Optimal. Leaf size=77 \[ -\frac {a \cot (x)}{a^2-b^2}+\frac {b \csc (x)}{a^2-b^2}-\frac {2 a^2 \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}} \]
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Rubi [A] time = 0.10, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2727, 3767, 8, 2606, 2659, 205} \[ -\frac {a \cot (x)}{a^2-b^2}+\frac {b \csc (x)}{a^2-b^2}-\frac {2 a^2 \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 205
Rule 2606
Rule 2659
Rule 2727
Rule 3767
Rubi steps
\begin {align*} \int \frac {\cot ^2(x)}{a+b \cos (x)} \, dx &=\frac {a \int \csc ^2(x) \, dx}{a^2-b^2}-\frac {a^2 \int \frac {1}{a+b \cos (x)} \, dx}{a^2-b^2}-\frac {b \int \cot (x) \csc (x) \, dx}{a^2-b^2}\\ &=-\frac {a \operatorname {Subst}(\int 1 \, dx,x,\cot (x))}{a^2-b^2}-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{a^2-b^2}+\frac {b \operatorname {Subst}(\int 1 \, dx,x,\csc (x))}{a^2-b^2}\\ &=-\frac {2 a^2 \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}}-\frac {a \cot (x)}{a^2-b^2}+\frac {b \csc (x)}{a^2-b^2}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 67, normalized size = 0.87 \[ \frac {b \csc (x)-a \cot (x)}{a^2-b^2}-\frac {2 a^2 \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {x}{2}\right )}{\sqrt {b^2-a^2}}\right )}{\left (b^2-a^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 230, normalized size = 2.99 \[ \left [\frac {\sqrt {-a^{2} + b^{2}} a^{2} \log \left (\frac {2 \, a b \cos \relax (x) + {\left (2 \, a^{2} - b^{2}\right )} \cos \relax (x)^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \relax (x) + b\right )} \sin \relax (x) - a^{2} + 2 \, b^{2}}{b^{2} \cos \relax (x)^{2} + 2 \, a b \cos \relax (x) + a^{2}}\right ) \sin \relax (x) + 2 \, a^{2} b - 2 \, b^{3} - 2 \, {\left (a^{3} - a b^{2}\right )} \cos \relax (x)}{2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \relax (x)}, -\frac {\sqrt {a^{2} - b^{2}} a^{2} \arctan \left (-\frac {a \cos \relax (x) + b}{\sqrt {a^{2} - b^{2}} \sin \relax (x)}\right ) \sin \relax (x) - a^{2} b + b^{3} + {\left (a^{3} - a b^{2}\right )} \cos \relax (x)}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \relax (x)}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 91, normalized size = 1.18 \[ \frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, x\right ) - b \tan \left (\frac {1}{2} \, x\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )} a^{2}}{{\left (a^{2} - b^{2}\right )}^{\frac {3}{2}}} + \frac {\tan \left (\frac {1}{2} \, x\right )}{2 \, {\left (a - b\right )}} - \frac {1}{2 \, {\left (a + b\right )} \tan \left (\frac {1}{2} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 78, normalized size = 1.01 \[ \frac {\tan \left (\frac {x}{2}\right )}{2 a -2 b}-\frac {2 a^{2} \arctan \left (\frac {\tan \left (\frac {x}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {1}{2 \left (a +b \right ) \tan \left (\frac {x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 86, normalized size = 1.12 \[ \frac {\mathrm {tan}\left (\frac {x}{2}\right )}{2\,a-2\,b}-\frac {2\,a^2\,\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^2-b^2\right )}{{\left (a+b\right )}^{3/2}\,\sqrt {a-b}}\right )}{{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}-\frac {a-b}{\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a+b\right )\,\left (2\,a-2\,b\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{2}{\relax (x )}}{a + b \cos {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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