Optimal. Leaf size=31 \[ \frac {1+e^2}{-19 e^{-x}+x-x \left (2-\frac {3}{-4+x^2}\right )} \]
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Rubi [F]
time = 7.61, 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 {e^x \left (-304+152 x^2-19 x^4+e^2 \left (-304+152 x^2-19 x^4\right )\right )+e^{2 x} \left (28-5 x^2+x^4+e^2 \left (28-5 x^2+x^4\right )\right )}{5776-2888 x^2+361 x^4+e^x \left (1064 x-418 x^3+38 x^5\right )+e^{2 x} \left (49 x^2-14 x^4+x^6\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 {e^x \left (1+e^2\right ) \left (-19 \left (-4+x^2\right )^2+e^x \left (28-5 x^2+x^4\right )\right )}{\left (e^x x \left (-7+x^2\right )+19 \left (-4+x^2\right )\right )^2} \, dx\\ &=\left (1+e^2\right ) \int \frac {e^x \left (-19 \left (-4+x^2\right )^2+e^x \left (28-5 x^2+x^4\right )\right )}{\left (e^x x \left (-7+x^2\right )+19 \left (-4+x^2\right )\right )^2} \, dx\\ &=\left (1+e^2\right ) \int \left (\frac {e^x \left (28-5 x^2+x^4\right )}{x \left (-7+x^2\right ) \left (-76-7 e^x x+19 x^2+e^x x^3\right )}-\frac {19 e^x \left (-112-112 x+48 x^2+72 x^3-9 x^4-15 x^5+x^6+x^7\right )}{x \left (-7+x^2\right ) \left (-76-7 e^x x+19 x^2+e^x x^3\right )^2}\right ) \, dx\\ &=\left (1+e^2\right ) \int \frac {e^x \left (28-5 x^2+x^4\right )}{x \left (-7+x^2\right ) \left (-76-7 e^x x+19 x^2+e^x x^3\right )} \, dx-\left (19 \left (1+e^2\right )\right ) \int \frac {e^x \left (-112-112 x+48 x^2+72 x^3-9 x^4-15 x^5+x^6+x^7\right )}{x \left (-7+x^2\right ) \left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx\\ &=\left (1+e^2\right ) \int \left (-\frac {4 e^x}{x \left (-76-7 e^x x+19 x^2+e^x x^3\right )}+\frac {e^x x}{-76-7 e^x x+19 x^2+e^x x^3}+\frac {6 e^x x}{\left (-7+x^2\right ) \left (-76-7 e^x x+19 x^2+e^x x^3\right )}\right ) \, dx-\left (19 \left (1+e^2\right )\right ) \int \left (\frac {16 e^x}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2}+\frac {16 e^x}{x \left (-76-7 e^x x+19 x^2+e^x x^3\right )^2}-\frac {2 e^x x}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2}-\frac {8 e^x x^2}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2}+\frac {e^x x^3}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2}+\frac {e^x x^4}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2}+\frac {18 e^x x}{\left (-7+x^2\right ) \left (-76-7 e^x x+19 x^2+e^x x^3\right )^2}\right ) \, dx\\ &=\left (1+e^2\right ) \int \frac {e^x x}{-76-7 e^x x+19 x^2+e^x x^3} \, dx-\left (4 \left (1+e^2\right )\right ) \int \frac {e^x}{x \left (-76-7 e^x x+19 x^2+e^x x^3\right )} \, dx+\left (6 \left (1+e^2\right )\right ) \int \frac {e^x x}{\left (-7+x^2\right ) \left (-76-7 e^x x+19 x^2+e^x x^3\right )} \, dx-\left (19 \left (1+e^2\right )\right ) \int \frac {e^x x^3}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx-\left (19 \left (1+e^2\right )\right ) \int \frac {e^x x^4}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx+\left (38 \left (1+e^2\right )\right ) \int \frac {e^x x}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx+\left (152 \left (1+e^2\right )\right ) \int \frac {e^x x^2}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx-\left (304 \left (1+e^2\right )\right ) \int \frac {e^x}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx-\left (304 \left (1+e^2\right )\right ) \int \frac {e^x}{x \left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx-\left (342 \left (1+e^2\right )\right ) \int \frac {e^x x}{\left (-7+x^2\right ) \left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx\\ &=\left (1+e^2\right ) \int \frac {e^x x}{-76-7 e^x x+19 x^2+e^x x^3} \, dx-\left (4 \left (1+e^2\right )\right ) \int \frac {e^x}{x \left (-76-7 e^x x+19 x^2+e^x x^3\right )} \, dx+\left (6 \left (1+e^2\right )\right ) \int \left (-\frac {e^x}{2 \left (\sqrt {7}-x\right ) \left (-76-7 e^x x+19 x^2+e^x x^3\right )}+\frac {e^x}{2 \left (\sqrt {7}+x\right ) \left (-76-7 e^x x+19 x^2+e^x x^3\right )}\right ) \, dx-\left (19 \left (1+e^2\right )\right ) \int \frac {e^x x^3}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx-\left (19 \left (1+e^2\right )\right ) \int \frac {e^x x^4}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx+\left (38 \left (1+e^2\right )\right ) \int \frac {e^x x}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx+\left (152 \left (1+e^2\right )\right ) \int \frac {e^x