3.2.46 \(\int \frac {\sinh (x)}{(1+\cosh (x))^3} \, dx\) [146]

Optimal. Leaf size=10 \[ -\frac {1}{2 (1+\cosh (x))^2} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 10, 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 = {2746, 32} \begin {gather*} -\frac {1}{2 (\cosh (x)+1)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[x]/(1 + Cosh[x])^3,x]

[Out]

-1/2*1/(1 + Cosh[x])^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]

Rule 2746

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))^3} \, dx &=\text {Subst}\left (\int \frac {1}{(1+x)^3} \, dx,x,\cosh (x)\right )\\ &=-\frac {1}{2 (1+\cosh (x))^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 12, normalized size = 1.20 \begin {gather*} -\frac {1}{8} \text {sech}^4\left (\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]/(1 + Cosh[x])^3,x]

[Out]

-1/8*Sech[x/2]^4

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Maple [A]
time = 0.24, size = 9, normalized size = 0.90

method	result	size
derivativedivides	\(-\frac {1}{2 \left (\cosh \left (x \right )+1\right )^{2}}\)	\(9\)
default	\(-\frac {1}{2 \left (\cosh \left (x \right )+1\right )^{2}}\)	\(9\)
risch	\(-\frac {2 \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{x}+1\right )^{4}}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 8, normalized size = 0.80 \begin {gather*} -\frac {1}{2 \, {\left (\cosh \left (x\right ) + 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(cosh(x) + 1)^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (8) = 16\).
time = 0.42, size = 55, normalized size = 5.50 \begin {gather*} -\frac {2 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{\cosh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right ) + 4\right )} \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + 4 \, \cosh \left (x\right )^{2} + {\left (3 \, \cosh \left (x\right )^{2} + 8 \, \cosh \left (x\right ) + 5\right )} \sinh \left (x\right ) + 7 \, \cosh \left (x\right ) + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(cosh(x) + sinh(x))/(cosh(x)^3 + (3*cosh(x) + 4)*sinh(x)^2 + sinh(x)^3 + 4*cosh(x)^2 + (3*cosh(x)^2 + 8*cos
h(x) + 5)*sinh(x) + 7*cosh(x) + 4)

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Sympy [A]
time = 0.26, size = 15, normalized size = 1.50 \begin {gather*} - \frac {1}{2 \cosh ^{2}{\left (x \right )} + 4 \cosh {\left (x \right )} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(2*cosh(x)**2 + 4*cosh(x) + 2)

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Giac [A]
time = 0.40, size = 12, normalized size = 1.20 \begin {gather*} -\frac {2 \, e^{\left (2 \, x\right )}}{{\left (e^{x} + 1\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*e^(2*x)/(e^x + 1)^4

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Mupad [B]
time = 0.92, size = 8, normalized size = 0.80 \begin {gather*} -\frac {1}{2\,{\left (\mathrm {cosh}\left (x\right )+1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/(cosh(x) + 1)^3,x)

[Out]

-1/(2*(cosh(x) + 1)^2)

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