Integrand size = 15, antiderivative size = 56 \[ \int x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {3 x^3 \cos \left (a+b \log \left (c x^n\right )\right )}{9+b^2 n^2}+\frac {b n x^3 \sin \left (a+b \log \left (c x^n\right )\right )}{9+b^2 n^2} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {4574} \[ \int x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b n x^3 \sin \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+9}+\frac {3 x^3 \cos \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+9} \]
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Rule 4574
Rubi steps \begin{align*} \text {integral}& = \frac {3 x^3 \cos \left (a+b \log \left (c x^n\right )\right )}{9+b^2 n^2}+\frac {b n x^3 \sin \left (a+b \log \left (c x^n\right )\right )}{9+b^2 n^2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.77 \[ \int x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^3 \left (3 \cos \left (a+b \log \left (c x^n\right )\right )+b n \sin \left (a+b \log \left (c x^n\right )\right )\right )}{9+b^2 n^2} \]
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Time = 2.53 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.79
method | result | size |
parallelrisch | \(\frac {x^{3} \left (\sin \left (a +b \ln \left (c \,x^{n}\right )\right ) b n +3 \cos \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{b^{2} n^{2}+9}\) | \(44\) |
parts | \(\frac {x^{2} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{n^{2} \left (\frac {1}{n^{2}}+b^{2}\right )}+\frac {x^{2} b \,{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{n \left (\frac {1}{n^{2}}+b^{2}\right )}-\frac {2 \left (\frac {n \left (\frac {3 c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3}}{b^{2} n^{2}+9}-\frac {3 c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} {\tan \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}{b^{2} n^{2}+9}+\frac {2 b n \,c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \tan \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}{b^{2} n^{2}+9}\right )}{\left (b^{2} n^{2}+1\right ) \left (1+{\tan \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}\right )}+\frac {b \,n^{2} \left (\frac {b n \,c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} {\tan \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}{b^{2} n^{2}+9}+\frac {6 c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \tan \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}{b^{2} n^{2}+9}-\frac {b n \,c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3}}{b^{2} n^{2}+9}\right )}{\left (b^{2} n^{2}+1\right ) \left (1+{\tan \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}\right )}\right )}{n}\) | \(477\) |
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none
Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b n x^{3} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 3 \, x^{3} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b^{2} n^{2} + 9} \]
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\[ \int x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \int x^{2} \cos {\left (a - \frac {3 i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = - \frac {3 i}{n} \\\int x^{2} \cos {\left (a + \frac {3 i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {3 i}{n} \\\frac {b n x^{3} \sin {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b^{2} n^{2} + 9} + \frac {3 x^{3} \cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b^{2} n^{2} + 9} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (56) = 112\).
Time = 0.23 (sec) , antiderivative size = 218, normalized size of antiderivative = 3.89 \[ \int x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {{\left ({\left (b \cos \left (b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) - b \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + b \sin \left (b \log \left (c\right )\right )\right )} n + 3 \, \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (b \log \left (c\right )\right ) + 3 \, \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + 3 \, \cos \left (b \log \left (c\right )\right )\right )} x^{3} \cos \left (b \log \left (x^{n}\right ) + a\right ) + {\left ({\left (b \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (b \log \left (c\right )\right ) + b \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + b \cos \left (b \log \left (c\right )\right )\right )} n - 3 \, \cos \left (b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + 3 \, \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) - 3 \, \sin \left (b \log \left (c\right )\right )\right )} x^{3} \sin \left (b \log \left (x^{n}\right ) + a\right )}{2 \, {\left ({\left (b^{2} \cos \left (b \log \left (c\right )\right )^{2} + b^{2} \sin \left (b \log \left (c\right )\right )^{2}\right )} n^{2} + 9 \, \cos \left (b \log \left (c\right )\right )^{2} + 9 \, \sin \left (b \log \left (c\right )\right )^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 923 vs. \(2 (56) = 112\).
Time = 0.35 (sec) , antiderivative size = 923, normalized size of antiderivative = 16.48 \[ \int x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
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Time = 26.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.77 \[ \int x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^3\,\left (3\,\cos \left (a+b\,\ln \left (c\,x^n\right )\right )+b\,n\,\sin \left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b^2\,n^2+9} \]
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