Integrand size = 17, antiderivative size = 39 \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\log (x)}{2}-\frac {\cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
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Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2715, 8} \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\log (x)}{2}-\frac {\sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
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Rule 8
Rule 2715
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sin ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {\cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\text {Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{2 n} \\ & = \frac {\log (x)}{2}-\frac {\cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{2 b n} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92 \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {-2 \left (a+b \log \left (c x^n\right )\right )+\sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )}{4 b n} \]
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Time = 0.84 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82
method | result | size |
parallelrisch | \(\frac {2 \ln \left (x \right ) b n -\sin \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )}{4 b n}\) | \(32\) |
derivativedivides | \(\frac {-\frac {\cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}+\frac {a}{2}}{n b}\) | \(45\) |
default | \(\frac {-\frac {\cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}+\frac {a}{2}}{n b}\) | \(45\) |
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Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.03 \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {b n \log \left (x\right ) - \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{2 \, b n} \]
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Time = 1.45 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.31 \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=- \frac {\begin {cases} \log {\left (x \right )} \cos {\left (2 a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \cos {\left (2 a + 2 b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\sin {\left (2 a + 2 b \log {\left (c x^{n} \right )} \right )}}{2 b n} & \text {otherwise} \end {cases}}{2} + \frac {\log {\left (x \right )}}{2} \]
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Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.41 \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2 \, b n \log \left (x\right ) - \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) \sin \left (2 \, b \log \left (c\right )\right ) - \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )}{4 \, b n} \]
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\[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{2}}{x} \,d x } \]
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Time = 26.71 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\ln \left (x^n\right )}{2\,n}-\frac {\sin \left (2\,a+2\,b\,\ln \left (c\,x^n\right )\right )}{4\,b\,n} \]
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