Optimal. Leaf size=81 \[ -\frac{2 \cos (e+f x)}{3 f (a+b)^2 \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\sin ^2(e+f x) \cos (e+f x)}{3 f (a+b) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}} \]
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Rubi [A] time = 0.0951856, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {3186, 378, 191} \[ -\frac{2 \cos (e+f x)}{3 f (a+b)^2 \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\sin ^2(e+f x) \cos (e+f x)}{3 f (a+b) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3186
Rule 378
Rule 191
Rubi steps
\begin{align*} \int \frac{\sin ^3(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1-x^2}{\left (a+b-b x^2\right )^{5/2}} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac{\cos (e+f x) \sin ^2(e+f x)}{3 (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\left (a+b-b x^2\right )^{3/2}} \, dx,x,\cos (e+f x)\right )}{3 (a+b) f}\\ &=-\frac{2 \cos (e+f x)}{3 (a+b)^2 f \sqrt{a+b-b \cos ^2(e+f x)}}-\frac{\cos (e+f x) \sin ^2(e+f x)}{3 (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.310118, size = 64, normalized size = 0.79 \[ \frac{\sqrt{2} \cos (e+f x) ((a+3 b) \cos (2 (e+f x))-5 a-3 b)}{3 f (a+b)^2 (2 a-b \cos (2 (e+f x))+b)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.161, size = 56, normalized size = 0.7 \begin{align*} -{\frac{ \left ( a \left ( \sin \left ( fx+e \right ) \right ) ^{2}+3\,b \left ( \sin \left ( fx+e \right ) \right ) ^{2}+2\,a \right ) \cos \left ( fx+e \right ) }{3\, \left ( a+b \right ) ^{2}f} \left ( a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99422, size = 163, normalized size = 2.01 \begin{align*} -\frac{\frac{2 \, \cos \left (f x + e\right )}{\sqrt{-b \cos \left (f x + e\right )^{2} + a + b}{\left (a + b\right )}^{2}} + \frac{\cos \left (f x + e\right )}{{\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{\frac{3}{2}}{\left (a + b\right )}} - \frac{\cos \left (f x + e\right )}{{\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{\frac{3}{2}} b} + \frac{\cos \left (f x + e\right )}{\sqrt{-b \cos \left (f x + e\right )^{2} + a + b}{\left (a + b\right )} b}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.67331, size = 323, normalized size = 3.99 \begin{align*} \frac{{\left ({\left (a + 3 \, b\right )} \cos \left (f x + e\right )^{3} - 3 \,{\left (a + b\right )} \cos \left (f x + e\right )\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}}{3 \,{\left ({\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} f \cos \left (f x + e\right )^{4} - 2 \,{\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} f \cos \left (f x + e\right )^{2} +{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} f\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 [B] time = 1.35355, size = 201, normalized size = 2.48 \begin{align*} -\frac{\sqrt{-{\left (\cos \left (f x + e\right )^{2} - 1\right )} b + a}{\left (\frac{3 \,{\left (a b f^{2} + b^{2} f^{2}\right )}}{a^{2} b f^{2} + 2 \, a b^{2} f^{2} + b^{3} f^{2}} - \frac{{\left (a b f^{4} + 3 \, b^{2} f^{4}\right )} \cos \left (f x + e\right )^{2}}{{\left (a^{2} b f^{2} + 2 \, a b^{2} f^{2} + b^{3} f^{2}\right )} f^{2}}\right )} \cos \left (f x + e\right )}{3 \,{\left ({\left (\cos \left (f x + e\right )^{2} - 1\right )} b - a\right )}^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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