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 = 0.05, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 14
Rule 2564
Rule 4288
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*}
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Mathematica [A] time = 0.09, 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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fricas [A] time = 0.47, size = 41, normalized size = 1.32 \[ \frac {8 \, {\left (5 \, \cos \left (b x + a\right )^{6} - 8 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2} + 2\right )} \sin \left (b x + a\right )}{35 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 54, normalized size = 1.74 \[ \frac {\sin \left (7 \, b x + 7 \, a\right )}{56 \, b} - \frac {\sin \left (5 \, b x + 5 \, a\right )}{40 \, b} - \frac {\sin \left (3 \, b x + 3 \, a\right )}{8 \, b} + \frac {3 \, \sin \left (b x + a\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.93, size = 55, normalized size = 1.77 \[ \frac {3 \sin \left (b x +a \right )}{8 b}-\frac {\sin \left (3 b x +3 a \right )}{8 b}-\frac {\sin \left (5 b x +5 a \right )}{40 b}+\frac {\sin \left (7 b x +7 a \right )}{56 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 47, normalized size = 1.52 \[ \frac {5 \, \sin \left (7 \, b x + 7 \, a\right ) - 7 \, \sin \left (5 \, b x + 5 \, a\right ) - 35 \, \sin \left (3 \, b x + 3 \, a\right ) + 105 \, \sin \left (b x + a\right )}{280 \, b} \]
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 \[ \frac {8\,\left (7\,{\sin \left (a+b\,x\right )}^5-5\,{\sin \left (a+b\,x\right )}^7\right )}{35\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.91, size = 126, normalized size = 4.06 \[ \begin {cases} - \frac {22 \sin {\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{35 b} - \frac {16 \sin {\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{35 b} + \frac {9 \sin ^{3}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{35 b} + \frac {8 \sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{35 b} & \text {for}\: b \neq 0 \\x \sin {\relax (a )} \sin ^{3}{\left (2 a \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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