Integrand size = 17, antiderivative size = 46 \[ \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx=-\frac {\cos ^8(a+b x)}{8 b}+\frac {\cos ^{10}(a+b x)}{5 b}-\frac {\cos ^{12}(a+b x)}{12 b} \]
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Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2645, 272, 45} \[ \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx=-\frac {\cos ^{12}(a+b x)}{12 b}+\frac {\cos ^{10}(a+b x)}{5 b}-\frac {\cos ^8(a+b x)}{8 b} \]
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Rule 45
Rule 272
Rule 2645
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int x^7 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {\text {Subst}\left (\int (1-x)^2 x^3 \, dx,x,\cos ^2(a+b x)\right )}{2 b} \\ & = -\frac {\text {Subst}\left (\int \left (x^3-2 x^4+x^5\right ) \, dx,x,\cos ^2(a+b x)\right )}{2 b} \\ & = -\frac {\cos ^8(a+b x)}{8 b}+\frac {\cos ^{10}(a+b x)}{5 b}-\frac {\cos ^{12}(a+b x)}{12 b} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.48 \[ \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx=-\frac {600 \cos (2 (a+b x))+75 \cos (4 (a+b x))-100 \cos (6 (a+b x))-30 \cos (8 (a+b x))+12 \cos (10 (a+b x))+5 \cos (12 (a+b x))}{122880 b} \]
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Time = 0.36 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(-\frac {\frac {\left (\sin ^{12}\left (b x +a \right )\right )}{12}-\frac {3 \left (\sin ^{10}\left (b x +a \right )\right )}{10}+\frac {3 \left (\sin ^{8}\left (b x +a \right )\right )}{8}-\frac {\left (\sin ^{6}\left (b x +a \right )\right )}{6}}{b}\) | \(47\) |
default | \(-\frac {\frac {\left (\sin ^{12}\left (b x +a \right )\right )}{12}-\frac {3 \left (\sin ^{10}\left (b x +a \right )\right )}{10}+\frac {3 \left (\sin ^{8}\left (b x +a \right )\right )}{8}-\frac {\left (\sin ^{6}\left (b x +a \right )\right )}{6}}{b}\) | \(47\) |
parallelrisch | \(\frac {30 \cos \left (8 b x +8 a \right )+100 \cos \left (6 b x +6 a \right )-600 \cos \left (2 b x +2 a \right )-75 \cos \left (4 b x +4 a \right )-5 \cos \left (12 b x +12 a \right )-12 \cos \left (10 b x +10 a \right )+562}{122880 b}\) | \(74\) |
risch | \(-\frac {\cos \left (12 b x +12 a \right )}{24576 b}-\frac {\cos \left (10 b x +10 a \right )}{10240 b}+\frac {\cos \left (8 b x +8 a \right )}{4096 b}+\frac {5 \cos \left (6 b x +6 a \right )}{6144 b}-\frac {5 \cos \left (4 b x +4 a \right )}{8192 b}-\frac {5 \cos \left (2 b x +2 a \right )}{1024 b}\) | \(86\) |
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Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx=-\frac {10 \, \cos \left (b x + a\right )^{12} - 24 \, \cos \left (b x + a\right )^{10} + 15 \, \cos \left (b x + a\right )^{8}}{120 \, b} \]
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Time = 2.63 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.41 \[ \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx=\begin {cases} - \frac {\sin ^{4}{\left (a + b x \right )} \cos ^{8}{\left (a + b x \right )}}{8 b} - \frac {\sin ^{2}{\left (a + b x \right )} \cos ^{10}{\left (a + b x \right )}}{20 b} - \frac {\cos ^{12}{\left (a + b x \right )}}{120 b} & \text {for}\: b \neq 0 \\x \sin ^{5}{\left (a \right )} \cos ^{7}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx=-\frac {10 \, \sin \left (b x + a\right )^{12} - 36 \, \sin \left (b x + a\right )^{10} + 45 \, \sin \left (b x + a\right )^{8} - 20 \, \sin \left (b x + a\right )^{6}}{120 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (40) = 80\).
Time = 0.34 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.85 \[ \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx=-\frac {\cos \left (12 \, b x + 12 \, a\right )}{24576 \, b} - \frac {\cos \left (10 \, b x + 10 \, a\right )}{10240 \, b} + \frac {\cos \left (8 \, b x + 8 \, a\right )}{4096 \, b} + \frac {5 \, \cos \left (6 \, b x + 6 \, a\right )}{6144 \, b} - \frac {5 \, \cos \left (4 \, b x + 4 \, a\right )}{8192 \, b} - \frac {5 \, \cos \left (2 \, b x + 2 \, a\right )}{1024 \, b} \]
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Time = 0.15 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.76 \[ \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx=-\frac {{\cos \left (a+b\,x\right )}^8\,\left (10\,{\cos \left (a+b\,x\right )}^4-24\,{\cos \left (a+b\,x\right )}^2+15\right )}{120\,b} \]
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