Optimal. Leaf size=33 \[ -\frac{2 b}{a f \sqrt{a \sin (e+f x)} \sqrt{b \sec (e+f x)}} \]
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Rubi [A] time = 0.0516478, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2578} \[ -\frac{2 b}{a f \sqrt{a \sin (e+f x)} \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2578
Rubi steps
\begin{align*} \int \frac{\sqrt{b \sec (e+f x)}}{(a \sin (e+f x))^{3/2}} \, dx &=-\frac{2 b}{a f \sqrt{b \sec (e+f x)} \sqrt{a \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.0740362, size = 37, normalized size = 1.12 \[ -\frac{\sin (2 (e+f x)) \sqrt{b \sec (e+f x)}}{f (a \sin (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.123, size = 40, normalized size = 1.2 \begin{align*} -2\,{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{f \left ( a\sin \left ( fx+e \right ) \right ) ^{3/2}}\sqrt{{\frac{b}{\cos \left ( fx+e \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \sec \left (f x + e\right )}}{\left (a \sin \left (f x + e\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.85517, size = 108, normalized size = 3.27 \begin{align*} -\frac{2 \, \sqrt{a \sin \left (f x + e\right )} \sqrt{\frac{b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a^{2} f \sin \left (f x + e\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \sec \left (f x + e\right )}}{\left (a \sin \left (f x + e\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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