Optimal. Leaf size=42 \[ -\frac{\cos ^5(a+b x)}{5 b}+\frac{2 \cos ^3(a+b x)}{3 b}-\frac{\cos (a+b x)}{b} \]
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Rubi [A] time = 0.0130112, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2633} \[ -\frac{\cos ^5(a+b x)}{5 b}+\frac{2 \cos ^3(a+b x)}{3 b}-\frac{\cos (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2633
Rubi steps
\begin{align*} \int \sin ^5(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{\cos (a+b x)}{b}+\frac{2 \cos ^3(a+b x)}{3 b}-\frac{\cos ^5(a+b x)}{5 b}\\ \end{align*}
Mathematica [A] time = 0.0121988, size = 44, normalized size = 1.05 \[ -\frac{5 \cos (a+b x)}{8 b}+\frac{5 \cos (3 (a+b x))}{48 b}-\frac{\cos (5 (a+b x))}{80 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 32, normalized size = 0.8 \begin{align*} -{\frac{\cos \left ( bx+a \right ) }{5\,b} \left ({\frac{8}{3}}+ \left ( \sin \left ( bx+a \right ) \right ) ^{4}+{\frac{4\, \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995149, size = 46, normalized size = 1.1 \begin{align*} -\frac{3 \, \cos \left (b x + a\right )^{5} - 10 \, \cos \left (b x + a\right )^{3} + 15 \, \cos \left (b x + a\right )}{15 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1845, size = 89, normalized size = 2.12 \begin{align*} -\frac{3 \, \cos \left (b x + a\right )^{5} - 10 \, \cos \left (b x + a\right )^{3} + 15 \, \cos \left (b x + a\right )}{15 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.83787, size = 60, normalized size = 1.43 \begin{align*} \begin{cases} - \frac{\sin ^{4}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{b} - \frac{4 \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac{8 \cos ^{5}{\left (a + b x \right )}}{15 b} & \text{for}\: b \neq 0 \\x \sin ^{5}{\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 = 1.12195, size = 51, normalized size = 1.21 \begin{align*} -\frac{\cos \left (b x + a\right )^{5}}{5 \, b} + \frac{2 \, \cos \left (b x + a\right )^{3}}{3 \, b} - \frac{\cos \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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