Integrand size = 17, antiderivative size = 95 \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {2 b^2 n^2}{\left (1+4 b^2 n^2\right ) x}-\frac {2 b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x}-\frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x} \]
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Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4575, 30} \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x \left (4 b^2 n^2+1\right )}-\frac {2 b n \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (4 b^2 n^2+1\right )}-\frac {2 b^2 n^2}{x \left (4 b^2 n^2+1\right )} \]
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Rule 30
Rule 4575
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x}-\frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x}+\frac {\left (2 b^2 n^2\right ) \int \frac {1}{x^2} \, dx}{1+4 b^2 n^2} \\ & = -\frac {2 b^2 n^2}{\left (1+4 b^2 n^2\right ) x}-\frac {2 b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x}-\frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.60 \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {-1-4 b^2 n^2+\cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-2 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )}{2 \left (x+4 b^2 n^2 x\right )} \]
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Time = 1.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.62
method | result | size |
parallelrisch | \(\frac {-4 b^{2} n^{2}-2 b n \sin \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )+\cos \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )-1}{8 b^{2} n^{2} x +2 x}\) | \(59\) |
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none
Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.75 \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {2 \, b^{2} n^{2} + 2 \, b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1}{{\left (4 \, b^{2} n^{2} + 1\right )} x} \]
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Result contains complex when optimal does not.
Time = 5.80 (sec) , antiderivative size = 303, normalized size of antiderivative = 3.19 \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\begin {cases} \frac {\cos {\left (2 a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{4 x} - \frac {1}{2 x} - \frac {i \log {\left (c x^{n} \right )} \sin {\left (2 a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{4 n x} - \frac {\log {\left (c x^{n} \right )} \cos {\left (2 a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{4 n x} & \text {for}\: b = - \frac {i}{2 n} \\\frac {i \sin {\left (2 a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{4 x} - \frac {1}{2 x} + \frac {i \log {\left (c x^{n} \right )} \sin {\left (2 a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{4 n x} - \frac {\log {\left (c x^{n} \right )} \cos {\left (2 a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{4 n x} & \text {for}\: b = \frac {i}{2 n} \\- \frac {2 b^{2} n^{2} \sin ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} n^{2} x + x} - \frac {2 b^{2} n^{2} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} n^{2} x + x} - \frac {2 b n \sin {\left (a + b \log {\left (c x^{n} \right )} \right )} \cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} n^{2} x + x} - \frac {\sin ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} n^{2} x + x} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (95) = 190\).
Time = 0.22 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.98 \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {8 \, {\left (b^{2} \cos \left (2 \, b \log \left (c\right )\right )^{2} + b^{2} \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} n^{2} + 2 \, \cos \left (2 \, b \log \left (c\right )\right )^{2} + {\left (2 \, {\left (b \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (4 \, b \log \left (c\right )\right ) - b \cos \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + b \sin \left (2 \, b \log \left (c\right )\right )\right )} n - \cos \left (4 \, b \log \left (c\right )\right ) \cos \left (2 \, b \log \left (c\right )\right ) - \sin \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) - \cos \left (2 \, b \log \left (c\right )\right )\right )} \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) + 2 \, \sin \left (2 \, b \log \left (c\right )\right )^{2} + {\left (2 \, {\left (b \cos \left (4 \, b \log \left (c\right )\right ) \cos \left (2 \, b \log \left (c\right )\right ) + b \sin \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + b \cos \left (2 \, b \log \left (c\right )\right )\right )} n + \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (4 \, b \log \left (c\right )\right ) - \cos \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + \sin \left (2 \, b \log \left (c\right )\right )\right )} \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )}{4 \, {\left (4 \, {\left (b^{2} \cos \left (2 \, b \log \left (c\right )\right )^{2} + b^{2} \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} n^{2} + \cos \left (2 \, b \log \left (c\right )\right )^{2} + \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} x} \]
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\[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{2}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^2} \,d x \]
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