3.28 \(\int \frac{\cos ^2(\frac{1}{4}+x+x^2)}{x^2} \, dx\)

Optimal. Leaf size=68 \[ -\text{Unintegrable}\left (\frac{\sin \left (2 x^2+2 x+\frac{1}{2}\right )}{x},x\right )-\sqrt{\pi } S\left (\frac{2 x+1}{\sqrt{\pi }}\right )-\frac{\cos \left (2 x^2+2 x+\frac{1}{2}\right )}{2 x}-\frac{1}{2 x} \]

[Out]

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

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Rubi [A]  time = 0.0523972, 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 = {} \[ \int \frac{\cos ^2\left (\frac{1}{4}+x+x^2\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cos[1/4 + x + x^2]^2/x^2,x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{\cos ^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 = 9.90145, size = 0, normalized size = 0. \[ \int \frac{\cos ^2\left (\frac{1}{4}+x+x^2\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Cos[1/4 + x + x^2]^2/x^2,x]

[Out]

Integrate[Cos[1/4 + x + x^2]^2/x^2, x]

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Maple [A]  time = 0.162, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( \cos \left ({\frac{1}{4}}+x+{x}^{2} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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.

[In]

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

[Out]

1/2*(x*integrate(cos(2*x^2 + 2*x + 1/2)/x^2, x) - 1)/x

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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}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cos(x^2 + x + 1/4)^2/x^2, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos ^{2}{\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(cos(1/4+x+x**2)**2/x**2,x)

[Out]

Integral(cos(x**2 + x + 1/4)**2/x**2, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \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.

[In]

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

[Out]

integrate(cos(x^2 + x + 1/4)^2/x^2, x)