3.15 \(\int \frac{\sin (\frac{1}{4}+x+x^2)}{x^2} \, dx\)

Optimal. Leaf size=52 \[ \text{Unintegrable}\left (\frac{\cos \left (x^2+x+\frac{1}{4}\right )}{x},x\right )+\sqrt{2 \pi } \text{FresnelC}\left (\frac{2 x+1}{\sqrt{2 \pi }}\right )-\frac{\sin \left (x^2+x+\frac{1}{4}\right )}{x} \]

[Out]

Sqrt[2*Pi]*FresnelC[(1 + 2*x)/Sqrt[2*Pi]] - Sin[1/4 + x + x^2]/x + Unintegrable[Cos[1/4 + x + x^2]/x, 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.

[In]

Int[Sin[1/4 + x + x^2]/x^2,x]

[Out]

Sqrt[2*Pi]*FresnelC[(1 + 2*x)/Sqrt[2*Pi]] - Sin[1/4 + x + x^2]/x + Defer[Int][Cos[1/4 + x + x^2]/x, x]

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.

[In]

Integrate[Sin[1/4 + x + x^2]/x^2,x]

[Out]

Integrate[Sin[1/4 + x + x^2]/x^2, x]

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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.

[In]

int(sin(1/4+x+x^2)/x^2,x)

[Out]

int(sin(1/4+x+x^2)/x^2,x)

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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.

[In]

integrate(sin(1/4+x+x^2)/x^2,x, algorithm="maxima")

[Out]

integrate(sin(x^2 + x + 1/4)/x^2, x)

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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.

[In]

integrate(sin(1/4+x+x^2)/x^2,x, algorithm="fricas")

[Out]

integral(sin(x^2 + x + 1/4)/x^2, x)

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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.

[In]

integrate(sin(1/4+x+x**2)/x**2,x)

[Out]

Integral(sin(x**2 + x + 1/4)/x**2, x)

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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.

[In]

integrate(sin(1/4+x+x^2)/x^2,x, algorithm="giac")

[Out]

integrate(sin(x^2 + x + 1/4)/x^2, x)