Integrand size = 24, antiderivative size = 68 \[ \int \sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {x}{2}-\frac {1}{8} e^{-2 a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{\frac {1}{n}}-\frac {1}{4} e^{2 a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{-1/n} \log (x) \]
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Time = 0.06 (sec) , antiderivative size = 68, 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 = {4571, 4577} \[ \int \sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {1}{8} x e^{-2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {1}{n}}-\frac {1}{4} x e^{2 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n}+\frac {x}{2} \]
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Rule 4571
Rule 4577
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n} \\ & = -\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \left (\frac {e^{2 a \sqrt {-\frac {1}{n^2}} n}}{x}-2 x^{-1+\frac {1}{n}}+e^{-2 a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {2}{n}}\right ) \, dx,x,c x^n\right )}{4 n} \\ & = \frac {x}{2}-\frac {1}{8} e^{-2 a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{\frac {1}{n}}-\frac {1}{4} e^{2 a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{-1/n} \log (x) \\ \end{align*}
\[ \int \sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int \sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
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\[\int {\sin \left (a +\frac {\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )}^{2}d x\]
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Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.84 \[ \int \sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {1}{8} \, {\left (x^{2} - 4 \, x e^{\left (\frac {2 i \, a n - \log \left (c\right )}{n}\right )} + 2 \, e^{\left (\frac {2 \, {\left (2 i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac {2 i \, a n - \log \left (c\right )}{n}\right )} \]
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\[ \int \sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int \sin ^{2}{\left (a + \frac {\sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )}}{2} \right )}\, dx \]
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none
Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.60 \[ \int \sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {c^{\frac {2}{n}} x^{2} \cos \left (2 \, a\right ) - 4 \, c^{\left (\frac {1}{n}\right )} x + 2 \, \cos \left (2 \, a\right ) \log \left (x\right )}{8 \, c^{\left (\frac {1}{n}\right )}} \]
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none
Time = 0.57 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.01 \[ \int \sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=+\infty \]
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Time = 27.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.26 \[ \int \sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {x}{2}-\frac {x\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}}\,1{}\mathrm {i}}{4\,n\,\sqrt {-\frac {1}{n^2}}+4{}\mathrm {i}}+\frac {x\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{4\,n\,\sqrt {-\frac {1}{n^2}}-4{}\mathrm {i}} \]
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