Optimal. Leaf size=20 \[ \frac {x+\frac {1}{x+x \left (25+\log \left (5+\frac {1}{x^2}\right )\right )}}{e^4} \]
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Rubi [F]
time = 0.41, 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 {-24+546 x^2+3380 x^4+\left (-1+47 x^2+260 x^4\right ) \log \left (\frac {1+5 x^2}{x^2}\right )+\left (x^2+5 x^4\right ) \log ^2\left (\frac {1+5 x^2}{x^2}\right )}{e^4 \left (676 x^2+3380 x^4\right )+e^4 \left (52 x^2+260 x^4\right ) \log \left (\frac {1+5 x^2}{x^2}\right )+e^4 \left (x^2+5 x^4\right ) \log ^2\left (\frac {1+5 x^2}{x^2}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-24+546 x^2+3380 x^4+\left (-1+47 x^2+260 x^4\right ) \log \left (5+\frac {1}{x^2}\right )+\left (x^2+5 x^4\right ) \log ^2\left (5+\frac {1}{x^2}\right )}{e^4 x^2 \left (1+5 x^2\right ) \left (26+\log \left (5+\frac {1}{x^2}\right )\right )^2} \, dx\\ &=\frac {\int \frac {-24+546 x^2+3380 x^4+\left (-1+47 x^2+260 x^4\right ) \log \left (5+\frac {1}{x^2}\right )+\left (x^2+5 x^4\right ) \log ^2\left (5+\frac {1}{x^2}\right )}{x^2 \left (1+5 x^2\right ) \left (26+\log \left (5+\frac {1}{x^2}\right )\right )^2} \, dx}{e^4}\\ &=\frac {\int \left (1+\frac {2}{x^2 \left (1+5 x^2\right ) \left (26+\log \left (5+\frac {1}{x^2}\right )\right )^2}-\frac {1}{x^2 \left (26+\log \left (5+\frac {1}{x^2}\right )\right )}\right ) \, dx}{e^4}\\ &=\frac {x}{e^4}-\frac {\int \frac {1}{x^2 \left (26+\log \left (5+\frac {1}{x^2}\right )\right )} \, dx}{e^4}+\frac {2 \int \frac {1}{x^2 \left (1+5 x^2\right ) \left (26+\log \left (5+\frac {1}{x^2}\right )\right )^2} \, dx}{e^4}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.18, size = 20, normalized size = 1.00 \begin {gather*} \frac {x+\frac {1}{x \left (26+\log \left (5+\frac {1}{x^2}\right )\right )}}{e^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 20.90, size = 28, normalized size = 1.40
method | result | size |
risch | \(x \,{\mathrm e}^{-4}+\frac {{\mathrm e}^{-4}}{x \left (\ln \left (\frac {5 x^{2}+1}{x^{2}}\right )+26\right )}\) | \(28\) |
norman | \(\frac {{\mathrm e}^{-4}+x^{2} {\mathrm e}^{-4} \ln \left (\frac {5 x^{2}+1}{x^{2}}\right )+26 x^{2} {\mathrm e}^{-4}}{x \left (\ln \left (\frac {5 x^{2}+1}{x^{2}}\right )+26\right )}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs.
\(2 (19) = 38\).
time = 0.31, size = 54, normalized size = 2.70 \begin {gather*} \frac {x^{2} \log \left (5 \, x^{2} + 1\right ) - 2 \, x^{2} \log \left (x\right ) + 26 \, x^{2} + 1}{x e^{4} \log \left (5 \, x^{2} + 1\right ) - 2 \, x e^{4} \log \left (x\right ) + 26 \, x e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs.
\(2 (19) = 38\).
time = 0.36, size = 48, normalized size = 2.40 \begin {gather*} \frac {x^{2} \log \left (\frac {5 \, x^{2} + 1}{x^{2}}\right ) + 26 \, x^{2} + 1}{x e^{4} \log \left (\frac {5 \, x^{2} + 1}{x^{2}}\right ) + 26 \, x e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 29, normalized size = 1.45 \begin {gather*} \frac {x}{e^{4}} + \frac {1}{x e^{4} \log {\left (\frac {5 x^{2} + 1}{x^{2}} \right )} + 26 x e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs.
\(2 (19) = 38\).
time = 0.49, size = 48, normalized size = 2.40 \begin {gather*} \frac {x^{2} \log \left (\frac {5 \, x^{2} + 1}{x^{2}}\right ) + 26 \, x^{2} + 1}{x e^{4} \log \left (\frac {5 \, x^{2} + 1}{x^{2}}\right ) + 26 \, x e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.48, size = 45, normalized size = 2.25 \begin {gather*} \frac {{\mathrm {e}}^{-4}\,\left (x^2\,\ln \left (\frac {5\,x^2+1}{x^2}\right )+26\,x^2+1\right )}{x\,\left (\ln \left (\frac {5\,x^2+1}{x^2}\right )+26\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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