Optimal. Leaf size=20 \[ \frac{2 d (d \sec (a+b x))^{5/2}}{5 b} \]
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Rubi [A] time = 0.0351707, 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} \[ \frac{2 d (d \sec (a+b x))^{5/2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 2622
Rule 30
Rubi steps
\begin{align*} \int (d \sec (a+b x))^{7/2} \sin (a+b x) \, dx &=\frac{d \operatorname{Subst}\left (\int x^{3/2} \, dx,x,d \sec (a+b x)\right )}{b}\\ &=\frac{2 d (d \sec (a+b x))^{5/2}}{5 b}\\ \end{align*}
Mathematica [A] time = 0.0545522, size = 20, normalized size = 1. \[ \frac{2 d (d \sec (a+b x))^{5/2}}{5 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 17, normalized size = 0.9 \begin{align*}{\frac{2\,d}{5\,b} \left ( d\sec \left ( bx+a \right ) \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1273, size = 31, normalized size = 1.55 \begin{align*} \frac{2 \, \left (\frac{d}{\cos \left (b x + a\right )}\right )^{\frac{7}{2}} \cos \left (b x + a\right )}{5 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63252, size = 66, normalized size = 3.3 \begin{align*} \frac{2 \, d^{3} \sqrt{\frac{d}{\cos \left (b x + a\right )}}}{5 \, b \cos \left (b x + a\right )^{2}} \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 [B] time = 1.27447, size = 45, normalized size = 2.25 \begin{align*} \frac{2 \, d^{4} \mathrm{sgn}\left (\cos \left (b x + a\right )\right )}{5 \, \sqrt{d \cos \left (b x + a\right )} b \cos \left (b x + a\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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