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

Optimal. Leaf size=23 \[ -\frac {\sinh (c+d x)}{d (1-\cosh (c+d x))} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2648} \[ -\frac {\sinh (c+d x)}{d (1-\cosh (c+d x))} \]

Antiderivative was successfully verified.

[In]

Int[(1 - Cosh[c + d*x])^(-1),x]

[Out]

-(Sinh[c + d*x]/(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]

Rubi steps

\begin {align*} \int \frac {1}{1-\cosh (c+d x)} \, dx &=-\frac {\sinh (c+d x)}{d (1-\cosh (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 14, normalized size = 0.61 \[ \frac {\coth \left (\frac {1}{2} (c+d x)\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Cosh[c + d*x])^(-1),x]

[Out]

Coth[(c + d*x)/2]/d

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fricas [A]  time = 0.58, size = 24, normalized size = 1.04 \[ \frac {2}{d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right ) - d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.14, size = 15, normalized size = 0.65 \[ \frac {2}{d {\left (e^{\left (d x + c\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(d*(e^(d*x + c) - 1))

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maple [A]  time = 0.06, size = 16, normalized size = 0.70 \[ \frac {1}{d \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d/tanh(1/2*d*x+1/2*c)

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maxima [A]  time = 0.39, size = 18, normalized size = 0.78 \[ -\frac {2}{d {\left (e^{\left (-d x - c\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(d*(e^(-d*x - c) - 1))

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mupad [B]  time = 0.89, size = 15, normalized size = 0.65 \[ \frac {2}{d\,\left ({\mathrm {e}}^{c+d\,x}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(cosh(c + d*x) - 1),x)

[Out]

2/(d*(exp(c + d*x) - 1))

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sympy [A]  time = 0.63, size = 32, normalized size = 1.39 \[ \begin {cases} \tilde {\infty } x & \text {for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\\frac {x}{1 - \cosh {\relax (c )}} & \text {for}\: d = 0 \\\frac {1}{d \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x, (Eq(c, 0) | Eq(c, -d*x)) & (Eq(d, 0) | Eq(c, -d*x))), (x/(1 - cosh(c)), Eq(d, 0)), (1/(d*tan
h(c/2 + d*x/2)), True))

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