x^2}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx-\left (304 \left (1+e^2\right )\right ) \int \frac {e^x}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx-\left (304 \left (1+e^2\right )\right ) \int \frac {e^x}{x \left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx-\left (342 \left (1+e^2\right )\right ) \int \left (-\frac {e^x}{2 \left (\sqrt {7}-x\right ) \left (-76-7 e^x x+19 x^2+e^x x^3\right )^2}+\frac {e^x}{2 \left (\sqrt {7}+x\right ) \left (-76-7 e^x x+19 x^2+e^x x^3\right )^2}\right ) \, dx\\ &=\left (1+e^2\right ) \int \frac {e^x x}{-76-7 e^x x+19 x^2+e^x x^3} \, dx-\left (3 \left (1+e^2\right )\right ) \int \frac {e^x}{\left (\sqrt {7}-x\right ) \left (-76-7 e^x x+19 x^2+e^x x^3\right )} \, dx+\left (3 \left (1+e^2\right )\right ) \int \frac {e^x}{\left (\sqrt {7}+x\right ) \left (-76-7 e^x x+19 x^2+e^x x^3\right )} \, dx-\left (4 \left (1+e^2\right )\right ) \int \frac {e^x}{x \left (-76-7 e^x x+19 x^2+e^x x^3\right )} \, dx-\left (19 \left (1+e^2\right )\right ) \int \frac {e^x x^3}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx-\left (19 \left (1+e^2\right )\right ) \int \frac {e^x x^4}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx+\left (38 \left (1+e^2\right )\right ) \int \frac {e^x x}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx+\left (152 \left (1+e^2\right )\right ) \int \frac {e^x x^2}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx+\left (171 \left (1+e^2\right )\right ) \int \frac {e^x}{\left (\sqrt {7}-x\right ) \left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx-\left (171 \left (1+e^2\right )\right ) \int \frac {e^x}{\left (\sqrt {7}+x\right ) \left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx-\left (304 \left (1+e^2\right )\right ) \int \frac {e^x}{\left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx-\left (304 \left (1+e^2\right )\right ) \int \frac {e^x}{x \left (-76-7 e^x x+19 x^2+e^x x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 10.16, size = 35, normalized size = 1.13 \begin {gather*} -\frac {e^x \left (1+e^2\right ) \left (-4+x^2\right )}{e^x x \left (-7+x^2\right )+19 \left (-4+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 44, normalized size = 1.42
method | result | size |
norman | \(\frac {\left (4 \,{\mathrm e}^{2}+4\right ) {\mathrm e}^{x}+\left (-{\mathrm e}^{2}-1\right ) x^{2} {\mathrm e}^{x}}{{\mathrm e}^{x} x^{3}-7 \,{\mathrm e}^{x} x +19 x^{2}-76}\) | \(44\) |
risch | \(\frac {\left (-{\mathrm e}^{2}-1\right ) x^{2}+4+4 \,{\mathrm e}^{2}}{\left (x^{2}-7\right ) x}+\frac {19 x^{4} {\mathrm e}^{2}+19 x^{4}-152 x^{2} {\mathrm e}^{2}-152 x^{2}+304 \,{\mathrm e}^{2}+304}{\left (x^{2}-7\right ) x \left ({\mathrm e}^{x} x^{3}-7 \,{\mathrm e}^{x} x +19 x^{2}-76\right )}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 37, normalized size = 1.19 \begin {gather*} -\frac {{\left (x^{2} {\left (e^{2} + 1\right )} - 4 \, e^{2} - 4\right )} e^{x}}{19 \, x^{2} + {\left (x^{3} - 7 \, x\right )} e^{x} - 76} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 36, normalized size = 1.16 \begin {gather*} -\frac {{\left (x^{2} + {\left (x^{2} - 4\right )} e^{2} - 4\right )} e^{x}}{19 \, x^{2} + {\left (x^{3} - 7 \, x\right )} e^{x} - 76} \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. 87 vs.
\(2 (20) = 40\).
time = 0.35, size = 87, normalized size = 2.81 \begin {gather*} \frac {19 x^{4} + 19 x^{4} e^{2} - 152 x^{2} e^{2} - 152 x^{2} + 304 + 304 e^{2}}{19 x^{5} - 209 x^{3} + 532 x + \left (x^{6} - 14 x^{4} + 49 x^{2}\right ) e^{x}} + \frac {x^{2} \left (- e^{2} - 1\right ) + 4 + 4 e^{2}}{x^{3} - 7 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 47, normalized size = 1.52 \begin {gather*} -\frac {x^{2} e^{\left (x + 2\right )} + x^{2} e^{x} - 4 \, e^{\left (x + 2\right )} - 4 \, e^{x}}{x^{3} e^{x} + 19 \, x^{2} - 7 \, x e^{x} - 76} \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}}^{2\,x}\,\left ({\mathrm {e}}^2\,\left (x^4-5\,x^2+28\right )-5\,x^2+x^4+28\right )-{\mathrm {e}}^x\,\left ({\mathrm {e}}^2\,\left (19\,x^4-152\,x^2+304\right )-152\,x^2+19\,x^4+304\right )}{{\mathrm {e}}^{2\,x}\,\left (x^6-14\,x^4+49\,x^2\right )-2888\,x^2+361\,x^4+{\mathrm {e}}^x\,\left (38\,x^5-418\,x^3+1064\,x\right )+5776} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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