Optimal. Leaf size=73 \[ -\frac{2 \cos (e+f x)}{3 f (a+b)^2 \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\cos (e+f x)}{3 f (a+b) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}} \]
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Rubi [A] time = 0.0580009, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3186, 192, 191} \[ -\frac{2 \cos (e+f x)}{3 f (a+b)^2 \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\cos (e+f x)}{3 f (a+b) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3186
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{\sin (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b-b x^2\right )^{5/2}} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac{\cos (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{\cos (e+f x)}{3 (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}-\frac{2 \cos (e+f x)}{3 (a+b)^2 f \sqrt{a+b-b \cos ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.171712, size = 60, normalized size = 0.82 \[ \frac{2 \sqrt{2} \cos (e+f x) (-3 a+b \cos (2 (e+f x))-2 b)}{3 f (a+b)^2 (2 a-b \cos (2 (e+f x))+b)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.041, size = 55, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,b \left ( \sin \left ( fx+e \right ) \right ) ^{2}+3\,a+b \right ) \cos \left ( fx+e \right ) }{ \left ( 3\,{a}^{2}+6\,ab+3\,{b}^{2} \right ) 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 = 0.958838, size = 85, normalized size = 1.16 \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 )}}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.53603, size = 315, normalized size = 4.32 \begin{align*} \frac{{\left (2 \, b \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 = 1.34598, size = 185, normalized size = 2.53 \begin{align*} \frac{{\left (\frac{2 \, b^{2} f^{2} \cos \left (f x + e\right )^{2}}{a^{2} b f^{2} + 2 \, a b^{2} f^{2} + b^{3} f^{2}} - \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}}\right )} \sqrt{-{\left (\cos \left (f x + e\right )^{2} - 1\right )} b + a} \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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