Optimal. Leaf size=23 \[ x+\frac {e^{-147+x}+x}{x \left (1+e^5+x^2\right )} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(233\) vs. \(2(23)=46\).
time = 0.63, antiderivative size = 233, normalized size of antiderivative = 10.13, number of steps
used = 16, number of rules used = 10, integrand size = 100, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6, 6873,
6874, 205, 209, 267, 294, 327, 393, 2326} \begin {gather*} \frac {\left (1+e^{10}\right ) \text {ArcTan}\left (\frac {x}{\sqrt {1+e^5}}\right )}{2 \left (1+e^5\right )^{3/2}}+\frac {e^5 \left (2+e^5\right ) \text {ArcTan}\left (\frac {x}{\sqrt {1+e^5}}\right )}{\left (1+e^5\right )^{3/2}}-\frac {3}{2} \sqrt {1+e^5} \text {ArcTan}\left (\frac {x}{\sqrt {1+e^5}}\right )+\frac {\text {ArcTan}\left (\frac {x}{\sqrt {1+e^5}}\right )}{\sqrt {1+e^5}}+\frac {\left (1+e^{10}\right ) x}{2 \left (1+e^5\right ) \left (x^2+e^5+1\right )}-\frac {e^{10} x}{\left (1+e^5\right ) \left (x^2+e^5+1\right )}-\frac {x}{x^2+e^5+1}+\frac {1}{x^2+e^5+1}-\frac {x^3}{2 \left (x^2+e^5+1\right )}+\frac {e^{x-147} \left (x^3+\left (1+e^5\right ) x\right )}{\left (x^2+e^5+1\right )^2 x^2}+\frac {3 x}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 205
Rule 209
Rule 267
Rule 294
Rule 327
Rule 393
Rule 2326
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2+e^{10} x^2-2 x^3+2 x^4+x^6+e^{-147+x} \left (-1+e^5 (-1+x)+x-3 x^2+x^3\right )+e^5 \left (2 x^2+2 x^4\right )}{\left (1+e^{10}\right ) x^2+2 x^4+x^6+e^5 \left (2 x^2+2 x^4\right )} \, dx\\ &=\int \frac {\left (1+e^{10}\right ) x^2-2 x^3+2 x^4+x^6+e^{-147+x} \left (-1+e^5 (-1+x)+x-3 x^2+x^3\right )+e^5 \left (2 x^2+2 x^4\right )}{\left (1+e^{10}\right ) x^2+2 x^4+x^6+e^5 \left (2 x^2+2 x^4\right )} \, dx\\ &=\int \frac {\left (1+e^{10}\right ) x^2-2 x^3+2 x^4+x^6+e^{-147+x} \left (-1+e^5 (-1+x)+x-3 x^2+x^3\right )+e^5 \left (2 x^2+2 x^4\right )}{x^2 \left (1+e^5+x^2\right )^2} \, dx\\ &=\int \left (\frac {1+e^{10}}{\left (1+e^5+x^2\right )^2}-\frac {2 x}{\left (1+e^5+x^2\right )^2}+\frac {2 x^2}{\left (1+e^5+x^2\right )^2}+\frac {x^4}{\left (1+e^5+x^2\right )^2}+\frac {2 e^5 \left (1+x^2\right )}{\left (1+e^5+x^2\right )^2}+\frac {e^{-147+x} \left (-1-e^5+\left (1+e^5\right ) x-3 x^2+x^3\right )}{x^2 \left (1+e^5+x^2\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {x}{\left (1+e^5+x^2\right )^2} \, dx\right )+2 \int \frac {x^2}{\left (1+e^5+x^2\right )^2} \, dx+\left (2 e^5\right ) \int \frac {1+x^2}{\left (1+e^5+x^2\right )^2} \, dx+\left (1+e^{10}\right ) \int \frac {1}{\left (1+e^5+x^2\right )^2} \, dx+\int \frac {x^4}{\left (1+e^5+x^2\right )^2} \, dx+\int \frac {e^{-147+x} \left (-1-e^5+\left (1+e^5\right ) x-3 x^2+x^3\right )}{x^2 \left (1+e^5+x^2\right )^2} \, dx\\ &=\frac {1}{1+e^5+x^2}-\frac {x}{1+e^5+x^2}-\frac {e^{10} x}{\left (1+e^5\right ) \left (1+e^5+x^2\right )}+\frac {\left (1+e^{10}\right ) x}{2 \left (1+e^5\right ) \left (1+e^5+x^2\right )}-\frac {x^3}{2 \left (1+e^5+x^2\right )}+\frac {e^{-147+x} \left (\left (1+e^5\right ) x+x^3\right )}{x^2 \left (1+e^5+x^2\right )^2}+\frac {3}{2} \int \frac {x^2}{1+e^5+x^2} \, dx+\frac {\left (e^5 \left (2+e^5\right )\right ) \int \frac {1}{1+e^5+x^2} \, dx}{1+e^5}+\frac {\left (1+e^{10}\right ) \int \frac {1}{1+e^5+x^2} \, dx}{2 \left (1+e^5\right )}+\int \frac {1}{1+e^5+x^2} \, dx\\ &=\frac {3 x}{2}+\frac {1}{1+e^5+x^2}-\frac {x}{1+e^5+x^2}-\frac {e^{10} x}{\left (1+e^5\right ) \left (1+e^5+x^2\right )}+\frac {\left (1+e^{10}\right ) x}{2 \left (1+e^5\right ) \left (1+e^5+x^2\right )}-\frac {x^3}{2 \left (1+e^5+x^2\right )}+\frac {e^{-147+x} \left (\left (1+e^5\right ) x+x^3\right )}{x^2 \left (1+e^5+x^2\right )^2}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {1+e^5}}\right )}{\sqrt {1+e^5}}+\frac {e^5 \left (2+e^5\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+e^5}}\right )}{\left (1+e^5\right )^{3/2}}+\frac {\left (1+e^{10}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+e^5}}\right )}{2 \left (1+e^5\right )^{3/2}}-\frac {1}{2} \left (3 \left (1+e^5\right )\right ) \int \frac {1}{1+e^5+x^2} \, dx\\ &=\frac {3 x}{2}+\frac {1}{1+e^5+x^2}-\frac {x}{1+e^5+x^2}-\frac {e^{10} x}{\left (1+e^5\right ) \left (1+e^5+x^2\right )}+\frac {\left (1+e^{10}\right ) x}{2 \left (1+e^5\right ) \left (1+e^5+x^2\right )}-\frac {x^3}{2 \left (1+e^5+x^2\right )}+\frac {e^{-147+x} \left (\left (1+e^5\right ) x+x^3\right )}{x^2 \left (1+e^5+x^2\right )^2}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {1+e^5}}\right )}{\sqrt {1+e^5}}-\frac {3}{2} \sqrt {1+e^5} \tan ^{-1}\left (\frac {x}{\sqrt {1+e^5}}\right )+\frac {e^5 \left (2+e^5\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+e^5}}\right )}{\left (1+e^5\right )^{3/2}}+\frac {\left (1+e^{10}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+e^5}}\right )}{2 \left (1+e^5\right )^{3/2}}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 3.49, size = 39, normalized size = 1.70 \begin {gather*} \frac {e^x+e^{152} x^2+e^{147} x \left (1+x+x^3\right )}{e^{147} x \left (1+e^5+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.88, size = 1381, normalized size = 60.04
method | result | size |
risch | \(x +\frac {1}{1+x^{2}+{\mathrm e}^{5}}+\frac {{\mathrm e}^{x -147}}{x \left (1+x^{2}+{\mathrm e}^{5}\right )}\) | \(29\) |
norman | \(\frac {x^{4}+x +\left ({\mathrm e}^{5}+1\right ) x^{2}+{\mathrm e}^{x -147}}{x \left (1+x^{2}+{\mathrm e}^{5}\right )}\) | \(31\) |
default | \(\text {Expression too large to display}\) | \(1381\) |
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. 242 vs.
\(2 (21) = 42\).
time = 0.69, size = 242, normalized size = 10.52 \begin {gather*} \frac {1}{2} \, {\left (\frac {x}{x^{2} {\left (e^{5} + 1\right )} + e^{10} + 2 \, e^{5} + 1} + \frac {\arctan \left (\frac {x}{\sqrt {e^{5} + 1}}\right )}{{\left (e^{5} + 1\right )}^{\frac {3}{2}}}\right )} e^{10} + {\left (\frac {\arctan \left (\frac {x}{\sqrt {e^{5} + 1}}\right )}{\sqrt {e^{5} + 1}} - \frac {x}{x^{2} + e^{5} + 1}\right )} e^{5} + {\left (\frac {x}{x^{2} {\left (e^{5} + 1\right )} + e^{10} + 2 \, e^{5} + 1} + \frac {\arctan \left (\frac {x}{\sqrt {e^{5} + 1}}\right )}{{\left (e^{5} + 1\right )}^{\frac {3}{2}}}\right )} e^{5} - \frac {3}{2} \, \sqrt {e^{5} + 1} \arctan \left (\frac {x}{\sqrt {e^{5} + 1}}\right ) + x + \frac {x {\left (e^{5} + 1\right )}}{2 \, {\left (x^{2} + e^{5} + 1\right )}} + \frac {\arctan \left (\frac {x}{\sqrt {e^{5} + 1}}\right )}{\sqrt {e^{5} + 1}} + \frac {x}{2 \, {\left (x^{2} {\left (e^{5} + 1\right )} + e^{10} + 2 \, e^{5} + 1\right )}} - \frac {x}{x^{2} + e^{5} + 1} + \frac {e^{x}}{x^{3} e^{147} + x {\left (e^{152} + e^{147}\right )}} + \frac {\arctan \left (\frac {x}{\sqrt {e^{5} + 1}}\right )}{2 \, {\left (e^{5} + 1\right )}^{\frac {3}{2}}} + \frac {1}{x^{2} + e^{5} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 30, normalized size = 1.30 \begin {gather*} \frac {x^{4} + x^{2} e^{5} + x^{2} + x + e^{\left (x - 147\right )}}{x^{3} + x e^{5} + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 26, normalized size = 1.13 \begin {gather*} x + \frac {e^{x - 147}}{x^{3} + x + x e^{5}} + \frac {1}{x^{2} + 1 + e^{5}} \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. 111 vs.
\(2 (21) = 42\).
time = 0.41, size = 111, normalized size = 4.83 \begin {gather*} \frac {x^{6} e^{147} + 2 \, x^{4} e^{152} + 2 \, x^{4} e^{147} + 2 \, x^{3} e^{147} + x^{2} e^{157} + 2 \, x^{2} e^{152} + x^{2} e^{147} + 2 \, x^{2} e^{x} + 2 \, x e^{152} + 2 \, x e^{147} + 2 \, e^{\left (x + 5\right )} + 2 \, e^{x}}{x^{5} e^{147} + 2 \, x^{3} e^{152} + 2 \, x^{3} e^{147} + x e^{157} + 2 \, x e^{152} + x e^{147}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.53, size = 21, normalized size = 0.91 \begin {gather*} x+\frac {x+{\mathrm {e}}^{x-147}}{x\,\left (x^2+{\mathrm {e}}^5+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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