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

Optimal. Leaf size=52 \[ \frac{x}{b}-\frac{2 a \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{b \sqrt{a-b} \sqrt{a+b}} \]

[Out]

x/b - (2*a*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b])

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Rubi [A]  time = 0.0525138, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2735, 2659, 208} \[ \frac{x}{b}-\frac{2 a \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{b \sqrt{a-b} \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]/(a + b*Cosh[x]),x]

[Out]

x/b - (2*a*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b])

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 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\cosh (x)}{a+b \cosh (x)} \, dx &=\frac{x}{b}-\frac{a \int \frac{1}{a+b \cosh (x)} \, dx}{b}\\ &=\frac{x}{b}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{b}\\ &=\frac{x}{b}-\frac{2 a \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} b \sqrt{a+b}}\\ \end{align*}

Mathematica [A]  time = 0.0448116, size = 48, normalized size = 0.92 \[ \frac{\frac{2 a \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}}+x}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]/(a + b*Cosh[x]),x]

[Out]

(x + (2*a*ArcTan[((a - b)*Tanh[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2])/b

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Maple [A]  time = 0.014, size = 64, normalized size = 1.2 \begin{align*} -2\,{\frac{a}{b\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tanh \left ( x/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }+{\frac{1}{b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{b}\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)/(a+b*cosh(x)),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.30895, size = 535, normalized size = 10.29 \begin{align*} \left [\frac{\sqrt{a^{2} - b^{2}} a \log \left (\frac{b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} - b^{2} + 2 \,{\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt{a^{2} - b^{2}}{\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \,{\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b}\right ) +{\left (a^{2} - b^{2}\right )} x}{a^{2} b - b^{3}}, \frac{2 \, \sqrt{-a^{2} + b^{2}} a \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{a^{2} - b^{2}}\right ) +{\left (a^{2} - b^{2}\right )} x}{a^{2} b - b^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(sqrt(a^2 - b^2)*a*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) + 2*a^2 - b^2 + 2*(b^2*cosh(x) + a*b)*s
inh(x) + 2*sqrt(a^2 - b^2)*(b*cosh(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x
) + a)*sinh(x) + b)) + (a^2 - b^2)*x)/(a^2*b - b^3), (2*sqrt(-a^2 + b^2)*a*arctan(-sqrt(-a^2 + b^2)*(b*cosh(x)
+ b*sinh(x) + a)/(a^2 - b^2)) + (a^2 - b^2)*x)/(a^2*b - b^3)]

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Sympy [A]  time = 135.171, size = 241, normalized size = 4.63 \begin{align*} \begin{cases} \tilde{\infty } x & \text{for}\: a = 0 \wedge b = 0 \\\frac{x}{b} - \frac{\tanh{\left (\frac{x}{2} \right )}}{b} & \text{for}\: a = b \\\frac{x}{b} - \frac{1}{b \tanh{\left (\frac{x}{2} \right )}} & \text{for}\: a = - b \\\frac{\sinh{\left (x \right )}}{a} & \text{for}\: b = 0 \\\frac{a x \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}}{a b \sqrt{\frac{a}{a - b} + \frac{b}{a - b}} - b^{2} \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}} + \frac{a \log{\left (- \sqrt{\frac{a}{a - b} + \frac{b}{a - b}} + \tanh{\left (\frac{x}{2} \right )} \right )}}{a b \sqrt{\frac{a}{a - b} + \frac{b}{a - b}} - b^{2} \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}} - \frac{a \log{\left (\sqrt{\frac{a}{a - b} + \frac{b}{a - b}} + \tanh{\left (\frac{x}{2} \right )} \right )}}{a b \sqrt{\frac{a}{a - b} + \frac{b}{a - b}} - b^{2} \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}} - \frac{b x \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}}{a b \sqrt{\frac{a}{a - b} + \frac{b}{a - b}} - b^{2} \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0)), (x/b - tanh(x/2)/b, Eq(a, b)), (x/b - 1/(b*tanh(x/2)), Eq(a, -b)), (si
nh(x)/a, Eq(b, 0)), (a*x*sqrt(a/(a - b) + b/(a - b))/(a*b*sqrt(a/(a - b) + b/(a - b)) - b**2*sqrt(a/(a - b) +
b/(a - b))) + a*log(-sqrt(a/(a - b) + b/(a - b)) + tanh(x/2))/(a*b*sqrt(a/(a - b) + b/(a - b)) - b**2*sqrt(a/(
a - b) + b/(a - b))) - a*log(sqrt(a/(a - b) + b/(a - b)) + tanh(x/2))/(a*b*sqrt(a/(a - b) + b/(a - b)) - b**2*
sqrt(a/(a - b) + b/(a - b))) - b*x*sqrt(a/(a - b) + b/(a - b))/(a*b*sqrt(a/(a - b) + b/(a - b)) - b**2*sqrt(a/
(a - b) + b/(a - b))), True))

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Giac [A]  time = 1.19825, size = 57, normalized size = 1.1 \begin{align*} -\frac{2 \, a \arctan \left (\frac{b e^{x} + a}{\sqrt{-a^{2} + b^{2}}}\right )}{\sqrt{-a^{2} + b^{2}} b} + \frac{x}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*arctan((b*e^x + a)/sqrt(-a^2 + b^2))/(sqrt(-a^2 + b^2)*b) + x/b