Optimal. Leaf size=30 \[ \frac {\sin (a+b x)}{2 b}-\frac {\sin (3 a+3 b x)}{6 b} \]
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Rubi [A] time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4282} \[ \frac {\sin (a+b x)}{2 b}-\frac {\sin (3 a+3 b x)}{6 b} \]
Antiderivative was successfully verified.
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Rule 4282
Rubi steps
\begin {align*} \int \sin (a+b x) \sin (2 a+2 b x) \, dx &=\frac {\sin (a+b x)}{2 b}-\frac {\sin (3 a+3 b x)}{6 b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 15, normalized size = 0.50 \[ \frac {2 \sin ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 21, normalized size = 0.70 \[ -\frac {2 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 26, normalized size = 0.87 \[ -\frac {\sin \left (3 \, b x + 3 \, a\right )}{6 \, b} + \frac {\sin \left (b x + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 27, normalized size = 0.90 \[ \frac {\sin \left (b x +a \right )}{2 b}-\frac {\sin \left (3 b x +3 a \right )}{6 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 26, normalized size = 0.87 \[ -\frac {\sin \left (3 \, b x + 3 \, a\right )}{6 \, b} + \frac {\sin \left (b x + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 44, normalized size = 1.47 \[ \left \{\begin {array}{cl} 2\,x\,\left (\cos \relax (a)-{\cos \relax (a)}^3\right ) & \text {\ if\ \ }b=0\\ \frac {3\,\sin \left (a+b\,x\right )-\sin \left (3\,a+3\,b\,x\right )}{6\,b} & \text {\ if\ \ }b\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.81, size = 51, normalized size = 1.70 \[ \begin {cases} - \frac {2 \sin {\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{3 b} + \frac {\sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x \sin {\relax (a )} \sin {\left (2 a \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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