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

Optimal. Leaf size=53 \[ \text {Int}\left (\frac {\cos \left (x^2+x+\frac {1}{4}\right )}{x},x\right )+\sqrt {2 \pi } C\left (\frac {2 x+1}{\sqrt {2 \pi }}\right )-\frac {\sin \left (x^2+x+\frac {1}{4}\right )}{x} \]

[Out]

-sin(1/4+x+x^2)/x+FresnelC(1/2*(1+2*x)*2^(1/2)/Pi^(1/2))*2^(1/2)*Pi^(1/2)+Unintegrable(cos(1/4+x+x^2)/x,x)

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Rubi [A]  time = 0.02, 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 = {} \[ \int \frac {\sin \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx \]

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*}

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Mathematica [A]  time = 11.40, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx \]

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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fricas [A]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sin \left (x^{2} + x + \frac {1}{4}\right )}{x^{2}}, x\right ) \]

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

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)

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maple [A]  time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (\frac {1}{4}+x +x^{2}\right )}{x^{2}}\, dx \]

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

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sin \left (x^2+x+\frac {1}{4}\right )}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x + x^2 + 1/4)/x^2,x)

[Out]

int(sin(x + x^2 + 1/4)/x^2, x)

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

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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