Integrand size = 24, antiderivative size = 88 \[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{12} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{3/n}-\frac {1}{2} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{-3/n} \log (x) \]
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Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4581, 4577} \[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{12} \sqrt {-\frac {1}{n^2}} n x^3 e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{3/n}-\frac {1}{2} \sqrt {-\frac {1}{n^2}} n x^3 e^{a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-3/n} \]
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Rule 4577
Rule 4581
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int x^{-1+\frac {3}{n}} \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n} \\ & = -\left (\frac {1}{2} \left (\sqrt {-\frac {1}{n^2}} x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \left (\frac {e^{a \sqrt {-\frac {1}{n^2}} n}}{x}-e^{-a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {6}{n}}\right ) \, dx,x,c x^n\right )\right ) \\ & = \frac {1}{12} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{3/n}-\frac {1}{2} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{-3/n} \log (x) \\ \end{align*}
\[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. \(618\) vs. \(2(77)=154\).
Time = 2.67 (sec) , antiderivative size = 619, normalized size of antiderivative = 7.03
method | result | size |
parts | \(\frac {3 n \,x^{2} \sqrt {-\frac {1}{n^{2}}}\, {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \cos \left (a +3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}{8}-\frac {x^{2} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \sin \left (a +3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}{8}-\frac {-\frac {n \left (-\frac {\sqrt {-\frac {1}{n^{2}}}\, n \,c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \ln \left (x \right )}{2}+\frac {\sqrt {-\frac {1}{n^{2}}}\, n \,c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3}}{6}+c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \ln \left (x \right ) \tan \left (\frac {a}{2}+\frac {3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )-\frac {n \sqrt {-\frac {1}{n^{2}}}\, {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} c^{-\frac {1}{n}} x^{3} {\tan \left (\frac {a}{2}+\frac {3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )}^{2}}{6}+\frac {\sqrt {-\frac {1}{n^{2}}}\, n \,c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \ln \left (x \right ) {\tan \left (\frac {a}{2}+\frac {3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )}^{2}}{2}\right )}{1+{\tan \left (\frac {a}{2}+\frac {3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )}^{2}}+\frac {3 \sqrt {-\frac {1}{n^{2}}}\, n^{2} \left (\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \ln \left (x \right )}{2}+\frac {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 {3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )}{3 \sqrt {-\frac {1}{n^{2}}}\, n}-\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \ln \left (x \right ) {\tan \left (\frac {a}{2}+\frac {3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )}^{2}}{2}-\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \ln \left (x \right ) \tan \left (\frac {a}{2}+\frac {3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )}{\sqrt {-\frac {1}{n^{2}}}\, n}\right )}{1+{\tan \left (\frac {a}{2}+\frac {3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )}^{2}}}{4 n}\) | \(619\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.48 \[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{12} \, {\left (i \, x^{6} - 6 i \, e^{\left (\frac {2 \, {\left (i \, a n - 3 \, \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac {i \, a n - 3 \, \log \left (c\right )}{n}\right )} \]
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\[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x^{2} \sin {\left (a + 3 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.35 \[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {c^{\frac {6}{n}} x^{6} \sin \left (a\right ) + 6 \, \log \left (x\right ) \sin \left (a\right )}{12 \, c^{\frac {3}{n}}} \]
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Time = 0.43 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.01 \[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=+\infty \]
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Time = 27.83 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.97 \[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {x^3\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,3{}\mathrm {i}}}}{6\,n\,\sqrt {-\frac {1}{n^2}}+6{}\mathrm {i}}-\frac {x^3\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,3{}\mathrm {i}}}{6\,n\,\sqrt {-\frac {1}{n^2}}-6{}\mathrm {i}} \]
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