Integrand size = 15, antiderivative size = 98 \[ \int x \cos ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b^2 n^2 x^2}{4 \left (1+b^2 n^2\right )}+\frac {x^2 \cos ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+b^2 n^2\right )}+\frac {b n x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{2 \left (1+b^2 n^2\right )} \]
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Time = 0.02 (sec) , antiderivative size = 98, 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 = {4576, 30} \[ \int x \cos ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^2 \cos ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (b^2 n^2+1\right )}+\frac {b n x^2 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{2 \left (b^2 n^2+1\right )}+\frac {b^2 n^2 x^2}{4 \left (b^2 n^2+1\right )} \]
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Rule 30
Rule 4576
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 \cos ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+b^2 n^2\right )}+\frac {b n x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{2 \left (1+b^2 n^2\right )}+\frac {\left (b^2 n^2\right ) \int x \, dx}{2 \left (1+b^2 n^2\right )} \\ & = \frac {b^2 n^2 x^2}{4 \left (1+b^2 n^2\right )}+\frac {x^2 \cos ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+b^2 n^2\right )}+\frac {b n x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{2 \left (1+b^2 n^2\right )} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.55 \[ \int x \cos ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^2 \left (1+b^2 n^2+\cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right )}{4+4 b^2 n^2} \]
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Time = 1.41 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.58
method | result | size |
parallelrisch | \(\frac {x^{2} \left (b^{2} n^{2}+b n \sin \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )+\cos \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )+1\right )}{4 b^{2} n^{2}+4}\) | \(57\) |
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none
Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.76 \[ \int x \cos ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b^{2} n^{2} x^{2} + 2 \, b n x^{2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 2 \, x^{2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}}{4 \, {\left (b^{2} n^{2} + 1\right )}} \]
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\[ \int x \cos ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \int x \cos ^{2}{\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = - \frac {i}{n} \\\int x \cos ^{2}{\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {i}{n} \\\frac {b^{2} n^{2} x^{2} \sin ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} n^{2} + 4} + \frac {b^{2} n^{2} x^{2} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} n^{2} + 4} + \frac {2 b n x^{2} \sin {\left (a + b \log {\left (c x^{n} \right )} \right )} \cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} n^{2} + 4} + \frac {2 x^{2} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} n^{2} + 4} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (92) = 184\).
Time = 0.23 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.88 \[ \int x \cos ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {{\left ({\left (b \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (4 \, b \log \left (c\right )\right ) - b \cos \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + b \sin \left (2 \, b \log \left (c\right )\right )\right )} n + \cos \left (4 \, b \log \left (c\right )\right ) \cos \left (2 \, b \log \left (c\right )\right ) + \sin \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + \cos \left (2 \, b \log \left (c\right )\right )\right )} x^{2} \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) + {\left ({\left (b \cos \left (4 \, b \log \left (c\right )\right ) \cos \left (2 \, b \log \left (c\right )\right ) + b \sin \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + b \cos \left (2 \, b \log \left (c\right )\right )\right )} n - \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (4 \, b \log \left (c\right )\right ) + \cos \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) - \sin \left (2 \, b \log \left (c\right )\right )\right )} x^{2} \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) + 2 \, {\left ({\left (b^{2} \cos \left (2 \, b \log \left (c\right )\right )^{2} + b^{2} \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} n^{2} + \cos \left (2 \, b \log \left (c\right )\right )^{2} + \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} x^{2}}{8 \, {\left ({\left (b^{2} \cos \left (2 \, b \log \left (c\right )\right )^{2} + b^{2} \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} n^{2} + \cos \left (2 \, b \log \left (c\right )\right )^{2} + \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 820 vs. \(2 (92) = 184\).
Time = 0.45 (sec) , antiderivative size = 820, normalized size of antiderivative = 8.37 \[ \int x \cos ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
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Time = 27.17 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.67 \[ \int x \cos ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^2}{4}+\frac {x^2\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}}\,1{}\mathrm {i}}{8\,b\,n+8{}\mathrm {i}}+\frac {x^2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}}{8+b\,n\,8{}\mathrm {i}} \]
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