Optimal. Leaf size=26 \[ -2+\left (\frac {2}{2-e}+x+25 x^4 \log ^2\left (x^2\right )\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(26)=52\).
time = 0.20, antiderivative size = 108, normalized size of antiderivative = 4.15, number of steps
used = 28, number of rules used = 8, integrand size = 99, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {6, 12, 1607,
45, 2371, 2395, 2342, 2341} \begin {gather*} x^2+625 x^8 \log ^4\left (x^2\right )+50 x^5 \log ^2\left (x^2\right )-40 x^5 \log \left (x^2\right )+\frac {100 x^4 \log ^2\left (x^2\right )}{2-e}-\frac {100 x^4 \log \left (x^2\right )}{2-e}+\frac {20 \left (2 (2-e) x^5+5 x^4\right ) \log \left (x^2\right )}{2-e}+\frac {4 x}{2-e} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 45
Rule 1607
Rule 2341
Rule 2342
Rule 2371
Rule 2395
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4+(-4+2 e) x+\left (-400 x^3-400 x^4+200 e x^4\right ) \log \left (x^2\right )+\left (-400 x^3-500 x^4+250 e x^4\right ) \log ^2\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^3\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^4\left (x^2\right )}{-2+e} \, dx\\ &=\frac {\int \left (-4+(-4+2 e) x+\left (-400 x^3-400 x^4+200 e x^4\right ) \log \left (x^2\right )+\left (-400 x^3-500 x^4+250 e x^4\right ) \log ^2\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^3\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^4\left (x^2\right )\right ) \, dx}{-2+e}\\ &=\frac {4 x}{2-e}+x^2+\frac {\int \left (-400 x^3-400 x^4+200 e x^4\right ) \log \left (x^2\right ) \, dx}{-2+e}+\frac {\int \left (-400 x^3-500 x^4+250 e x^4\right ) \log ^2\left (x^2\right ) \, dx}{-2+e}+\frac {\int \left (-10000 x^7+5000 e x^7\right ) \log ^3\left (x^2\right ) \, dx}{-2+e}+\frac {\int \left (-10000 x^7+5000 e x^7\right ) \log ^4\left (x^2\right ) \, dx}{-2+e}\\ &=\frac {4 x}{2-e}+x^2+\frac {\int \left (-400 x^3+(-400+200 e) x^4\right ) \log \left (x^2\right ) \, dx}{-2+e}+\frac {\int \left (-400 x^3+(-500+250 e) x^4\right ) \log ^2\left (x^2\right ) \, dx}{-2+e}+\frac {\int (-10000+5000 e) x^7 \log ^3\left (x^2\right ) \, dx}{-2+e}+\frac {\int (-10000+5000 e) x^7 \log ^4\left (x^2\right ) \, dx}{-2+e}\\ &=\frac {4 x}{2-e}+x^2+5000 \int x^7 \log ^3\left (x^2\right ) \, dx+5000 \int x^7 \log ^4\left (x^2\right ) \, dx+\frac {\int x^3 (-400+(-400+200 e) x) \log \left (x^2\right ) \, dx}{-2+e}+\frac {\int x^3 (-400+(-500+250 e) x) \log ^2\left (x^2\right ) \, dx}{-2+e}\\ &=\frac {4 x}{2-e}+x^2+\frac {20 \left (5 x^4+2 (2-e) x^5\right ) \log \left (x^2\right )}{2-e}+625 x^8 \log ^3\left (x^2\right )+625 x^8 \log ^4\left (x^2\right )-3750 \int x^7 \log ^2\left (x^2\right ) \, dx-5000 \int x^7 \log ^3\left (x^2\right ) \, dx+\frac {2 \int 20 x^3 (-5+2 (-2+e) x) \, dx}{2-e}+\frac {\int \left (-400 x^3 \log ^2\left (x^2\right )+250 (-2+e) x^4 \log ^2\left (x^2\right )\right ) \, dx}{-2+e}\\ &=\frac {4 x}{2-e}+x^2+\frac {20 \left (5 x^4+2 (2-e) x^5\right ) \log \left (x^2\right )}{2-e}-\frac {1875}{4} x^8 \log ^2\left (x^2\right )+625 x^8 \log ^4\left (x^2\right )+250 \int x^4 \log ^2\left (x^2\right ) \, dx+1875 \int x^7 \log \left (x^2\right ) \, dx+3750 \int x^7 \log ^2\left (x^2\right ) \, dx+\frac {40 \int x^3 (-5+2 (-2+e) x) \, dx}{2-e}+\frac {400 \int x^3 \log ^2\left (x^2\right ) \, dx}{2-e}\\ &=\frac {4 x}{2-e}+x^2-\frac {1875 x^8}{32}+\frac {1875}{8} x^8 \log \left (x^2\right )+\frac {20 \left (5 x^4+2 (2-e) x^5\right ) \log \left (x^2\right )}{2-e}+\frac {100 x^4 \log ^2\left (x^2\right )}{2-e}+50 x^5 \log ^2\left (x^2\right )+625 x^8 \log ^4\left (x^2\right )-200 \int x^4 \log \left (x^2\right ) \, dx-1875 \int x^7 \log \left (x^2\right ) \, dx+\frac {40 \int \left (-5 x^3+2 (-2+e) x^4\right ) \, dx}{2-e}-\frac {400 \int x^3 \log \left (x^2\right ) \, dx}{2-e}\\ &=\frac {4 x}{2-e}+x^2-\frac {100 x^4 \log \left (x^2\right )}{2-e}-40 x^5 \log \left (x^2\right )+\frac {20 \left (5 x^4+2 (2-e) x^5\right ) \log \left (x^2\right )}{2-e}+\frac {100 x^4 \log ^2\left (x^2\right )}{2-e}+50 x^5 \log ^2\left (x^2\right )+625 x^8 \log ^4\left (x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 29, normalized size = 1.12 \begin {gather*} \frac {\left (-2+(-2+e) x+25 (-2+e) x^4 \log ^2\left (x^2\right )\right )^2}{(-2+e)^2} \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. \(75\) vs.
