Integrand size = 17, antiderivative size = 61 \[ \int \cos ^7(a+b x) \sin ^2(a+b x) \, dx=\frac {\sin ^3(a+b x)}{3 b}-\frac {3 \sin ^5(a+b x)}{5 b}+\frac {3 \sin ^7(a+b x)}{7 b}-\frac {\sin ^9(a+b x)}{9 b} \]
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Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2644, 276} \[ \int \cos ^7(a+b x) \sin ^2(a+b x) \, dx=-\frac {\sin ^9(a+b x)}{9 b}+\frac {3 \sin ^7(a+b x)}{7 b}-\frac {3 \sin ^5(a+b x)}{5 b}+\frac {\sin ^3(a+b x)}{3 b} \]
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Rule 276
Rule 2644
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^2 \left (1-x^2\right )^3 \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {\text {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*}
Time = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77 \[ \int \cos ^7(a+b x) \sin ^2(a+b x) \, dx=\frac {(1606+1389 \cos (2 (a+b x))+330 \cos (4 (a+b x))+35 \cos (6 (a+b x))) \sin ^3(a+b x)}{10080 b} \]
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Time = 0.22 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(-\frac {\frac {\left (\sin ^{9}\left (b x +a \right )\right )}{9}-\frac {3 \left (\sin ^{7}\left (b x +a \right )\right )}{7}+\frac {3 \left (\sin ^{5}\left (b x +a \right )\right )}{5}-\frac {\left (\sin ^{3}\left (b x +a \right )\right )}{3}}{b}\) | \(47\) |
default | \(-\frac {\frac {\left (\sin ^{9}\left (b x +a \right )\right )}{9}-\frac {3 \left (\sin ^{7}\left (b x +a \right )\right )}{7}+\frac {3 \left (\sin ^{5}\left (b x +a \right )\right )}{5}-\frac {\left (\sin ^{3}\left (b x +a \right )\right )}{3}}{b}\) | \(47\) |
risch | \(\frac {7 \sin \left (b x +a \right )}{128 b}-\frac {\sin \left (9 b x +9 a \right )}{2304 b}-\frac {5 \sin \left (7 b x +7 a \right )}{1792 b}-\frac {\sin \left (5 b x +5 a \right )}{160 b}\) | \(55\) |
parallelrisch | \(-\frac {\left (\sin \left (\frac {3 b x}{2}+\frac {3 a}{2}\right )-3 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \left (35 \cos \left (6 b x +6 a \right )+1389 \cos \left (2 b x +2 a \right )+330 \cos \left (4 b x +4 a \right )+1606\right ) \left (\cos \left (\frac {3 b x}{2}+\frac {3 a}{2}\right )+3 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{20160 b}\) | \(83\) |
norman | \(\frac {\frac {8 \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}-\frac {16 \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 b}+\frac {632 \left (\tan ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{35 b}-\frac {2848 \left (\tan ^{9}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{315 b}+\frac {632 \left (\tan ^{11}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{35 b}-\frac {16 \left (\tan ^{13}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 b}+\frac {8 \left (\tan ^{15}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{9}}\) | \(130\) |
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Time = 0.35 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \cos ^7(a+b x) \sin ^2(a+b x) \, dx=-\frac {{\left (35 \, \cos \left (b x + a\right )^{8} - 5 \, \cos \left (b x + a\right )^{6} - 6 \, \cos \left (b x + a\right )^{4} - 8 \, \cos \left (b x + a\right )^{2} - 16\right )} \sin \left (b x + a\right )}{315 \, b} \]
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Time = 0.91 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.44 \[ \int \cos ^7(a+b x) \sin ^2(a+b x) \, dx=\begin {cases} \frac {16 \sin ^{9}{\left (a + b x \right )}}{315 b} + \frac {8 \sin ^{7}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{35 b} + \frac {2 \sin ^{5}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{5 b} + \frac {\sin ^{3}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x \sin ^{2}{\left (a \right )} \cos ^{7}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \cos ^7(a+b x) \sin ^2(a+b x) \, dx=-\frac {35 \, \sin \left (b x + a\right )^{9} - 135 \, \sin \left (b x + a\right )^{7} + 189 \, \sin \left (b x + a\right )^{5} - 105 \, \sin \left (b x + a\right )^{3}}{315 \, b} \]
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Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89 \[ \int \cos ^7(a+b x) \sin ^2(a+b x) \, dx=-\frac {\sin \left (9 \, b x + 9 \, a\right )}{2304 \, b} - \frac {5 \, \sin \left (7 \, b x + 7 \, a\right )}{1792 \, b} - \frac {\sin \left (5 \, b x + 5 \, a\right )}{160 \, b} + \frac {7 \, \sin \left (b x + a\right )}{128 \, b} \]
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Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.74 \[ \int \cos ^7(a+b x) \sin ^2(a+b x) \, dx=\frac {-\frac {{\sin \left (a+b\,x\right )}^9}{9}+\frac {3\,{\sin \left (a+b\,x\right )}^7}{7}-\frac {3\,{\sin \left (a+b\,x\right )}^5}{5}+\frac {{\sin \left (a+b\,x\right )}^3}{3}}{b} \]
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