Optimal. Leaf size=29 \[ \frac{\sin (e+f x)}{a f \sqrt{a+b \sin ^2(e+f x)}} \]
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Rubi [A] time = 0.0439352, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3190, 191} \[ \frac{\sin (e+f x)}{a f \sqrt{a+b \sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3190
Rule 191
Rubi steps
\begin{align*} \int \frac{\cos (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\sin (e+f x)}{a f \sqrt{a+b \sin ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.0302894, size = 29, normalized size = 1. \[ \frac{\sin (e+f x)}{a f \sqrt{a+b \sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.085, size = 28, normalized size = 1. \begin{align*}{\frac{\sin \left ( fx+e \right ) }{af}{\frac{1}{\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.969579, size = 36, normalized size = 1.24 \begin{align*} \frac{\sin \left (f x + e\right )}{\sqrt{b \sin \left (f x + e\right )^{2} + a} a f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.43475, size = 116, normalized size = 4. \begin{align*} -\frac{\sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sin \left (f x + e\right )}{a b f \cos \left (f x + e\right )^{2} -{\left (a^{2} + a b\right )} f} \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 = 1.36706, size = 39, normalized size = 1.34 \begin{align*} \frac{\sin \left (f x + e\right )}{\sqrt{b \sin \left (f x + e\right )^{2} + a} a f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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