \[ -\frac {2 \, e^{x}}{{\left (e^{x} + 1\right )}^{2}} \]

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

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maple [A]  time = 0.03, size = 9, normalized size = 1.12 \[ -\frac {1}{1+\cosh \relax (x )} \]

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 = 0.30, size = 8, normalized size = 1.00 \[ -\frac {1}{\cosh \relax (x) + 1} \]

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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mupad [B]  time = 0.07, size = 8, normalized size = 1.00 \[ -\frac {1}{\mathrm {cosh}\relax (x)+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/(cosh(x) + 1)^2,x)

[Out]

-1/(cosh(x) + 1)

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sympy [A]  time = 0.34, size = 7, normalized size = 0.88 \[ - \frac {1}{\cosh {\relax (x )} + 1} \]

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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