Optimal. Leaf size=24 \[ e^{\frac {1}{x}+x} \left (4+\frac {1}{x^2}-\frac {2}{x}-2 x+x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 0.24, 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^{\frac {1}{x}+x} \left (-1-x^2+x^4+x^6\right )}{x^4} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\int \frac {e^{\frac {1}{x}+x} \left (-1+x^2\right ) \left (1+x^2\right )^2}{x^4} \, dx\\ &=\int \left (e^{\frac {1}{x}+x}-\frac {e^{\frac {1}{x}+x}}{x^4}-\frac {e^{\frac {1}{x}+x}}{x^2}+e^{\frac {1}{x}+x} x^2\right ) \, dx\\ &=\int e^{\frac {1}{x}+x} \, dx-\int \frac {e^{\frac {1}{x}+x}}{x^4} \, dx-\int \frac {e^{\frac {1}{x}+x}}{x^2} \, dx+\int e^{\frac {1}{x}+x} x^2 \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 24, normalized size = 1.00 \begin {gather*} e^{\frac {1}{x}+x} \left (4+\frac {1}{x^2}-\frac {2}{x}-2 x+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 33, normalized size = 1.38
method | result | size |
gosper | \(\frac {{\mathrm e}^{\frac {x^{2}+1}{x}} \left (x^{4}-2 x^{3}+4 x^{2}-2 x +1\right )}{x^{2}}\) | \(33\) |
risch | \(\frac {{\mathrm e}^{\frac {x^{2}+1}{x}} \left (x^{4}-2 x^{3}+4 x^{2}-2 x +1\right )}{x^{2}}\) | \(33\) |
norman | \(\frac {x \,{\mathrm e}^{x +\frac {1}{x}}+x^{5} {\mathrm e}^{x +\frac {1}{x}}+4 x^{3} {\mathrm e}^{x +\frac {1}{x}}-2 x^{4} {\mathrm e}^{x +\frac {1}{x}}-2 \,{\mathrm e}^{x +\frac {1}{x}} x^{2}}{x^{3}}\) | \(57\) |
parallelrisch | \(\frac {{\mathrm e}^{\frac {x^{2}+1}{x}} x^{4}-2 \,{\mathrm e}^{\frac {x^{2}+1}{x}} x^{3}+4 \,{\mathrm e}^{\frac {x^{2}+1}{x}} x^{2}-2 \,{\mathrm e}^{\frac {x^{2}+1}{x}} x +{\mathrm e}^{\frac {x^{2}+1}{x}}}{x^{2}}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.39, size = 28, normalized size = 1.17 \begin {gather*} \frac {{\left (x^{4} - 2 \, x^{3} + 4 \, x^{2} - 2 \, x + 1\right )} e^{\left (x + \frac {1}{x}\right )}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.58, size = 32, normalized size = 1.33 \begin {gather*} \frac {{\left (x^{4} - 2 \, x^{3} + 4 \, x^{2} - 2 \, x + 1\right )} e^{\left (\frac {x^{2} + 1}{x}\right )}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.05, size = 27, normalized size = 1.12 \begin {gather*} \frac {\left (x^{4} - 2 x^{3} + 4 x^{2} - 2 x + 1\right ) e^{x + \frac {1}{x}}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 72 vs.
\(2 (23) = 46\).
time = 0.48, size = 72, normalized size = 3.00 \begin {gather*} \frac {x^{4} e^{\left (\frac {x^{2} + 1}{x}\right )} - 2 \, x^{3} e^{\left (\frac {x^{2} + 1}{x}\right )} + 4 \, x^{2} e^{\left (\frac {x^{2} + 1}{x}\right )} - 2 \, x e^{\left (\frac {x^{2} + 1}{x}\right )} + e^{\left (\frac {x^{2} + 1}{x}\right )}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.17, size = 28, normalized size = 1.17 \begin {gather*} \frac {{\mathrm {e}}^{x+\frac {1}{x}}\,\left (x^4-2\,x^3+4\,x^2-2\,x+1\right )}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Chatgpt [F] Failed to verify
time = 1.00, size = 35, normalized size = 1.46 \begin {gather*} \frac {{\mathrm e}^{x +\frac {1}{x}} \left (x^{2}+1\right )}{2}+{\mathrm e}^{x +\frac {1}{x}} \left (\frac {1}{5} x^{5}-\frac {1}{3} x^{3}-x \right ) \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________