Optimal. Leaf size=56 \[ \frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right )}{b^{5/2} \sqrt{a+b}}-\frac{(a-b) \sin (x)}{b^2}-\frac{\sin ^3(x)}{3 b} \]
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Rubi [A] time = 0.071169, antiderivative size = 56, 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 = {3186, 390, 208} \[ \frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right )}{b^{5/2} \sqrt{a+b}}-\frac{(a-b) \sin (x)}{b^2}-\frac{\sin ^3(x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 3186
Rule 390
Rule 208
Rubi steps
\begin{align*} \int \frac{\cos ^5(x)}{a+b \cos ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{a+b-b x^2} \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{a-b}{b^2}-\frac{x^2}{b}+\frac{a^2}{b^2 \left (a+b-b x^2\right )}\right ) \, dx,x,\sin (x)\right )\\ &=-\frac{(a-b) \sin (x)}{b^2}-\frac{\sin ^3(x)}{3 b}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\sin (x)\right )}{b^2}\\ &=\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right )}{b^{5/2} \sqrt{a+b}}-\frac{(a-b) \sin (x)}{b^2}-\frac{\sin ^3(x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.184982, size = 86, normalized size = 1.54 \[ \frac{\frac{6 a^2 \left (\log \left (\sqrt{a+b}+\sqrt{b} \sin (x)\right )-\log \left (\sqrt{a+b}-\sqrt{b} \sin (x)\right )\right )}{\sqrt{a+b}}+3 \sqrt{b} (3 b-4 a) \sin (x)+b^{3/2} \sin (3 x)}{12 b^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 50, normalized size = 0.9 \begin{align*} -{\frac{1}{{b}^{2}} \left ({\frac{ \left ( \sin \left ( x \right ) \right ) ^{3}b}{3}}+\sin \left ( x \right ) a-\sin \left ( x \right ) b \right ) }+{\frac{{a}^{2}}{{b}^{2}}{\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 = 1.87134, size = 441, normalized size = 7.88 \begin{align*} \left [\frac{3 \, \sqrt{a b + b^{2}} a^{2} \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 (3 \, a^{2} b + a b^{2} - 2 \, b^{3} -{\left (a b^{2} + b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{6 \,{\left (a b^{3} + b^{4}\right )}}, -\frac{3 \, \sqrt{-a b - b^{2}} a^{2} \arctan \left (\frac{\sqrt{-a b - b^{2}} \sin \left (x\right )}{a + b}\right ) +{\left (3 \, a^{2} b + a b^{2} - 2 \, b^{3} -{\left (a b^{2} + b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{3 \,{\left (a b^{3} + b^{4}\right )}}\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 = 1.16901, size = 88, normalized size = 1.57 \begin{align*} -\frac{a^{2} \arctan \left (\frac{b \sin \left (x\right )}{\sqrt{-a b - b^{2}}}\right )}{\sqrt{-a b - b^{2}} b^{2}} - \frac{b^{2} \sin \left (x\right )^{3} + 3 \, a b \sin \left (x\right ) - 3 \, 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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