Integrand size = 15, antiderivative size = 15 \[ \int \cos ^3(a+b x) \sin (a+b x) \, dx=-\frac {\cos ^4(a+b x)}{4 b} \]
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Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2645, 30} \[ \int \cos ^3(a+b x) \sin (a+b x) \, dx=-\frac {\cos ^4(a+b x)}{4 b} \]
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Rule 30
Rule 2645
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int x^3 \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {\cos ^4(a+b x)}{4 b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \cos ^3(a+b x) \sin (a+b x) \, dx=-\frac {\cos ^4(a+b x)}{4 b} \]
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Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(-\frac {\cos ^{4}\left (b x +a \right )}{4 b}\) | \(14\) |
default | \(-\frac {\cos ^{4}\left (b x +a \right )}{4 b}\) | \(14\) |
risch | \(-\frac {\cos \left (4 b x +4 a \right )}{32 b}-\frac {\cos \left (2 b x +2 a \right )}{8 b}\) | \(30\) |
parallelrisch | \(\frac {2 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{b \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{4}}\) | \(45\) |
norman | \(\frac {\frac {2 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+\frac {2 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{4}}\) | \(50\) |
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Time = 0.31 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \cos ^3(a+b x) \sin (a+b x) \, dx=-\frac {\cos \left (b x + a\right )^{4}}{4 \, b} \]
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Time = 0.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.47 \[ \int \cos ^3(a+b x) \sin (a+b x) \, dx=\begin {cases} - \frac {\cos ^{4}{\left (a + b x \right )}}{4 b} & \text {for}\: b \neq 0 \\x \sin {\left (a \right )} \cos ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \cos ^3(a+b x) \sin (a+b x) \, dx=-\frac {\cos \left (b x + a\right )^{4}}{4 \, b} \]
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Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.60 \[ \int \cos ^3(a+b x) \sin (a+b x) \, dx=-\frac {\sin \left (b x + a\right )^{4} - 2 \, \sin \left (b x + a\right )^{2}}{4 \, b} \]
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Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \cos ^3(a+b x) \sin (a+b x) \, dx=-\frac {{\cos \left (a+b\,x\right )}^4}{4\,b} \]
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