Integrand size = 17, antiderivative size = 69 \[ \int \cos ^2(a+b x) \sin ^4(a+b x) \, dx=\frac {x}{16}+\frac {\cos (a+b x) \sin (a+b x)}{16 b}-\frac {\cos ^3(a+b x) \sin (a+b x)}{8 b}-\frac {\cos ^3(a+b x) \sin ^3(a+b x)}{6 b} \]
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Time = 0.04 (sec) , antiderivative size = 69, 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 = {2648, 2715, 8} \[ \int \cos ^2(a+b x) \sin ^4(a+b x) \, dx=-\frac {\sin ^3(a+b x) \cos ^3(a+b x)}{6 b}-\frac {\sin (a+b x) \cos ^3(a+b x)}{8 b}+\frac {\sin (a+b x) \cos (a+b x)}{16 b}+\frac {x}{16} \]
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Rule 8
Rule 2648
Rule 2715
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^3(a+b x) \sin ^3(a+b x)}{6 b}+\frac {1}{2} \int \cos ^2(a+b x) \sin ^2(a+b x) \, dx \\ & = -\frac {\cos ^3(a+b x) \sin (a+b x)}{8 b}-\frac {\cos ^3(a+b x) \sin ^3(a+b x)}{6 b}+\frac {1}{8} \int \cos ^2(a+b x) \, dx \\ & = \frac {\cos (a+b x) \sin (a+b x)}{16 b}-\frac {\cos ^3(a+b x) \sin (a+b x)}{8 b}-\frac {\cos ^3(a+b x) \sin ^3(a+b x)}{6 b}+\frac {\int 1 \, dx}{16} \\ & = \frac {x}{16}+\frac {\cos (a+b x) \sin (a+b x)}{16 b}-\frac {\cos ^3(a+b x) \sin (a+b x)}{8 b}-\frac {\cos ^3(a+b x) \sin ^3(a+b x)}{6 b} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.58 \[ \int \cos ^2(a+b x) \sin ^4(a+b x) \, dx=\frac {12 b x-3 \sin (2 (a+b x))-3 \sin (4 (a+b x))+\sin (6 (a+b x))}{192 b} \]
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Time = 0.14 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.61
method | result | size |
parallelrisch | \(\frac {12 b x +\sin \left (6 b x +6 a \right )-3 \sin \left (4 b x +4 a \right )-3 \sin \left (2 b x +2 a \right )}{192 b}\) | \(42\) |
risch | \(\frac {x}{16}+\frac {\sin \left (6 b x +6 a \right )}{192 b}-\frac {\sin \left (4 b x +4 a \right )}{64 b}-\frac {\sin \left (2 b x +2 a \right )}{64 b}\) | \(47\) |
derivativedivides | \(\frac {-\frac {\left (\cos ^{3}\left (b x +a \right )\right ) \left (\sin ^{3}\left (b x +a \right )\right )}{6}-\frac {\left (\cos ^{3}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{8}+\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{16}+\frac {b x}{16}+\frac {a}{16}}{b}\) | \(61\) |
default | \(\frac {-\frac {\left (\cos ^{3}\left (b x +a \right )\right ) \left (\sin ^{3}\left (b x +a \right )\right )}{6}-\frac {\left (\cos ^{3}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{8}+\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{16}+\frac {b x}{16}+\frac {a}{16}}{b}\) | \(61\) |
norman | \(\frac {\frac {x}{16}-\frac {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{8 b}-\frac {17 \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{24 b}+\frac {19 \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b}-\frac {19 \left (\tan ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b}+\frac {17 \left (\tan ^{9}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{24 b}+\frac {\tan ^{11}\left (\frac {b x}{2}+\frac {a}{2}\right )}{8 b}+\frac {3 x \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8}+\frac {15 x \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{16}+\frac {5 x \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4}+\frac {15 x \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{16}+\frac {3 x \left (\tan ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8}+\frac {x \left (\tan ^{12}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{16}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{6}}\) | \(199\) |
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Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.67 \[ \int \cos ^2(a+b x) \sin ^4(a+b x) \, dx=\frac {3 \, b x + {\left (8 \, \cos \left (b x + a\right )^{5} - 14 \, \cos \left (b x + a\right )^{3} + 3 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{48 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (58) = 116\).
Time = 0.33 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.97 \[ \int \cos ^2(a+b x) \sin ^4(a+b x) \, dx=\begin {cases} \frac {x \sin ^{6}{\left (a + b x \right )}}{16} + \frac {3 x \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16} + \frac {3 x \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{16} + \frac {x \cos ^{6}{\left (a + b x \right )}}{16} + \frac {\sin ^{5}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{16 b} - \frac {\sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{6 b} - \frac {\sin {\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{16 b} & \text {for}\: b \neq 0 \\x \sin ^{4}{\left (a \right )} \cos ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.54 \[ \int \cos ^2(a+b x) \sin ^4(a+b x) \, dx=-\frac {4 \, \sin \left (2 \, b x + 2 \, a\right )^{3} - 12 \, b x - 12 \, a + 3 \, \sin \left (4 \, b x + 4 \, a\right )}{192 \, b} \]
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Time = 0.32 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.67 \[ \int \cos ^2(a+b x) \sin ^4(a+b x) \, dx=\frac {1}{16} \, x + \frac {\sin \left (6 \, b x + 6 \, a\right )}{192 \, b} - \frac {\sin \left (4 \, b x + 4 \, a\right )}{64 \, b} - \frac {\sin \left (2 \, b x + 2 \, a\right )}{64 \, b} \]
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Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.62 \[ \int \cos ^2(a+b x) \sin ^4(a+b x) \, dx=\frac {x}{16}-\frac {\frac {\sin \left (2\,a+2\,b\,x\right )}{64}+\frac {\sin \left (4\,a+4\,b\,x\right )}{64}-\frac {\sin \left (6\,a+6\,b\,x\right )}{192}}{b} \]
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