Integrand size = 17, antiderivative size = 46 \[ \int \cos ^2(a+b x) \sin ^5(a+b x) \, dx=-\frac {\cos ^3(a+b x)}{3 b}+\frac {2 \cos ^5(a+b x)}{5 b}-\frac {\cos ^7(a+b x)}{7 b} \]
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Time = 0.03 (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} \[ \int \cos ^2(a+b x) \sin ^5(a+b x) \, dx=-\frac {\cos ^7(a+b x)}{7 b}+\frac {2 \cos ^5(a+b x)}{5 b}-\frac {\cos ^3(a+b x)}{3 b} \]
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Rule 276
Rule 2645
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {\text {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {\cos ^3(a+b x)}{3 b}+\frac {2 \cos ^5(a+b x)}{5 b}-\frac {\cos ^7(a+b x)}{7 b} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80 \[ \int \cos ^2(a+b x) \sin ^5(a+b x) \, dx=\frac {\cos ^3(a+b x) (-157+108 \cos (2 (a+b x))-15 \cos (4 (a+b x)))}{840 b} \]
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Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(-\frac {\frac {\left (\cos ^{7}\left (b x +a \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (b x +a \right )\right )}{5}+\frac {\left (\cos ^{3}\left (b x +a \right )\right )}{3}}{b}\) | \(37\) |
default | \(-\frac {\frac {\left (\cos ^{7}\left (b x +a \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (b x +a \right )\right )}{5}+\frac {\left (\cos ^{3}\left (b x +a \right )\right )}{3}}{b}\) | \(37\) |
parallelrisch | \(\frac {-512-525 \cos \left (b x +a \right )-35 \cos \left (3 b x +3 a \right )+63 \cos \left (5 b x +5 a \right )-15 \cos \left (7 b x +7 a \right )}{6720 b}\) | \(49\) |
risch | \(-\frac {5 \cos \left (b x +a \right )}{64 b}-\frac {\cos \left (7 b x +7 a \right )}{448 b}+\frac {3 \cos \left (5 b x +5 a \right )}{320 b}-\frac {\cos \left (3 b x +3 a \right )}{192 b}\) | \(55\) |
norman | \(\frac {-\frac {16}{105 b}-\frac {32 \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}-\frac {16 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{15 b}-\frac {16 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 b}+\frac {16 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{7}}\) | \(87\) |
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Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^2(a+b x) \sin ^5(a+b x) \, dx=-\frac {15 \, \cos \left (b x + a\right )^{7} - 42 \, \cos \left (b x + a\right )^{5} + 35 \, \cos \left (b x + a\right )^{3}}{105 \, b} \]
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Time = 0.44 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.48 \[ \int \cos ^2(a+b x) \sin ^5(a+b x) \, dx=\begin {cases} - \frac {\sin ^{4}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {4 \sin ^{2}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{15 b} - \frac {8 \cos ^{7}{\left (a + b x \right )}}{105 b} & \text {for}\: b \neq 0 \\x \sin ^{5}{\left (a \right )} \cos ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^2(a+b x) \sin ^5(a+b x) \, dx=-\frac {15 \, \cos \left (b x + a\right )^{7} - 42 \, \cos \left (b x + a\right )^{5} + 35 \, \cos \left (b x + a\right )^{3}}{105 \, b} \]
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Time = 0.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.17 \[ \int \cos ^2(a+b x) \sin ^5(a+b x) \, dx=-\frac {\cos \left (7 \, b x + 7 \, a\right )}{448 \, b} + \frac {3 \, \cos \left (5 \, b x + 5 \, a\right )}{320 \, b} - \frac {\cos \left (3 \, b x + 3 \, a\right )}{192 \, b} - \frac {5 \, \cos \left (b x + a\right )}{64 \, b} \]
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Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^2(a+b x) \sin ^5(a+b x) \, dx=-\frac {15\,{\cos \left (a+b\,x\right )}^7-42\,{\cos \left (a+b\,x\right )}^5+35\,{\cos \left (a+b\,x\right )}^3}{105\,b} \]
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