Optimal. Leaf size=90 \[ -\frac{\sin ^3(a+b x) \cos ^5(a+b x)}{8 b}-\frac{\sin (a+b x) \cos ^5(a+b x)}{16 b}+\frac{\sin (a+b x) \cos ^3(a+b x)}{64 b}+\frac{3 \sin (a+b x) \cos (a+b x)}{128 b}+\frac{3 x}{128} \]
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Rubi [A] time = 0.0832022, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ -\frac{\sin ^3(a+b x) \cos ^5(a+b x)}{8 b}-\frac{\sin (a+b x) \cos ^5(a+b x)}{16 b}+\frac{\sin (a+b x) \cos ^3(a+b x)}{64 b}+\frac{3 \sin (a+b x) \cos (a+b x)}{128 b}+\frac{3 x}{128} \]
Antiderivative was successfully verified.
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Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^4(a+b x) \sin ^4(a+b x) \, dx &=-\frac{\cos ^5(a+b x) \sin ^3(a+b x)}{8 b}+\frac{3}{8} \int \cos ^4(a+b x) \sin ^2(a+b x) \, dx\\ &=-\frac{\cos ^5(a+b x) \sin (a+b x)}{16 b}-\frac{\cos ^5(a+b x) \sin ^3(a+b x)}{8 b}+\frac{1}{16} \int \cos ^4(a+b x) \, dx\\ &=\frac{\cos ^3(a+b x) \sin (a+b x)}{64 b}-\frac{\cos ^5(a+b x) \sin (a+b x)}{16 b}-\frac{\cos ^5(a+b x) \sin ^3(a+b x)}{8 b}+\frac{3}{64} \int \cos ^2(a+b x) \, dx\\ &=\frac{3 \cos (a+b x) \sin (a+b x)}{128 b}+\frac{\cos ^3(a+b x) \sin (a+b x)}{64 b}-\frac{\cos ^5(a+b x) \sin (a+b x)}{16 b}-\frac{\cos ^5(a+b x) \sin ^3(a+b x)}{8 b}+\frac{3 \int 1 \, dx}{128}\\ &=\frac{3 x}{128}+\frac{3 \cos (a+b x) \sin (a+b x)}{128 b}+\frac{\cos ^3(a+b x) \sin (a+b x)}{64 b}-\frac{\cos ^5(a+b x) \sin (a+b x)}{16 b}-\frac{\cos ^5(a+b x) \sin ^3(a+b x)}{8 b}\\ \end{align*}
Mathematica [A] time = 0.0437347, size = 33, normalized size = 0.37 \[ \frac{24 (a+b x)-8 \sin (4 (a+b x))+\sin (8 (a+b x))}{1024 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 72, normalized size = 0.8 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{5} \left ( \sin \left ( bx+a \right ) \right ) ^{3}}{8}}-{\frac{\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{5}}{16}}+{\frac{\sin \left ( bx+a \right ) }{64} \left ( \left ( \cos \left ( bx+a \right ) \right ) ^{3}+{\frac{3\,\cos \left ( bx+a \right ) }{2}} \right ) }+{\frac{3\,bx}{128}}+{\frac{3\,a}{128}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.997071, size = 45, normalized size = 0.5 \begin{align*} \frac{24 \, b x + 24 \, a + \sin \left (8 \, b x + 8 \, a\right ) - 8 \, \sin \left (4 \, b x + 4 \, a\right )}{1024 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60475, size = 146, normalized size = 1.62 \begin{align*} \frac{3 \, b x +{\left (16 \, \cos \left (b x + a\right )^{7} - 24 \, \cos \left (b x + a\right )^{5} + 2 \, \cos \left (b x + a\right )^{3} + 3 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{128 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.3905, size = 189, normalized size = 2.1 \begin{align*} \begin{cases} \frac{3 x \sin ^{8}{\left (a + b x \right )}}{128} + \frac{3 x \sin ^{6}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32} + \frac{9 x \sin ^{4}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{64} + \frac{3 x \sin ^{2}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{32} + \frac{3 x \cos ^{8}{\left (a + b x \right )}}{128} + \frac{3 \sin ^{7}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{128 b} + \frac{11 \sin ^{5}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{128 b} - \frac{11 \sin ^{3}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{128 b} - \frac{3 \sin{\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{128 b} & \text{for}\: b \neq 0 \\x \sin ^{4}{\left (a \right )} \cos ^{4}{\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 = 1.16124, size = 43, normalized size = 0.48 \begin{align*} \frac{3}{128} \, x + \frac{\sin \left (8 \, b x + 8 \, a\right )}{1024 \, b} - \frac{\sin \left (4 \, b x + 4 \, a\right )}{128 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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