Integrand size = 17, antiderivative size = 46 \[ \int \cos ^4(a+b x) \sin ^5(a+b x) \, dx=-\frac {\cos ^5(a+b x)}{5 b}+\frac {2 \cos ^7(a+b x)}{7 b}-\frac {\cos ^9(a+b x)}{9 b} \]
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Time = 0.02 (sec) , antiderivative size = 46, 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 = {2645, 276} \[ \int \cos ^4(a+b x) \sin ^5(a+b x) \, dx=-\frac {\cos ^9(a+b x)}{9 b}+\frac {2 \cos ^7(a+b x)}{7 b}-\frac {\cos ^5(a+b x)}{5 b} \]
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Rule 276
Rule 2645
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {\text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {\cos ^5(a+b x)}{5 b}+\frac {2 \cos ^7(a+b x)}{7 b}-\frac {\cos ^9(a+b x)}{9 b} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80 \[ \int \cos ^4(a+b x) \sin ^5(a+b x) \, dx=\frac {\cos ^5(a+b x) (-249+220 \cos (2 (a+b x))-35 \cos (4 (a+b x)))}{2520 b} \]
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Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(-\frac {\frac {\left (\cos ^{9}\left (b x +a \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (b x +a \right )\right )}{7}+\frac {\left (\cos ^{5}\left (b x +a \right )\right )}{5}}{b}\) | \(37\) |
default | \(-\frac {\frac {\left (\cos ^{9}\left (b x +a \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (b x +a \right )\right )}{7}+\frac {\left (\cos ^{5}\left (b x +a \right )\right )}{5}}{b}\) | \(37\) |
parallelrisch | \(\frac {-2048-35 \cos \left (9 b x +9 a \right )-1890 \cos \left (b x +a \right )+252 \cos \left (5 b x +5 a \right )-420 \cos \left (3 b x +3 a \right )+45 \cos \left (7 b x +7 a \right )}{80640 b}\) | \(60\) |
risch | \(-\frac {3 \cos \left (b x +a \right )}{128 b}-\frac {\cos \left (9 b x +9 a \right )}{2304 b}+\frac {\cos \left (7 b x +7 a \right )}{1792 b}+\frac {\cos \left (5 b x +5 a \right )}{320 b}-\frac {\cos \left (3 b x +3 a \right )}{192 b}\) | \(69\) |
norman | \(\frac {-\frac {16}{315 b}-\frac {112 \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 b}-\frac {32 \left (\tan ^{12}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}+\frac {16 \left (\tan ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+\frac {32 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 b}-\frac {16 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{35 b}-\frac {64 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{35 b}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{9}}\) | \(119\) |
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Time = 0.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^4(a+b x) \sin ^5(a+b x) \, dx=-\frac {35 \, \cos \left (b x + a\right )^{9} - 90 \, \cos \left (b x + a\right )^{7} + 63 \, \cos \left (b x + a\right )^{5}}{315 \, b} \]
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Time = 0.91 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.48 \[ \int \cos ^4(a+b x) \sin ^5(a+b x) \, dx=\begin {cases} - \frac {\sin ^{4}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{5 b} - \frac {4 \sin ^{2}{\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{35 b} - \frac {8 \cos ^{9}{\left (a + b x \right )}}{315 b} & \text {for}\: b \neq 0 \\x \sin ^{5}{\left (a \right )} \cos ^{4}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^4(a+b x) \sin ^5(a+b x) \, dx=-\frac {35 \, \cos \left (b x + a\right )^{9} - 90 \, \cos \left (b x + a\right )^{7} + 63 \, \cos \left (b x + a\right )^{5}}{315 \, b} \]
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Time = 0.33 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.48 \[ \int \cos ^4(a+b x) \sin ^5(a+b x) \, dx=-\frac {\cos \left (9 \, b x + 9 \, a\right )}{2304 \, b} + \frac {\cos \left (7 \, b x + 7 \, a\right )}{1792 \, b} + \frac {\cos \left (5 \, b x + 5 \, a\right )}{320 \, b} - \frac {\cos \left (3 \, b x + 3 \, a\right )}{192 \, b} - \frac {3 \, \cos \left (b x + a\right )}{128 \, b} \]
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Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^4(a+b x) \sin ^5(a+b x) \, dx=-\frac {35\,{\cos \left (a+b\,x\right )}^9-90\,{\cos \left (a+b\,x\right )}^7+63\,{\cos \left (a+b\,x\right )}^5}{315\,b} \]
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