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

Optimal. Leaf size=20 \[ \frac{\sinh (c+d x)}{d (\cosh (c+d x)+1)} \]

[Out]

Sinh[c + d*x]/(d*(1 + Cosh[c + d*x]))

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Rubi [A]  time = 0.0100075, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2648} \[ \frac{\sinh (c+d x)}{d (\cosh (c+d x)+1)} \]

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 = 0.0141596, size = 14, normalized size = 0.7 \[ \frac{\tanh \left (\frac{1}{2} (c+d x)\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cosh[c + d*x])^(-1),x]

[Out]

Tanh[(c + d*x)/2]/d

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Maple [A]  time = 0.01, size = 14, normalized size = 0.7 \begin{align*}{\frac{1}{d}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) } \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 = 1.16488, size = 24, normalized size = 1.2 \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 = 1.73141, size = 59, normalized size = 2.95 \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 = 0.592076, size = 17, normalized size = 0.85 \begin{align*} \begin{cases} \frac{\tanh{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{d} & \text{for}\: d \neq 0 \\\frac{x}{\cosh{\left (c \right )} + 1} & \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((tanh(c/2 + d*x/2)/d, Ne(d, 0)), (x/(cosh(c) + 1), True))

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Giac [A]  time = 1.16337, size = 20, normalized size = 1. \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))