3.27 \(\int \frac {\cosh (x)}{a+a \cosh (x)} \, dx\)

Optimal. Leaf size=18 \[ \frac {x}{a}-\frac {\sinh (x)}{a \cosh (x)+a} \]

[Out]

x/a-sinh(x)/(a+a*cosh(x))

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Rubi [A]  time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2735, 2648} \[ \frac {x}{a}-\frac {\sinh (x)}{a \cosh (x)+a} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]/(a + a*Cosh[x]),x]

[Out]

x/a - Sinh[x]/(a + a*Cosh[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 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 14, normalized size = 0.78 \[ \frac {x-\tanh \left (\frac {x}{2}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]/(a + a*Cosh[x]),x]

[Out]

(x - Tanh[x/2])/a

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fricas [A]  time = 0.38, size = 24, normalized size = 1.33 \[ \frac {x \cosh \relax (x) + x \sinh \relax (x) + x + 2}{a \cosh \relax (x) + a \sinh \relax (x) + a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)/(a+a*cosh(x)),x, algorithm="fricas")

[Out]

(x*cosh(x) + x*sinh(x) + x + 2)/(a*cosh(x) + a*sinh(x) + a)

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giac [A]  time = 0.12, size = 17, normalized size = 0.94 \[ \frac {x}{a} + \frac {2}{a {\left (e^{x} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)/(a+a*cosh(x)),x, algorithm="giac")

[Out]

x/a + 2/(a*(e^x + 1))

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maple [A]  time = 0.05, size = 34, normalized size = 1.89 \[ -\frac {\tanh \left (\frac {x}{2}\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)/(a+a*cosh(x)),x)

[Out]

-1/a*tanh(1/2*x)-1/a*ln(tanh(1/2*x)-1)+1/a*ln(tanh(1/2*x)+1)

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maxima [A]  time = 0.31, size = 18, normalized size = 1.00 \[ \frac {x}{a} - \frac {2}{a e^{\left (-x\right )} + a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)/(a+a*cosh(x)),x, algorithm="maxima")

[Out]

x/a - 2/(a*e^(-x) + a)

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mupad [B]  time = 0.87, size = 17, normalized size = 0.94 \[ \frac {x}{a}+\frac {2}{a\,\left ({\mathrm {e}}^x+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)/(a + a*cosh(x)),x)

[Out]

x/a + 2/(a*(exp(x) + 1))

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sympy [A]  time = 0.33, size = 8, normalized size = 0.44 \[ \frac {x}{a} - \frac {\tanh {\left (\frac {x}{2} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)/(a+a*cosh(x)),x)

[Out]

x/a - tanh(x/2)/a

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