\(\int \frac {\cos ^2(\frac {1}{4}+x+x^2)}{x^2} \, dx\) [28]

Optimal result

Integrand size = 15, antiderivative size = 15 \[ \int \frac {\cos ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=-\frac {1}{2 x}-\frac {\cos \left (\frac {1}{2}+2 x+2 x^2\right )}{2 x}-\sqrt {\pi } \operatorname {FresnelS}\left (\frac {1+2 x}{\sqrt {\pi }}\right )-\text {Int}\left (\frac {\sin \left (\frac {1}{2}+2 x+2 x^2\right )}{x},x\right ) \]

[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)

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cos ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int \frac {\cos ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx \]

[In]

Int[Cos[1/4 + x + x^2]^2/x^2,x]

[Out]

-1/2*1/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*} \text {integral}& = \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 } \operatorname {FresnelS}\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 [N/A]

Not integrable

Time = 7.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\cos ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int \frac {\cos ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx \]

[In]

Integrate[Cos[1/4 + x + x^2]^2/x^2,x]

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.40 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87

\[\int \frac {\cos ^{2}\left (\frac {1}{4}+x +x^{2}\right )}{x^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int { \frac {\cos \left (x^{2} + x + \frac {1}{4}\right )^{2}}{x^{2}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int \frac {\cos ^{2}{\left (x^{2} + x + \frac {1}{4} \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73 \[ \int \frac {\cos ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int { \frac {\cos \left (x^{2} + x + \frac {1}{4}\right )^{2}}{x^{2}} \,d x } \]

[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

Giac [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int { \frac {\cos \left (x^{2} + x + \frac {1}{4}\right )^{2}}{x^{2}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 13.31 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int \frac {{\cos \left (x^2+x+\frac {1}{4}\right )}^2}{x^2} \,d x \]

[In]

int(cos(x + x^2 + 1/4)^2/x^2,x)

[Out]

int(cos(x + x^2 + 1/4)^2/x^2, x)