Optimal. Leaf size=57 \[ -\frac {\sqrt {1+x+x^2}}{2 x^2}+\frac {3 \sqrt {1+x+x^2}}{4 x}+\frac {1}{8} \tanh ^{-1}\left (\frac {2+x}{2 \sqrt {1+x+x^2}}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {758, 820, 738,
212} \begin {gather*} \frac {3 \sqrt {x^2+x+1}}{4 x}-\frac {\sqrt {x^2+x+1}}{2 x^2}+\frac {1}{8} \tanh ^{-1}\left (\frac {x+2}{2 \sqrt {x^2+x+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 738
Rule 758
Rule 820
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {1+x+x^2}} \, dx &=-\frac {\sqrt {1+x+x^2}}{2 x^2}-\frac {1}{2} \int \frac {\frac {3}{2}+x}{x^2 \sqrt {1+x+x^2}} \, dx\\ &=-\frac {\sqrt {1+x+x^2}}{2 x^2}+\frac {3 \sqrt {1+x+x^2}}{4 x}-\frac {1}{8} \int \frac {1}{x \sqrt {1+x+x^2}} \, dx\\ &=-\frac {\sqrt {1+x+x^2}}{2 x^2}+\frac {3 \sqrt {1+x+x^2}}{4 x}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {2+x}{\sqrt {1+x+x^2}}\right )\\ &=-\frac {\sqrt {1+x+x^2}}{2 x^2}+\frac {3 \sqrt {1+x+x^2}}{4 x}+\frac {1}{8} \tanh ^{-1}\left (\frac {2+x}{2 \sqrt {1+x+x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 42, normalized size = 0.74 \begin {gather*} \frac {(-2+3 x) \sqrt {1+x+x^2}}{4 x^2}-\frac {1}{4} \tanh ^{-1}\left (x-\sqrt {1+x+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 44, normalized size = 0.77
method | result | size |
trager | \(\frac {\left (-2+3 x \right ) \sqrt {x^{2}+x +1}}{4 x^{2}}+\frac {\ln \left (\frac {2 \sqrt {x^{2}+x +1}+2+x}{x}\right )}{8}\) | \(40\) |
risch | \(\frac {3 x^{3}+x^{2}+x -2}{4 x^{2} \sqrt {x^{2}+x +1}}+\frac {\arctanh \left (\frac {2+x}{2 \sqrt {x^{2}+x +1}}\right )}{8}\) | \(42\) |
default | \(\frac {\arctanh \left (\frac {2+x}{2 \sqrt {x^{2}+x +1}}\right )}{8}-\frac {\sqrt {x^{2}+x +1}}{2 x^{2}}+\frac {3 \sqrt {x^{2}+x +1}}{4 x}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.60, size = 50, normalized size = 0.88 \begin {gather*} \frac {3 \, \sqrt {x^{2} + x + 1}}{4 \, x} - \frac {\sqrt {x^{2} + x + 1}}{2 \, x^{2}} + \frac {1}{8} \, \operatorname {arsinh}\left (\frac {\sqrt {3} x}{3 \, {\left | x \right |}} + \frac {2 \, \sqrt {3}}{3 \, {\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.54, size = 63, normalized size = 1.11 \begin {gather*} \frac {x^{2} \log \left (-x + \sqrt {x^{2} + x + 1} + 1\right ) - x^{2} \log \left (-x + \sqrt {x^{2} + x + 1} - 1\right ) + 6 \, x^{2} + 2 \, \sqrt {x^{2} + x + 1} {\left (3 \, x - 2\right )}}{8 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \sqrt {x^{2} + x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.65, size = 84, normalized size = 1.47 \begin {gather*} \frac {{\left (x - \sqrt {x^{2} + x + 1}\right )}^{3} + 9 \, x - 9 \, \sqrt {x^{2} + x + 1} + 8}{4 \, {\left ({\left (x - \sqrt {x^{2} + x + 1}\right )}^{2} - 1\right )}^{2}} + \frac {1}{8} \, \log \left ({\left | -x + \sqrt {x^{2} + x + 1} + 1 \right |}\right ) - \frac {1}{8} \, \log \left ({\left | -x + \sqrt {x^{2} + x + 1} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x^3\,\sqrt {x^2+x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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