Optimal. Leaf size=45 \[ \frac{2}{7 b d (d \cos (a+b x))^{7/2}}-\frac{2}{3 b d^3 (d \cos (a+b x))^{3/2}} \]
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Rubi [A] time = 0.0502524, antiderivative size = 45, 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} \[ \frac{2}{7 b d (d \cos (a+b x))^{7/2}}-\frac{2}{3 b d^3 (d \cos (a+b x))^{3/2}} \]
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))^{9/2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1-\frac{x^2}{d^2}}{x^{9/2}} \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x^{9/2}}-\frac{1}{d^2 x^{5/2}}\right ) \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=\frac{2}{7 b d (d \cos (a+b x))^{7/2}}-\frac{2}{3 b d^3 (d \cos (a+b x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.279019, size = 70, normalized size = 1.56 \[ \frac{2 \tan ^2(a+b x) \left (-4 \cos ^2(a+b x)^{3/4}+4 \left (\cos ^2(a+b x)^{3/4}-1\right ) \csc ^2(a+b x)+7\right )}{21 b d^3 (d \cos (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.149, size = 111, normalized size = 2.5 \begin{align*} -{\frac{8}{21\,{d}^{5}b}\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d+d} \left ( 7\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}-7\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+1 \right ) \left ( 16\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{8}-32\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{6}+24\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}-8\, \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 = 0.972873, size = 50, normalized size = 1.11 \begin{align*} -\frac{2 \,{\left (7 \, d^{2} \cos \left (b x + a\right )^{2} - 3 \, d^{2}\right )}}{21 \, \left (d \cos \left (b x + a\right )\right )^{\frac{7}{2}} b d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06844, size = 100, normalized size = 2.22 \begin{align*} -\frac{2 \, \sqrt{d \cos \left (b x + a\right )}{\left (7 \, \cos \left (b x + a\right )^{2} - 3\right )}}{21 \, b d^{5} \cos \left (b x + a\right )^{4}} \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 = 1.18774, size = 61, normalized size = 1.36 \begin{align*} -\frac{2 \,{\left (7 \, d^{4} \cos \left (b x + a\right )^{2} - 3 \, d^{4}\right )}}{21 \, \sqrt{d \cos \left (b x + a\right )} b d^{8} \cos \left (b x + a\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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