Optimal. Leaf size=41 \[ \frac{2 d^3}{b \sqrt{d \sec (a+b x)}}+\frac{2 d (d \sec (a+b x))^{3/2}}{3 b} \]
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Rubi [A] time = 0.048539, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2622, 14} \[ \frac{2 d^3}{b \sqrt{d \sec (a+b x)}}+\frac{2 d (d \sec (a+b x))^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 2622
Rule 14
Rubi steps
\begin{align*} \int (d \sec (a+b x))^{5/2} \sin ^3(a+b x) \, dx &=\frac{d^3 \operatorname{Subst}\left (\int \frac{-1+\frac{x^2}{d^2}}{x^{3/2}} \, dx,x,d \sec (a+b x)\right )}{b}\\ &=\frac{d^3 \operatorname{Subst}\left (\int \left (-\frac{1}{x^{3/2}}+\frac{\sqrt{x}}{d^2}\right ) \, dx,x,d \sec (a+b x)\right )}{b}\\ &=\frac{2 d^3}{b \sqrt{d \sec (a+b x)}}+\frac{2 d (d \sec (a+b x))^{3/2}}{3 b}\\ \end{align*}
Mathematica [A] time = 0.219513, size = 32, normalized size = 0.78 \[ \frac{d (3 \cos (2 (a+b x))+5) (d \sec (a+b x))^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.199, size = 357, normalized size = 8.7 \begin{align*} -{\frac{ \left ( -1+\cos \left ( bx+a \right ) \right ) \cos \left ( bx+a \right ) }{6\,b \left ( \sin \left ( bx+a \right ) \right ) ^{2}} \left ( 12\,\sqrt{-{\frac{\cos \left ( bx+a \right ) }{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}}} \left ( \cos \left ( bx+a \right ) \right ) ^{3}+12\,\sqrt{-{\frac{\cos \left ( bx+a \right ) }{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}}} \left ( \cos \left ( bx+a \right ) \right ) ^{2}-3\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}\ln \left ( -{\frac{1}{ \left ( \sin \left ( bx+a \right ) \right ) ^{2}} \left ( 2\,\sqrt{-{\frac{\cos \left ( bx+a \right ) }{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}}} \left ( \cos \left ( bx+a \right ) \right ) ^{2}- \left ( \cos \left ( bx+a \right ) \right ) ^{2}-2\,\sqrt{-{\frac{\cos \left ( bx+a \right ) }{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}}}+2\,\cos \left ( bx+a \right ) -1 \right ) } \right ) +3\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}\ln \left ( -2\,{\frac{1}{ \left ( \sin \left ( bx+a \right ) \right ) ^{2}} \left ( 2\,\sqrt{-{\frac{\cos \left ( bx+a \right ) }{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}}} \left ( \cos \left ( bx+a \right ) \right ) ^{2}- \left ( \cos \left ( bx+a \right ) \right ) ^{2}-2\,\sqrt{-{\frac{\cos \left ( bx+a \right ) }{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}}}+2\,\cos \left ( bx+a \right ) -1 \right ) } \right ) +4\,\cos \left ( bx+a \right ) \sqrt{-{\frac{\cos \left ( bx+a \right ) }{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}}}+4\,\sqrt{-{\frac{\cos \left ( bx+a \right ) }{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}}} \right ) \left ({\frac{d}{\cos \left ( bx+a \right ) }} \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{-{\frac{\cos \left ( bx+a \right ) }{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09725, size = 49, normalized size = 1.2 \begin{align*} \frac{2 \,{\left (\frac{3 \, d^{2}}{\sqrt{\frac{d}{\cos \left (b x + a\right )}}} + \left (\frac{d}{\cos \left (b x + a\right )}\right )^{\frac{3}{2}}\right )} d}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65811, size = 97, normalized size = 2.37 \begin{align*} \frac{2 \,{\left (3 \, d^{2} \cos \left (b x + a\right )^{2} + d^{2}\right )} \sqrt{\frac{d}{\cos \left (b x + a\right )}}}{3 \, b \cos \left (b x + a\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 [A] time = 1.18904, size = 66, normalized size = 1.61 \begin{align*} \frac{2 \,{\left (3 \, \sqrt{d \cos \left (b x + a\right )} d + \frac{d^{2}}{\sqrt{d \cos \left (b x + a\right )} \cos \left (b x + a\right )}\right )} d \mathrm{sgn}\left (\cos \left (b x + a\right )\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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