Optimal. Leaf size=26 \[ \left (e^2+e^5-x+\log \left (-1-\frac {5}{x}\right )+x \log (-1+x)\right )^2 \]
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Rubi [A]
time = 0.33, antiderivative size = 30, normalized size of antiderivative = 1.15, number of steps
used = 5, number of rules used = 5, integrand size = 172, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6, 1608, 6820,
12, 6818} \begin {gather*} \left (-x+x \log (x-1)+\log \left (-\frac {x+5}{x}\right )+e^2 \left (1+e^3\right )\right )^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 1608
Rule 6818
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-10 x-2 x^3+\left (e^2+e^5\right ) \left (10+2 x^2\right )+\left (-10 x^2+8 x^3+2 x^4\right ) \log ^2(-1+x)+\left (10+2 x^2\right ) \log \left (\frac {-5-x}{x}\right )+\log (-1+x) \left (10 x+10 x^2-6 x^3-2 x^4+e^2 \left (-10 x+8 x^2+2 x^3\right )+e^5 \left (-10 x+8 x^2+2 x^3\right )+\left (-10 x+8 x^2+2 x^3\right ) \log \left (\frac {-5-x}{x}\right )\right )}{-5 x+4 x^2+x^3} \, dx\\ &=\int \frac {-10 x-2 x^3+\left (e^2+e^5\right ) \left (10+2 x^2\right )+\left (-10 x^2+8 x^3+2 x^4\right ) \log ^2(-1+x)+\left (10+2 x^2\right ) \log \left (\frac {-5-x}{x}\right )+\log (-1+x) \left (10 x+10 x^2-6 x^3-2 x^4+e^2 \left (-10 x+8 x^2+2 x^3\right )+e^5 \left (-10 x+8 x^2+2 x^3\right )+\left (-10 x+8 x^2+2 x^3\right ) \log \left (\frac {-5-x}{x}\right )\right )}{x \left (-5+4 x+x^2\right )} \, dx\\ &=\int \frac {2 \left (5+x^2+x \left (-5+4 x+x^2\right ) \log (-1+x)\right ) \left (-e^2 \left (1+e^3\right )+x-x \log (-1+x)-\log \left (-\frac {5+x}{x}\right )\right )}{x \left (5-4 x-x^2\right )} \, dx\\ &=2 \int \frac {\left (5+x^2+x \left (-5+4 x+x^2\right ) \log (-1+x)\right ) \left (-e^2 \left (1+e^3\right )+x-x \log (-1+x)-\log \left (-\frac {5+x}{x}\right )\right )}{x \left (5-4 x-x^2\right )} \, dx\\ &=\left (e^2 \left (1+e^3\right )-x+x \log (-1+x)+\log \left (-\frac {5+x}{x}\right )\right )^2\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 27, normalized size = 1.04 \begin {gather*} \left (e^2+e^5-x+x \log (-1+x)+\log \left (-\frac {5+x}{x}\right )\right )^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 2.94, size = 1472, normalized size = 56.62
method | result | size |
risch | \(-2 \,{\mathrm e}^{2} \ln \left (x \right )-2 \,{\mathrm e}^{5} \ln \left (x \right )+2 x \ln \left (x \right )-2 x \ln \left (5+x \right )-2 x \,{\mathrm e}^{5}-2 \,{\mathrm e}^{2} x +\ln \left (5+x \right )^{2}+\ln \left (x \right )^{2}+x^{2}+i \pi x \,\mathrm {csgn}\left (i \left (5+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (5+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right ) \ln \left (x \right )-i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (5+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right ) \ln \left (5+x \right )+\left (2 x \ln \left (5+x \right )+i \pi x \,\mathrm {csgn}\left (i \left (5+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{2}-i \pi x \,\mathrm {csgn}\left (i \left (5+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )+i \pi x \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )-2 i \pi x \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{2}+i \pi x \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{3}+2 x \,{\mathrm e}^{5}+2 i x \pi +2 \,{\mathrm e}^{2} x -2 x^{2}-2 x \ln \left (x \right )\right ) \ln \left (x -1\right )-2 \ln \left (x \right ) \ln \left (5+x \right )+2 i \pi \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{2} \ln \left (x \right )+2 \,{\mathrm e}^{5} \ln \left (5+x \right )+2 \ln \left (5+x \right ) {\mathrm e}^{2}+2 i \pi \ln \left (5+x \right )-2 i \pi \ln \left (x \right )+2 i \pi x \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{3} \ln \left (x \right )+\ln \left (x -1\right )^{2} x^{2}-2 i x \pi -i \pi x \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{3}-2 i \pi \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{2} \ln \left (5+x \right )+i \pi \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{3} \ln \left (5+x \right )+i \pi \,\mathrm {csgn}\left (i \left (5+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{2} \ln \left (5+x \right )-i \pi x \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )-i \pi x \,\mathrm {csgn}\left (i \left (5+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (5+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{2} \ln \left (x \right )+i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{2} \ln \left (5+x \right )-i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{2} \ln \left (x \right )\) | \(604\) |
default | \(\text {Expression too large to display}\) | \(1472\) |
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. 141 vs.
