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

Optimal. Leaf size=25 \[ \frac{\sinh (x)}{3 (\cosh (x)+1)}+\frac{\sinh (x)}{3 (\cosh (x)+1)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0166621, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2650, 2648} \[ \frac{\sinh (x)}{3 (\cosh (x)+1)}+\frac{\sinh (x)}{3 (\cosh (x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cosh[x])^(-2),x]

[Out]

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

Rule 2650

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(a*
d*(2*n + 1)), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d},
x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{1}{(1+\cosh (x))^2} \, dx &=\frac{\sinh (x)}{3 (1+\cosh (x))^2}+\frac{1}{3} \int \frac{1}{1+\cosh (x)} \, dx\\ &=\frac{\sinh (x)}{3 (1+\cosh (x))^2}+\frac{\sinh (x)}{3 (1+\cosh (x))}\\ \end{align*}

Mathematica [A]  time = 0.0114978, size = 16, normalized size = 0.64 \[ \frac{\sinh (x) (\cosh (x)+2)}{3 (\cosh (x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cosh[x])^(-2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 16, normalized size = 0.6 \begin{align*} -{\frac{1}{6} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{1}{2}\tanh \left ({\frac{x}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*tanh(1/2*x)^3+1/2*tanh(1/2*x)

________________________________________________________________________________________

Maxima [B]  time = 0.938846, size = 66, normalized size = 2.64 \begin{align*} \frac{2 \, e^{\left (-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(1/(1+cosh(x))^2,x, algorithm="maxima")

[Out]

2*e^(-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)

________________________________________________________________________________________

Fricas [B]  time = 2.00102, size = 211, normalized size = 8.44 \begin{align*} -\frac{2 \,{\left (3 \, \cosh \left (x\right ) + 3 \, \sinh \left (x\right ) + 1\right )}}{3 \,{\left (\cosh \left (x\right )^{3} + 3 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} + 3 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.409911, size = 14, normalized size = 0.56 \begin{align*} - \frac{\tanh ^{3}{\left (\frac{x}{2} \right )}}{6} + \frac{\tanh{\left (\frac{x}{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-tanh(x/2)**3/6 + tanh(x/2)/2

________________________________________________________________________________________

Giac [A]  time = 1.1271, size = 19, normalized size = 0.76 \begin{align*} -\frac{2 \,{\left (3 \, e^{x} + 1\right )}}{3 \,{\left (e^{x} + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*e^x + 1)/(e^x + 1)^3