Optimal. Leaf size=69 \[ -\frac {1}{2 x}-\frac {\cos \left (\frac {1}{2}+2 x+2 x^2\right )}{2 x}-\sqrt {\pi } S\left (\frac {1+2 x}{\sqrt {\pi }}\right )-\text {Int}\left (\frac {\sin \left (\frac {1}{2}+2 x+2 x^2\right )}{x},x\right ) \]
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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 = {}
\begin {gather*} \int \frac {\cos ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\cos ^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*}
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Mathematica [A]
time = 10.57, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^2\left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\cos ^{2}\left (\frac {1}{4}+x +x^{2}\right )}{x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{2}{\left (x^{2} + x + \frac {1}{4} \right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\cos \left (x^2+x+\frac {1}{4}\right )}^2}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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