Integrand size = 15, antiderivative size = 15 \[ \int \frac {\sin ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=-\frac {1}{2 x}+\frac {\cos \left (\frac {1}{2}+2 x+2 x^2\right )}{2 x}+\sqrt {\pi } \operatorname {FresnelS}\left (\frac {1+2 x}{\sqrt {\pi }}\right )+\text {Int}\left (\frac {\sin \left (\frac {1}{2}+2 x+2 x^2\right )}{x},x\right ) \]
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Not integrable
Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sin ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int \frac {\sin ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2 x^2}-\frac {\cos \left (\frac {1}{2}+2 x+2 x^2\right )}{2 x^2}\right ) \, dx \\ & = -\frac {1}{2 x}-\frac {1}{2} \int \frac {\cos \left (\frac {1}{2}+2 x+2 x^2\right )}{x^2} \, dx \\ & = -\frac {1}{2 x}+\frac {\cos \left (\frac {1}{2}+2 x+2 x^2\right )}{2 x}+2 \int \sin \left (\frac {1}{2}+2 x+2 x^2\right ) \, dx+\int \frac {\sin \left (\frac {1}{2}+2 x+2 x^2\right )}{x} \, dx \\ & = -\frac {1}{2 x}+\frac {\cos \left (\frac {1}{2}+2 x+2 x^2\right )}{2 x}+2 \int \sin \left (\frac {1}{8} (2+4 x)^2\right ) \, dx+\int \frac {\sin \left (\frac {1}{2}+2 x+2 x^2\right )}{x} \, dx \\ & = -\frac {1}{2 x}+\frac {\cos \left (\frac {1}{2}+2 x+2 x^2\right )}{2 x}+\sqrt {\pi } \operatorname {FresnelS}\left (\frac {1+2 x}{\sqrt {\pi }}\right )+\int \frac {\sin \left (\frac {1}{2}+2 x+2 x^2\right )}{x} \, dx \\ \end{align*}
Not integrable
Time = 7.37 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\sin ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int \frac {\sin ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx \]
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Not integrable
Time = 0.15 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87
\[\int \frac {\sin ^{2}\left (\frac {1}{4}+x +x^{2}\right )}{x^{2}}d x\]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20 \[ \int \frac {\sin ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int { \frac {\sin \left (x^{2} + x + \frac {1}{4}\right )^{2}}{x^{2}} \,d x } \]
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Not integrable
Time = 0.61 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int \frac {\sin ^{2}{\left (x^{2} + x + \frac {1}{4} \right )}}{x^{2}}\, dx \]
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Not integrable
Time = 0.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73 \[ \int \frac {\sin ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int { \frac {\sin \left (x^{2} + x + \frac {1}{4}\right )^{2}}{x^{2}} \,d x } \]
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Not integrable
Time = 0.33 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int { \frac {\sin \left (x^{2} + x + \frac {1}{4}\right )^{2}}{x^{2}} \,d x } \]
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Not integrable
Time = 5.67 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int \frac {{\sin \left (x^2+x+\frac {1}{4}\right )}^2}{x^2} \,d x \]
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