Optimal. Leaf size=46 \[ -\frac{32 \cos ^{11}(a+b x)}{11 b}+\frac{64 \cos ^9(a+b x)}{9 b}-\frac{32 \cos ^7(a+b x)}{7 b} \]
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Rubi [A] time = 0.0554593, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4287, 2565, 270} \[ -\frac{32 \cos ^{11}(a+b x)}{11 b}+\frac{64 \cos ^9(a+b x)}{9 b}-\frac{32 \cos ^7(a+b x)}{7 b} \]
Antiderivative was successfully verified.
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Rule 4287
Rule 2565
Rule 270
Rubi steps
\begin{align*} \int \cos (a+b x) \sin ^5(2 a+2 b x) \, dx &=32 \int \cos ^6(a+b x) \sin ^5(a+b x) \, dx\\ &=-\frac{32 \operatorname{Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{32 \operatorname{Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{32 \cos ^7(a+b x)}{7 b}+\frac{64 \cos ^9(a+b x)}{9 b}-\frac{32 \cos ^{11}(a+b x)}{11 b}\\ \end{align*}
Mathematica [A] time = 0.264445, size = 37, normalized size = 0.8 \[ \frac{4 \cos ^7(a+b x) (364 \cos (2 (a+b x))-63 \cos (4 (a+b x))-365)}{693 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 83, normalized size = 1.8 \begin{align*} -{\frac{5\,\cos \left ( bx+a \right ) }{16\,b}}-{\frac{5\,\cos \left ( 3\,bx+3\,a \right ) }{48\,b}}+{\frac{\cos \left ( 5\,bx+5\,a \right ) }{32\,b}}+{\frac{5\,\cos \left ( 7\,bx+7\,a \right ) }{224\,b}}-{\frac{\cos \left ( 9\,bx+9\,a \right ) }{288\,b}}-{\frac{\cos \left ( 11\,bx+11\,a \right ) }{352\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02742, size = 93, normalized size = 2.02 \begin{align*} -\frac{63 \, \cos \left (11 \, b x + 11 \, a\right ) + 77 \, \cos \left (9 \, b x + 9 \, a\right ) - 495 \, \cos \left (7 \, b x + 7 \, a\right ) - 693 \, \cos \left (5 \, b x + 5 \, a\right ) + 2310 \, \cos \left (3 \, b x + 3 \, a\right ) + 6930 \, \cos \left (b x + a\right )}{22176 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.50379, size = 99, normalized size = 2.15 \begin{align*} -\frac{32 \,{\left (63 \, \cos \left (b x + a\right )^{11} - 154 \, \cos \left (b x + a\right )^{9} + 99 \, \cos \left (b x + a\right )^{7}\right )}}{693 \, 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 [B] time = 1.38627, size = 111, normalized size = 2.41 \begin{align*} -\frac{\cos \left (11 \, b x + 11 \, a\right )}{352 \, b} - \frac{\cos \left (9 \, b x + 9 \, a\right )}{288 \, b} + \frac{5 \, \cos \left (7 \, b x + 7 \, a\right )}{224 \, b} + \frac{\cos \left (5 \, b x + 5 \, a\right )}{32 \, b} - \frac{5 \, \cos \left (3 \, b x + 3 \, a\right )}{48 \, b} - \frac{5 \, \cos \left (b x + a\right )}{16 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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