Optimal. Leaf size=43 \[ \frac{2 (d \cos (a+b x))^{3/2}}{3 b d^3}+\frac{2}{b d \sqrt{d \cos (a+b x)}} \]
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Rubi [A] time = 0.0513659, antiderivative size = 43, 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 = {2565, 14} \[ \frac{2 (d \cos (a+b x))^{3/2}}{3 b d^3}+\frac{2}{b d \sqrt{d \cos (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2565
Rule 14
Rubi steps
\begin{align*} \int \frac{\sin ^3(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1-\frac{x^2}{d^2}}{x^{3/2}} \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x^{3/2}}-\frac{\sqrt{x}}{d^2}\right ) \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=\frac{2}{b d \sqrt{d \cos (a+b x)}}+\frac{2 (d \cos (a+b x))^{3/2}}{3 b d^3}\\ \end{align*}
Mathematica [A] time = 0.0747173, size = 46, normalized size = 1.07 \[ -\frac{2 \left (\sin ^2(a+b x)+4 \sqrt [4]{\cos ^2(a+b x)}-4\right )}{3 b d \sqrt{d \cos (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.1, size = 70, normalized size = 1.6 \begin{align*} -{\frac{8}{3\,{d}^{2}b}\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d+d} \left ( \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{4}- \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}+1 \right ) \left ( 2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983387, size = 47, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (\frac{3}{\sqrt{d \cos \left (b x + a\right )}} + \frac{\left (d \cos \left (b x + a\right )\right )^{\frac{3}{2}}}{d^{2}}\right )}}{3 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87155, size = 92, normalized size = 2.14 \begin{align*} \frac{2 \, \sqrt{d \cos \left (b x + a\right )}{\left (\cos \left (b x + a\right )^{2} + 3\right )}}{3 \, b d^{2} \cos \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.16376, size = 61, normalized size = 1.42 \begin{align*} \begin{cases} \frac{2 \sin ^{2}{\left (a + b x \right )}}{b d^{\frac{3}{2}} \sqrt{\cos{\left (a + b x \right )}}} + \frac{8 \cos ^{\frac{3}{2}}{\left (a + b x \right )}}{3 b d^{\frac{3}{2}}} & \text{for}\: b \neq 0 \\\frac{x \sin ^{3}{\left (a \right )}}{\left (d \cos{\left (a \right )}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20139, size = 57, normalized size = 1.33 \begin{align*} \frac{2 \,{\left (\sqrt{d \cos \left (b x + a\right )} d \cos \left (b x + a\right ) + \frac{3 \, d^{2}}{\sqrt{d \cos \left (b x + a\right )}}\right )}}{3 \, b d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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