\(2 (24) = 48\).
time = 0.33, size = 141, normalized size = 5.42 \begin {gather*} x^{2} \log \left (x - 1\right )^{2} + x^{2} - 2 \, x {\left (e^{5} + e^{2}\right )} + \frac {1}{3} \, {\left (\log \left (x + 5\right ) + 5 \, \log \left (x - 1\right ) - 6 \, \log \left (x\right )\right )} e^{5} + \frac {1}{3} \, {\left (\log \left (x + 5\right ) + 5 \, \log \left (x - 1\right ) - 6 \, \log \left (x\right )\right )} e^{2} - \frac {1}{3} \, {\left (6 \, x^{2} - 6 \, x {\left (e^{5} + e^{2}\right )} + 6 \, x \log \left (x\right ) + 5 \, e^{5} + 5 \, e^{2}\right )} \log \left (x - 1\right ) + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2} + \frac {1}{3} \, {\left (6 \, x \log \left (x - 1\right ) - 6 \, x + 5 \, e^{5} + 5 \, e^{2} - 6 \, \log \left (x\right )\right )} \log \left (-x - 5\right ) + \log \left (-x - 5\right )^{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. 88 vs.
\(2 (24) = 48\).
time = 0.38, size = 88, normalized size = 3.38 \begin {gather*} x^{2} \log \left (x - 1\right )^{2} + x^{2} - 2 \, x e^{5} - 2 \, x e^{2} - 2 \, {\left (x^{2} - x e^{5} - x e^{2} - x \log \left (-\frac {x + 5}{x}\right )\right )} \log \left (x - 1\right ) - 2 \, {\left (x - e^{5} - e^{2}\right )} \log \left (-\frac {x + 5}{x}\right ) + \log \left (-\frac {x + 5}{x}\right )^{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. 202 vs.
\(2 (24) = 48\).
time = 0.47, size = 202, normalized size = 7.77 \begin {gather*} x^{2} \log {\left (x - 1 \right )}^{2} + x^{2} + x \left (- 2 e^{5} - 2 e^{2}\right ) + \left (2 x \log {\left (x - 1 \right )} - 2 x\right ) \log {\left (\frac {- x - 5}{x} \right )} + \left (- 2 x^{2} + 2 x e^{2} + 2 x e^{5}\right ) \log {\left (x - 1 \right )} + \log {\left (\frac {- x - 5}{x} \right )}^{2} - 2 \cdot \left (1 + e\right ) \left (- e + 1 + e^{2}\right ) e^{2} \log {\left (x + \frac {- 10 \cdot \left (1 + e\right ) \left (- e + 1 + e^{2}\right ) e^{2} + 10 e^{2} + 10 e^{5}}{4 e^{2} + 4 e^{5}} \right )} + 2 \cdot \left (1 + e\right ) \left (- e + 1 + e^{2}\right ) e^{2} \log {\left (x + \frac {10 e^{2} + 10 e^{5} + 10 \cdot \left (1 + e\right ) \left (- e + 1 + e^{2}\right ) e^{2}}{4 e^{2} + 4 e^{5}} \right )} \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. 149 vs.
\(2 (24) = 48\).
time = 0.44, size = 149, normalized size = 5.73 \begin {gather*} x^{2} \log \left (x - 1\right )^{2} - 2 \, x^{2} \log \left (x - 1\right ) + 2 \, x e^{5} \log \left (x - 1\right ) + 2 \, x e^{2} \log \left (x - 1\right ) - 2 \, x \log \left (x - 1\right ) \log \left (x\right ) + 2 \, x \log \left (x - 1\right ) \log \left (-x - 5\right ) + x^{2} - 2 \, x e^{5} - 2 \, x e^{2} + 2 \, e^{5} \log \left (x + 5\right ) + 2 \, e^{2} \log \left (x + 5\right ) - \log \left (x + 5\right )^{2} + 2 \, x \log \left (x\right ) - 2 \, e^{5} \log \left (x\right ) - 2 \, e^{2} \log \left (x\right ) + \log \left (x\right )^{2} - 2 \, x \log \left (-x - 5\right ) + 2 \, \log \left (x + 5\right ) \log \left (-x - 5\right ) - 2 \, \log \left (x\right ) \log \left (-x - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.42, size = 97, normalized size = 3.73 \begin {gather*} x^2-\ln \left (x-1\right )\,\left (2\,x^2-2\,x\,{\mathrm {e}}^2\,\left ({\mathrm {e}}^3+1\right )\right )-x\,\left (2\,{\mathrm {e}}^2+2\,{\mathrm {e}}^5\right )-\ln \left (-\frac {x+5}{x}\right )\,\left (2\,x-2\,x\,\ln \left (x-1\right )\right )+x^2\,{\ln \left (x-1\right )}^2+{\ln \left (-\frac {x+5}{x}\right )}^2-{\mathrm {e}}^2\,\mathrm {atan}\left (\frac {x\,2{}\mathrm {i}}{5}+1{}\mathrm {i}\right )\,\left ({\mathrm {e}}^3+1\right )\,4{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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