3.583 \(\int \frac{1}{a+b \cosh (x)} \, dx\)

Optimal. Leaf size=41 \[ \frac{2 \tanh ^{-1}\left (\frac{(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}} \]

[Out]

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

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Rubi [A]  time = 0.0843094, antiderivative size = 42, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Cosh[x])^(-1),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{a + b \cosh{\left (x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*cosh(x)),x)

[Out]

Integral(1/(a + b*cosh(x)), x)

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Mathematica [A]  time = 0.0379954, size = 41, normalized size = 1. \[ -\frac{2 \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Cosh[x])^(-1),x]

[Out]

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

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Maple [A]  time = 0.017, size = 36, normalized size = 0.9 \[ 2\,{\frac{1}{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tanh \left ( x/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*cosh(x)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*cosh(x) + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226393, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{2 \, a^{3} - 2 \, a b^{2} + 2 \,{\left (a^{2} b - b^{3}\right )} \cosh \left (x\right ) + 2 \,{\left (a^{2} b - b^{3}\right )} \sinh \left (x\right ) -{\left (b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} - b^{2} + 2 \,{\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )\right )} \sqrt{a^{2} - b^{2}}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \,{\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b}\right )}{\sqrt{a^{2} - b^{2}}}, \frac{2 \, \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{a^{2} - b^{2}}\right )}{\sqrt{-a^{2} + b^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*cosh(x) + a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 20.1161, size = 126, normalized size = 3.07 \[ \begin{cases} \tilde{\infty } \operatorname{atan}{\left (\tanh{\left (\frac{x}{2} \right )} \right )} & \text{for}\: a = 0 \wedge b = 0 \\\frac{\tanh{\left (\frac{x}{2} \right )}}{b} & \text{for}\: a = b \\- \frac{1}{b \tanh{\left (\frac{x}{2} \right )}} & \text{for}\: a = - b \\- \frac{\log{\left (- \sqrt{\frac{a}{a - b} + \frac{b}{a - b}} + \tanh{\left (\frac{x}{2} \right )} \right )}}{a \sqrt{\frac{a}{a - b} + \frac{b}{a - b}} - b \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}} + \frac{\log{\left (\sqrt{\frac{a}{a - b} + \frac{b}{a - b}} + \tanh{\left (\frac{x}{2} \right )} \right )}}{a \sqrt{\frac{a}{a - b} + \frac{b}{a - b}} - b \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*cosh(x)),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.215767, size = 43, normalized size = 1.05 \[ \frac{2 \, \arctan \left (\frac{b e^{x} + a}{\sqrt{-a^{2} + b^{2}}}\right )}{\sqrt{-a^{2} + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*cosh(x) + a),x, algorithm="giac")

[Out]

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