Optimal. Leaf size=81 \[ \frac{4 \sin (a+b x)}{21 b \sqrt{\sin (2 a+2 b x)}}-\frac{\cos ^3(a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}-\frac{2 \cos (a+b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)} \]
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Rubi [A] time = 0.0678956, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4295, 4303, 4292} \[ \frac{4 \sin (a+b x)}{21 b \sqrt{\sin (2 a+2 b x)}}-\frac{\cos ^3(a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}-\frac{2 \cos (a+b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)} \]
Antiderivative was successfully verified.
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Rule 4295
Rule 4303
Rule 4292
Rubi steps
\begin{align*} \int \frac{\cos ^3(a+b x)}{\sin ^{\frac{9}{2}}(2 a+2 b x)} \, dx &=-\frac{\cos ^3(a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}+\frac{2}{7} \int \frac{\cos (a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{\cos ^3(a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}-\frac{2 \cos (a+b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{4}{21} \int \frac{\sin (a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{\cos ^3(a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}-\frac{2 \cos (a+b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{4 \sin (a+b x)}{21 b \sqrt{\sin (2 a+2 b x)}}\\ \end{align*}
Mathematica [A] time = 0.114477, size = 55, normalized size = 0.68 \[ \frac{\sqrt{\sin (2 (a+b x))} (-12 \cos (2 (a+b x))+4 \cos (4 (a+b x))+5) \csc ^4(a+b x) \sec (a+b x)}{336 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cos \left ( bx+a \right ) \right ) ^{3} \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{-{\frac{9}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x + a\right )^{3}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.505902, size = 282, normalized size = 3.48 \begin{align*} \frac{32 \, \cos \left (b x + a\right )^{5} - 64 \, \cos \left (b x + a\right )^{3} + \sqrt{2}{\left (32 \, \cos \left (b x + a\right )^{4} - 56 \, \cos \left (b x + a\right )^{2} + 21\right )} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 32 \, \cos \left (b x + a\right )}{336 \,{\left (b \cos \left (b x + a\right )^{5} - 2 \, b \cos \left (b x + a\right )^{3} + b \cos \left (b x + a\right )\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x + a\right )^{3}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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