Optimal. Leaf size=46 \[ \frac{32 \sin ^{13}(a+b x)}{13 b}-\frac{64 \sin ^{11}(a+b x)}{11 b}+\frac{32 \sin ^9(a+b x)}{9 b} \]
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Rubi [A] time = 0.0631673, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4288, 2564, 270} \[ \frac{32 \sin ^{13}(a+b x)}{13 b}-\frac{64 \sin ^{11}(a+b x)}{11 b}+\frac{32 \sin ^9(a+b x)}{9 b} \]
Antiderivative was successfully verified.
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Rule 4288
Rule 2564
Rule 270
Rubi steps
\begin{align*} \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx &=32 \int \cos ^5(a+b x) \sin ^8(a+b x) \, dx\\ &=\frac{32 \operatorname{Subst}\left (\int x^8 \left (1-x^2\right )^2 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{32 \operatorname{Subst}\left (\int \left (x^8-2 x^{10}+x^{12}\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{32 \sin ^9(a+b x)}{9 b}-\frac{64 \sin ^{11}(a+b x)}{11 b}+\frac{32 \sin ^{13}(a+b x)}{13 b}\\ \end{align*}
Mathematica [A] time = 0.373848, size = 37, normalized size = 0.8 \[ \frac{4 \sin ^9(a+b x) (540 \cos (2 (a+b x))+99 \cos (4 (a+b x))+505)}{1287 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 97, normalized size = 2.1 \begin{align*}{\frac{5\,\sin \left ( bx+a \right ) }{32\,b}}-{\frac{25\,\sin \left ( 3\,bx+3\,a \right ) }{384\,b}}-{\frac{\sin \left ( 5\,bx+5\,a \right ) }{128\,b}}+{\frac{\sin \left ( 7\,bx+7\,a \right ) }{64\,b}}-{\frac{\sin \left ( 9\,bx+9\,a \right ) }{576\,b}}-{\frac{3\,\sin \left ( 11\,bx+11\,a \right ) }{1408\,b}}+{\frac{\sin \left ( 13\,bx+13\,a \right ) }{1664\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06181, size = 108, normalized size = 2.35 \begin{align*} \frac{99 \, \sin \left (13 \, b x + 13 \, a\right ) - 351 \, \sin \left (11 \, b x + 11 \, a\right ) - 286 \, \sin \left (9 \, b x + 9 \, a\right ) + 2574 \, \sin \left (7 \, b x + 7 \, a\right ) - 1287 \, \sin \left (5 \, b x + 5 \, a\right ) - 10725 \, \sin \left (3 \, b x + 3 \, a\right ) + 25740 \, \sin \left (b x + a\right )}{164736 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.515469, size = 204, normalized size = 4.43 \begin{align*} \frac{32 \,{\left (99 \, \cos \left (b x + a\right )^{12} - 360 \, \cos \left (b x + a\right )^{10} + 458 \, \cos \left (b x + a\right )^{8} - 212 \, \cos \left (b x + a\right )^{6} + 3 \, \cos \left (b x + a\right )^{4} + 4 \, \cos \left (b x + a\right )^{2} + 8\right )} \sin \left (b x + a\right )}{1287 \, 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 = 1.59182, size = 130, normalized size = 2.83 \begin{align*} \frac{\sin \left (13 \, b x + 13 \, a\right )}{1664 \, b} - \frac{3 \, \sin \left (11 \, b x + 11 \, a\right )}{1408 \, b} - \frac{\sin \left (9 \, b x + 9 \, a\right )}{576 \, b} + \frac{\sin \left (7 \, b x + 7 \, a\right )}{64 \, b} - \frac{\sin \left (5 \, b x + 5 \, a\right )}{128 \, b} - \frac{25 \, \sin \left (3 \, b x + 3 \, a\right )}{384 \, b} + \frac{5 \, \sin \left (b x + a\right )}{32 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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