Optimal. Leaf size=78 \[ -\frac{\left (a^2+3 a b+3 b^2\right ) \sin (x)}{b^3}+\frac{(a+3 b) \sin ^3(x)}{3 b^2}+\frac{(a+b)^3 \tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} b^{7/2}}-\frac{\sin ^5(x)}{5 b} \]
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Rubi [A] time = 0.0895978, 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 = {3190, 390, 205} \[ -\frac{\left (a^2+3 a b+3 b^2\right ) \sin (x)}{b^3}+\frac{(a+3 b) \sin ^3(x)}{3 b^2}+\frac{(a+b)^3 \tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} b^{7/2}}-\frac{\sin ^5(x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 3190
Rule 390
Rule 205
Rubi steps
\begin{align*} \int \frac{\cos ^7(x)}{a+b \sin ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{a+b x^2} \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{a^2+3 a b+3 b^2}{b^3}+\frac{(a+3 b) x^2}{b^2}-\frac{x^4}{b}+\frac{a^3+3 a^2 b+3 a b^2+b^3}{b^3 \left (a+b x^2\right )}\right ) \, dx,x,\sin (x)\right )\\ &=-\frac{\left (a^2+3 a b+3 b^2\right ) \sin (x)}{b^3}+\frac{(a+3 b) \sin ^3(x)}{3 b^2}-\frac{\sin ^5(x)}{5 b}+\frac{(a+b)^3 \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sin (x)\right )}{b^3}\\ &=\frac{(a+b)^3 \tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} b^{7/2}}-\frac{\left (a^2+3 a b+3 b^2\right ) \sin (x)}{b^3}+\frac{(a+3 b) \sin ^3(x)}{3 b^2}-\frac{\sin ^5(x)}{5 b}\\ \end{align*}
Mathematica [A] time = 0.275232, size = 109, normalized size = 1.4 \[ \frac{-2 \sqrt{a} \sqrt{b} \sin (x) \left (120 a^2+4 b (5 a+12 b) \cos (2 x)+340 a b+3 b^2 \cos (4 x)+309 b^2\right )+120 (a+b)^3 \tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )-120 (a+b)^3 \tan ^{-1}\left (\frac{\sqrt{a} \csc (x)}{\sqrt{b}}\right )}{240 \sqrt{a} b^{7/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.043, size = 136, normalized size = 1.7 \begin{align*} -{\frac{ \left ( \sin \left ( x \right ) \right ) ^{5}}{5\,b}}+{\frac{a \left ( \sin \left ( x \right ) \right ) ^{3}}{3\,{b}^{2}}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{3}}{b}}-{\frac{{a}^{2}\sin \left ( x \right ) }{{b}^{3}}}-3\,{\frac{a\sin \left ( x \right ) }{{b}^{2}}}-3\,{\frac{\sin \left ( x \right ) }{b}}+{\frac{{a}^{3}}{{b}^{3}}\arctan \left ({\sin \left ( x \right ) b{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+3\,{\frac{{a}^{2}}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{\sin \left ( x \right ) b}{\sqrt{ab}}} \right ) }+3\,{\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 = 2.4446, size = 566, normalized size = 7.26 \begin{align*} \left [-\frac{15 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\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 (3 \, a b^{3} \cos \left (x\right )^{4} + 15 \, a^{3} b + 40 \, a^{2} b^{2} + 33 \, a b^{3} +{\left (5 \, a^{2} b^{2} + 9 \, a b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{30 \, a b^{4}}, \frac{15 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} \sin \left (x\right )}{a}\right ) -{\left (3 \, a b^{3} \cos \left (x\right )^{4} + 15 \, a^{3} b + 40 \, a^{2} b^{2} + 33 \, a b^{3} +{\left (5 \, a^{2} b^{2} + 9 \, a b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{15 \, a b^{4}}\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 = 1.11695, size = 132, normalized size = 1.69 \begin{align*} \frac{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac{b \sin \left (x\right )}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} - \frac{3 \, b^{4} \sin \left (x\right )^{5} - 5 \, a b^{3} \sin \left (x\right )^{3} - 15 \, b^{4} \sin \left (x\right )^{3} + 15 \, a^{2} b^{2} \sin \left (x\right ) + 45 \, a b^{3} \sin \left (x\right ) + 45 \, 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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