3.33 \(\int \frac{1}{(1+\cosh (c+d x))^2} \, dx\)

Optimal. Leaf size=47 \[ \frac{\sinh (c+d x)}{3 d (\cosh (c+d x)+1)}+\frac{\sinh (c+d x)}{3 d (\cosh (c+d x)+1)^2} \]

[Out]

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

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Rubi [A]  time = 0.0223655, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2650, 2648} \[ \frac{\sinh (c+d x)}{3 d (\cosh (c+d x)+1)}+\frac{\sinh (c+d x)}{3 d (\cosh (c+d x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cosh[c + d*x])^(-2),x]

[Out]

Sinh[c + d*x]/(3*d*(1 + Cosh[c + d*x])^2) + Sinh[c + d*x]/(3*d*(1 + Cosh[c + d*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 (c+d x))^2} \, dx &=\frac{\sinh (c+d x)}{3 d (1+\cosh (c+d x))^2}+\frac{1}{3} \int \frac{1}{1+\cosh (c+d x)} \, dx\\ &=\frac{\sinh (c+d x)}{3 d (1+\cosh (c+d x))^2}+\frac{\sinh (c+d x)}{3 d (1+\cosh (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.0287307, size = 34, normalized size = 0.72 \[ \frac{4 \sinh (c+d x)+\sinh (2 (c+d x))}{6 d (\cosh (c+d x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cosh[c + d*x])^(-2),x]

[Out]

(4*Sinh[c + d*x] + Sinh[2*(c + d*x)])/(6*d*(1 + Cosh[c + d*x])^2)

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Maple [A]  time = 0.01, size = 30, normalized size = 0.6 \begin{align*}{\frac{1}{d} \left ( -{\frac{1}{6} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{1}{2}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+cosh(d*x+c))^2,x)

[Out]

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

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Maxima [B]  time = 1.07127, size = 122, normalized size = 2.6 \begin{align*} \frac{2 \, e^{\left (-d x - c\right )}}{d{\left (3 \, e^{\left (-d x - c\right )} + 3 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-3 \, d x - 3 \, c\right )} + 1\right )}} + \frac{2}{3 \, d{\left (3 \, e^{\left (-d x - c\right )} + 3 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-3 \, d x - 3 \, c\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cosh(d*x+c))^2,x, algorithm="maxima")

[Out]

2*e^(-d*x - c)/(d*(3*e^(-d*x - c) + 3*e^(-2*d*x - 2*c) + e^(-3*d*x - 3*c) + 1)) + 2/3/(d*(3*e^(-d*x - c) + 3*e
^(-2*d*x - 2*c) + e^(-3*d*x - 3*c) + 1))

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Fricas [B]  time = 1.79341, size = 319, normalized size = 6.79 \begin{align*} -\frac{2 \,{\left (3 \, \cosh \left (d x + c\right ) + 3 \, \sinh \left (d x + c\right ) + 1\right )}}{3 \,{\left (d \cosh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} + 3 \,{\left (d \cosh \left (d x + c\right ) + d\right )} \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right ) + 3 \,{\left (d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) + d\right )} \sinh \left (d x + c\right ) + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.34671, size = 36, normalized size = 0.77 \begin{align*} \begin{cases} - \frac{\tanh ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 d} + \frac{\tanh{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{2 d} & \text{for}\: d \neq 0 \\\frac{x}{\left (\cosh{\left (c \right )} + 1\right )^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-tanh(c/2 + d*x/2)**3/(6*d) + tanh(c/2 + d*x/2)/(2*d), Ne(d, 0)), (x/(cosh(c) + 1)**2, True))

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Giac [A]  time = 1.15745, size = 34, normalized size = 0.72 \begin{align*} -\frac{2 \,{\left (3 \, e^{\left (d x + c\right )} + 1\right )}}{3 \, d{\left (e^{\left (d x + c\right )} + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*e^(d*x + c) + 1)/(d*(e^(d*x + c) + 1)^3)