Optimal. Leaf size=39 \[ \frac {x^2}{\log ^2(2) \log ^2\left (\frac {\left (-e^{1-x}+2 x\right ) \left (\frac {2}{x}-x^2\right )}{x^2}\right )} \]
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Rubi [F]
time = 3.16, 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 {16 x^2+4 x^5+e^{1-x} \left (-12 x-4 x^2+2 x^5\right )+\left (8 x^2-4 x^5+e^{1-x} \left (-4 x+2 x^4\right )\right ) \log \left (\frac {4 x-2 x^4+e^{1-x} \left (-2+x^3\right )}{x^3}\right )}{\left (e^{1-x} \left (-2+x^3\right ) \log ^2(2)+\left (4 x-2 x^4\right ) \log ^2(2)\right ) \log ^3\left (\frac {4 x-2 x^4+e^{1-x} \left (-2+x^3\right )}{x^3}\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 {2 x \left (-2 e^x x \left (4+x^3\right )-e \left (-6-2 x+x^4\right )-\left (e-2 e^x x\right ) \left (-2+x^3\right ) \log \left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )\right )}{\left (e-2 e^x x\right ) \left (2-x^3\right ) \log ^2(2) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx\\ &=\frac {2 \int \frac {x \left (-2 e^x x \left (4+x^3\right )-e \left (-6-2 x+x^4\right )-\left (e-2 e^x x\right ) \left (-2+x^3\right ) \log \left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )\right )}{\left (e-2 e^x x\right ) \left (2-x^3\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}\\ &=\frac {2 \int \left (\frac {e x (1+x)}{\left (e-2 e^x x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )}+\frac {x \left (-4-x^3-2 \log \left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )+x^3 \log \left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )\right )}{\left (-2+x^3\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )}\right ) \, dx}{\log ^2(2)}\\ &=\frac {2 \int \frac {x \left (-4-x^3-2 \log \left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )+x^3 \log \left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )\right )}{\left (-2+x^3\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}+\frac {(2 e) \int \frac {x (1+x)}{\left (e-2 e^x x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}\\ &=\frac {2 \int \frac {x \left (4+x^3-\left (-2+x^3\right ) \log \left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )\right )}{\left (2-x^3\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}+\frac {(2 e) \int \left (-\frac {x}{\left (-e+2 e^x x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )}-\frac {x^2}{\left (-e+2 e^x x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )}\right ) \, dx}{\log ^2(2)}\\ &=\frac {2 \int \left (-\frac {x \left (4+x^3\right )}{\left (-2+x^3\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )}+\frac {x}{\log ^2\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )}\right ) \, dx}{\log ^2(2)}-\frac {(2 e) \int \frac {x}{\left (-e+2 e^x x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}-\frac {(2 e) \int \frac {x^2}{\left (-e+2 e^x x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}\\ &=-\frac {2 \int \frac {x \left (4+x^3\right )}{\left (-2+x^3\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}+\frac {2 \int \frac {x}{\log ^2\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}-\frac {(2 e) \int \frac {x}{\left (-e+2 e^x x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}-\frac {(2 e) \int \frac {x^2}{\left (-e+2 e^x x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}\\ &=-\frac {2 \int \left (\frac {x}{\log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )}+\frac {6 x}{\left (-2+x^3\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )}\right ) \, dx}{\log ^2(2)}+\frac {2 \int \frac {x}{\log ^2\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}-\frac {(2 e) \int \frac {x}{\left (-e+2 e^x x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}-\frac {(2 e) \int \frac {x^2}{\left (-e+2 e^x x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}\\ &=-\frac {2 \int \frac {x}{\log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}+\frac {2 \int \frac {x}{\log ^2\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}-\frac {12 \int \frac {x}{\left (-2+x^3\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}-\frac {(2 e) \int \frac {x}{\left (-e+2 e^x x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}-\frac {(2 e) \int \frac {x^2}{\left (-e+2 e^x x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}\\ &=-\frac {2 \int \frac {x}{\log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}+\frac {2 \int \frac {x}{\log ^2\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}-\frac {12 \int \left (-\frac {1}{3 \sqrt [3]{2} \left (\sqrt [3]{2}-x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )}-\frac {(-1)^{2/3}}{3 \sqrt [3]{2} \left (\sqrt [3]{2}+\sqrt [3]{-1} x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )}+\frac {\sqrt [3]{-\frac {1}{2}}}{3 \left (\sqrt [3]{2}-(-1)^{2/3} x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )}\right ) \, dx}{\log ^2(2)}-\frac {(2 e) \int \frac {x}{\left (-e+2 e^x x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}-\frac {(2 e) \int \frac {x^2}{\left (-e+2 e^x x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}\\ &=-\frac {2 \int \frac {x}{\log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}+\frac {2 \int \frac {x}{\log ^2\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}+\frac {\left (2 (-2)^{2/3}\right ) \int \frac {1}{\left (\sqrt [3]{2}+\sqrt [3]{-1} x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}+\frac {\left (2\ 2^{2/3}\right ) \int \frac {1}{\left (\sqrt [3]{2}-x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}-\frac {\left (2 \sqrt [3]{-1} 2^{2/3}\right ) \int \frac {1}{\left (\sqrt [3]{2}-(-1)^{2/3} x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}-\frac {(2 e) \int \frac {x}{\left (-e+2 e^x x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}-\frac {(2 e) \int \frac {x^2}{\left (-e+2 e^x x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \, dx}{\log ^2(2)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 33, normalized size = 0.85 \begin {gather*} \frac {x^2}{\log ^2(2) \log ^2\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \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 = 0.40, size = 411, normalized size = 10.54
method | result | size |
risch | \(-\frac {4 x^{2}}{\left (\pi \mathrm {csgn}\left (i x^{3}\right )^{3}-\pi \mathrm {csgn}\left (i x^{3}\right )^{2} \mathrm {csgn}\left (i x \right )-\pi \mathrm {csgn}\left (i x^{3}\right )^{2} \mathrm {csgn}\left (i x^{2}\right )+\pi \,\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )-2 \pi \mathrm {csgn}\left (\frac {i \left (x^{4}-\frac {{\mathrm e}^{1-x} x^{3}}{2}-2 x +{\mathrm e}^{1-x}\right )}{x^{3}}\right )^{2}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}-\pi \,\mathrm {csgn}\left (\frac {i}{x^{3}}\right ) \mathrm {csgn}\left (i \left (x^{4}-\frac {{\mathrm e}^{1-x} x^{3}}{2}-2 x +{\mathrm e}^{1-x}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{4}-\frac {{\mathrm e}^{1-x} x^{3}}{2}-2 x +{\mathrm e}^{1-x}\right )}{x^{3}}\right )+\pi \,\mathrm {csgn}\left (\frac {i}{x^{3}}\right ) \mathrm {csgn}\left (\frac {i \left (x^{4}-\frac {{\mathrm e}^{1-x} x^{3}}{2}-2 x +{\mathrm e}^{1-x}\right )}{x^{3}}\right )^{2}+\pi \,\mathrm {csgn}\left (i \left (x^{4}-\frac {{\mathrm e}^{1-x} x^{3}}{2}-2 x +{\mathrm e}^{1-x}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{4}-\frac {{\mathrm e}^{1-x} x^{3}}{2}-2 x +{\mathrm e}^{1-x}\right )}{x^{3}}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (x^{4}-\frac {{\mathrm e}^{1-x} x^{3}}{2}-2 x +{\mathrm e}^{1-x}\right )}{x^{3}}\right )^{3}+2 \pi +6 i \ln \left (x \right )-2 i \ln \left (2\right )-2 i \ln \left (x^{4}-\frac {{\mathrm e}^{1-x} x^{3}}{2}-2 x +{\mathrm e}^{1-x}\right )\right )^{2} \ln \left (2\right )^{2}}\) | \(411\) |
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. 124 vs.
\(2 (37) = 74\).
time = 0.73, size = 124, normalized size = 3.18 \begin {gather*} \frac {x^{2}}{x^{2} \log \left (2\right )^{2} + \log \left (2\right )^{2} \log \left (x^{3} - 2\right )^{2} + \log \left (2\right )^{2} \log \left (-2 \, x e^{x} + e\right )^{2} + 6 \, x \log \left (2\right )^{2} \log \left (x\right ) + 9 \, \log \left (2\right )^{2} \log \left (x\right )^{2} - 2 \, {\left (x \log \left (2\right )^{2} + 3 \, \log \left (2\right )^{2} \log \left (x\right )\right )} \log \left (x^{3} - 2\right ) - 2 \, {\left (x \log \left (2\right )^{2} - \log \left (2\right )^{2} \log \left (x^{3} - 2\right ) + 3 \, \log \left (2\right )^{2} \log \left (x\right )\right )} \log \left (-2 \, x e^{x} + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 38, normalized size = 0.97 \begin {gather*} \frac {x^{2}}{\log \left (2\right )^{2} \log \left (-\frac {2 \, x^{4} - {\left (x^{3} - 2\right )} e^{\left (-x + 1\right )} - 4 \, x}{x^{3}}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 32, normalized size = 0.82 \begin {gather*} \frac {x^{2}}{\log {\left (2 \right )}^{2} \log {\left (\frac {- 2 x^{4} + 4 x + \left (x^{3} - 2\right ) e^{1 - x}}{x^{3}} \right )}^{2}} \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. 91 vs.
\(2 (37) = 74\).
time = 0.76, size = 91, normalized size = 2.33 \begin {gather*} \frac {x^{2}}{\log \left (2\right )^{2} \log \left (-2 \, x^{4} + x^{3} e^{\left (-x + 1\right )} + 4 \, x - 2 \, e^{\left (-x + 1\right )}\right )^{2} - 2 \, \log \left (2\right )^{2} \log \left (-2 \, x^{4} + x^{3} e^{\left (-x + 1\right )} + 4 \, x - 2 \, e^{\left (-x + 1\right )}\right ) \log \left (x^{3}\right ) + \log \left (2\right )^{2} \log \left (x^{3}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {{\mathrm {e}}^{1-x}\,\left (-2\,x^5+4\,x^2+12\,x\right )-16\,x^2-4\,x^5+\ln \left (\frac {4\,x+{\mathrm {e}}^{1-x}\,\left (x^3-2\right )-2\,x^4}{x^3}\right )\,\left ({\mathrm {e}}^{1-x}\,\left (4\,x-2\,x^4\right )-8\,x^2+4\,x^5\right )}{{\ln \left (\frac {4\,x+{\mathrm {e}}^{1-x}\,\left (x^3-2\right )-2\,x^4}{x^3}\right )}^3\,\left ({\ln \left (2\right )}^2\,\left (4\,x-2\,x^4\right )+{\mathrm {e}}^{1-x}\,{\ln \left (2\right )}^2\,\left (x^3-2\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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