Integrand size = 8, antiderivative size = 46 \[ \int \cos ^4(a+b x) \, dx=\frac {3 x}{8}+\frac {3 \cos (a+b x) \sin (a+b x)}{8 b}+\frac {\cos ^3(a+b x) \sin (a+b x)}{4 b} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2715, 8} \[ \int \cos ^4(a+b x) \, dx=\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3 \sin (a+b x) \cos (a+b x)}{8 b}+\frac {3 x}{8} \]
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Rule 8
Rule 2715
Rubi steps \begin{align*} \text {integral}& = \frac {\cos ^3(a+b x) \sin (a+b x)}{4 b}+\frac {3}{4} \int \cos ^2(a+b x) \, dx \\ & = \frac {3 \cos (a+b x) \sin (a+b x)}{8 b}+\frac {\cos ^3(a+b x) \sin (a+b x)}{4 b}+\frac {3 \int 1 \, dx}{8} \\ & = \frac {3 x}{8}+\frac {3 \cos (a+b x) \sin (a+b x)}{8 b}+\frac {\cos ^3(a+b x) \sin (a+b x)}{4 b} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.72 \[ \int \cos ^4(a+b x) \, dx=\frac {12 (a+b x)+8 \sin (2 (a+b x))+\sin (4 (a+b x))}{32 b} \]
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Time = 0.75 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.67
method | result | size |
parallelrisch | \(\frac {12 b x +\sin \left (4 b x +4 a \right )+8 \sin \left (2 b x +2 a \right )}{32 b}\) | \(31\) |
risch | \(\frac {3 x}{8}+\frac {\sin \left (4 b x +4 a \right )}{32 b}+\frac {\sin \left (2 b x +2 a \right )}{4 b}\) | \(33\) |
derivativedivides | \(\frac {\frac {\left (\cos ^{3}\left (b x +a \right )+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}}{b}\) | \(38\) |
default | \(\frac {\frac {\left (\cos ^{3}\left (b x +a \right )+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}}{b}\) | \(38\) |
norman | \(\frac {\frac {3 x}{8}+\frac {5 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{4 b}-\frac {3 \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b}+\frac {3 \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b}-\frac {5 \left (\tan ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b}+\frac {3 x \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2}+\frac {9 x \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4}+\frac {3 x \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2}+\frac {3 x \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{4}}\) | \(139\) |
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Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^4(a+b x) \, dx=\frac {3 \, b x + {\left (2 \, \cos \left (b x + a\right )^{3} + 3 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{8 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (41) = 82\).
Time = 0.17 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.07 \[ \int \cos ^4(a+b x) \, dx=\begin {cases} \frac {3 x \sin ^{4}{\left (a + b x \right )}}{8} + \frac {3 x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} + \frac {3 x \cos ^{4}{\left (a + b x \right )}}{8} + \frac {3 \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} + \frac {5 \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} & \text {for}\: b \neq 0 \\x \cos ^{4}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.72 \[ \int \cos ^4(a+b x) \, dx=\frac {12 \, b x + 12 \, a + \sin \left (4 \, b x + 4 \, a\right ) + 8 \, \sin \left (2 \, b x + 2 \, a\right )}{32 \, b} \]
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Time = 0.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.70 \[ \int \cos ^4(a+b x) \, dx=\frac {3}{8} \, x + \frac {\sin \left (4 \, b x + 4 \, a\right )}{32 \, b} + \frac {\sin \left (2 \, b x + 2 \, a\right )}{4 \, b} \]
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Time = 13.69 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.67 \[ \int \cos ^4(a+b x) \, dx=\frac {3\,x}{8}+\frac {\frac {\sin \left (2\,a+2\,b\,x\right )}{4}+\frac {\sin \left (4\,a+4\,b\,x\right )}{32}}{b} \]
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