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

Optimal. Leaf size=65 \[ \text{Unintegrable}\left (\frac{\sin \left (2 x^2+2 x+\frac{1}{2}\right )}{x},x\right )+\sqrt{\pi } S\left (\frac{2 x+1}{\sqrt{\pi }}\right )+\frac{\cos \left (2 x^2+2 x+\frac{1}{2}\right )}{2 x}-\frac{1}{2 x} \]

[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]

________________________________________________________________________________________

Rubi [A]  time = 0.0494996, 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 = {} \[ \int \frac{\sin ^2\left (\frac{1}{4}+x+x^2\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sin[1/4 + x + x^2]^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{\sin ^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 = 11.0357, size = 0, normalized size = 0. \[ \int \frac{\sin ^2\left (\frac{1}{4}+x+x^2\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sin[1/4 + x + x^2]^2/x^2,x]

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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(sin(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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(cos(x^2 + x + 1/4)^2 - 1)/x^2, x)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \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(sin(1/4+x+x^2)^2/x^2,x, algorithm="giac")

[Out]

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