Optimal. Leaf size=54 \[ -\frac{(a+2 b) \sin (x)}{b^2}+\frac{(a+b)^2 \tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2}}+\frac{\sin ^3(x)}{3 b} \]
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Rubi [A] time = 0.0732421, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3190, 390, 205} \[ -\frac{(a+2 b) \sin (x)}{b^2}+\frac{(a+b)^2 \tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2}}+\frac{\sin ^3(x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 3190
Rule 390
Rule 205
Rubi steps
\begin{align*} \int \frac{\cos ^5(x)}{a+b \sin ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{a+b x^2} \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{a+2 b}{b^2}+\frac{x^2}{b}+\frac{a^2+2 a b+b^2}{b^2 \left (a+b x^2\right )}\right ) \, dx,x,\sin (x)\right )\\ &=-\frac{(a+2 b) \sin (x)}{b^2}+\frac{\sin ^3(x)}{3 b}+\frac{(a+b)^2 \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sin (x)\right )}{b^2}\\ &=\frac{(a+b)^2 \tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2}}-\frac{(a+2 b) \sin (x)}{b^2}+\frac{\sin ^3(x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.173709, size = 84, normalized size = 1.56 \[ \frac{6 (a+b)^2 \tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )-2 \sqrt{a} \sqrt{b} \sin (x) (6 a+b \cos (2 x)+11 b)-6 (a+b)^2 \tan ^{-1}\left (\frac{\sqrt{a} \csc (x)}{\sqrt{b}}\right )}{12 \sqrt{a} b^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 85, normalized size = 1.6 \begin{align*}{\frac{ \left ( \sin \left ( x \right ) \right ) ^{3}}{3\,b}}-{\frac{a\sin \left ( x \right ) }{{b}^{2}}}-2\,{\frac{\sin \left ( x \right ) }{b}}+{\frac{{a}^{2}}{{b}^{2}}\arctan \left ({\sin \left ( x \right ) b{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+2\,{\frac{a}{\sqrt{ab}b}\arctan \left ({\frac{\sin \left ( x \right ) b}{\sqrt{ab}}} \right ) }+{\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 = 2.30336, size = 393, normalized size = 7.28 \begin{align*} \left [-\frac{3 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt{-a b} \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 \,{\left (a b^{2} \cos \left (x\right )^{2} + 3 \, a^{2} b + 5 \, a b^{2}\right )} \sin \left (x\right )}{6 \, a b^{3}}, \frac{3 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} \sin \left (x\right )}{a}\right ) -{\left (a b^{2} \cos \left (x\right )^{2} + 3 \, a^{2} b + 5 \, a b^{2}\right )} \sin \left (x\right )}{3 \, a 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 = 1.13079, size = 78, normalized size = 1.44 \begin{align*} \frac{{\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (\frac{b \sin \left (x\right )}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{b^{2} \sin \left (x\right )^{3} - 3 \, a b \sin \left (x\right ) - 6 \, b^{2} \sin \left (x\right )}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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