Integrand size = 15, antiderivative size = 18 \[ \int \frac {\cos \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\sin \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2717} \[ \int \frac {\cos \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\sin \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Rule 2717
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \cos (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\sin \left (a+b \log \left (c x^n\right )\right )}{b n} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(18)=36\).
Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06 \[ \int \frac {\cos \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\cos \left (b \log \left (c x^n\right )\right ) \sin (a)}{b n}+\frac {\cos (a) \sin \left (b \log \left (c x^n\right )\right )}{b n} \]
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Time = 0.52 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\) | \(19\) |
default | \(\frac {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\) | \(19\) |
parallelrisch | \(\frac {\sin \left (a +2 b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{b n}\) | \(22\) |
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Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\cos \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b n} \]
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Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (14) = 28\).
Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.89 \[ \int \frac {\cos \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\begin {cases} \log {\left (x \right )} \cos {\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \cos {\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\sin {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b n} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )}{b n} \]
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\[ \int \frac {\cos \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\cos \left (b \log \left (c x^{n}\right ) + a\right )}{x} \,d x } \]
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Time = 27.47 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}{b\,n} \]
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