Integrand size = 13, antiderivative size = 56 \[ \int x \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 x^2 \cos \left (a+b \log \left (c x^n\right )\right )}{4+b^2 n^2}+\frac {b n x^2 \sin \left (a+b \log \left (c x^n\right )\right )}{4+b^2 n^2} \]
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Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {4574} \[ \int x \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b n x^2 \sin \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+4}+\frac {2 x^2 \cos \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+4} \]
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Rule 4574
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^2 \cos \left (a+b \log \left (c x^n\right )\right )}{4+b^2 n^2}+\frac {b n x^2 \sin \left (a+b \log \left (c x^n\right )\right )}{4+b^2 n^2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.77 \[ \int x \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^2 \left (2 \cos \left (a+b \log \left (c x^n\right )\right )+b n \sin \left (a+b \log \left (c x^n\right )\right )\right )}{4+b^2 n^2} \]
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Time = 1.98 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.79
method | result | size |
parallelrisch | \(\frac {x^{2} \left (\sin \left (a +b \ln \left (c \,x^{n}\right )\right ) b n +2 \cos \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{b^{2} n^{2}+4}\) | \(44\) |
parts | \(\frac {x \,{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{n^{2} \left (\frac {1}{n^{2}}+b^{2}\right )}+\frac {x b \,{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{n \left (\frac {1}{n^{2}}+b^{2}\right )}-\frac {\frac {b \left (\frac {b n \,c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{2} {\tan \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}{b^{2} n^{2}+4}+\frac {4 c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{2} \tan \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}{b^{2} n^{2}+4}-\frac {b n \,c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{2}}{b^{2} n^{2}+4}\right )}{\left (\frac {1}{n^{2}}+b^{2}\right ) \left (1+{\tan \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}\right )}+\frac {\frac {2 c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{2}}{b^{2} n^{2}+4}-\frac {2 c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{2} {\tan \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}{b^{2} n^{2}+4}+\frac {2 b n \,c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{2} \tan \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}{b^{2} n^{2}+4}}{n \left (\frac {1}{n^{2}}+b^{2}\right ) \left (1+{\tan \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}\right )}}{n}\) | \(468\) |
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none
Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int x \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b n x^{2} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 2 \, x^{2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b^{2} n^{2} + 4} \]
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\[ \int x \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \int x \cos {\left (a - \frac {2 i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = - \frac {2 i}{n} \\\int x \cos {\left (a + \frac {2 i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {2 i}{n} \\\frac {b n x^{2} \sin {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b^{2} n^{2} + 4} + \frac {2 x^{2} \cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b^{2} n^{2} + 4} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (56) = 112\).
Time = 0.25 (sec) , antiderivative size = 218, normalized size of antiderivative = 3.89 \[ \int x \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {{\left ({\left (b \cos \left (b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) - b \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + b \sin \left (b \log \left (c\right )\right )\right )} n + 2 \, \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (b \log \left (c\right )\right ) + 2 \, \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + 2 \, \cos \left (b \log \left (c\right )\right )\right )} x^{2} \cos \left (b \log \left (x^{n}\right ) + a\right ) + {\left ({\left (b \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (b \log \left (c\right )\right ) + b \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + b \cos \left (b \log \left (c\right )\right )\right )} n - 2 \, \cos \left (b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + 2 \, \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) - 2 \, \sin \left (b \log \left (c\right )\right )\right )} x^{2} \sin \left (b \log \left (x^{n}\right ) + a\right )}{2 \, {\left ({\left (b^{2} \cos \left (b \log \left (c\right )\right )^{2} + b^{2} \sin \left (b \log \left (c\right )\right )^{2}\right )} n^{2} + 4 \, \cos \left (b \log \left (c\right )\right )^{2} + 4 \, \sin \left (b \log \left (c\right )\right )^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 915 vs. \(2 (56) = 112\).
Time = 0.33 (sec) , antiderivative size = 915, normalized size of antiderivative = 16.34 \[ \int x \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
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Time = 26.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.77 \[ \int x \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^2\,\left (2\,\cos \left (a+b\,\ln \left (c\,x^n\right )\right )+b\,n\,\sin \left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b^2\,n^2+4} \]
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