Optimal. Leaf size=24 \[ \frac {x}{-1+x-\frac {x}{3 \log \left (5 e^{x+x^2}\right )}} \]
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Rubi [F]
time = 1.25, 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 {-3 x^2-6 x^3-9 \log ^2\left (5 e^{x+x^2}\right )}{x^2+\left (6 x-6 x^2\right ) \log \left (5 e^{x+x^2}\right )+\left (9-18 x+9 x^2\right ) \log ^2\left (5 e^{x+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 {3 \left (-x^2-2 x^3-3 \log ^2\left (5 e^{x+x^2}\right )\right )}{\left (x-3 (-1+x) \log \left (5 e^{x+x^2}\right )\right )^2} \, dx\\ &=3 \int \frac {-x^2-2 x^3-3 \log ^2\left (5 e^{x+x^2}\right )}{\left (x-3 (-1+x) \log \left (5 e^{x+x^2}\right )\right )^2} \, dx\\ &=3 \int \left (-\frac {1}{3 (-1+x)^2}+\frac {x^2 \left (-4+9 x^2-6 x^3\right )}{3 (1-x)^2 \left (x+3 \log \left (5 e^{x+x^2}\right )-3 x \log \left (5 e^{x+x^2}\right )\right )^2}-\frac {2 x}{3 (-1+x)^2 \left (-x-3 \log \left (5 e^{x+x^2}\right )+3 x \log \left (5 e^{x+x^2}\right )\right )}\right ) \, dx\\ &=-\frac {1}{1-x}-2 \int \frac {x}{(-1+x)^2 \left (-x-3 \log \left (5 e^{x+x^2}\right )+3 x \log \left (5 e^{x+x^2}\right )\right )} \, dx+\int \frac {x^2 \left (-4+9 x^2-6 x^3\right )}{(1-x)^2 \left (x+3 \log \left (5 e^{x+x^2}\right )-3 x \log \left (5 e^{x+x^2}\right )\right )^2} \, dx\\ &=-\frac {1}{1-x}-2 \int \left (\frac {1}{(-1+x)^2 \left (-x-3 \log \left (5 e^{x+x^2}\right )+3 x \log \left (5 e^{x+x^2}\right )\right )}+\frac {1}{(-1+x) \left (-x-3 \log \left (5 e^{x+x^2}\right )+3 x \log \left (5 e^{x+x^2}\right )\right )}\right ) \, dx+\int \frac {x^2 \left (-4+9 x^2-6 x^3\right )}{(1-x)^2 \left (x-3 (-1+x) \log \left (5 e^{x+x^2}\right )\right )^2} \, dx\\ &=-\frac {1}{1-x}-2 \int \frac {1}{(-1+x)^2 \left (-x-3 \log \left (5 e^{x+x^2}\right )+3 x \log \left (5 e^{x+x^2}\right )\right )} \, dx-2 \int \frac {1}{(-1+x) \left (-x-3 \log \left (5 e^{x+x^2}\right )+3 x \log \left (5 e^{x+x^2}\right )\right )} \, dx+\int \left (-\frac {1}{\left (-x-3 \log \left (5 e^{x+x^2}\right )+3 x \log \left (5 e^{x+x^2}\right )\right )^2}-\frac {1}{(-1+x)^2 \left (-x-3 \log \left (5 e^{x+x^2}\right )+3 x \log \left (5 e^{x+x^2}\right )\right )^2}-\frac {2}{(-1+x) \left (-x-3 \log \left (5 e^{x+x^2}\right )+3 x \log \left (5 e^{x+x^2}\right )\right )^2}-\frac {3 x^2}{\left (-x-3 \log \left (5 e^{x+x^2}\right )+3 x \log \left (5 e^{x+x^2}\right )\right )^2}-\frac {6 x^3}{\left (-x-3 \log \left (5 e^{x+x^2}\right )+3 x \log \left (5 e^{x+x^2}\right )\right )^2}\right ) \, dx\\ &=-\frac {1}{1-x}-2 \int \frac {1}{(1-x) \left (x-3 (-1+x) \log \left (5 e^{x+x^2}\right )\right )} \, dx-2 \int \frac {1}{(-1+x) \left (-x-3 \log \left (5 e^{x+x^2}\right )+3 x \log \left (5 e^{x+x^2}\right )\right )^2} \, dx-2 \int \frac {1}{(-1+x)^2 \left (-x-3 \log \left (5 e^{x+x^2}\right )+3 x \log \left (5 e^{x+x^2}\right )\right )} \, dx-3 \int \frac {x^2}{\left (-x-3 \log \left (5 e^{x+x^2}\right )+3 x \log \left (5 e^{x+x^2}\right )\right )^2} \, dx-6 \int \frac {x^3}{\left (-x-3 \log \left (5 e^{x+x^2}\right )+3 x \log \left (5 e^{x+x^2}\right )\right )^2} \, dx-\int \frac {1}{\left (-x-3 \log \left (5 e^{x+x^2}\right )+3 x \log \left (5 e^{x+x^2}\right )\right )^2} \, dx-\int \frac {1}{(-1+x)^2 \left (-x-3 \log \left (5 e^{x+x^2}\right )+3 x \log \left (5 e^{x+x^2}\right )\right )^2} \, dx\\ &=-\frac {1}{1-x}-2 \int \frac {1}{(-1+x) \left (x-3 (-1+x) \log \left (5 e^{x+x^2}\right )\right )^2} \, dx-2 \int \frac {1}{(1-x) \left (x-3 (-1+x) \log \left (5 e^{x+x^2}\right )\right )} \, dx-2 \int \frac {1}{(-1+x)^2 \left (-x-3 \log \left (5 e^{x+x^2}\right )+3 x \log \left (5 e^{x+x^2}\right )\right )} \, dx-3 \int \frac {x^2}{\left (x-3 (-1+x) \log \left (5 e^{x+x^2}\right )\right )^2} \, dx-6 \int \frac {x^3}{\left (x-3 (-1+x) \log \left (5 e^{x+x^2}\right )\right )^2} \, dx-\int \frac {1}{\left (x-3 (-1+x) \log \left (5 e^{x+x^2}\right )\right )^2} \, dx-\int \frac {1}{(1-x)^2 \left (x-3 (-1+x) \log \left (5 e^{x+x^2}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.10, size = 37, normalized size = 1.54 \begin {gather*} \frac {3 \left (x+3 \log \left (5 e^{x+x^2}\right )\right )}{-3 x+9 (-1+x) \log \left (5 e^{x+x^2}\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. \(60\) vs.
\(2(21)=42\).
time = 0.24, size = 61, normalized size = 2.54
method | result | size |
risch | \(\frac {1}{x -1}-\frac {2 i x^{2}}{\left (x -1\right ) \left (-6 i x \ln \left (5\right )-6 i x \ln \left ({\mathrm e}^{\left (x +1\right ) x}\right )+6 i \ln \left (5\right )+2 i x +6 i \ln \left ({\mathrm e}^{\left (x +1\right ) x}\right )\right )}\) | \(57\) |
default | \(-\frac {3 \left (-\frac {x}{9}-\frac {\ln \left (5 \,{\mathrm e}^{x^{2}+x}\right )}{3}\right )}{x^{3}+x \left (\ln \left (5 \,{\mathrm e}^{x^{2}+x}\right )-x^{2}-x \right )-\frac {x}{3}-\ln \left (5 \,{\mathrm e}^{x^{2}+x}\right )+x^{2}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 34, normalized size = 1.42 \begin {gather*} \frac {3 \, x^{2} + 4 \, x + 3 \, \log \left (5\right )}{3 \, x^{3} + x {\left (3 \, \log \left (5\right ) - 4\right )} - 3 \, \log \left (5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 32, normalized size = 1.33 \begin {gather*} \frac {3 \, x^{2} + 4 \, x + 3 \, \log \left (5\right )}{3 \, x^{3} + 3 \, {\left (x - 1\right )} \log \left (5\right ) - 4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.70, size = 34, normalized size = 1.42 \begin {gather*} - \frac {- 3 x^{2} - 4 x - 3 \log {\left (5 \right )}}{3 x^{3} + x \left (-4 + 3 \log {\left (5 \right )}\right ) - 3 \log {\left (5 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 34, normalized size = 1.42 \begin {gather*} \frac {3 \, x^{2} + 4 \, x + 3 \, \log \left (5\right )}{3 \, x^{3} + 3 \, x \log \left (5\right ) - 4 \, x - 3 \, \log \left (5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.72, size = 30, normalized size = 1.25 \begin {gather*} \frac {3\,x^2+4\,x+\ln \left (125\right )}{3\,x^3+\left (\ln \left (125\right )-4\right )\,x-\ln \left (125\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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