Optimal. Leaf size=23 \[ \frac {x^2+e^2 (5+x)}{4+e^6+\frac {1}{x}+x} \]
________________________________________________________________________________________
Rubi [B] time = 0.23, antiderivative size = 81, normalized size of antiderivative = 3.52, number of steps used = 5, number of rules used = 5, integrand size = 91, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.055, Rules used = {6, 1680, 1814, 21, 8} \begin {gather*} \frac {\left (2+e^6\right ) \left (6+e^6\right ) \left (15+e^2+8 e^6-e^8+e^{12}\right ) x+\left (2+e^6\right ) \left (6+e^6\right ) \left (4-e^2+e^6\right )}{\left (12+8 e^6+e^{12}\right ) \left (x^2+\left (4+e^6\right ) x+1\right )}+x \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 6
Rule 8
Rule 21
Rule 1680
Rule 1814
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 x^2+8 x^3+x^4+e^2 \left (5+2 x-x^2\right )+e^6 \left (e^2 x^2+2 x^3\right )}{1+8 x+\left (18+e^{12}\right ) x^2+8 x^3+x^4+e^6 \left (2 x+8 x^2+2 x^3\right )} \, dx\\ &=\operatorname {Subst}\left (\int \frac {-\left (\left (2+e^6\right ) \left (6+e^6\right ) \left (48+4 e^2+24 e^6-4 e^8+3 e^{12}\right )\right )+16 \left (52+6 e^2+45 e^6-3 e^8+12 e^{12}-e^{14}+e^{18}\right ) x-8 \left (42+2 e^2+24 e^6-2 e^8+3 e^{12}\right ) x^2+16 x^4}{\left (12+8 e^6+e^{12}-4 x^2\right )^2} \, dx,x,\frac {1}{4} \left (8+2 e^6\right )+x\right )\\ &=\frac {\left (2+e^6\right ) \left (6+e^6\right ) \left (4-e^2+e^6\right )+\left (2+e^6\right ) \left (6+e^6\right ) \left (15+e^2+8 e^6-e^8+e^{12}\right ) x}{\left (12+8 e^6+e^{12}\right ) \left (1+\left (4+e^6\right ) x+x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-2 \left (12+8 e^6+e^{12}\right )^2+8 \left (12+8 e^6+e^{12}\right ) x^2}{12+8 e^6+e^{12}-4 x^2} \, dx,x,\frac {1}{4} \left (8+2 e^6\right )+x\right )}{2 \left (12+8 e^6+e^{12}\right )}\\ &=\frac {\left (2+e^6\right ) \left (6+e^6\right ) \left (4-e^2+e^6\right )+\left (2+e^6\right ) \left (6+e^6\right ) \left (15+e^2+8 e^6-e^8+e^{12}\right ) x}{\left (12+8 e^6+e^{12}\right ) \left (1+\left (4+e^6\right ) x+x^2\right )}+\operatorname {Subst}\left (\int 1 \, dx,x,\frac {1}{4} \left (8+2 e^6\right )+x\right )\\ &=x+\frac {\left (2+e^6\right ) \left (6+e^6\right ) \left (4-e^2+e^6\right )+\left (2+e^6\right ) \left (6+e^6\right ) \left (15+e^2+8 e^6-e^8+e^{12}\right ) x}{\left (12+8 e^6+e^{12}\right ) \left (1+\left (4+e^6\right ) x+x^2\right )}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [B] time = 0.04, size = 49, normalized size = 2.13 \begin {gather*} x+\frac {4+e^2 (-1+x)+15 x-e^8 x+e^{12} x+e^6 (1+8 x)}{1+\left (4+e^6\right ) x+x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.64, size = 54, normalized size = 2.35 \begin {gather*} \frac {x^{3} + 4 \, x^{2} + x e^{12} - x e^{8} + {\left (x^{2} + 8 \, x + 1\right )} e^{6} + {\left (x - 1\right )} e^{2} + 16 \, x + 4}{x^{2} + x e^{6} + 4 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.12, size = 39, normalized size = 1.70
method | result | size |
norman | \(\frac {x^{3}+\left (-{\mathrm e}^{6} {\mathrm e}^{2}+{\mathrm e}^{2}\right ) x -{\mathrm e}^{2}}{1+x \,{\mathrm e}^{6}+x^{2}+4 x}\) | \(39\) |
gosper | \(-\frac {{\mathrm e}^{2} {\mathrm e}^{6} x -x^{3}-{\mathrm e}^{2} x +{\mathrm e}^{2}}{1+x \,{\mathrm e}^{6}+x^{2}+4 x}\) | \(40\) |
risch | \(x +\frac {\left (8 \,{\mathrm e}^{6}+{\mathrm e}^{12}+15+{\mathrm e}^{2}-{\mathrm e}^{8}\right ) x +{\mathrm e}^{6}-{\mathrm e}^{2}+4}{1+x \,{\mathrm e}^{6}+x^{2}+4 x}\) | \(42\) |
default | \(x -\frac {-\frac {\left ({\mathrm e}^{2} {\mathrm e}^{12}+8 \,{\mathrm e}^{18}+\left ({\mathrm e}^{12}\right )^{2}-{\mathrm e}^{12} {\mathrm e}^{8}+10 \,{\mathrm e}^{6} {\mathrm e}^{2}+91 \,{\mathrm e}^{12}+8 \,{\mathrm e}^{6} {\mathrm e}^{12}-8 \,{\mathrm e}^{6} {\mathrm e}^{8}+12 \,{\mathrm e}^{2}+216 \,{\mathrm e}^{6}-14 \,{\mathrm e}^{8}+180\right ) x}{{\mathrm e}^{12}+8 \,{\mathrm e}^{6}+12}+\frac {4 \,{\mathrm e}^{6} {\mathrm e}^{2}-12 \,{\mathrm e}^{12}-{\mathrm e}^{6} {\mathrm e}^{12}+{\mathrm e}^{6} {\mathrm e}^{8}+12 \,{\mathrm e}^{2}-44 \,{\mathrm e}^{6}+4 \,{\mathrm e}^{8}-48}{{\mathrm e}^{12}+8 \,{\mathrm e}^{6}+12}}{1+x \,{\mathrm e}^{6}+x^{2}+4 x}-\frac {4 \left ({\mathrm e}^{6} {\mathrm e}^{2}-{\mathrm e}^{8}\right ) \arctanh \left (\frac {{\mathrm e}^{6}+2 x +4}{\sqrt {{\mathrm e}^{12}+8 \,{\mathrm e}^{6}+12}}\right )}{\left ({\mathrm e}^{12}+8 \,{\mathrm e}^{6}+12\right )^{\frac {3}{2}}}\) | \(212\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.37, size = 40, normalized size = 1.74 \begin {gather*} x + \frac {x {\left (e^{12} - e^{8} + 8 \, e^{6} + e^{2} + 15\right )} + e^{6} - e^{2} + 4}{x^{2} + x {\left (e^{6} + 4\right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.18, size = 40, normalized size = 1.74 \begin {gather*} x+\frac {{\mathrm {e}}^6-{\mathrm {e}}^2+x\,\left ({\mathrm {e}}^2+8\,{\mathrm {e}}^6-{\mathrm {e}}^8+{\mathrm {e}}^{12}+15\right )+4}{x^2+\left ({\mathrm {e}}^6+4\right )\,x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.50, size = 39, normalized size = 1.70 \begin {gather*} x + \frac {x \left (- e^{8} + e^{2} + 15 + 8 e^{6} + e^{12}\right ) - e^{2} + 4 + e^{6}}{x^{2} + x \left (4 + e^{6}\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________