∫ 1___ 1+a cos(x) dx [141]

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

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Rubi [A]
time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2738, 211} \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {\sqrt {1-a} \tan \left (\frac {x}{2}\right )}{\sqrt {a+1}}\right )}{\sqrt {1-a^2}} \end {gather*}

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

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}

\begin {align*} \int \frac {1}{1+a \cos (x)} \, dx &=2 \text {Subst}\left (\int \frac {1}{1+a+(1-a) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-a} \tan \left (\frac {x}{2}\right )}{\sqrt {1+a}}\right )}{\sqrt {1-a^2}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 31, normalized size = 0.84 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {(-1+a) \tan \left (\frac {x}{2}\right )}{\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}} \end {gather*}

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method	result	size

default	\(\frac {2 \arctanh \left (\frac {\left (a -1\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (1+a \right ) \left (a -1\right )}}\right )}{\sqrt {\left (1+a \right ) \left (a -1\right )}}\)	\(30\)
risch	\(\frac {\ln \left ({\mathrm e}^{i x}+\frac {i a^{2}-i+\sqrt {a^{2}-1}}{a \sqrt {a^{2}-1}}\right )}{\sqrt {a^{2}-1}}-\frac {\ln \left ({\mathrm e}^{i x}-\frac {i a^{2}-i-\sqrt {a^{2}-1}}{a \sqrt {a^{2}-1}}\right )}{\sqrt {a^{2}-1}}\)	\(90\)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a

ssume' command before evaluation *may* help (example of legal syntax is 'assume(a^2-1.0>0)', see `assume?` for

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Fricas [A]
time = 0.69, size = 111, normalized size = 3.00 \begin {gather*} \left [\frac {\log \left (-\frac {{\left (a^{2} - 2\right )} \cos \left (x\right )^{2} - 2 \, \sqrt {a^{2} - 1} {\left (a + \cos \left (x\right )\right )} \sin \left (x\right ) - 2 \, a^{2} - 2 \, a \cos \left (x\right ) + 1}{a^{2} \cos \left (x\right )^{2} + 2 \, a \cos \left (x\right ) + 1}\right )}{2 \, \sqrt {a^{2} - 1}}, -\frac {\sqrt {-a^{2} + 1} \arctan \left (\frac {\sqrt {-a^{2} + 1} {\left (a + \cos \left (x\right )\right )}}{{\left (a^{2} - 1\right )} \sin \left (x\right )}\right )}{a^{2} - 1}\right ] \end {gather*}

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

2*a*cos(x) + 1))/sqrt(a^2 - 1), -sqrt(-a^2 + 1)*arctan(sqrt(-a^2 + 1)*(a + cos(x))/((a^2 - 1)*sin(x)))/(a^2 -

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (29) = 58\).
time = 1.41, size = 110, normalized size = 2.97 \begin {gather*} \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} \end {gather*}

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*sq

rt(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

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Giac [A]
time = 0.46, size = 53, normalized size = 1.43 \begin {gather*} -\frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\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}} \end {gather*}

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

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Mupad [B]
time = 0.32, size = 28, normalized size = 0.76 \begin {gather*} \frac {2\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {a-1}}{\sqrt {a+1}}\right )}{\sqrt {a-1}\,\sqrt {a+1}} \end {gather*}

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3.2.41 \(\int \frac {1}{1+a \cos (x)} \, dx\) [141]