Optimal. Leaf size=45 \[ \frac{2}{9 b d (d \cos (a+b x))^{9/2}}-\frac{2}{5 b d^3 (d \cos (a+b x))^{5/2}} \]
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Rubi [A] time = 0.0513467, 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} \[ \frac{2}{9 b d (d \cos (a+b x))^{9/2}}-\frac{2}{5 b d^3 (d \cos (a+b x))^{5/2}} \]
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))^{11/2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1-\frac{x^2}{d^2}}{x^{11/2}} \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x^{11/2}}-\frac{1}{d^2 x^{7/2}}\right ) \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=\frac{2}{9 b d (d \cos (a+b x))^{9/2}}-\frac{2}{5 b d^3 (d \cos (a+b x))^{5/2}}\\ \end{align*}
Mathematica [B] time = 0.547615, size = 94, normalized size = 2.09 \[ \frac{2 \tan ^4(a+b x) \left (4 \sqrt [4]{\cos ^2(a+b x)}+4 \left (\sqrt [4]{\cos ^2(a+b x)}-1\right ) \csc ^4(a+b x)+\left (9-8 \sqrt [4]{\cos ^2(a+b x)}\right ) \csc ^2(a+b x)\right )}{45 b d^5 \sqrt{d \cos (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.178, size = 124, normalized size = 2.8 \begin{align*}{\frac{8}{45\,{d}^{6}b}\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d+d} \left ( 9\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}-9\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+1 \right ) \left ( 32\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{10}-80\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{8}+80\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{6}-40\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+10\, \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 = 0.977458, size = 50, normalized size = 1.11 \begin{align*} -\frac{2 \,{\left (9 \, d^{2} \cos \left (b x + a\right )^{2} - 5 \, d^{2}\right )}}{45 \, \left (d \cos \left (b x + a\right )\right )^{\frac{9}{2}} b d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17817, size = 100, normalized size = 2.22 \begin{align*} -\frac{2 \, \sqrt{d \cos \left (b x + a\right )}{\left (9 \, \cos \left (b x + a\right )^{2} - 5\right )}}{45 \, b d^{6} \cos \left (b x + a\right )^{5}} \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 = 1.21318, size = 61, normalized size = 1.36 \begin{align*} -\frac{2 \,{\left (9 \, d^{5} \cos \left (b x + a\right )^{2} - 5 \, d^{5}\right )}}{45 \, \sqrt{d \cos \left (b x + a\right )} b d^{10} \cos \left (b x + a\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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