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

Optimal. Leaf size=51 \[ -\frac{\sinh (c+d x)}{3 d (1-\cosh (c+d x))}-\frac{\sinh (c+d x)}{3 d (1-\cosh (c+d x))^2} \]

[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 = 0.0248098, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2650, 2648} \[ -\frac{\sinh (c+d x)}{3 d (1-\cosh (c+d x))}-\frac{\sinh (c+d x)}{3 d (1-\cosh (c+d x))^2} \]

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 = 0.027212, size = 31, normalized size = 0.61 \[ \frac{\sinh (c+d x) (\cosh (c+d x)-2)}{3 d (\cosh (c+d x)-1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Cosh[c + d*x])^(-2),x]

[Out]

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

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Maple [A]  time = 0.015, size = 32, normalized size = 0.6 \begin{align*}{\frac{1}{d} \left ({\frac{1}{2} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{1}{6} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}} \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*(1/2/tanh(1/2*d*x+1/2*c)-1/6/tanh(1/2*d*x+1/2*c)^3)

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Maxima [B]  time = 1.09933, size = 122, normalized size = 2.39 \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 = 1.85765, size = 319, normalized size = 6.25 \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 = 1.61781, size = 48, normalized size = 0.94 \begin{align*} \begin{cases} \tilde{\infty } x & \text{for}\: c = 0 \wedge d = 0 \\\frac{x}{\left (1 - \cosh{\left (c \right )}\right )^{2}} & \text{for}\: d = 0 \\\tilde{\infty } x & \text{for}\: c = - d x \\\frac{1}{2 d \tanh{\left (\frac{c}{2} + \frac{d x}{2} \right )}} - \frac{1}{6 d \tanh ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}} & \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((zoo*x, Eq(c, 0) & Eq(d, 0)), (x/(1 - cosh(c))**2, Eq(d, 0)), (zoo*x, Eq(c, -d*x)), (1/(2*d*tanh(c/2
+ d*x/2)) - 1/(6*d*tanh(c/2 + d*x/2)**3), True))

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Giac [A]  time = 1.19703, size = 34, normalized size = 0.67 \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)