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[c + d*x]/(d*(1 - Cosh[c + d*x])))

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Rubi [A]  time = 0.0109634, antiderivative size = 23, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0229011, 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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Maple [A]  time = 0.01, size = 16, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}} \end{align*}

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 = 1.06642, size = 24, normalized size = 1.04 \begin{align*} -\frac{2}{d{\left (e^{\left (-d x - c\right )} - 1\right )}} \end{align*}

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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Fricas [A]  time = 1.82494, size = 58, normalized size = 2.52 \begin{align*} \frac{2}{d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right ) - d} \end{align*}

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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Sympy [A]  time = 0.712213, size = 32, normalized size = 1.39 \begin{align*} \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{\left (c \right )}} & \text{for}\: d = 0 \\\frac{1}{d \tanh{\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)),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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Giac [A]  time = 1.18187, size = 20, normalized size = 0.87 \begin{align*} \frac{2}{d{\left (e^{\left (d x + c\right )} - 1\right )}} \end{align*}

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))