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

Optimal. Leaf size=69 \[ -\text {Int}\left (\frac {\sin \left (2 x^2+2 x+\frac {1}{2}\right )}{x},x\right )-\sqrt {\pi } S\left (\frac {2 x+1}{\sqrt {\pi }}\right )-\frac {\cos \left (2 x^2+2 x+\frac {1}{2}\right )}{2 x}-\frac {1}{2 x} \]

[Out]

-1/2/x-1/2*cos(1/2+2*x+2*x^2)/x-FresnelS((1+2*x)/Pi^(1/2))*Pi^(1/2)-Unintegrable(sin(1/2+2*x+2*x^2)/x,x)

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Rubi [A]  time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 10.65, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx \]

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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fricas [A]  time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (x^{2} + x + \frac {1}{4}\right )^{2}}{x^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (x^{2} + x + \frac {1}{4}\right )^{2}}{x^{2}}\,{d x} \]

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)

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maple [A]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^{2}\left (\frac {1}{4}+x +x^{2}\right )}{x^{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \frac {x \int \frac {\cos \left (2 \, x^{2} + 2 \, x + \frac {1}{2}\right )}{x^{2}}\,{d x} - 1}{2 \, x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (x^2+x+\frac {1}{4}\right )}^2}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x + x^2 + 1/4)^2/x^2,x)

[Out]

int(cos(x + x^2 + 1/4)^2/x^2, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^{2}{\left (x^{2} + x + \frac {1}{4} \right )}}{x^{2}}\, dx \]

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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