Optimal. Leaf size=67 \[ \frac{\sin (a+b x) \cos ^5(a+b x)}{6 b}+\frac{5 \sin (a+b x) \cos ^3(a+b x)}{24 b}+\frac{5 \sin (a+b x) \cos (a+b x)}{16 b}+\frac{5 x}{16} \]
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Rubi [A] time = 0.0331819, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2635, 8} \[ \frac{\sin (a+b x) \cos ^5(a+b x)}{6 b}+\frac{5 \sin (a+b x) \cos ^3(a+b x)}{24 b}+\frac{5 \sin (a+b x) \cos (a+b x)}{16 b}+\frac{5 x}{16} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^6(a+b x) \, dx &=\frac{\cos ^5(a+b x) \sin (a+b x)}{6 b}+\frac{5}{6} \int \cos ^4(a+b x) \, dx\\ &=\frac{5 \cos ^3(a+b x) \sin (a+b x)}{24 b}+\frac{\cos ^5(a+b x) \sin (a+b x)}{6 b}+\frac{5}{8} \int \cos ^2(a+b x) \, dx\\ &=\frac{5 \cos (a+b x) \sin (a+b x)}{16 b}+\frac{5 \cos ^3(a+b x) \sin (a+b x)}{24 b}+\frac{\cos ^5(a+b x) \sin (a+b x)}{6 b}+\frac{5 \int 1 \, dx}{16}\\ &=\frac{5 x}{16}+\frac{5 \cos (a+b x) \sin (a+b x)}{16 b}+\frac{5 \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*}
Mathematica [A] time = 0.0375434, size = 43, normalized size = 0.64 \[ \frac{45 \sin (2 (a+b x))+9 \sin (4 (a+b x))+\sin (6 (a+b x))+60 a+60 b x}{192 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 48, normalized size = 0.7 \begin{align*}{\frac{1}{b} \left ({\frac{\sin \left ( bx+a \right ) }{6} \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{5\,bx}{16}}+{\frac{5\,a}{16}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44311, size = 65, normalized size = 0.97 \begin{align*} -\frac{4 \, \sin \left (2 \, b x + 2 \, a\right )^{3} - 60 \, b x - 60 \, a - 9 \, \sin \left (4 \, b x + 4 \, a\right ) - 48 \, \sin \left (2 \, b x + 2 \, a\right )}{192 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64133, size = 120, normalized size = 1.79 \begin{align*} \frac{15 \, b x +{\left (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 )}{48 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.8272, size = 139, normalized size = 2.07 \begin{align*} \begin{cases} \frac{5 x \sin ^{6}{\left (a + b x \right )}}{16} + \frac{15 x \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16} + \frac{15 x \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{16} + \frac{5 x \cos ^{6}{\left (a + b x \right )}}{16} + \frac{5 \sin ^{5}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{16 b} + \frac{5 \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{6 b} + \frac{11 \sin{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{16 b} & \text{for}\: b \neq 0 \\x \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 = 1.18822, size = 62, normalized size = 0.93 \begin{align*} \frac{5}{16} \, x + \frac{\sin \left (6 \, b x + 6 \, a\right )}{192 \, b} + \frac{3 \, \sin \left (4 \, b x + 4 \, a\right )}{64 \, b} + \frac{15 \, \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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