Optimal. Leaf size=20 \[ \frac {x^2}{\log \left (-15+5 \left (1+\frac {x^2}{625}\right )\right )} \]
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Rubi [A]
time = 0.32, antiderivative size = 36, normalized size of antiderivative = 1.80, number of steps
used = 14, number of rules used = 11, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.239, Rules used = {6857, 2525,
2458, 12, 2395, 2334, 2335, 2339, 30, 2504, 2436} \begin {gather*} \frac {1250}{\log \left (\frac {x^2}{125}-10\right )}-\frac {1250-x^2}{\log \left (\frac {x^2}{125}-10\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2334
Rule 2335
Rule 2339
Rule 2395
Rule 2436
Rule 2458
Rule 2504
Rule 2525
Rule 6857
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 x^3}{\left (-1250+x^2\right ) \log ^2\left (-10+\frac {x^2}{125}\right )}+\frac {2 x}{\log \left (-10+\frac {x^2}{125}\right )}\right ) \, dx\\ &=-\left (2 \int \frac {x^3}{\left (-1250+x^2\right ) \log ^2\left (-10+\frac {x^2}{125}\right )} \, dx\right )+2 \int \frac {x}{\log \left (-10+\frac {x^2}{125}\right )} \, dx\\ &=-\text {Subst}\left (\int \frac {x}{(-1250+x) \log ^2\left (-10+\frac {x}{125}\right )} \, dx,x,x^2\right )+\text {Subst}\left (\int \frac {1}{\log \left (-10+\frac {x}{125}\right )} \, dx,x,x^2\right )\\ &=-\left (125 \text {Subst}\left (\int \frac {1250+125 x}{125 x \log ^2(x)} \, dx,x,\frac {1}{125} \left (-1250+x^2\right )\right )\right )+125 \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,\frac {1}{125} \left (-1250+x^2\right )\right )\\ &=125 \text {li}\left (-10+\frac {x^2}{125}\right )-\text {Subst}\left (\int \frac {1250+125 x}{x \log ^2(x)} \, dx,x,\frac {1}{125} \left (-1250+x^2\right )\right )\\ &=125 \text {li}\left (-10+\frac {x^2}{125}\right )-\text {Subst}\left (\int \left (\frac {125}{\log ^2(x)}+\frac {1250}{x \log ^2(x)}\right ) \, dx,x,\frac {1}{125} \left (-1250+x^2\right )\right )\\ &=125 \text {li}\left (-10+\frac {x^2}{125}\right )-125 \text {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,\frac {1}{125} \left (-1250+x^2\right )\right )-1250 \text {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,\frac {1}{125} \left (-1250+x^2\right )\right )\\ &=-\frac {1250-x^2}{\log \left (-10+\frac {x^2}{125}\right )}+125 \text {li}\left (-10+\frac {x^2}{125}\right )-125 \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,\frac {1}{125} \left (-1250+x^2\right )\right )-1250 \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (-10+\frac {x^2}{125}\right )\right )\\ &=\frac {1250}{\log \left (-10+\frac {x^2}{125}\right )}-\frac {1250-x^2}{\log \left (-10+\frac {x^2}{125}\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.07, size = 16, normalized size = 0.80 \begin {gather*} \frac {x^2}{\log \left (-10+\frac {x^2}{125}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(37\) vs.
\(2(14)=28\).
time = 3.12, size = 38, normalized size = 1.90
method | result | size |
norman | \(\frac {x^{2}}{\ln \left (\frac {x^{2}}{125}-10\right )}\) | \(15\) |
risch | \(\frac {x^{2}}{\ln \left (\frac {x^{2}}{125}-10\right )}\) | \(15\) |
default | \(\frac {x^{2}-1250}{-3 \ln \left (5\right )+\ln \left (x^{2}-1250\right )}-\frac {1250}{3 \ln \left (5\right )-\ln \left (x^{2}-1250\right )}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 20, normalized size = 1.00 \begin {gather*} -\frac {x^{2}}{3 \, \log \left (5\right ) - \log \left (x^{2} - 1250\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 14, normalized size = 0.70 \begin {gather*} \frac {x^{2}}{\log \left (\frac {1}{125} \, x^{2} - 10\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 10, normalized size = 0.50 \begin {gather*} \frac {x^{2}}{\log {\left (\frac {x^{2}}{125} - 10 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 14, normalized size = 0.70 \begin {gather*} \frac {x^{2}}{\log \left (\frac {1}{125} \, x^{2} - 10\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.93, size = 14, normalized size = 0.70 \begin {gather*} \frac {x^2}{\ln \left (\frac {x^2}{125}-10\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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