Optimal. Leaf size=41 \[ -\frac {\cot (x)}{a+b}-\frac {b \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3191, 388, 205} \[ -\frac {\cot (x)}{a+b}-\frac {b \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rule 3191
Rubi steps
\begin {align*} \int \frac {\csc ^2(x)}{a+b \cos ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1+x^2}{a+(a+b) x^2} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{a+b}-\frac {b \operatorname {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\cot (x)\right )}{a+b}\\ &=-\frac {b \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^{3/2}}-\frac {\cot (x)}{a+b}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 40, normalized size = 0.98 \[ \frac {b \tan ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a+b}}\right )}{\sqrt {a} (a+b)^{3/2}}-\frac {\cot (x)}{a+b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 228, normalized size = 5.56 \[ \left [-\frac {\sqrt {-a^{2} - a b} b \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \relax (x)^{4} - 2 \, {\left (4 \, a^{2} + 3 \, a b\right )} \cos \relax (x)^{2} + 4 \, {\left ({\left (2 \, a + b\right )} \cos \relax (x)^{3} - a \cos \relax (x)\right )} \sqrt {-a^{2} - a b} \sin \relax (x) + a^{2}}{b^{2} \cos \relax (x)^{4} + 2 \, a b \cos \relax (x)^{2} + a^{2}}\right ) \sin \relax (x) + 4 \, {\left (a^{2} + a b\right )} \cos \relax (x)}{4 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \sin \relax (x)}, -\frac {\sqrt {a^{2} + a b} b \arctan \left (\frac {{\left (2 \, a + b\right )} \cos \relax (x)^{2} - a}{2 \, \sqrt {a^{2} + a b} \cos \relax (x) \sin \relax (x)}\right ) \sin \relax (x) + 2 \, {\left (a^{2} + a b\right )} \cos \relax (x)}{2 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \sin \relax (x)}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 55, normalized size = 1.34 \[ \frac {{\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \relax (x)}{\sqrt {a^{2} + a b}}\right )\right )} b}{\sqrt {a^{2} + a b} {\left (a + b\right )}} - \frac {1}{{\left (a + b\right )} \tan \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 39, normalized size = 0.95 \[ \frac {b \arctan \left (\frac {a \tan \relax (x )}{\sqrt {\left (a +b \right ) a}}\right )}{\left (a +b \right ) \sqrt {\left (a +b \right ) a}}-\frac {1}{\left (a +b \right ) \tan \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 38, normalized size = 0.93 \[ \frac {b \arctan \left (\frac {a \tan \relax (x)}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} {\left (a + b\right )}} - \frac {1}{{\left (a + b\right )} \tan \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.30, size = 34, normalized size = 0.83 \[ \frac {b\,\mathrm {atan}\left (\frac {\sqrt {a}\,\mathrm {tan}\relax (x)}{\sqrt {a+b}}\right )}{\sqrt {a}\,{\left (a+b\right )}^{3/2}}-\frac {1}{\mathrm {tan}\relax (x)\,\left (a+b\right )} \]
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 \[ \int \frac {\csc ^{2}{\relax (x )}}{a + b \cos ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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