Optimal. Leaf size=46 \[ -\frac{\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac{\sin (a+b x) \cos (a+b x)}{8 b}+\frac{x}{8} \]
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Rubi [A] time = 0.0396221, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ -\frac{\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac{\sin (a+b x) \cos (a+b x)}{8 b}+\frac{x}{8} \]
Antiderivative was successfully verified.
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Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^2(a+b x) \sin ^2(a+b x) \, dx &=-\frac{\cos ^3(a+b x) \sin (a+b x)}{4 b}+\frac{1}{4} \int \cos ^2(a+b x) \, dx\\ &=\frac{\cos (a+b x) \sin (a+b x)}{8 b}-\frac{\cos ^3(a+b x) \sin (a+b x)}{4 b}+\frac{\int 1 \, dx}{8}\\ &=\frac{x}{8}+\frac{\cos (a+b x) \sin (a+b x)}{8 b}-\frac{\cos ^3(a+b x) \sin (a+b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0325703, size = 23, normalized size = 0.5 \[ -\frac{\sin (4 (a+b x))-4 (a+b x)}{32 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 43, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{3}\sin \left ( bx+a \right ) }{4}}+{\frac{\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{8}}+{\frac{bx}{8}}+{\frac{a}{8}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.97436, size = 32, normalized size = 0.7 \begin{align*} \frac{4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )}{32 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57523, size = 84, normalized size = 1.83 \begin{align*} \frac{b x -{\left (2 \, \cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.01716, size = 92, normalized size = 2. \begin{align*} \begin{cases} \frac{x \sin ^{4}{\left (a + b x \right )}}{8} + \frac{x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} + \frac{x \cos ^{4}{\left (a + b x \right )}}{8} + \frac{\sin ^{3}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{8 b} - \frac{\sin{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} & \text{for}\: b \neq 0 \\x \sin ^{2}{\left (a \right )} \cos ^{2}{\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 = 1.15917, size = 24, normalized size = 0.52 \begin{align*} \frac{1}{8} \, x - \frac{\sin \left (4 \, b x + 4 \, a\right )}{32 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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