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 = 0.06, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 14
Rule 2564
Rule 4288
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*}
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Mathematica [A] time = 0.19, 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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fricas [A] time = 0.41, size = 53, normalized size = 1.71 \[ -\frac {8 \, {\left (7 \, \cos \left (b x + a\right )^{8} - 19 \, \cos \left (b x + a\right )^{6} + 15 \, \cos \left (b x + a\right )^{4} - \cos \left (b x + a\right )^{2} - 2\right )} \sin \left (b x + a\right )}{63 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 54, normalized size = 1.74 \[ -\frac {\sin \left (9 \, b x + 9 \, a\right )}{288 \, b} + \frac {3 \, \sin \left (7 \, b x + 7 \, a\right )}{224 \, b} - \frac {\sin \left (3 \, b x + 3 \, a\right )}{12 \, b} + \frac {3 \, \sin \left (b x + a\right )}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.88, size = 55, normalized size = 1.77 \[ \frac {3 \sin \left (b x +a \right )}{16 b}-\frac {\sin \left (3 b x +3 a \right )}{12 b}+\frac {3 \sin \left (7 b x +7 a \right )}{224 b}-\frac {\sin \left (9 b x +9 a \right )}{288 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 47, normalized size = 1.52 \[ -\frac {7 \, \sin \left (9 \, b x + 9 \, a\right ) - 27 \, \sin \left (7 \, b x + 7 \, a\right ) + 168 \, \sin \left (3 \, b x + 3 \, a\right ) - 378 \, \sin \left (b x + a\right )}{2016 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 26, normalized size = 0.84 \[ \frac {8\,\left (9\,{\sin \left (a+b\,x\right )}^7-7\,{\sin \left (a+b\,x\right )}^9\right )}{63\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 115.46, size = 284, normalized size = 9.16 \[ \begin {cases} - \frac {46 \sin ^{3}{\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{105 b} - \frac {16 \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{63 b} - \frac {13 \sin ^{2}{\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{105 b} - \frac {8 \sin ^{2}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{35 b} - \frac {4 \sin {\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{7 b} - \frac {64 \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{105 b} + \frac {94 \sin ^{3}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )}}{315 b} + \frac {32 \sin {\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{105 b} & \text {for}\: b \neq 0 \\x \sin ^{3}{\relax (a )} \sin ^{3}{\left (2 a \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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