Optimal. Leaf size=36 \[ \frac{(a+b) \tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}-\frac{\sin (x)}{b} \]
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Rubi [A] time = 0.0563766, 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} \[ \frac{(a+b) \tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}-\frac{\sin (x)}{b} \]
Antiderivative was successfully verified.
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Rule 3190
Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{\cos ^3(x)}{a+b \sin ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1-x^2}{a+b x^2} \, dx,x,\sin (x)\right )\\ &=-\frac{\sin (x)}{b}+\frac{(a+b) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sin (x)\right )}{b}\\ &=\frac{(a+b) \tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}-\frac{\sin (x)}{b}\\ \end{align*}
Mathematica [A] time = 0.023088, size = 36, normalized size = 1. \[ \frac{(a+b) \tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}-\frac{\sin (x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 45, normalized size = 1.3 \begin{align*} -{\frac{\sin \left ( x \right ) }{b}}+{\frac{a}{b}\arctan \left ({\sin \left ( x \right ) b{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\arctan \left ({\sin \left ( x \right ) b{\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 = 2.00546, size = 262, normalized size = 7.28 \begin{align*} \left [-\frac{2 \, a b \sin \left (x\right ) + \sqrt{-a b}{\left (a + b\right )} \log \left (-\frac{b \cos \left (x\right )^{2} + 2 \, \sqrt{-a b} \sin \left (x\right ) + a - b}{b \cos \left (x\right )^{2} - a - b}\right )}{2 \, a b^{2}}, -\frac{a b \sin \left (x\right ) - \sqrt{a b}{\left (a + b\right )} \arctan \left (\frac{\sqrt{a b} \sin \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 = 1.11156, size = 41, normalized size = 1.14 \begin{align*} \frac{{\left (a + b\right )} \arctan \left (\frac{b \sin \left (x\right )}{\sqrt{a b}}\right )}{\sqrt{a b} b} - \frac{\sin \left (x\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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