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

Optimal. Leaf size=31 \[ \frac {\log (x)}{2}-\frac {1}{2} \text {Int}\left (\frac {\cos \left (2 x^2+2 x+\frac {1}{2}\right )}{x},x\right ) \]

[Out]

1/2*ln(x)-1/2*Unintegrable(cos(1/2+2*x+2*x^2)/x,x)

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Rubi [A]  time = 0.03, 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 = {} \[ \int \frac {\sin ^2\left (\frac {1}{4}+x+x^2\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sin[1/4 + x + x^2]^2/x,x]

[Out]

Log[x]/2 - Defer[Int][Cos[1/2 + 2*x + 2*x^2]/x, x]/2

Rubi steps

\begin {align*} \int \frac {\sin ^2\left (\frac {1}{4}+x+x^2\right )}{x} \, dx &=\int \left (\frac {1}{2 x}-\frac {\cos \left (\frac {1}{2}+2 x+2 x^2\right )}{2 x}\right ) \, dx\\ &=\frac {\log (x)}{2}-\frac {1}{2} \int \frac {\cos \left (\frac {1}{2}+2 x+2 x^2\right )}{x} \, dx\\ \end {align*}

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Mathematica [A]  time = 10.49, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^2\left (\frac {1}{4}+x+x^2\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sin[1/4 + x + x^2]^2/x,x]

[Out]

Integrate[Sin[1/4 + x + x^2]^2/x, x]

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\cos \left (x^{2} + x + \frac {1}{4}\right )^{2} - 1}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(x^2 + x + 1/4)^2/x, x)

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maple [A]  time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{2}\left (\frac {1}{4}+x +x^{2}\right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*integrate(cos(2*x^2 + 2*x + 1/2)/x, x) + 1/2*log(x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\sin \left (x^2+x+\frac {1}{4}\right )}^2}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x + x^2 + 1/4)^2/x,x)

[Out]

int(sin(x + x^2 + 1/4)^2/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(x**2 + x + 1/4)**2/x, x)

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