Optimal. Leaf size=65 \[ \text{Unintegrable}\left (\frac{\sin \left (2 x^2+2 x+\frac{1}{2}\right )}{x},x\right )+\sqrt{\pi } S\left (\frac{2 x+1}{\sqrt{\pi }}\right )+\frac{\cos \left (2 x^2+2 x+\frac{1}{2}\right )}{2 x}-\frac{1}{2 x} \]
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Rubi [A] time = 0.0494996, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sin ^2\left (\frac{1}{4}+x+x^2\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sin ^2\left (\frac{1}{4}+x+x^2\right )}{x^2} \, dx &=\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 } S\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*}
Mathematica [A] time = 11.0357, size = 0, normalized size = 0. \[ \int \frac{\sin ^2\left (\frac{1}{4}+x+x^2\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.162, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( \sin \left ({\frac{1}{4}}+x+{x}^{2} \right ) \right ) ^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{x \int \frac{\cos \left (2 \, x^{2} + 2 \, x + \frac{1}{2}\right )}{x^{2}}\,{d x} + 1}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\cos \left (x^{2} + x + \frac{1}{4}\right )^{2} - 1}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin ^{2}{\left (x^{2} + x + \frac{1}{4} \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (x^{2} + x + \frac{1}{4}\right )^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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