Optimal. Leaf size=25 \[ 4+e^{2 x}+\frac {e^5}{(1-2 x)^2}+2 x-\log (x) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(25)=50\).
time = 0.40, antiderivative size = 61, normalized size of antiderivative = 2.44, number of steps
used = 12, number of rules used = 7, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.096, Rules used = {6, 6873, 6874,
2225, 46, 37, 45} \begin {gather*} -\frac {12 x^2}{(1-2 x)^2}+2 x+e^{2 x}-\frac {5}{1-2 x}+\frac {1}{2 x-1}+\frac {2+e^5}{(1-2 x)^2}+\frac {1}{(1-2 x)^2}-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 37
Rule 45
Rule 46
Rule 2225
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1+\left (-8-4 e^5\right ) x+24 x^2-32 x^3+16 x^4+e^{2 x} \left (-2 x+12 x^2-24 x^3+16 x^4\right )}{-x+6 x^2-12 x^3+8 x^4} \, dx\\ &=\int \frac {-1-\left (-8-4 e^5\right ) x-24 x^2+32 x^3-16 x^4-e^{2 x} \left (-2 x+12 x^2-24 x^3+16 x^4\right )}{x \left (1-6 x+12 x^2-8 x^3\right )} \, dx\\ &=\int \left (2 e^{2 x}-\frac {4 \left (2+e^5\right )}{(-1+2 x)^3}+\frac {1}{x (-1+2 x)^3}+\frac {24 x}{(-1+2 x)^3}-\frac {32 x^2}{(-1+2 x)^3}+\frac {16 x^3}{(-1+2 x)^3}\right ) \, dx\\ &=\frac {2+e^5}{(1-2 x)^2}+2 \int e^{2 x} \, dx+16 \int \frac {x^3}{(-1+2 x)^3} \, dx+24 \int \frac {x}{(-1+2 x)^3} \, dx-32 \int \frac {x^2}{(-1+2 x)^3} \, dx+\int \frac {1}{x (-1+2 x)^3} \, dx\\ &=e^{2 x}+\frac {2+e^5}{(1-2 x)^2}-\frac {12 x^2}{(1-2 x)^2}+16 \int \left (\frac {1}{8}+\frac {1}{8 (-1+2 x)^3}+\frac {3}{8 (-1+2 x)^2}+\frac {3}{8 (-1+2 x)}\right ) \, dx-32 \int \left (\frac {1}{4 (-1+2 x)^3}+\frac {1}{2 (-1+2 x)^2}+\frac {1}{4 (-1+2 x)}\right ) \, dx+\int \left (-\frac {1}{x}+\frac {2}{(-1+2 x)^3}-\frac {2}{(-1+2 x)^2}+\frac {2}{-1+2 x}\right ) \, dx\\ &=e^{2 x}+\frac {1}{(1-2 x)^2}+\frac {2+e^5}{(1-2 x)^2}-\frac {5}{1-2 x}+2 x-\frac {12 x^2}{(1-2 x)^2}+\frac {1}{-1+2 x}-\log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.55, size = 24, normalized size = 0.96 \begin {gather*} e^{2 x}+2 x+\frac {e^5}{(-1+2 x)^2}-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 25, normalized size = 1.00
method | result | size |
derivativedivides | \(-\ln \left (2 x \right )+2 x +{\mathrm e}^{2 x}+\frac {{\mathrm e}^{5}}{\left (2 x -1\right )^{2}}\) | \(25\) |
default | \(-\ln \left (2 x \right )+2 x +{\mathrm e}^{2 x}+\frac {{\mathrm e}^{5}}{\left (2 x -1\right )^{2}}\) | \(25\) |
risch | \(2 x +\frac {{\mathrm e}^{5}}{4 x^{2}-4 x +1}-\ln \left (x \right )+{\mathrm e}^{2 x}\) | \(27\) |
norman | \(\frac {-6 x +8 x^{3}-4 x \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{2 x} x^{2}+2+{\mathrm e}^{5}+{\mathrm e}^{2 x}}{\left (2 x -1\right )^{2}}-\ln \left (x \right )\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 58 vs.
\(2 (23) = 46\).
time = 0.36, size = 58, normalized size = 2.32 \begin {gather*} \frac {8 \, x^{3} - 8 \, x^{2} + {\left (4 \, x^{2} - 4 \, x + 1\right )} e^{\left (2 \, x\right )} - {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (x\right ) + 2 \, x + e^{5}}{4 \, x^{2} - 4 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 24, normalized size = 0.96 \begin {gather*} 2 x + e^{2 x} - \log {\left (x \right )} + \frac {e^{5}}{4 x^{2} - 4 x + 1} \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. 65 vs.
\(2 (23) = 46\).
time = 0.40, size = 65, normalized size = 2.60 \begin {gather*} \frac {8 \, x^{3} + 4 \, x^{2} e^{\left (2 \, x\right )} - 4 \, x^{2} \log \left (x\right ) - 8 \, x^{2} - 4 \, x e^{\left (2 \, x\right )} + 4 \, x \log \left (x\right ) + 2 \, x + e^{5} + e^{\left (2 \, x\right )} - \log \left (x\right )}{4 \, x^{2} - 4 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.17, size = 22, normalized size = 0.88 \begin {gather*} 2\,x+{\mathrm {e}}^{2\,x}-\ln \left (x\right )+\frac {{\mathrm {e}}^5}{{\left (2\,x-1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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