Optimal. Leaf size=52 \[ \text{Unintegrable}\left (\frac{\cos \left (x^2+x+\frac{1}{4}\right )}{x},x\right )+\sqrt{2 \pi } \text{FresnelC}\left (\frac{2 x+1}{\sqrt{2 \pi }}\right )-\frac{\sin \left (x^2+x+\frac{1}{4}\right )}{x} \]
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Rubi [A] time = 0.0246402, 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 = {} \[ \int \frac{\sin \left (\frac{1}{4}+x+x^2\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sin \left (\frac{1}{4}+x+x^2\right )}{x^2} \, dx &=-\frac{\sin \left (\frac{1}{4}+x+x^2\right )}{x}+2 \int \cos \left (\frac{1}{4}+x+x^2\right ) \, dx+\int \frac{\cos \left (\frac{1}{4}+x+x^2\right )}{x} \, dx\\ &=-\frac{\sin \left (\frac{1}{4}+x+x^2\right )}{x}+2 \int \cos \left (\frac{1}{4} (1+2 x)^2\right ) \, dx+\int \frac{\cos \left (\frac{1}{4}+x+x^2\right )}{x} \, dx\\ &=\sqrt{2 \pi } C\left (\frac{1+2 x}{\sqrt{2 \pi }}\right )-\frac{\sin \left (\frac{1}{4}+x+x^2\right )}{x}+\int \frac{\cos \left (\frac{1}{4}+x+x^2\right )}{x} \, dx\\ \end{align*}
Mathematica [A] time = 11.5538, size = 0, normalized size = 0. \[ \int \frac{\sin \left (\frac{1}{4}+x+x^2\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\sin \left ({\frac{1}{4}}+x+{x}^{2} \right ) }\, 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*} \int \frac{\sin \left (x^{2} + x + \frac{1}{4}\right )}{x^{2}}\,{d 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{\sin \left (x^{2} + x + \frac{1}{4}\right )}{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{\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 )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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