Optimal. Leaf size=29 \[ -e^4+3 x+x^2-\frac {e^{e^{x/4}} x}{16+x} \]
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Rubi [F]
time = 0.35, 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 {3072+2432 x+268 x^2+8 x^3+e^{e^{x/4}} \left (-64+e^{x/4} \left (-16 x-x^2\right )\right )}{1024+128 x+4 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3072+2432 x+268 x^2+8 x^3+e^{e^{x/4}} \left (-64+e^{x/4} \left (-16 x-x^2\right )\right )}{4 (16+x)^2} \, dx\\ &=\frac {1}{4} \int \frac {3072+2432 x+268 x^2+8 x^3+e^{e^{x/4}} \left (-64+e^{x/4} \left (-16 x-x^2\right )\right )}{(16+x)^2} \, dx\\ &=\frac {1}{4} \int \left (\frac {e^{\frac {1}{4} \left (4 e^{x/4}+x\right )} x}{-16-x}+\frac {3072}{(16+x)^2}-\frac {64 e^{e^{x/4}}}{(16+x)^2}+\frac {2432 x}{(16+x)^2}+\frac {268 x^2}{(16+x)^2}+\frac {8 x^3}{(16+x)^2}\right ) \, dx\\ &=-\frac {768}{16+x}+\frac {1}{4} \int \frac {e^{\frac {1}{4} \left (4 e^{x/4}+x\right )} x}{-16-x} \, dx+2 \int \frac {x^3}{(16+x)^2} \, dx-16 \int \frac {e^{e^{x/4}}}{(16+x)^2} \, dx+67 \int \frac {x^2}{(16+x)^2} \, dx+608 \int \frac {x}{(16+x)^2} \, dx\\ &=-\frac {768}{16+x}+\frac {1}{4} \int \left (-e^{\frac {1}{4} \left (4 e^{x/4}+x\right )}+\frac {16 e^{\frac {1}{4} \left (4 e^{x/4}+x\right )}}{16+x}\right ) \, dx+2 \int \left (-32+x-\frac {4096}{(16+x)^2}+\frac {768}{16+x}\right ) \, dx-16 \int \frac {e^{e^{x/4}}}{(16+x)^2} \, dx+67 \int \left (1+\frac {256}{(16+x)^2}-\frac {32}{16+x}\right ) \, dx+608 \int \left (-\frac {16}{(16+x)^2}+\frac {1}{16+x}\right ) \, dx\\ &=3 x+x^2-\frac {1}{4} \int e^{\frac {1}{4} \left (4 e^{x/4}+x\right )} \, dx+4 \int \frac {e^{\frac {1}{4} \left (4 e^{x/4}+x\right )}}{16+x} \, dx-16 \int \frac {e^{e^{x/4}}}{(16+x)^2} \, dx\\ &=3 x+x^2+4 \int \frac {e^{\frac {1}{4} \left (4 e^{x/4}+x\right )}}{16+x} \, dx-16 \int \frac {e^{e^{x/4}}}{(16+x)^2} \, dx-\text {Subst}\left (\int e^{e^x+x} \, dx,x,\frac {x}{4}\right )\\ &=3 x+x^2+4 \int \frac {e^{\frac {1}{4} \left (4 e^{x/4}+x\right )}}{16+x} \, dx-16 \int \frac {e^{e^{x/4}}}{(16+x)^2} \, dx-\text {Subst}\left (\int e^x \, dx,x,e^{x/4}\right )\\ &=-e^{e^{x/4}}+3 x+x^2+4 \int \frac {e^{\frac {1}{4} \left (4 e^{x/4}+x\right )}}{16+x} \, dx-16 \int \frac {e^{e^{x/4}}}{(16+x)^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.25, size = 26, normalized size = 0.90 \begin {gather*} \frac {x \left (48-e^{e^{x/4}}+19 x+x^2\right )}{16+x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.52, size = 21, normalized size = 0.72
method | result | size |
risch | \(x^{2}+3 x -\frac {{\mathrm e}^{{\mathrm e}^{\frac {x}{4}}} x}{x +16}\) | \(21\) |
norman | \(\frac {x^{3}+19 x^{2}-x \,{\mathrm e}^{{\mathrm e}^{\frac {x}{4}}}-768}{x +16}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 20, normalized size = 0.69 \begin {gather*} x^{2} + 3 \, x - \frac {x e^{\left (e^{\left (\frac {1}{4} \, x\right )}\right )}}{x + 16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 26, normalized size = 0.90 \begin {gather*} \frac {x^{3} + 19 \, x^{2} - x e^{\left (e^{\left (\frac {1}{4} \, x\right )}\right )} + 48 \, x}{x + 16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 17, normalized size = 0.59 \begin {gather*} x^{2} + 3 x - \frac {x e^{e^{\frac {x}{4}}}}{x + 16} \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. 53 vs.
\(2 (24) = 48\).
time = 0.42, size = 53, normalized size = 1.83 \begin {gather*} \frac {x^{3} e^{\left (\frac {1}{4} \, x\right )} + 19 \, x^{2} e^{\left (\frac {1}{4} \, x\right )} + 48 \, x e^{\left (\frac {1}{4} \, x\right )} - x e^{\left (\frac {1}{4} \, x + e^{\left (\frac {1}{4} \, x\right )}\right )}}{x e^{\left (\frac {1}{4} \, x\right )} + 16 \, e^{\left (\frac {1}{4} \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.22, size = 22, normalized size = 0.76 \begin {gather*} \frac {x\,\left (19\,x-{\mathrm {e}}^{{\mathrm {e}}^{x/4}}+x^2+48\right )}{x+16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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