Optimal. Leaf size=78 \[ \frac{\left (a^2-a b+b^2\right ) \sin (x)}{b^3}-\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right )}{b^{7/2} \sqrt{a+b}}+\frac{(a-2 b) \sin ^3(x)}{3 b^2}+\frac{\sin ^5(x)}{5 b} \]
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Rubi [A] time = 0.0840385, antiderivative size = 78, 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} \[ \frac{\left (a^2-a b+b^2\right ) \sin (x)}{b^3}-\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right )}{b^{7/2} \sqrt{a+b}}+\frac{(a-2 b) \sin ^3(x)}{3 b^2}+\frac{\sin ^5(x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 3186
Rule 390
Rule 208
Rubi steps
\begin{align*} \int \frac{\cos ^7(x)}{a+b \cos ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{a+b-b x^2} \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{a^2-a b+b^2}{b^3}+\frac{(a-2 b) x^2}{b^2}+\frac{x^4}{b}-\frac{a^3}{b^3 \left (a+b-b x^2\right )}\right ) \, dx,x,\sin (x)\right )\\ &=\frac{\left (a^2-a b+b^2\right ) \sin (x)}{b^3}+\frac{(a-2 b) \sin ^3(x)}{3 b^2}+\frac{\sin ^5(x)}{5 b}-\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\sin (x)\right )}{b^3}\\ &=-\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right )}{b^{7/2} \sqrt{a+b}}+\frac{\left (a^2-a b+b^2\right ) \sin (x)}{b^3}+\frac{(a-2 b) \sin ^3(x)}{3 b^2}+\frac{\sin ^5(x)}{5 b}\\ \end{align*}
Mathematica [A] time = 0.385071, size = 111, normalized size = 1.42 \[ \frac{\left (8 a^2-6 a b+5 b^2\right ) \sin (x)}{8 b^3}+\frac{a^3 \left (\log \left (\sqrt{a+b}-\sqrt{b} \sin (x)\right )-\log \left (\sqrt{a+b}+\sqrt{b} \sin (x)\right )\right )}{2 b^{7/2} \sqrt{a+b}}+\frac{(5 b-4 a) \sin (3 x)}{48 b^2}+\frac{\sin (5 x)}{80 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 78, normalized size = 1. \begin{align*}{\frac{1}{{b}^{3}} \left ({\frac{ \left ( \sin \left ( x \right ) \right ) ^{5}{b}^{2}}{5}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{3}ba}{3}}-{\frac{2\, \left ( \sin \left ( x \right ) \right ) ^{3}{b}^{2}}{3}}+{a}^{2}\sin \left ( x \right ) -ab\sin \left ( x \right ) +{b}^{2}\sin \left ( x \right ) \right ) }-{\frac{{a}^{3}}{{b}^{3}}{\it Artanh} \left ({b\sin \left ( x \right ){\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 = 1.91061, size = 597, normalized size = 7.65 \begin{align*} \left [\frac{15 \, \sqrt{a b + b^{2}} a^{3} \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 \,{\left (a b^{3} + b^{4}\right )} \cos \left (x\right )^{4} + 15 \, a^{3} b + 5 \, a^{2} b^{2} - 2 \, a b^{3} + 8 \, b^{4} -{\left (5 \, a^{2} b^{2} + a b^{3} - 4 \, b^{4}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{30 \,{\left (a b^{4} + b^{5}\right )}}, \frac{15 \, \sqrt{-a b - b^{2}} a^{3} \arctan \left (\frac{\sqrt{-a b - b^{2}} \sin \left (x\right )}{a + b}\right ) +{\left (3 \,{\left (a b^{3} + b^{4}\right )} \cos \left (x\right )^{4} + 15 \, a^{3} b + 5 \, a^{2} b^{2} - 2 \, a b^{3} + 8 \, b^{4} -{\left (5 \, a^{2} b^{2} + a b^{3} - 4 \, b^{4}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{15 \,{\left (a b^{4} + b^{5}\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 = 1.16356, size = 130, normalized size = 1.67 \begin{align*} \frac{a^{3} \arctan \left (\frac{b \sin \left (x\right )}{\sqrt{-a b - b^{2}}}\right )}{\sqrt{-a b - b^{2}} b^{3}} + \frac{3 \, b^{4} \sin \left (x\right )^{5} + 5 \, a b^{3} \sin \left (x\right )^{3} - 10 \, b^{4} \sin \left (x\right )^{3} + 15 \, a^{2} b^{2} \sin \left (x\right ) - 15 \, a b^{3} \sin \left (x\right ) + 15 \, b^{4} \sin \left (x\right )}{15 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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