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

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Rubi [A]  time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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*}

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Mathematica [A]  time = 0.02, size = 14, normalized size = 0.70 \[ \frac {\tanh \left (\frac {1}{2} (c+d x)\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cosh[c + d*x])^(-1),x]

[Out]

Tanh[(c + d*x)/2]/d

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fricas [A]  time = 0.43, size = 22, normalized size = 1.10 \[ -\frac {2}{d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right ) + d} \]

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 = 0.75 \[ -\frac {2}{d {\left (e^{\left (d x + c\right )} + 1\right )}} \]

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 = 0.05, size = 14, normalized size = 0.70 \[ \frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.31, size = 18, normalized size = 0.90 \[ \frac {2}{d {\left (e^{\left (-d x - c\right )} + 1\right )}} \]

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 = 0.75 \[ -\frac {2}{d\,\left ({\mathrm {e}}^{c+d\,x}+1\right )} \]

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 = 0.52, size = 17, normalized size = 0.85 \[ \begin {cases} \frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{d} & \text {for}\: d \neq 0 \\\frac {x}{\cosh {\relax (c )} + 1} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((tanh(c/2 + d*x/2)/d, Ne(d, 0)), (x/(cosh(c) + 1), True))

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