Optimal. Leaf size=69 \[ -\frac{\sin ^3(a+b x) \cos ^3(a+b x)}{6 b}-\frac{\sin (a+b x) \cos ^3(a+b x)}{8 b}+\frac{\sin (a+b x) \cos (a+b x)}{16 b}+\frac{x}{16} \]
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Rubi [A] time = 0.0683697, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ -\frac{\sin ^3(a+b x) \cos ^3(a+b x)}{6 b}-\frac{\sin (a+b x) \cos ^3(a+b x)}{8 b}+\frac{\sin (a+b x) \cos (a+b x)}{16 b}+\frac{x}{16} \]
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 ^4(a+b x) \, dx &=-\frac{\cos ^3(a+b x) \sin ^3(a+b x)}{6 b}+\frac{1}{2} \int \cos ^2(a+b x) \sin ^2(a+b x) \, dx\\ &=-\frac{\cos ^3(a+b x) \sin (a+b x)}{8 b}-\frac{\cos ^3(a+b x) \sin ^3(a+b x)}{6 b}+\frac{1}{8} \int \cos ^2(a+b x) \, dx\\ &=\frac{\cos (a+b x) \sin (a+b x)}{16 b}-\frac{\cos ^3(a+b x) \sin (a+b x)}{8 b}-\frac{\cos ^3(a+b x) \sin ^3(a+b x)}{6 b}+\frac{\int 1 \, dx}{16}\\ &=\frac{x}{16}+\frac{\cos (a+b x) \sin (a+b x)}{16 b}-\frac{\cos ^3(a+b x) \sin (a+b x)}{8 b}-\frac{\cos ^3(a+b x) \sin ^3(a+b x)}{6 b}\\ \end{align*}
Mathematica [A] time = 0.0672379, size = 40, normalized size = 0.58 \[ \frac{-3 \sin (2 (a+b x))-3 \sin (4 (a+b x))+\sin (6 (a+b x))+12 b x}{192 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 61, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{3} \left ( \sin \left ( bx+a \right ) \right ) ^{3}}{6}}-{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{3}\sin \left ( bx+a \right ) }{8}}+{\frac{\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{16}}+{\frac{bx}{16}}+{\frac{a}{16}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10965, size = 50, normalized size = 0.72 \begin{align*} -\frac{4 \, \sin \left (2 \, b x + 2 \, a\right )^{3} - 12 \, b x - 12 \, a + 3 \, \sin \left (4 \, b x + 4 \, a\right )}{192 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66524, size = 117, normalized size = 1.7 \begin{align*} \frac{3 \, b x +{\left (8 \, \cos \left (b x + a\right )^{5} - 14 \, \cos \left (b x + a\right )^{3} + 3 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{48 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.8449, size = 136, normalized size = 1.97 \begin{align*} \begin{cases} \frac{x \sin ^{6}{\left (a + b x \right )}}{16} + \frac{3 x \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16} + \frac{3 x \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{16} + \frac{x \cos ^{6}{\left (a + b x \right )}}{16} + \frac{\sin ^{5}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{16 b} - \frac{\sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{6 b} - \frac{\sin{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{16 b} & \text{for}\: b \neq 0 \\x \sin ^{4}{\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 = 1.17517, size = 62, normalized size = 0.9 \begin{align*} \frac{1}{16} \, x + \frac{\sin \left (6 \, b x + 6 \, a\right )}{192 \, b} - \frac{\sin \left (4 \, b x + 4 \, a\right )}{64 \, b} - \frac{\sin \left (2 \, b x + 2 \, a\right )}{64 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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