Optimal. Leaf size=111 \[ -\frac{\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}-\frac{3 \sin (a+b x) \cos ^7(a+b x)}{80 b}+\frac{\sin (a+b x) \cos ^5(a+b x)}{160 b}+\frac{\sin (a+b x) \cos ^3(a+b x)}{128 b}+\frac{3 \sin (a+b x) \cos (a+b x)}{256 b}+\frac{3 x}{256} \]
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Rubi [A] time = 0.096975, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ -\frac{\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}-\frac{3 \sin (a+b x) \cos ^7(a+b x)}{80 b}+\frac{\sin (a+b x) \cos ^5(a+b x)}{160 b}+\frac{\sin (a+b x) \cos ^3(a+b x)}{128 b}+\frac{3 \sin (a+b x) \cos (a+b x)}{256 b}+\frac{3 x}{256} \]
Antiderivative was successfully verified.
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Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx &=-\frac{\cos ^7(a+b x) \sin ^3(a+b x)}{10 b}+\frac{3}{10} \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx\\ &=-\frac{3 \cos ^7(a+b x) \sin (a+b x)}{80 b}-\frac{\cos ^7(a+b x) \sin ^3(a+b x)}{10 b}+\frac{3}{80} \int \cos ^6(a+b x) \, dx\\ &=\frac{\cos ^5(a+b x) \sin (a+b x)}{160 b}-\frac{3 \cos ^7(a+b x) \sin (a+b x)}{80 b}-\frac{\cos ^7(a+b x) \sin ^3(a+b x)}{10 b}+\frac{1}{32} \int \cos ^4(a+b x) \, dx\\ &=\frac{\cos ^3(a+b x) \sin (a+b x)}{128 b}+\frac{\cos ^5(a+b x) \sin (a+b x)}{160 b}-\frac{3 \cos ^7(a+b x) \sin (a+b x)}{80 b}-\frac{\cos ^7(a+b x) \sin ^3(a+b x)}{10 b}+\frac{3}{128} \int \cos ^2(a+b x) \, dx\\ &=\frac{3 \cos (a+b x) \sin (a+b x)}{256 b}+\frac{\cos ^3(a+b x) \sin (a+b x)}{128 b}+\frac{\cos ^5(a+b x) \sin (a+b x)}{160 b}-\frac{3 \cos ^7(a+b x) \sin (a+b x)}{80 b}-\frac{\cos ^7(a+b x) \sin ^3(a+b x)}{10 b}+\frac{3 \int 1 \, dx}{256}\\ &=\frac{3 x}{256}+\frac{3 \cos (a+b x) \sin (a+b x)}{256 b}+\frac{\cos ^3(a+b x) \sin (a+b x)}{128 b}+\frac{\cos ^5(a+b x) \sin (a+b x)}{160 b}-\frac{3 \cos ^7(a+b x) \sin (a+b x)}{80 b}-\frac{\cos ^7(a+b x) \sin ^3(a+b x)}{10 b}\\ \end{align*}
Mathematica [A] time = 0.195666, size = 62, normalized size = 0.56 \[ \frac{20 \sin (2 (a+b x))-40 \sin (4 (a+b x))-10 \sin (6 (a+b x))+5 \sin (8 (a+b x))+2 \sin (10 (a+b x))+120 b x}{10240 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 82, normalized size = 0.7 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{3} \left ( \cos \left ( bx+a \right ) \right ) ^{7}}{10}}-{\frac{3\,\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{7}}{80}}+{\frac{\sin \left ( bx+a \right ) }{160} \left ( \left ( \cos \left ( bx+a \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( bx+a \right ) }{8}} \right ) }+{\frac{3\,bx}{256}}+{\frac{3\,a}{256}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00769, size = 65, normalized size = 0.59 \begin{align*} \frac{32 \, \sin \left (2 \, b x + 2 \, a\right )^{5} + 120 \, b x + 120 \, a + 5 \, \sin \left (8 \, b x + 8 \, a\right ) - 40 \, \sin \left (4 \, b x + 4 \, a\right )}{10240 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71907, size = 180, normalized size = 1.62 \begin{align*} \frac{15 \, b x +{\left (128 \, \cos \left (b x + a\right )^{9} - 176 \, \cos \left (b x + a\right )^{7} + 8 \, \cos \left (b x + a\right )^{5} + 10 \, \cos \left (b x + a\right )^{3} + 15 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{1280 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 32.5377, size = 231, normalized size = 2.08 \begin{align*} \begin{cases} \frac{3 x \sin ^{10}{\left (a + b x \right )}}{256} + \frac{15 x \sin ^{8}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{256} + \frac{15 x \sin ^{6}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{128} + \frac{15 x \sin ^{4}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{128} + \frac{15 x \sin ^{2}{\left (a + b x \right )} \cos ^{8}{\left (a + b x \right )}}{256} + \frac{3 x \cos ^{10}{\left (a + b x \right )}}{256} + \frac{3 \sin ^{9}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{256 b} + \frac{7 \sin ^{7}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{128 b} + \frac{\sin ^{5}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{10 b} - \frac{7 \sin ^{3}{\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{128 b} - \frac{3 \sin{\left (a + b x \right )} \cos ^{9}{\left (a + b x \right )}}{256 b} & \text{for}\: b \neq 0 \\x \sin ^{4}{\left (a \right )} \cos ^{6}{\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 = 1.14561, size = 100, normalized size = 0.9 \begin{align*} \frac{3}{256} \, x + \frac{\sin \left (10 \, b x + 10 \, a\right )}{5120 \, b} + \frac{\sin \left (8 \, b x + 8 \, a\right )}{2048 \, b} - \frac{\sin \left (6 \, b x + 6 \, a\right )}{1024 \, b} - \frac{\sin \left (4 \, b x + 4 \, a\right )}{256 \, b} + \frac{\sin \left (2 \, b x + 2 \, a\right )}{512 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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