\(2(27)=54\).
time = 0.61, size = 76, normalized size = 2.92
method | result | size |
risch | \(625 x^{8} \ln \left (x^{2}\right )^{4}+\frac {\left (50 x^{5} {\mathrm e}-100 x^{5}-100 x^{4}\right ) \ln \left (x^{2}\right )^{2}}{{\mathrm e}-2}+\frac {x^{2} {\mathrm e}}{{\mathrm e}-2}-\frac {2 x^{2}}{{\mathrm e}-2}-\frac {4 x}{{\mathrm e}-2}\) | \(76\) |
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. 149 vs.
\(2 (25) = 50\).
time = 0.27, size = 149, normalized size = 5.73 \begin {gather*} -\frac {625 \, x^{8} {\left (e - 2\right )} \log \left (x^{2}\right )^{3} - 625 \, {\left (x^{8} e - 2 \, x^{8}\right )} \log \left (x^{2}\right )^{4} - 625 \, {\left (x^{8} e - 2 \, x^{8}\right )} \log \left (x^{2}\right )^{3} - x^{2} e - 50 \, {\left (x^{5} e - 2 \, x^{5} - 2 \, x^{4}\right )} \log \left (x^{2}\right )^{2} + 2 \, x^{2} + 20 \, {\left (2 \, x^{5} {\left (e - 2\right )} - 5 \, x^{4}\right )} \log \left (x^{2}\right ) - 20 \, {\left (2 \, x^{5} e - 4 \, x^{5} - 5 \, x^{4}\right )} \log \left (x^{2}\right ) + 4 \, x}{e - 2} \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. 67 vs.
\(2 (25) = 50\).
time = 0.38, size = 67, normalized size = 2.58 \begin {gather*} \frac {625 \, {\left (x^{8} e - 2 \, x^{8}\right )} \log \left (x^{2}\right )^{4} + x^{2} e + 50 \, {\left (x^{5} e - 2 \, x^{5} - 2 \, x^{4}\right )} \log \left (x^{2}\right )^{2} - 2 \, x^{2} - 4 \, x}{e - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (22) = 44\).
time = 0.26, size = 53, normalized size = 2.04 \begin {gather*} 625 x^{8} \log {\left (x^{2} \right )}^{4} + x^{2} - \frac {4 x}{-2 + e} + \frac {\left (- 100 x^{5} + 50 e x^{5} - 100 x^{4}\right ) \log {\left (x^{2} \right )}^{2}}{-2 + e} \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. 221 vs.
\(2 (25) = 50\).
time = 0.42, size = 221, normalized size = 8.50 \begin {gather*} -\frac {40000 \, x^{8} \log \left (x^{2}\right )^{4} + 512 \, x^{5} e - 2560 \, x^{5} \log \left (x^{2}\right ) - 3200 \, x^{4} \log \left (x^{2}\right ) - 32 \, x^{2} e - 1600 \, {\left (x^{5} e - 2 \, x^{5} - 2 \, x^{4}\right )} \log \left (x^{2}\right )^{2} + 64 \, x^{2} - 625 \, {\left (32 \, x^{8} \log \left (x^{2}\right )^{4} - 32 \, x^{8} \log \left (x^{2}\right )^{3} + 24 \, x^{8} \log \left (x^{2}\right )^{2} - 12 \, x^{8} \log \left (x^{2}\right ) + 3 \, x^{8}\right )} e - 625 \, {\left (32 \, x^{8} \log \left (x^{2}\right )^{3} - 24 \, x^{8} \log \left (x^{2}\right )^{2} + 12 \, x^{8} \log \left (x^{2}\right ) - 3 \, x^{8}\right )} e + 256 \, {\left (5 \, x^{5} \log \left (x^{2}\right ) - 2 \, x^{5}\right )} e - 640 \, {\left (2 \, x^{5} e - 4 \, x^{5} - 5 \, x^{4}\right )} \log \left (x^{2}\right ) + 128 \, x}{32 \, {\left (e - 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.25, size = 56, normalized size = 2.15 \begin {gather*} -\frac {x\,\left (25\,x^3\,{\ln \left (x^2\right )}^2+1\right )\,\left (2\,x-x\,\mathrm {e}+50\,x^4\,{\ln \left (x^2\right )}^2-25\,x^4\,{\ln \left (x^2\right )}^2\,\mathrm {e}+4\right )}{\mathrm {e}-2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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