Optimal. Leaf size=31 \[ \frac{8 \sin ^5(a+b x)}{5 b}-\frac{8 \sin ^7(a+b x)}{7 b} \]
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Rubi [A] time = 0.0496947, antiderivative size = 31, 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 = {4288, 2564, 14} \[ \frac{8 \sin ^5(a+b x)}{5 b}-\frac{8 \sin ^7(a+b x)}{7 b} \]
Antiderivative was successfully verified.
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Rule 4288
Rule 2564
Rule 14
Rubi steps
\begin{align*} \int \sin (a+b x) \sin ^3(2 a+2 b x) \, dx &=8 \int \cos ^3(a+b x) \sin ^4(a+b x) \, dx\\ &=\frac{8 \operatorname{Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{8 \operatorname{Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{8 \sin ^5(a+b x)}{5 b}-\frac{8 \sin ^7(a+b x)}{7 b}\\ \end{align*}
Mathematica [A] time = 0.0902121, size = 27, normalized size = 0.87 \[ \frac{4 \sin ^5(a+b x) (5 \cos (2 (a+b x))+9)}{35 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 55, normalized size = 1.8 \begin{align*}{\frac{3\,\sin \left ( bx+a \right ) }{8\,b}}-{\frac{\sin \left ( 3\,bx+3\,a \right ) }{8\,b}}-{\frac{\sin \left ( 5\,bx+5\,a \right ) }{40\,b}}+{\frac{\sin \left ( 7\,bx+7\,a \right ) }{56\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18821, size = 63, normalized size = 2.03 \begin{align*} \frac{5 \, \sin \left (7 \, b x + 7 \, a\right ) - 7 \, \sin \left (5 \, b x + 5 \, a\right ) - 35 \, \sin \left (3 \, b x + 3 \, a\right ) + 105 \, \sin \left (b x + a\right )}{280 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.481122, size = 108, normalized size = 3.48 \begin{align*} \frac{8 \,{\left (5 \, \cos \left (b x + a\right )^{6} - 8 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2} + 2\right )} \sin \left (b x + a\right )}{35 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 16.9376, size = 126, normalized size = 4.06 \begin{align*} \begin{cases} - \frac{22 \sin{\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos{\left (2 a + 2 b x \right )}}{35 b} - \frac{16 \sin{\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{35 b} + \frac{9 \sin ^{3}{\left (2 a + 2 b x \right )} \cos{\left (a + b x \right )}}{35 b} + \frac{8 \sin{\left (2 a + 2 b x \right )} \cos{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{35 b} & \text{for}\: b \neq 0 \\x \sin{\left (a \right )} \sin ^{3}{\left (2 a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27638, size = 73, normalized size = 2.35 \begin{align*} \frac{\sin \left (7 \, b x + 7 \, a\right )}{56 \, b} - \frac{\sin \left (5 \, b x + 5 \, a\right )}{40 \, b} - \frac{\sin \left (3 \, b x + 3 \, a\right )}{8 \, b} + \frac{3 \, \sin \left (b x + a\right )}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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