Integrand size = 23, antiderivative size = 76 \[ \int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {x^2}{4}-\frac {1}{16} e^{-2 a \sqrt {-\frac {1}{n^2}} n} x^2 \left (c x^n\right )^{2/n}-\frac {1}{4} e^{2 a \sqrt {-\frac {1}{n^2}} n} x^2 \left (c x^n\right )^{-2/n} \log (x) \]
[Out]
Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {4581, 4577} \[ \int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} x^2 e^{-2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{2/n}-\frac {1}{4} x^2 e^{2 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-2/n}+\frac {x^2}{4} \]
[In]
[Out]
Rule 4577
Rule 4581
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int x^{-1+\frac {2}{n}} \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n} \\ & = -\frac {\left (x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int \left (\frac {e^{2 a \sqrt {-\frac {1}{n^2}} n}}{x}-2 x^{-1+\frac {2}{n}}+e^{-2 a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {4}{n}}\right ) \, dx,x,c x^n\right )}{4 n} \\ & = \frac {x^2}{4}-\frac {1}{16} e^{-2 a \sqrt {-\frac {1}{n^2}} n} x^2 \left (c x^n\right )^{2/n}-\frac {1}{4} e^{2 a \sqrt {-\frac {1}{n^2}} n} x^2 \left (c x^n\right )^{-2/n} \log (x) \\ \end{align*}
\[ \int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
[In]
[Out]
\[\int x {\sin \left (a +\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}^{2}d x\]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.79 \[ \int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} \, {\left (x^{4} - 4 \, x^{2} e^{\left (\frac {2 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} + 4 \, e^{\left (\frac {4 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac {2 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} \]
[In]
[Out]
\[ \int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x \sin ^{2}{\left (a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}\, dx \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.62 \[ \int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {c^{\frac {4}{n}} x^{4} \cos \left (2 \, a\right ) - 4 \, c^{\frac {2}{n}} x^{2} + 4 \, \cos \left (2 \, a\right ) \log \left (x\right )}{16 \, c^{\frac {2}{n}}} \]
[In]
[Out]
none
Time = 0.74 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.01 \[ \int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=+\infty \]
[In]
[Out]
Time = 28.61 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.21 \[ \int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {x^2}{4}-\frac {x^2\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}}\,1{}\mathrm {i}}{8\,n\,\sqrt {-\frac {1}{n^2}}+8{}\mathrm {i}}+\frac {x^2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}\,1{}\mathrm {i}}{8\,n\,\sqrt {-\frac {1}{n^2}}-8{}\mathrm {i}} \]
[In]
[Out]