3.141 \(\int \frac{1}{1+a \cos (x)} \, dx\)

Optimal. Leaf size=37 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{1-a} \tan \left (\frac{x}{2}\right )}{\sqrt{a+1}}\right )}{\sqrt{1-a^2}} \]

[Out]

(2*ArcTan[(Sqrt[1 - a]*Tan[x/2])/Sqrt[1 + a]])/Sqrt[1 - a^2]

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Rubi [A]  time = 0.0639, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{1-a} \tan \left (\frac{x}{2}\right )}{\sqrt{a+1}}\right )}{\sqrt{1-a^2}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + a*Cos[x])^(-1),x]

[Out]

(2*ArcTan[(Sqrt[1 - a]*Tan[x/2])/Sqrt[1 + a]])/Sqrt[1 - a^2]

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Rubi in Sympy [A]  time = 2.36815, size = 34, normalized size = 0.92 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{- a + 1} \tan{\left (\frac{x}{2} \right )}}{\sqrt{a + 1}} \right )}}{\sqrt{- a + 1} \sqrt{a + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*atan(sqrt(-a + 1)*tan(x/2)/sqrt(a + 1))/(sqrt(-a + 1)*sqrt(a + 1))

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Mathematica [A]  time = 0.0297242, size = 31, normalized size = 0.84 \[ \frac{2 \tanh ^{-1}\left (\frac{(a-1) \tan \left (\frac{x}{2}\right )}{\sqrt{a^2-1}}\right )}{\sqrt{a^2-1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*ArcTanh[((-1 + a)*Tan[x/2])/Sqrt[-1 + a^2]])/Sqrt[-1 + a^2]

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Maple [A]  time = 0.02, size = 30, normalized size = 0.8 \[ 2\,{\frac{1}{\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) }}{\it Artanh} \left ({\frac{ \left ( a-1 \right ) \tan \left ( x/2 \right ) }{\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+a*cos(x)),x)

[Out]

2/((1+a)*(a-1))^(1/2)*arctanh((a-1)*tan(1/2*x)/((1+a)*(a-1))^(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/(a*cos(x) + 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227078, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{2 \,{\left (a^{3} +{\left (a^{2} - 1\right )} \cos \left (x\right ) - a\right )} \sin \left (x\right ) -{\left ({\left (a^{2} - 2\right )} \cos \left (x\right )^{2} - 2 \, a^{2} - 2 \, a \cos \left (x\right ) + 1\right )} \sqrt{a^{2} - 1}}{a^{2} \cos \left (x\right )^{2} + 2 \, a \cos \left (x\right ) + 1}\right )}{2 \, \sqrt{a^{2} - 1}}, \frac{\arctan \left (\frac{\sqrt{-a^{2} + 1}{\left (a + \cos \left (x\right )\right )}}{{\left (a^{2} - 1\right )} \sin \left (x\right )}\right )}{\sqrt{-a^{2} + 1}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*log((2*(a^3 + (a^2 - 1)*cos(x) - a)*sin(x) - ((a^2 - 2)*cos(x)^2 - 2*a^2 -
2*a*cos(x) + 1)*sqrt(a^2 - 1))/(a^2*cos(x)^2 + 2*a*cos(x) + 1))/sqrt(a^2 - 1), a
rctan(sqrt(-a^2 + 1)*(a + cos(x))/((a^2 - 1)*sin(x)))/sqrt(-a^2 + 1)]

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Sympy [A]  time = 11.1774, size = 110, normalized size = 2.97 \[ \begin{cases} - \frac{1}{\tan{\left (\frac{x}{2} \right )}} & \text{for}\: a = -1 \\\tan{\left (\frac{x}{2} \right )} & \text{for}\: a = 1 \\- \frac{\log{\left (- \sqrt{\frac{a}{a - 1} + \frac{1}{a - 1}} + \tan{\left (\frac{x}{2} \right )} \right )}}{a \sqrt{\frac{a}{a - 1} + \frac{1}{a - 1}} - \sqrt{\frac{a}{a - 1} + \frac{1}{a - 1}}} + \frac{\log{\left (\sqrt{\frac{a}{a - 1} + \frac{1}{a - 1}} + \tan{\left (\frac{x}{2} \right )} \right )}}{a \sqrt{\frac{a}{a - 1} + \frac{1}{a - 1}} - \sqrt{\frac{a}{a - 1} + \frac{1}{a - 1}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+a*cos(x)),x)

[Out]

Piecewise((-1/tan(x/2), Eq(a, -1)), (tan(x/2), Eq(a, 1)), (-log(-sqrt(a/(a - 1)
+ 1/(a - 1)) + tan(x/2))/(a*sqrt(a/(a - 1) + 1/(a - 1)) - sqrt(a/(a - 1) + 1/(a
- 1))) + log(sqrt(a/(a - 1) + 1/(a - 1)) + tan(x/2))/(a*sqrt(a/(a - 1) + 1/(a -
1)) - sqrt(a/(a - 1) + 1/(a - 1))), True))

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GIAC/XCAS [A]  time = 0.215294, size = 72, normalized size = 1.95 \[ -\frac{2 \,{\left (\pi \left \lfloor \frac{x}{2 \, \pi } + \frac{1}{2} \right \rfloor{\rm sign}\left (2 \, a - 2\right ) + \arctan \left (\frac{a \tan \left (\frac{1}{2} \, x\right ) - \tan \left (\frac{1}{2} \, x\right )}{\sqrt{-a^{2} + 1}}\right )\right )}}{\sqrt{-a^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(pi*floor(1/2*x/pi + 1/2)*sign(2*a - 2) + arctan((a*tan(1/2*x) - tan(1/2*x))/
sqrt(-a^2 + 1)))/sqrt(-a^2 + 1)