Optimal. Leaf size=46 \[ \frac{\sin ^9(a+b x)}{9 b}-\frac{2 \sin ^7(a+b x)}{7 b}+\frac{\sin ^5(a+b x)}{5 b} \]
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Rubi [A] time = 0.0356168, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2564, 270} \[ \frac{\sin ^9(a+b x)}{9 b}-\frac{2 \sin ^7(a+b x)}{7 b}+\frac{\sin ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 2564
Rule 270
Rubi steps
\begin{align*} \int \cos ^5(a+b x) \sin ^4(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{\sin ^5(a+b x)}{5 b}-\frac{2 \sin ^7(a+b x)}{7 b}+\frac{\sin ^9(a+b x)}{9 b}\\ \end{align*}
Mathematica [A] time = 0.110989, size = 37, normalized size = 0.8 \[ \frac{\sin ^5(a+b x) (220 \cos (2 (a+b x))+35 \cos (4 (a+b x))+249)}{2520 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 68, normalized size = 1.5 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{3} \left ( \cos \left ( bx+a \right ) \right ) ^{6}}{9}}-{\frac{\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{6}}{21}}+{\frac{\sin \left ( bx+a \right ) }{105} \left ({\frac{8}{3}}+ \left ( \cos \left ( bx+a \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.990053, size = 49, normalized size = 1.07 \begin{align*} \frac{35 \, \sin \left (b x + a\right )^{9} - 90 \, \sin \left (b x + a\right )^{7} + 63 \, \sin \left (b x + a\right )^{5}}{315 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68277, size = 140, normalized size = 3.04 \begin{align*} \frac{{\left (35 \, \cos \left (b x + a\right )^{8} - 50 \, \cos \left (b x + a\right )^{6} + 3 \, \cos \left (b x + a\right )^{4} + 4 \, \cos \left (b x + a\right )^{2} + 8\right )} \sin \left (b x + a\right )}{315 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 21.5265, size = 66, normalized size = 1.43 \begin{align*} \begin{cases} \frac{8 \sin ^{9}{\left (a + b x \right )}}{315 b} + \frac{4 \sin ^{7}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{35 b} + \frac{\sin ^{5}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{5 b} & \text{for}\: b \neq 0 \\x \sin ^{4}{\left (a \right )} \cos ^{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 = 1.14985, size = 92, normalized size = 2. \begin{align*} \frac{\sin \left (9 \, b x + 9 \, a\right )}{2304 \, b} + \frac{\sin \left (7 \, b x + 7 \, a\right )}{1792 \, b} - \frac{\sin \left (5 \, b x + 5 \, a\right )}{320 \, b} - \frac{\sin \left (3 \, b x + 3 \, a\right )}{192 \, b} + \frac{3 \, \sin \left (b x + a\right )}{128 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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