Optimal. Leaf size=18 \[ \frac{2 b \sqrt{b \sec (e+f x)}}{f} \]
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Rubi [A] time = 0.0343327, antiderivative size = 18, 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} \[ \frac{2 b \sqrt{b \sec (e+f x)}}{f} \]
Antiderivative was successfully verified.
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Rule 2622
Rule 30
Rubi steps
\begin{align*} \int (b \sec (e+f x))^{3/2} \sin (e+f x) \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{x}} \, dx,x,b \sec (e+f x)\right )}{f}\\ &=\frac{2 b \sqrt{b \sec (e+f x)}}{f}\\ \end{align*}
Mathematica [A] time = 0.0271977, size = 18, normalized size = 1. \[ \frac{2 b \sqrt{b \sec (e+f x)}}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 17, normalized size = 0.9 \begin{align*} 2\,{\frac{b\sqrt{b\sec \left ( fx+e \right ) }}{f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995149, size = 31, normalized size = 1.72 \begin{align*} \frac{2 \, \left (\frac{b}{\cos \left (f x + e\right )}\right )^{\frac{3}{2}} \cos \left (f x + e\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19677, size = 38, normalized size = 2.11 \begin{align*} \frac{2 \, b \sqrt{\frac{b}{\cos \left (f x + e\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 = 1.32588, size = 36, normalized size = 2. \begin{align*} \frac{2 \, b^{2} \mathrm{sgn}\left (\cos \left (f x + e\right )\right )}{\sqrt{b \cos \left (f x + e\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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