∫ 1___ a+b cos(x) dx [3]

Optimal. Leaf size=42 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}} \]

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Rubi [A]
time = 0.03, antiderivative size = 42, 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 \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}} \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}{a+b \cos (x)} \, dx &=2 \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 41, normalized size = 0.98 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}} \end {gather*}

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 3.77, size = 228, normalized size = 5.43 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [\text {Log}\left [-1+\text {Tan}\left [\frac {x}{2}\right ]\right ]-\text {Log}\left [1+\text {Tan}\left [\frac {x}{2}\right ]\right ]\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{\frac {1}{b \text {Tan}\left [\frac {x}{2}\right ]},a\text {==}-b\right \},\left \{\frac {\text {Tan}\left [\frac {x}{2}\right ]}{b},a\text {==}b\right \}\right \},-\frac {\text {Log}\left [\sqrt {-\frac {a}{a-b}-\frac {b}{a-b}}+\text {Tan}\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 {\text {Log}\left [\text {Tan}\left [\frac {x}{2}\right ]-\sqrt {-\frac {a}{a-b}-\frac {b}{a-b}}\right ]}{a \sqrt {-\frac {a}{a-b}-\frac {b}{a-b}}-b \sqrt {-\frac {a}{a-b}-\frac {b}{a-b}}}\right ] \end {gather*}

Piecewise[{{DirectedInfinity[Log[-1 + Tan[x / 2]] - Log[1 + Tan[x / 2]]], a == 0 && b == 0}, {1 / (b Tan[x / 2

]), a == -b}, {Tan[x / 2] / b, a == b}}, -Log[Sqrt[-a / (a - b) - b / (a - b)] + Tan[x / 2]] / (a Sqrt[-a / (a

- b) - b / (a - b)] - b Sqrt[-a / (a - b) - b / (a - b)]) + Log[Tan[x / 2] - Sqrt[-a / (a - b) - b / (a - b)]

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method	result	size

default	\(\frac {2 \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\)	\(36\)
risch	\(-\frac {\ln \left ({\mathrm e}^{i x}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}}+\frac {\ln \left ({\mathrm e}^{i x}-\frac {i a^{2}-i b^{2}-a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}}\)	\(125\)

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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(4*b^2-4*a^2>0)', see `assume?`

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Fricas [A]
time = 0.33, size = 137, normalized size = 3.26 \begin {gather*} \left [-\frac {\sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (x\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (x\right ) + b\right )} \sin \left (x\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (x\right )^{2} + 2 \, a b \cos \left (x\right ) + a^{2}}\right )}{2 \, {\left (a^{2} - b^{2}\right )}}, \frac {\arctan \left (-\frac {a \cos \left (x\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (x\right )}\right )}{\sqrt {a^{2} - b^{2}}}\right ] \end {gather*}

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

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

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Sympy [A]
time = 1.82, size = 144, normalized size = 3.43 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )} + \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {1}{b \tan {\left (\frac {x}{2} \right )}} & \text {for}\: a = - b \\\frac {\tan {\left (\frac {x}{2} \right )}}{b} & \text {for}\: a = b \\\frac {\log {\left (- \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} + \tan {\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}} + \tan {\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} \end {gather*}

Piecewise((zoo*(-log(tan(x/2) - 1) + log(tan(x/2) + 1)), Eq(a, 0) & Eq(b, 0)), (1/(b*tan(x/2)), Eq(a, -b)), (t

an(x/2)/b, Eq(a, b)), (log(-sqrt(-a/(a - b) - b/(a - b)) + tan(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)) + tan(x/2))/(a*sqrt(-a/(a - b) - b/(a - b)) - b*sq

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Giac [A]
time = 0.00, size = 72, normalized size = 1.71 \begin {gather*} -\frac {2\cdot 2 \left (\arctan \left (\frac {-a \tan \left (\frac {x}{2}\right )+b \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2}-b^{2}}}\right )+\pi \mathrm {sign}\left (2 b-2 a\right ) \left \lfloor \frac {x}{2 \pi }+\frac 1{2}\right \rfloor \right )}{2 \sqrt {a^{2}-b^{2}}} \end {gather*}

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

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Mupad [B]
time = 0.59, size = 38, normalized size = 0.90 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\left (2\,a-2\,b\right )}{2\,\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}} \end {gather*}

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3.1.3 \(\int \frac {1}{a+b \cos (x)} \, dx\) [3]