Optimal. Leaf size=61 \[ -\frac {2 \sqrt {a-b} \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^2}-\frac {b \tanh ^{-1}(\sin (x))}{a^2}+\frac {\tan (x)}{a} \]
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Rubi [A] time = 0.22, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2723, 3056, 3001, 3770, 2659, 205} \[ -\frac {2 \sqrt {a-b} \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^2}-\frac {b \tanh ^{-1}(\sin (x))}{a^2}+\frac {\tan (x)}{a} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2659
Rule 2723
Rule 3001
Rule 3056
Rule 3770
Rubi steps
\begin {align*} \int \frac {\tan ^2(x)}{a+b \cos (x)} \, dx &=\int \frac {\left (1-\cos ^2(x)\right ) \sec ^2(x)}{a+b \cos (x)} \, dx\\ &=\frac {\tan (x)}{a}+\frac {\int \frac {(-b-a \cos (x)) \sec (x)}{a+b \cos (x)} \, dx}{a}\\ &=\frac {\tan (x)}{a}-\frac {b \int \sec (x) \, dx}{a^2}+\frac {\left (-a^2+b^2\right ) \int \frac {1}{a+b \cos (x)} \, dx}{a^2}\\ &=-\frac {b \tanh ^{-1}(\sin (x))}{a^2}+\frac {\tan (x)}{a}+\frac {\left (2 \left (-a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{a^2}\\ &=-\frac {2 \sqrt {a-b} \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^2}-\frac {b \tanh ^{-1}(\sin (x))}{a^2}+\frac {\tan (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 85, normalized size = 1.39 \[ \frac {-2 \sqrt {b^2-a^2} \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {x}{2}\right )}{\sqrt {b^2-a^2}}\right )+a \tan (x)+b \left (\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right )}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 203, normalized size = 3.33 \[ \left [-\frac {b \cos \relax (x) \log \left (\sin \relax (x) + 1\right ) - b \cos \relax (x) \log \left (-\sin \relax (x) + 1\right ) - \sqrt {-a^{2} + b^{2}} \cos \relax (x) \log \left (\frac {2 \, a b \cos \relax (x) + {\left (2 \, a^{2} - b^{2}\right )} \cos \relax (x)^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \relax (x) + b\right )} \sin \relax (x) - a^{2} + 2 \, b^{2}}{b^{2} \cos \relax (x)^{2} + 2 \, a b \cos \relax (x) + a^{2}}\right ) - 2 \, a \sin \relax (x)}{2 \, a^{2} \cos \relax (x)}, -\frac {b \cos \relax (x) \log \left (\sin \relax (x) + 1\right ) - b \cos \relax (x) \log \left (-\sin \relax (x) + 1\right ) + 2 \, \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \relax (x) + b}{\sqrt {a^{2} - b^{2}} \sin \relax (x)}\right ) \cos \relax (x) - 2 \, a \sin \relax (x)}{2 \, a^{2} \cos \relax (x)}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.48, size = 111, normalized size = 1.82 \[ -\frac {b \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right )}{a^{2}} + \frac {b \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) - 1 \right |}\right )}{a^{2}} + \frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, x\right ) - b \tan \left (\frac {1}{2} \, x\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )} \sqrt {a^{2} - b^{2}}}{a^{2}} - \frac {2 \, \tan \left (\frac {1}{2} \, x\right )}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 1\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 129, normalized size = 2.11 \[ -\frac {2 \arctan \left (\frac {\tan \left (\frac {x}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {2 \arctan \left (\frac {\tan \left (\frac {x}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) b^{2}}{a^{2} \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {1}{a \left (\tan \left (\frac {x}{2}\right )-1\right )}+\frac {b \ln \left (\tan \left (\frac {x}{2}\right )-1\right )}{a^{2}}-\frac {1}{a \left (\tan \left (\frac {x}{2}\right )+1\right )}-\frac {b \ln \left (\tan \left (\frac {x}{2}\right )+1\right )}{a^{2}} \]
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 = 0.55, size = 77, normalized size = 1.26 \[ \frac {2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )\,\sqrt {b^2-a^2}}{a\,\cos \left (\frac {x}{2}\right )+b\,\cos \left (\frac {x}{2}\right )}\right )\,\sqrt {b^2-a^2}}{a^2}-\frac {2\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{a^2}+\frac {\sin \relax (x)}{a\,\cos \relax (x)} \]
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 \[ \int \frac {\tan ^{2}{\relax (x )}}{a + b \cos {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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