Optimal. Leaf size=20 \[ -\frac{2 b}{5 f (b \sec (e+f x))^{5/2}} \]
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Rubi [A] time = 0.0361541, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2622, 30} \[ -\frac{2 b}{5 f (b \sec (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2622
Rule 30
Rubi steps
\begin{align*} \int \frac{\sin (e+f x)}{(b \sec (e+f x))^{3/2}} \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{1}{x^{7/2}} \, dx,x,b \sec (e+f x)\right )}{f}\\ &=-\frac{2 b}{5 f (b \sec (e+f x))^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0536722, size = 20, normalized size = 1. \[ -\frac{2 b}{5 f (b \sec (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 17, normalized size = 0.9 \begin{align*} -{\frac{2\,b}{5\,f} \left ( b\sec \left ( fx+e \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984656, size = 31, normalized size = 1.55 \begin{align*} -\frac{2 \, \cos \left (f x + e\right )}{5 \, f \left (\frac{b}{\cos \left (f x + e\right )}\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.48782, size = 68, normalized size = 3.4 \begin{align*} -\frac{2 \, \sqrt{\frac{b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{3}}{5 \, b^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (e + f x \right )}}{\left (b \sec{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.46603, size = 41, normalized size = 2.05 \begin{align*} -\frac{2 \, \cos \left (f x + e\right )^{2}}{5 \, b f \sqrt{\frac{b}{\cos \left (f x + e\right )}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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