Optimal. Leaf size=26 \[ 1+\frac {9 e^4 (-4+x)}{(2-x) \left (-8+4 x^2\right )} \]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 34, normalized size of antiderivative = 1.31, number of steps used = 6, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {12, 2074, 639, 207} \begin {gather*} \frac {9 e^4 (x+3)}{4 \left (2-x^2\right )}-\frac {9 e^4}{4 (2-x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 207
Rule 639
Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=e^4 \int \frac {18+72 x-63 x^2+9 x^3}{32-32 x-24 x^2+32 x^3-8 x^5+2 x^6} \, dx\\ &=e^4 \int \left (-\frac {9}{4 (-2+x)^2}+\frac {9 (2+3 x)}{2 \left (-2+x^2\right )^2}+\frac {9}{4 \left (-2+x^2\right )}\right ) \, dx\\ &=-\frac {9 e^4}{4 (2-x)}+\frac {1}{4} \left (9 e^4\right ) \int \frac {1}{-2+x^2} \, dx+\frac {1}{2} \left (9 e^4\right ) \int \frac {2+3 x}{\left (-2+x^2\right )^2} \, dx\\ &=-\frac {9 e^4}{4 (2-x)}+\frac {9 e^4 (3+x)}{4 \left (2-x^2\right )}-\frac {9 e^4 \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{4 \sqrt {2}}-\frac {1}{4} \left (9 e^4\right ) \int \frac {1}{-2+x^2} \, dx\\ &=-\frac {9 e^4}{4 (2-x)}+\frac {9 e^4 (3+x)}{4 \left (2-x^2\right )}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 27, normalized size = 1.04 \begin {gather*} \frac {9 e^4 (4-x)}{4 \left (4-2 x-2 x^2+x^3\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.71, size = 22, normalized size = 0.85 \begin {gather*} -\frac {9 \, {\left (x - 4\right )} e^{4}}{4 \, {\left (x^{3} - 2 \, x^{2} - 2 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.13, size = 22, normalized size = 0.85 \begin {gather*} -\frac {9 \, {\left (x - 4\right )} e^{4}}{4 \, {\left (x^{3} - 2 \, x^{2} - 2 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 24, normalized size = 0.92
method | result | size |
risch | \(\frac {{\mathrm e}^{4} \left (-\frac {9 x}{4}+9\right )}{x^{3}-2 x^{2}-2 x +4}\) | \(24\) |
gosper | \(-\frac {9 \left (x -4\right ) {\mathrm e}^{4}}{4 \left (x^{3}-2 x^{2}-2 x +4\right )}\) | \(25\) |
default | \(\frac {9 \,{\mathrm e}^{4} \left (\frac {-3-x}{2 x^{2}-4}+\frac {1}{2 x -4}\right )}{2}\) | \(29\) |
norman | \(\frac {-\frac {9 x \,{\mathrm e}^{4}}{4}+9 \,{\mathrm e}^{4}}{x^{3}-2 x^{2}-2 x +4}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 22, normalized size = 0.85 \begin {gather*} -\frac {9 \, {\left (x - 4\right )} e^{4}}{4 \, {\left (x^{3} - 2 \, x^{2} - 2 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.08, size = 19, normalized size = 0.73 \begin {gather*} -\frac {9\,{\mathrm {e}}^4\,\left (x-4\right )}{4\,\left (x^2-2\right )\,\left (x-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.25, size = 26, normalized size = 1.00 \begin {gather*} \frac {- 9 x e^{4} + 36 e^{4}}{4 x^{3} - 8 x^{2} - 8 x + 16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________