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 = 0.0560492, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 3191
Rule 388
Rule 205
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*}
Mathematica [A] time = 0.0894784, 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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Maple [A] time = 0.03, size = 39, normalized size = 1. \begin{align*}{\frac{b}{a+b}\arctan \left ({a\tan \left ( x \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}}-{\frac{1}{ \left ( a+b \right ) \tan \left ( x \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.91062, size = 571, normalized size = 13.93 \begin{align*} \left [-\frac{\sqrt{-a^{2} - a b} b \log \left (\frac{{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (x\right )^{4} - 2 \,{\left (4 \, a^{2} + 3 \, a b\right )} \cos \left (x\right )^{2} + 4 \,{\left ({\left (2 \, a + b\right )} \cos \left (x\right )^{3} - a \cos \left (x\right )\right )} \sqrt{-a^{2} - a b} \sin \left (x\right ) + a^{2}}{b^{2} \cos \left (x\right )^{4} + 2 \, a b \cos \left (x\right )^{2} + a^{2}}\right ) \sin \left (x\right ) + 4 \,{\left (a^{2} + a b\right )} \cos \left (x\right )}{4 \,{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \sin \left (x\right )}, -\frac{\sqrt{a^{2} + a b} b \arctan \left (\frac{{\left (2 \, a + b\right )} \cos \left (x\right )^{2} - a}{2 \, \sqrt{a^{2} + a b} \cos \left (x\right ) \sin \left (x\right )}\right ) \sin \left (x\right ) + 2 \,{\left (a^{2} + a b\right )} \cos \left (x\right )}{2 \,{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \sin \left (x\right )}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{2}{\left (x \right )}}{a + b \cos ^{2}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17665, size = 74, normalized size = 1.8 \begin{align*} \frac{{\left (\pi \left \lfloor \frac{x}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (a\right ) + \arctan \left (\frac{a \tan \left (x\right )}{\sqrt{a^{2} + a b}}\right )\right )} b}{\sqrt{a^{2} + a b}{\left (a + b\right )}} - \frac{1}{{\left (a + b\right )} \tan \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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