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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, 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 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]

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]

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 = 0.03, 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)

________________________________________________________________________________________

fricas [B]  time = 0.77, size = 117, normalized size = 2.29 \[ -\frac {2 \, {\left (3 \, \cosh \left (d x + c\right ) + 3 \, \sinh \left (d x + c\right ) - 1\right )}}{3 \, {\left (d \cosh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{3} - 3 \, d \cosh \left (d x + c\right )^{2} + 3 \, {\left (d \cosh \left (d x + c\right ) - d\right )} \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right ) + 3 \, {\left (d \cosh \left (d x + c\right )^{2} - 2 \, d \cosh \left (d x + c\right ) + d\right )} \sinh \left (d x + c\right ) - d\right )}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.12, size = 25, normalized size = 0.49 \[ -\frac {2 \, {\left (3 \, e^{\left (d x + c\right )} - 1\right )}}{3 \, d {\left (e^{\left (d x + c\right )} - 1\right )}^{3}} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.07, size = 32, normalized size = 0.63 \[ \frac {-\frac {1}{6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {1}{2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d} \]

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))

________________________________________________________________________________________

maxima [B]  time = 0.88, size = 90, normalized size = 1.76 \[ \frac {2 \, e^{\left (-d x - c\right )}}{d {\left (3 \, e^{\left (-d x - c\right )} - 3 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-3 \, d x - 3 \, c\right )} - 1\right )}} - \frac {2}{3 \, d {\left (3 \, e^{\left (-d x - c\right )} - 3 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-3 \, d x - 3 \, c\right )} - 1\right )}} \]

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))

________________________________________________________________________________________

mupad [B]  time = 0.06, size = 25, normalized size = 0.49 \[ -\frac {2\,\left (3\,{\mathrm {e}}^{c+d\,x}-1\right )}{3\,d\,{\left ({\mathrm {e}}^{c+d\,x}-1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(c + d*x) - 1)^2,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 1.23, size = 53, normalized size = 1.04 \[ \begin {cases} \tilde {\infty } x & \text {for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\\frac {x}{\left (1 - \cosh {\relax (c )}\right )^{2}} & \text {for}\: d = 0 \\\frac {1}{2 d \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}} - \frac {1}{6 d \tanh ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}} & \text {otherwise} \end {cases} \]

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(c, -d*x)) & (Eq(d, 0) | Eq(c, -d*x))), (x/(1 - cosh(c))**2, Eq(d, 0)), (1/(2*
d*tanh(c/2 + d*x/2)) - 1/(6*d*tanh(c/2 + d*x/2)**3), True))

________________________________________________________________________________________