Optimal. Leaf size=29 \[ \frac {4 \sin ^6(a+b x)}{3 b}-\frac {\sin ^8(a+b x)}{b} \]
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Rubi [A] time = 0.06, antiderivative size = 29, 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} \[ \frac {4 \sin ^6(a+b x)}{3 b}-\frac {\sin ^8(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2564
Rule 4288
Rubi steps
\begin {align*} \int \sin ^2(a+b x) \sin ^3(2 a+2 b x) \, dx &=8 \int \cos ^3(a+b x) \sin ^5(a+b x) \, dx\\ &=\frac {8 \operatorname {Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {8 \operatorname {Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {4 \sin ^6(a+b x)}{3 b}-\frac {\sin ^8(a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 48, normalized size = 1.66 \[ \frac {-72 \cos (2 (a+b x))+12 \cos (4 (a+b x))+8 \cos (6 (a+b x))-3 \cos (8 (a+b x))}{384 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 36, normalized size = 1.24 \[ -\frac {3 \, \cos \left (b x + a\right )^{8} - 8 \, \cos \left (b x + a\right )^{6} + 6 \, \cos \left (b x + a\right )^{4}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.50, size = 57, normalized size = 1.97 \[ -\frac {\cos \left (8 \, b x + 8 \, a\right )}{128 \, b} + \frac {\cos \left (6 \, b x + 6 \, a\right )}{48 \, b} + \frac {\cos \left (4 \, b x + 4 \, a\right )}{32 \, b} - \frac {3 \, \cos \left (2 \, b x + 2 \, a\right )}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.29, size = 58, normalized size = 2.00 \[ -\frac {3 \cos \left (2 b x +2 a \right )}{16 b}+\frac {\cos \left (4 b x +4 a \right )}{32 b}+\frac {\cos \left (6 b x +6 a \right )}{48 b}-\frac {\cos \left (8 b x +8 a \right )}{128 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 50, normalized size = 1.72 \[ -\frac {3 \, \cos \left (8 \, b x + 8 \, a\right ) - 8 \, \cos \left (6 \, b x + 6 \, a\right ) - 12 \, \cos \left (4 \, b x + 4 \, a\right ) + 72 \, \cos \left (2 \, b x + 2 \, a\right )}{384 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 33, normalized size = 1.14 \[ -\frac {{\cos \left (a+b\,x\right )}^4\,\left ({\cos \left (a+b\,x\right )}^4-\frac {8\,{\cos \left (a+b\,x\right )}^2}{3}+2\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 40.10, size = 362, normalized size = 12.48 \[ \begin {cases} \frac {3 x \sin ^{2}{\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )}}{16} + \frac {3 x \sin ^{2}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{16} + \frac {3 x \sin {\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{8} + \frac {3 x \sin {\left (a + b x \right )} \cos {\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{8} - \frac {3 x \sin ^{3}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )}}{16} - \frac {3 x \sin {\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{16} + \frac {17 \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{96 b} - \frac {13 \sin {\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{16 b} - \frac {7 \sin {\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{8 b} - \frac {\sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{2 b} - \frac {49 \cos ^{2}{\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{96 b} & \text {for}\: b \neq 0 \\x \sin ^{2}{\relax (a )} \sin ^{3}{\left (2 a \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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