Optimal. Leaf size=46 \[ -\frac{4 \cos ^9(a+b x)}{9 b}+\frac{8 \cos ^7(a+b x)}{7 b}-\frac{4 \cos ^5(a+b x)}{5 b} \]
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Rubi [A] time = 0.0966612, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {4312, 2565, 270} \[ -\frac{4 \cos ^9(a+b x)}{9 b}+\frac{8 \cos ^7(a+b x)}{7 b}-\frac{4 \cos ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 4312
Rule 2565
Rule 270
Rubi steps
\begin{align*} \int \cos ^2(a+b x) \sin ^3(a+b x) \sin ^2(2 a+2 b x) \, dx &=4 \int \cos ^4(a+b x) \sin ^5(a+b x) \, dx\\ &=-\frac{4 \operatorname{Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{4 \operatorname{Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{4 \cos ^5(a+b x)}{5 b}+\frac{8 \cos ^7(a+b x)}{7 b}-\frac{4 \cos ^9(a+b x)}{9 b}\\ \end{align*}
Mathematica [A] time = 0.163096, size = 37, normalized size = 0.8 \[ \frac{\cos ^5(a+b x) (220 \cos (2 (a+b x))-35 \cos (4 (a+b x))-249)}{630 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 69, normalized size = 1.5 \begin{align*} -{\frac{3\,\cos \left ( bx+a \right ) }{32\,b}}-{\frac{\cos \left ( 3\,bx+3\,a \right ) }{48\,b}}+{\frac{\cos \left ( 5\,bx+5\,a \right ) }{80\,b}}+{\frac{\cos \left ( 7\,bx+7\,a \right ) }{448\,b}}-{\frac{\cos \left ( 9\,bx+9\,a \right ) }{576\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.2228, size = 78, normalized size = 1.7 \begin{align*} -\frac{35 \, \cos \left (9 \, b x + 9 \, a\right ) - 45 \, \cos \left (7 \, b x + 7 \, a\right ) - 252 \, \cos \left (5 \, b x + 5 \, a\right ) + 420 \, \cos \left (3 \, b x + 3 \, a\right ) + 1890 \, \cos \left (b x + a\right )}{20160 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.490691, size = 95, normalized size = 2.07 \begin{align*} -\frac{4 \,{\left (35 \, \cos \left (b x + a\right )^{9} - 90 \, \cos \left (b x + a\right )^{7} + 63 \, \cos \left (b x + a\right )^{5}\right )}}{315 \, b} \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 = 1.85772, size = 92, normalized size = 2. \begin{align*} -\frac{\cos \left (9 \, b x + 9 \, a\right )}{576 \, b} + \frac{\cos \left (7 \, b x + 7 \, a\right )}{448 \, b} + \frac{\cos \left (5 \, b x + 5 \, a\right )}{80 \, b} - \frac{\cos \left (3 \, b x + 3 \, a\right )}{48 \, b} - \frac{3 \, \cos \left (b x + a\right )}{32 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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