Optimal. Leaf size=67 \[ -\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}+\frac {\sin (a+b x) \cos ^3(a+b x)}{24 b}+\frac {\sin (a+b x) \cos (a+b x)}{16 b}+\frac {x}{16} \]
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Rubi [A] time = 0.05, antiderivative size = 67, normalized size of antiderivative = 1.00, 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} \[ -\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}+\frac {\sin (a+b x) \cos ^3(a+b x)}{24 b}+\frac {\sin (a+b x) \cos (a+b x)}{16 b}+\frac {x}{16} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2568
Rule 2635
Rubi steps
\begin {align*} \int \cos ^4(a+b x) \sin ^2(a+b x) \, dx &=-\frac {\cos ^5(a+b x) \sin (a+b x)}{6 b}+\frac {1}{6} \int \cos ^4(a+b x) \, dx\\ &=\frac {\cos ^3(a+b x) \sin (a+b x)}{24 b}-\frac {\cos ^5(a+b x) \sin (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)}{24 b}-\frac {\cos ^5(a+b x) \sin (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)}{24 b}-\frac {\cos ^5(a+b x) \sin (a+b x)}{6 b}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 40, normalized size = 0.60 \[ -\frac {-3 \sin (2 (a+b x))+3 \sin (4 (a+b x))+\sin (6 (a+b x))-12 b x}{192 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 47, normalized size = 0.70 \[ \frac {3 \, b x - {\left (8 \, \cos \left (b x + a\right )^{5} - 2 \, \cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{48 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 46, normalized size = 0.69 \[ \frac {1}{16} \, x - \frac {\sin \left (6 \, b x + 6 \, a\right )}{192 \, b} - \frac {\sin \left (4 \, b x + 4 \, a\right )}{64 \, b} + \frac {\sin \left (2 \, b x + 2 \, a\right )}{64 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 54, normalized size = 0.81 \[ \frac {-\frac {\sin \left (b x +a \right ) \left (\cos ^{5}\left (b x +a \right )\right )}{6}+\frac {\left (\cos ^{3}\left (b x +a \right )+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{24}+\frac {b x}{16}+\frac {a}{16}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 37, normalized size = 0.55 \[ \frac {4 \, \sin \left (2 \, b x + 2 \, a\right )^{3} + 12 \, b x + 12 \, a - 3 \, \sin \left (4 \, b x + 4 \, a\right )}{192 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.56, size = 43, normalized size = 0.64 \[ \frac {x}{16}-\frac {\frac {\sin \left (4\,a+4\,b\,x\right )}{64}-\frac {\sin \left (2\,a+2\,b\,x\right )}{64}+\frac {\sin \left (6\,a+6\,b\,x\right )}{192}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.06, size = 136, normalized size = 2.03 \[ \begin {cases} \frac {x \sin ^{6}{\left (a + b x \right )}}{16} + \frac {3 x \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16} + \frac {3 x \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{16} + \frac {x \cos ^{6}{\left (a + b x \right )}}{16} + \frac {\sin ^{5}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{16 b} + \frac {\sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{6 b} - \frac {\sin {\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{16 b} & \text {for}\: b \neq 0 \\x \sin ^{2}{\relax (a )} \cos ^{4}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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