Optimal. Leaf size=38 \[ \frac{\sin (x)}{b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right )}{b^{3/2} \sqrt{a+b}} \]
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Rubi [A] time = 0.0550803, antiderivative size = 38, 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 = {3186, 388, 208} \[ \frac{\sin (x)}{b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right )}{b^{3/2} \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Rule 3186
Rule 388
Rule 208
Rubi steps
\begin{align*} \int \frac{\cos ^3(x)}{a+b \cos ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1-x^2}{a+b-b x^2} \, dx,x,\sin (x)\right )\\ &=\frac{\sin (x)}{b}-\frac{a \operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\sin (x)\right )}{b}\\ &=-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right )}{b^{3/2} \sqrt{a+b}}+\frac{\sin (x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0294949, size = 38, normalized size = 1. \[ \frac{\sin (x)}{b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right )}{b^{3/2} \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 33, normalized size = 0.9 \begin{align*}{\frac{\sin \left ( x \right ) }{b}}-{\frac{a}{b}{\it Artanh} \left ({\sin \left ( x \right ) b{\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \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 [A] time = 1.87761, size = 319, normalized size = 8.39 \begin{align*} \left [\frac{\sqrt{a b + b^{2}} a \log \left (-\frac{b \cos \left (x\right )^{2} + 2 \, \sqrt{a b + b^{2}} \sin \left (x\right ) - a - 2 \, b}{b \cos \left (x\right )^{2} + a}\right ) + 2 \,{\left (a b + b^{2}\right )} \sin \left (x\right )}{2 \,{\left (a b^{2} + b^{3}\right )}}, \frac{\sqrt{-a b - b^{2}} a \arctan \left (\frac{\sqrt{-a b - b^{2}} \sin \left (x\right )}{a + b}\right ) +{\left (a b + b^{2}\right )} \sin \left (x\right )}{a b^{2} + b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15699, size = 55, normalized size = 1.45 \begin{align*} \frac{a \arctan \left (\frac{b \sin \left (x\right )}{\sqrt{-a b - b^{2}}}\right )}{\sqrt{-a b - b^{2}} b} + \frac{\sin \left (x\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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