Optimal. Leaf size=18 \[ \frac{2 d \sqrt{d \sec (a+b x)}}{b} \]
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Rubi [A] time = 0.0362843, 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} \[ \frac{2 d \sqrt{d \sec (a+b x)}}{b} \]
Antiderivative was successfully verified.
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Rule 2622
Rule 30
Rubi steps
\begin{align*} \int (d \sec (a+b x))^{3/2} \sin (a+b x) \, dx &=\frac{d \operatorname{Subst}\left (\int \frac{1}{\sqrt{x}} \, dx,x,d \sec (a+b x)\right )}{b}\\ &=\frac{2 d \sqrt{d \sec (a+b x)}}{b}\\ \end{align*}
Mathematica [A] time = 0.032755, size = 18, normalized size = 1. \[ \frac{2 d \sqrt{d \sec (a+b x)}}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 17, normalized size = 0.9 \begin{align*} 2\,{\frac{d\sqrt{d\sec \left ( bx+a \right ) }}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1578, size = 31, normalized size = 1.72 \begin{align*} \frac{2 \, \left (\frac{d}{\cos \left (b x + a\right )}\right )^{\frac{3}{2}} \cos \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66254, size = 38, normalized size = 2.11 \begin{align*} \frac{2 \, d \sqrt{\frac{d}{\cos \left (b x + a\right )}}}{b} \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 = 1.33896, size = 34, normalized size = 1.89 \begin{align*} \frac{2 \, d^{2} \mathrm{sgn}\left (\cos \left (b x + a\right )\right )}{\sqrt{d \cos \left (b x + a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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