Optimal. Leaf size=31 \[ \frac{\sin ^4(a+b x)}{4 b}-\frac{\sin ^6(a+b x)}{6 b} \]
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Rubi [A] time = 0.0333805, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2564, 14} \[ \frac{\sin ^4(a+b x)}{4 b}-\frac{\sin ^6(a+b x)}{6 b} \]
Antiderivative was successfully verified.
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Rule 2564
Rule 14
Rubi steps
\begin{align*} \int \cos ^3(a+b x) \sin ^3(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^3 \left (1-x^2\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (x^3-x^5\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{\sin ^4(a+b x)}{4 b}-\frac{\sin ^6(a+b x)}{6 b}\\ \end{align*}
Mathematica [A] time = 0.0145893, size = 35, normalized size = 1.13 \[ \frac{1}{8} \left (\frac{\cos (6 (a+b x))}{24 b}-\frac{3 \cos (2 (a+b x))}{8 b}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 34, normalized size = 1.1 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{4} \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{6}}-{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{12}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98458, size = 35, normalized size = 1.13 \begin{align*} -\frac{2 \, \sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4}}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61483, size = 62, normalized size = 2. \begin{align*} \frac{2 \, \cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4}}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.582, size = 42, normalized size = 1.35 \begin{align*} \begin{cases} \frac{\sin ^{6}{\left (a + b x \right )}}{12 b} + \frac{\sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4 b} & \text{for}\: b \neq 0 \\x \sin ^{3}{\left (a \right )} \cos ^{3}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23579, size = 35, normalized size = 1.13 \begin{align*} -\frac{2 \, \sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4}}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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