Optimal. Leaf size=31 \[ \frac{8 \sin ^7(a+b x)}{7 b}-\frac{8 \sin ^9(a+b x)}{9 b} \]
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Rubi [A] time = 0.059593, antiderivative size = 31, 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, 14} \[ \frac{8 \sin ^7(a+b x)}{7 b}-\frac{8 \sin ^9(a+b x)}{9 b} \]
Antiderivative was successfully verified.
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Rule 4288
Rule 2564
Rule 14
Rubi steps
\begin{align*} \int \sin ^3(a+b x) \sin ^3(2 a+2 b x) \, dx &=8 \int \cos ^3(a+b x) \sin ^6(a+b x) \, dx\\ &=\frac{8 \operatorname{Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{8 \operatorname{Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{8 \sin ^7(a+b x)}{7 b}-\frac{8 \sin ^9(a+b x)}{9 b}\\ \end{align*}
Mathematica [A] time = 0.141587, size = 27, normalized size = 0.87 \[ \frac{4 \sin ^7(a+b x) (7 \cos (2 (a+b x))+11)}{63 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 55, normalized size = 1.8 \begin{align*}{\frac{3\,\sin \left ( bx+a \right ) }{16\,b}}-{\frac{\sin \left ( 3\,bx+3\,a \right ) }{12\,b}}+{\frac{3\,\sin \left ( 7\,bx+7\,a \right ) }{224\,b}}-{\frac{\sin \left ( 9\,bx+9\,a \right ) }{288\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.176, size = 63, normalized size = 2.03 \begin{align*} -\frac{7 \, \sin \left (9 \, b x + 9 \, a\right ) - 27 \, \sin \left (7 \, b x + 7 \, a\right ) + 168 \, \sin \left (3 \, b x + 3 \, a\right ) - 378 \, \sin \left (b x + a\right )}{2016 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.488846, size = 138, normalized size = 4.45 \begin{align*} -\frac{8 \,{\left (7 \, \cos \left (b x + a\right )^{8} - 19 \, \cos \left (b x + a\right )^{6} + 15 \, \cos \left (b x + a\right )^{4} - \cos \left (b x + a\right )^{2} - 2\right )} \sin \left (b x + a\right )}{63 \, 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 = 1.28407, size = 73, normalized size = 2.35 \begin{align*} -\frac{\sin \left (9 \, b x + 9 \, a\right )}{288 \, b} + \frac{3 \, \sin \left (7 \, b x + 7 \, a\right )}{224 \, b} - \frac{\sin \left (3 \, b x + 3 \, a\right )}{12 \, b} + \frac{3 \, \sin \left (b x + a\right )}{16 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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