Optimal. Leaf size=36 \[ \frac{\cos (x)}{b}-\frac{(a+b) \tan ^{-1}\left (\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}} \]
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Rubi [A] time = 0.0531164, antiderivative size = 36, 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 = {3190, 388, 205} \[ \frac{\cos (x)}{b}-\frac{(a+b) \tan ^{-1}\left (\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3190
Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{\sin ^3(x)}{a+b \cos ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1-x^2}{a+b x^2} \, dx,x,\cos (x)\right )\\ &=\frac{\cos (x)}{b}-\frac{(a+b) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\cos (x)\right )}{b}\\ &=-\frac{(a+b) \tan ^{-1}\left (\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{\cos (x)}{b}\\ \end{align*}
Mathematica [B] time = 0.161007, size = 90, normalized size = 2.5 \[ \frac{\sqrt{a} \sqrt{b} \cos (x)-(a+b) \tan ^{-1}\left (\frac{\sqrt{b}-\sqrt{a+b} \tan \left (\frac{x}{2}\right )}{\sqrt{a}}\right )-(a+b) \tan ^{-1}\left (\frac{\sqrt{a+b} \tan \left (\frac{x}{2}\right )+\sqrt{b}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 46, normalized size = 1.3 \begin{align*}{\frac{\cos \left ( x \right ) }{b}}-{\frac{a}{b}\arctan \left ({b\cos \left ( x \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\arctan \left ({b\cos \left ( x \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 1.74506, size = 248, normalized size = 6.89 \begin{align*} \left [\frac{2 \, a b \cos \left (x\right ) - \sqrt{-a b}{\left (a + b\right )} \log \left (-\frac{b \cos \left (x\right )^{2} + 2 \, \sqrt{-a b} \cos \left (x\right ) - a}{b \cos \left (x\right )^{2} + a}\right )}{2 \, a b^{2}}, \frac{a b \cos \left (x\right ) - \sqrt{a b}{\left (a + b\right )} \arctan \left (\frac{\sqrt{a b} \cos \left (x\right )}{a}\right )}{a b^{2}}\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 = 1.12843, size = 41, normalized size = 1.14 \begin{align*} -\frac{{\left (a + b\right )} \arctan \left (\frac{b \cos \left (x\right )}{\sqrt{a b}}\right )}{\sqrt{a b} b} + \frac{\cos \left (x\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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