3.26 \(\int \frac{\cosh ^2(x)}{a+a \cosh (x)} \, dx\)

Optimal. Leaf size=25 \[ -\frac{x}{a}+\frac{\sinh (x)}{a}+\frac{\sinh (x)}{a (\cosh (x)+1)} \]

[Out]

-(x/a) + Sinh[x]/a + Sinh[x]/(a*(1 + Cosh[x]))

________________________________________________________________________________________

Rubi [A]  time = 0.0693868, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2746, 12, 2735, 2648} \[ -\frac{x}{a}+\frac{\sinh (x)}{a}+\frac{\sinh (x)}{a (\cosh (x)+1)} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^2/(a + a*Cosh[x]),x]

[Out]

-(x/a) + Sinh[x]/a + Sinh[x]/(a*(1 + Cosh[x]))

Rule 2746

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b^2
*Cos[e + f*x])/(d*f), x] + Dist[1/d, Int[Simp[a^2*d - b*(b*c - 2*a*d)*Sin[e + f*x], x]/(c + d*Sin[e + f*x]), x
], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

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{\cosh ^2(x)}{a+a \cosh (x)} \, dx &=\frac{\sinh (x)}{a}-\frac{\int \frac{a \cosh (x)}{a+a \cosh (x)} \, dx}{a}\\ &=\frac{\sinh (x)}{a}-\int \frac{\cosh (x)}{a+a \cosh (x)} \, dx\\ &=-\frac{x}{a}+\frac{\sinh (x)}{a}+\int \frac{1}{a+a \cosh (x)} \, dx\\ &=-\frac{x}{a}+\frac{\sinh (x)}{a}+\frac{\sinh (x)}{a+a \cosh (x)}\\ \end{align*}

Mathematica [A]  time = 0.0544024, size = 32, normalized size = 1.28 \[ \frac{-2 x+3 \tanh \left (\frac{x}{2}\right )+\sinh \left (\frac{3 x}{2}\right ) \text{sech}\left (\frac{x}{2}\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^2/(a + a*Cosh[x]),x]

[Out]

(-2*x + Sech[x/2]*Sinh[(3*x)/2] + 3*Tanh[x/2])/(2*a)

________________________________________________________________________________________

Maple [B]  time = 0.02, size = 59, normalized size = 2.4 \begin{align*}{\frac{1}{a}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{1}{a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^2/(a+a*cosh(x)),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.17736, size = 55, normalized size = 2.2 \begin{align*} -\frac{x}{a} + \frac{5 \, e^{\left (-x\right )} + 1}{2 \,{\left (a e^{\left (-x\right )} + a e^{\left (-2 \, x\right )}\right )}} - \frac{e^{\left (-x\right )}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^2/(a+a*cosh(x)),x, algorithm="maxima")

[Out]

-x/a + 1/2*(5*e^(-x) + 1)/(a*e^(-x) + a*e^(-2*x)) - 1/2*e^(-x)/a

________________________________________________________________________________________

Fricas [A]  time = 1.90001, size = 151, normalized size = 6.04 \begin{align*} -\frac{2 \, x \cosh \left (x\right ) - \cosh \left (x\right )^{2} + 2 \,{\left (x - \cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) - \sinh \left (x\right )^{2} + 2 \, x + 5}{2 \,{\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^2/(a+a*cosh(x)),x, algorithm="fricas")

[Out]

-1/2*(2*x*cosh(x) - cosh(x)^2 + 2*(x - cosh(x) - 1)*sinh(x) - sinh(x)^2 + 2*x + 5)/(a*cosh(x) + a*sinh(x) + a)

________________________________________________________________________________________

Sympy [B]  time = 0.896657, size = 63, normalized size = 2.52 \begin{align*} - \frac{x \tanh ^{2}{\left (\frac{x}{2} \right )}}{a \tanh ^{2}{\left (\frac{x}{2} \right )} - a} + \frac{x}{a \tanh ^{2}{\left (\frac{x}{2} \right )} - a} + \frac{\tanh ^{3}{\left (\frac{x}{2} \right )}}{a \tanh ^{2}{\left (\frac{x}{2} \right )} - a} - \frac{3 \tanh{\left (\frac{x}{2} \right )}}{a \tanh ^{2}{\left (\frac{x}{2} \right )} - a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)**2/(a+a*cosh(x)),x)

[Out]

-x*tanh(x/2)**2/(a*tanh(x/2)**2 - a) + x/(a*tanh(x/2)**2 - a) + tanh(x/2)**3/(a*tanh(x/2)**2 - a) - 3*tanh(x/2
)/(a*tanh(x/2)**2 - a)

________________________________________________________________________________________

Giac [A]  time = 1.16933, size = 47, normalized size = 1.88 \begin{align*} -\frac{x}{a} - \frac{{\left (5 \, e^{x} + 1\right )} e^{\left (-x\right )}}{2 \, a{\left (e^{x} + 1\right )}} + \frac{e^{x}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^2/(a+a*cosh(x)),x, algorithm="giac")

[Out]

-x/a - 1/2*(5*e^x + 1)*e^(-x)/(a*(e^x + 1)) + 1/2*e^x/a