\(\int \frac {\sin (\frac {1}{4}+x+x^2)}{x^2} \, dx\) [15]

Optimal result

Integrand size = 13, antiderivative size = 13 \[ \int \frac {\sin \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {1+2 x}{\sqrt {2 \pi }}\right )-\frac {\sin \left (\frac {1}{4}+x+x^2\right )}{x}+\text {Int}\left (\frac {\cos \left (\frac {1}{4}+x+x^2\right )}{x},x\right ) \]

[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)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 13, 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 = {} \[ \int \frac {\sin \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int \frac {\sin \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx \]

[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*} \text {integral}& = -\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 } \operatorname {FresnelC}\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*}

Mathematica [N/A]

Not integrable

Time = 10.64 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {\sin \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int \frac {\sin \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx \]

[In]

Integrate[Sin[1/4 + x + x^2]/x^2,x]

[Out]

Integrate[Sin[1/4 + x + x^2]/x^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.09 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\sin \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int { \frac {\sin \left (x^{2} + x + \frac {1}{4}\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

integral(sin(x^2 + x + 1/4)/x^2, x)

Sympy [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {\sin \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int \frac {\sin {\left (x^{2} + x + \frac {1}{4} \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

Integral(sin(x**2 + x + 1/4)/x**2, x)

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\sin \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int { \frac {\sin \left (x^{2} + x + \frac {1}{4}\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(sin(x^2 + x + 1/4)/x^2, x)

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\sin \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int { \frac {\sin \left (x^{2} + x + \frac {1}{4}\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(sin(x^2 + x + 1/4)/x^2, x)

Mupad [N/A]

Not integrable

Time = 6.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\sin \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int \frac {\sin \left (x^2+x+\frac {1}{4}\right )}{x^2} \,d x \]

[In]

int(sin(x + x^2 + 1/4)/x^2,x)

[Out]

int(sin(x + x^2 + 1/4)/x^2, x)