Optimal. Leaf size=61 \[ -\frac {\sin ^9(a+b x)}{9 b}+\frac {3 \sin ^7(a+b x)}{7 b}-\frac {3 \sin ^5(a+b x)}{5 b}+\frac {\sin ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, 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} \[ -\frac {\sin ^9(a+b x)}{9 b}+\frac {3 \sin ^7(a+b x)}{7 b}-\frac {3 \sin ^5(a+b x)}{5 b}+\frac {\sin ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2564
Rubi steps
\begin {align*} \int \cos ^7(a+b x) \sin ^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int x^2 \left (1-x^2\right )^3 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (x^2-3 x^4+3 x^6-x^8\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {\sin ^3(a+b x)}{3 b}-\frac {3 \sin ^5(a+b x)}{5 b}+\frac {3 \sin ^7(a+b x)}{7 b}-\frac {\sin ^9(a+b x)}{9 b}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 47, normalized size = 0.77 \[ \frac {\sin ^3(a+b x) (1389 \cos (2 (a+b x))+330 \cos (4 (a+b x))+35 \cos (6 (a+b x))+1606)}{10080 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 53, normalized size = 0.87 \[ -\frac {{\left (35 \, \cos \left (b x + a\right )^{8} - 5 \, \cos \left (b x + a\right )^{6} - 6 \, \cos \left (b x + a\right )^{4} - 8 \, \cos \left (b x + a\right )^{2} - 16\right )} \sin \left (b x + a\right )}{315 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 54, normalized size = 0.89 \[ -\frac {\sin \left (9 \, b x + 9 \, a\right )}{2304 \, b} - \frac {5 \, \sin \left (7 \, b x + 7 \, a\right )}{1792 \, b} - \frac {\sin \left (5 \, b x + 5 \, a\right )}{160 \, b} + \frac {7 \, \sin \left (b x + a\right )}{128 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 60, normalized size = 0.98 \[ \frac {-\frac {\sin \left (b x +a \right ) \left (\cos ^{8}\left (b x +a \right )\right )}{9}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (b x +a \right )+\frac {6 \left (\cos ^{4}\left (b x +a \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (b x +a \right )\right )}{5}\right ) \sin \left (b x +a \right )}{63}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 46, normalized size = 0.75 \[ -\frac {35 \, \sin \left (b x + a\right )^{9} - 135 \, \sin \left (b x + a\right )^{7} + 189 \, \sin \left (b x + a\right )^{5} - 105 \, \sin \left (b x + a\right )^{3}}{315 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 45, normalized size = 0.74 \[ \frac {-\frac {{\sin \left (a+b\,x\right )}^9}{9}+\frac {3\,{\sin \left (a+b\,x\right )}^7}{7}-\frac {3\,{\sin \left (a+b\,x\right )}^5}{5}+\frac {{\sin \left (a+b\,x\right )}^3}{3}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.10, size = 88, normalized size = 1.44 \[ \begin {cases} \frac {16 \sin ^{9}{\left (a + b x \right )}}{315 b} + \frac {8 \sin ^{7}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{35 b} + \frac {2 \sin ^{5}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{5 b} + \frac {\sin ^{3}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x \sin ^{2}{\relax (a )} \cos ^{7}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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