Optimal. Leaf size=25 \[ \left (x+\frac {1-2 x}{16 x \left (-e^x+x\right )}\right )^2 \]
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Rubi [F]
time = 1.38, 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 {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, 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-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 \left (e^x-x\right )^3 x^3} \, dx\\ &=\frac {1}{128} \int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{\left (e^x-x\right )^3 x^3} \, dx\\ &=\frac {1}{128} \int \left (256 x-\frac {16 (-3+2 x)}{e^x-x}+\frac {(-1+x) (-1+2 x)^2}{x^2 \left (-e^x+x\right )^3}-\frac {1-x-4 x^2+20 x^3-48 x^4+32 x^5}{\left (e^x-x\right )^2 x^3}\right ) \, dx\\ &=x^2+\frac {1}{128} \int \frac {(-1+x) (-1+2 x)^2}{x^2 \left (-e^x+x\right )^3} \, dx-\frac {1}{128} \int \frac {1-x-4 x^2+20 x^3-48 x^4+32 x^5}{\left (e^x-x\right )^2 x^3} \, dx-\frac {1}{8} \int \frac {-3+2 x}{e^x-x} \, dx\\ &=x^2-\frac {1}{128} \int \left (\frac {20}{\left (e^x-x\right )^2}+\frac {1}{\left (e^x-x\right )^2 x^3}-\frac {1}{\left (e^x-x\right )^2 x^2}-\frac {4}{\left (e^x-x\right )^2 x}-\frac {48 x}{\left (e^x-x\right )^2}+\frac {32 x^2}{\left (e^x-x\right )^2}\right ) \, dx+\frac {1}{128} \int \left (\frac {8}{\left (e^x-x\right )^3}-\frac {4 x}{\left (e^x-x\right )^3}-\frac {1}{x^2 \left (-e^x+x\right )^3}+\frac {5}{x \left (-e^x+x\right )^3}\right ) \, dx-\frac {1}{8} \int \left (-\frac {3}{e^x-x}+\frac {2 x}{e^x-x}\right ) \, dx\\ &=x^2-\frac {1}{128} \int \frac {1}{\left (e^x-x\right )^2 x^3} \, dx+\frac {1}{128} \int \frac {1}{\left (e^x-x\right )^2 x^2} \, dx-\frac {1}{128} \int \frac {1}{x^2 \left (-e^x+x\right )^3} \, dx+\frac {1}{32} \int \frac {1}{\left (e^x-x\right )^2 x} \, dx-\frac {1}{32} \int \frac {x}{\left (e^x-x\right )^3} \, dx+\frac {5}{128} \int \frac {1}{x \left (-e^x+x\right )^3} \, dx+\frac {1}{16} \int \frac {1}{\left (e^x-x\right )^3} \, dx-\frac {5}{32} \int \frac {1}{\left (e^x-x\right )^2} \, dx-\frac {1}{4} \int \frac {x}{e^x-x} \, dx-\frac {1}{4} \int \frac {x^2}{\left (e^x-x\right )^2} \, dx+\frac {3}{8} \int \frac {1}{e^x-x} \, dx+\frac {3}{8} \int \frac {x}{\left (e^x-x\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 6.65, size = 36, normalized size = 1.44 \begin {gather*} \frac {\left (1-2 x-16 e^x x^2+16 x^3\right )^2}{256 \left (e^x-x\right )^2 x^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. \(51\) vs.
\(2(23)=46\).
time = 0.18, size = 52, normalized size = 2.08
method | result | size |
risch | \(x^{2}-\frac {64 x^{4}-64 \,{\mathrm e}^{x} x^{3}-32 x^{3}+32 \,{\mathrm e}^{x} x^{2}-4 x^{2}+4 x -1}{256 x^{2} \left (x -{\mathrm e}^{x}\right )^{2}}\) | \(52\) |
norman | \(\frac {\frac {1}{256}+x^{6}-\frac {x^{4}}{4}+{\mathrm e}^{2 x} x^{4}+\frac {{\mathrm e}^{x} x^{3}}{4}-\frac {x}{64}+\frac {x^{2}}{64}+\frac {x^{3}}{8}-2 x^{5} {\mathrm e}^{x}-\frac {{\mathrm e}^{x} x^{2}}{8}}{x^{2} \left (x -{\mathrm e}^{x}\right )^{2}}\) | \(65\) |
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. 75 vs.
\(2 (26) = 52\).
time = 0.31, size = 75, normalized size = 3.00 \begin {gather*} \frac {256 \, x^{6} + 256 \, x^{4} e^{\left (2 \, x\right )} - 64 \, x^{4} + 32 \, x^{3} + 4 \, x^{2} - 32 \, {\left (16 \, x^{5} - 2 \, x^{3} + x^{2}\right )} e^{x} - 4 \, x + 1}{256 \, {\left (x^{4} - 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )}\right )}} \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. 75 vs.
\(2 (26) = 52\).
time = 0.41, size = 75, normalized size = 3.00 \begin {gather*} \frac {256 \, x^{6} + 256 \, x^{4} e^{\left (2 \, x\right )} - 64 \, x^{4} + 32 \, x^{3} + 4 \, x^{2} - 32 \, {\left (16 \, x^{5} - 2 \, x^{3} + x^{2}\right )} e^{x} - 4 \, x + 1}{256 \, {\left (x^{4} - 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )}\right )}} \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. 58 vs.
\(2 (17) = 34\).
time = 0.21, size = 58, normalized size = 2.32 \begin {gather*} x^{2} + \frac {- 16 x^{4} + 8 x^{3} + x^{2} - x + \left (16 x^{3} - 8 x^{2}\right ) e^{x} + \frac {1}{4}}{64 x^{4} - 128 x^{3} e^{x} + 64 x^{2} e^{2 x}} \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. 78 vs.
\(2 (26) = 52\).
time = 0.41, size = 78, normalized size = 3.12 \begin {gather*} \frac {256 \, x^{6} - 512 \, x^{5} e^{x} + 256 \, x^{4} e^{\left (2 \, x\right )} - 64 \, x^{4} + 64 \, x^{3} e^{x} + 32 \, x^{3} - 32 \, x^{2} e^{x} + 4 \, x^{2} - 4 \, x + 1}{256 \, {\left (x^{4} - 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.58, size = 32, normalized size = 1.28 \begin {gather*} \frac {{\left (2\,x+16\,x^2\,{\mathrm {e}}^x-16\,x^3-1\right )}^2}{256\,x^2\,{\left (x-{\mathrm {e}}^x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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