Optimal. Leaf size=32 \[ \frac {\left (e^{-1-x}+\frac {2 e^x}{x}+\frac {x}{4}\right ) (5+x (5+x))}{x} \]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 64, normalized size of antiderivative = 2.00, number of steps used = 18, number of rules used = 7, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {12, 14, 2199, 2194, 2177, 2178, 2176} \begin {gather*} \frac {x^2}{4}+\frac {10 e^x}{x^2}+e^{-x-1} x+\frac {5 x}{4}+5 e^{-x-1}+2 e^x+\frac {5 e^{-x-1}}{x}+\frac {10 e^x}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 2176
Rule 2177
Rule 2178
Rule 2194
Rule 2199
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {5 x^3+2 x^4+e^x \left (-80+40 x^2+8 x^3\right )+e^{-1-x} \left (-20 x-20 x^2-16 x^3-4 x^4\right )}{x^3} \, dx\\ &=\frac {1}{4} \int \left (5+2 x-\frac {4 e^{-1-x} \left (5+5 x+4 x^2+x^3\right )}{x^2}+\frac {8 e^x \left (-10+5 x^2+x^3\right )}{x^3}\right ) \, dx\\ &=\frac {5 x}{4}+\frac {x^2}{4}+2 \int \frac {e^x \left (-10+5 x^2+x^3\right )}{x^3} \, dx-\int \frac {e^{-1-x} \left (5+5 x+4 x^2+x^3\right )}{x^2} \, dx\\ &=\frac {5 x}{4}+\frac {x^2}{4}+2 \int \left (e^x-\frac {10 e^x}{x^3}+\frac {5 e^x}{x}\right ) \, dx-\int \left (4 e^{-1-x}+\frac {5 e^{-1-x}}{x^2}+\frac {5 e^{-1-x}}{x}+e^{-1-x} x\right ) \, dx\\ &=\frac {5 x}{4}+\frac {x^2}{4}+2 \int e^x \, dx-4 \int e^{-1-x} \, dx-5 \int \frac {e^{-1-x}}{x^2} \, dx-5 \int \frac {e^{-1-x}}{x} \, dx+10 \int \frac {e^x}{x} \, dx-20 \int \frac {e^x}{x^3} \, dx-\int e^{-1-x} x \, dx\\ &=4 e^{-1-x}+2 e^x+\frac {10 e^x}{x^2}+\frac {5 e^{-1-x}}{x}+\frac {5 x}{4}+e^{-1-x} x+\frac {x^2}{4}-\frac {5 \text {Ei}(-x)}{e}+10 \text {Ei}(x)+5 \int \frac {e^{-1-x}}{x} \, dx-10 \int \frac {e^x}{x^2} \, dx-\int e^{-1-x} \, dx\\ &=5 e^{-1-x}+2 e^x+\frac {10 e^x}{x^2}+\frac {5 e^{-1-x}}{x}+\frac {10 e^x}{x}+\frac {5 x}{4}+e^{-1-x} x+\frac {x^2}{4}+10 \text {Ei}(x)-10 \int \frac {e^x}{x} \, dx\\ &=5 e^{-1-x}+2 e^x+\frac {10 e^x}{x^2}+\frac {5 e^{-1-x}}{x}+\frac {10 e^x}{x}+\frac {5 x}{4}+e^{-1-x} x+\frac {x^2}{4}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 59, normalized size = 1.84 \begin {gather*} \frac {1}{4} \left (8 e^x+\frac {40 e^x}{x^2}+\frac {40 e^x}{x}+5 x+x^2-4 e^{-x} \left (-\frac {5}{e}-\frac {5}{e x}-\frac {x}{e}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.71, size = 55, normalized size = 1.72 \begin {gather*} \frac {{\left (4 \, x^{3} + 20 \, x^{2} + 8 \, {\left (x^{2} + 5 \, x + 5\right )} e^{\left (2 \, x + 1\right )} + {\left (x^{4} + 5 \, x^{3}\right )} e^{\left (x + 1\right )} + 20 \, x\right )} e^{\left (-x - 1\right )}}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.22, size = 68, normalized size = 2.12 \begin {gather*} \frac {{\left (x^{4} e + 5 \, x^{3} e + 4 \, x^{3} e^{\left (-x\right )} + 20 \, x^{2} e^{\left (-x\right )} + 8 \, x^{2} e^{\left (x + 1\right )} + 20 \, x e^{\left (-x\right )} + 40 \, x e^{\left (x + 1\right )} + 40 \, e^{\left (x + 1\right )}\right )} e^{\left (-1\right )}}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.11, size = 43, normalized size = 1.34
method | result | size |
risch | \(\frac {x^{2}}{4}+\frac {5 x}{4}+\frac {2 \left (x^{2}+5 x +5\right ) {\mathrm e}^{x}}{x^{2}}+\frac {\left (x^{2}+5 x +5\right ) {\mathrm e}^{-x -1}}{x}\) | \(43\) |
norman | \(\frac {\left ({\mathrm e}^{-1} x^{3}+10 \,{\mathrm e}^{2 x}+5 \,{\mathrm e}^{-1} x +10 x \,{\mathrm e}^{2 x}+5 x^{2} {\mathrm e}^{-1}+\frac {5 \,{\mathrm e}^{x} x^{3}}{4}+\frac {{\mathrm e}^{x} x^{4}}{4}+2 \,{\mathrm e}^{2 x} x^{2}\right ) {\mathrm e}^{-x}}{x^{2}}\) | \(70\) |
default | \(\frac {x^{2}}{4}+\frac {5 x}{4}+4 \,{\mathrm e}^{-x} {\mathrm e}^{-1}+\frac {10 \,{\mathrm e}^{x}}{x^{2}}+\frac {10 \,{\mathrm e}^{x}}{x}-5 \,{\mathrm e}^{-1} \left (-\frac {{\mathrm e}^{-x}}{x}+\expIntegralEi \left (1, x\right )\right )+5 \,{\mathrm e}^{-1} \expIntegralEi \left (1, x\right )-{\mathrm e}^{-1} \left (-x \,{\mathrm e}^{-x}-{\mathrm e}^{-x}\right )+2 \,{\mathrm e}^{x}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 0.40, size = 57, normalized size = 1.78 \begin {gather*} \frac {1}{4} \, x^{2} - 5 \, {\rm Ei}\left (-x\right ) e^{\left (-1\right )} + {\left (x + 1\right )} e^{\left (-x - 1\right )} + 5 \, e^{\left (-1\right )} \Gamma \left (-1, x\right ) + \frac {5}{4} \, x + 10 \, {\rm Ei}\relax (x) + 2 \, e^{x} + 4 \, e^{\left (-x - 1\right )} + 20 \, \Gamma \left (-2, -x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.12, size = 54, normalized size = 1.69 \begin {gather*} \frac {5\,x}{4}+5\,{\mathrm {e}}^{-x-1}+2\,{\mathrm {e}}^x+\frac {10\,{\mathrm {e}}^x}{x}+\frac {10\,{\mathrm {e}}^x}{x^2}+x\,{\mathrm {e}}^{-x-1}+\frac {5\,{\mathrm {e}}^{-x-1}}{x}+\frac {x^2}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.18, size = 60, normalized size = 1.88 \begin {gather*} \frac {x^{2}}{4} + \frac {5 x}{4} + \frac {\left (x^{4} + 5 x^{3} + 5 x^{2}\right ) e^{- x} + \left (2 e x^{3} + 10 e x^{2} + 10 e x\right ) e^{x}}{e x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________