\(\int \frac {\cosh (x)}{a+a \cosh (x)} \, dx\) [27]

Optimal result

Integrand size = 11, antiderivative size = 18 \[ \int \frac {\cosh (x)}{a+a \cosh (x)} \, dx=\frac {x}{a}-\frac {\sinh (x)}{a+a \cosh (x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2814, 2727} \[ \int \frac {\cosh (x)}{a+a \cosh (x)} \, dx=\frac {x}{a}-\frac {\sinh (x)}{a \cosh (x)+a} \]

[In]

Int[Cosh[x]/(a + a*Cosh[x]),x]

[Out]

x/a - Sinh[x]/(a + a*Cosh[x])

Rule 2727

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 2814

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.61 \[ \int \frac {\cosh (x)}{a+a \cosh (x)} \, dx=\frac {\arcsin (\cosh (x)) \text {csch}(x) \sqrt {-\sinh ^2(x)}-\tanh \left (\frac {x}{2}\right )}{a} \]

[In]

Integrate[Cosh[x]/(a + a*Cosh[x]),x]

[Out]

(ArcSin[Cosh[x]]*Csch[x]*Sqrt[-Sinh[x]^2] - Tanh[x/2])/a

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72

method	result	size
parallelrisch	\(\frac {x -\tanh \left (\frac {x}{2}\right )}{a}\)	\(13\)
risch	\(\frac {x}{a}+\frac {2}{\left ({\mathrm e}^{x}+1\right ) a}\)	\(18\)
default	\(\frac {-\tanh \left (\frac {x}{2}\right )-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}\)	\(28\)

[In]

int(cosh(x)/(a+a*cosh(x)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \[ \int \frac {\cosh (x)}{a+a \cosh (x)} \, dx=\frac {x \cosh \left (x\right ) + x \sinh \left (x\right ) + x + 2}{a \cosh \left (x\right ) + a \sinh \left (x\right ) + a} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.44 \[ \int \frac {\cosh (x)}{a+a \cosh (x)} \, dx=\frac {x}{a} - \frac {\tanh {\left (\frac {x}{2} \right )}}{a} \]

[In]

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

[Out]

x/a - tanh(x/2)/a

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh (x)}{a+a \cosh (x)} \, dx=\frac {x}{a} - \frac {2}{a e^{\left (-x\right )} + a} \]

[In]

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

[Out]

x/a - 2/(a*e^(-x) + a)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {\cosh (x)}{a+a \cosh (x)} \, dx=\frac {x}{a} + \frac {2}{a {\left (e^{x} + 1\right )}} \]

[In]

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

[Out]

x/a + 2/(a*(e^x + 1))

Mupad [B] (verification not implemented)

Time = 1.67 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {\cosh (x)}{a+a \cosh (x)} \, dx=\frac {x}{a}+\frac {2}{a\,\left ({\mathrm {e}}^x+1\right )} \]

[In]

int(cosh(x)/(a + a*cosh(x)),x)

[Out]

x/a + 2/(a*(exp(x) + 1))