Optimal. Leaf size=26 \[ \frac {\frac {6}{-1+x}-\frac {e^2}{x}}{25 e^2 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 42, normalized size of antiderivative = 1.62, number of steps
used = 6, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {12, 1608, 27,
1834} \begin {gather*} -\frac {1}{25 x^3}-\frac {6}{25 e^2 x^2}-\frac {6}{25 e^2 x}-\frac {6}{25 e^2 (1-x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 27
Rule 1608
Rule 1834
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {12 x-18 x^2+e^2 \left (3-6 x+3 x^2\right )}{25 x^4-50 x^5+25 x^6} \, dx}{e^2}\\ &=\frac {\int \frac {12 x-18 x^2+e^2 \left (3-6 x+3 x^2\right )}{x^4 \left (25-50 x+25 x^2\right )} \, dx}{e^2}\\ &=\frac {\int \frac {12 x-18 x^2+e^2 \left (3-6 x+3 x^2\right )}{25 (-1+x)^2 x^4} \, dx}{e^2}\\ &=\frac {\int \frac {12 x-18 x^2+e^2 \left (3-6 x+3 x^2\right )}{(-1+x)^2 x^4} \, dx}{25 e^2}\\ &=\frac {\int \left (-\frac {6}{(-1+x)^2}+\frac {3 e^2}{x^4}+\frac {12}{x^3}+\frac {6}{x^2}\right ) \, dx}{25 e^2}\\ &=-\frac {6}{25 e^2 (1-x)}-\frac {1}{25 x^3}-\frac {6}{25 e^2 x^2}-\frac {6}{25 e^2 x}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 20, normalized size = 0.77 \begin {gather*} \frac {-1+\frac {6 x}{e^2 (-1+x)}}{25 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 32, normalized size = 1.23
method | result | size |
risch | \(\frac {{\mathrm e}^{-2} \left (\left (-\frac {{\mathrm e}^{2}}{25}+\frac {6}{25}\right ) x +\frac {{\mathrm e}^{2}}{25}\right )}{x^{3} \left (x -1\right )}\) | \(25\) |
gosper | \(-\frac {\left ({\mathrm e}^{2} x -{\mathrm e}^{2}-6 x \right ) {\mathrm e}^{-2}}{25 x^{3} \left (x -1\right )}\) | \(27\) |
default | \(\frac {{\mathrm e}^{-2} \left (-\frac {{\mathrm e}^{2}}{x^{3}}-\frac {6}{x^{2}}-\frac {6}{x}+\frac {6}{x -1}\right )}{25}\) | \(32\) |
norman | \(\frac {\left (\frac {{\mathrm e}^{-1} {\mathrm e}^{2}}{25}-\frac {\left ({\mathrm e}^{2}-6\right ) {\mathrm e}^{-1} x}{25}\right ) {\mathrm e}^{-1}}{x^{3} \left (x -1\right )}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 26, normalized size = 1.00 \begin {gather*} -\frac {{\left (x {\left (e^{2} - 6\right )} - e^{2}\right )} e^{\left (-2\right )}}{25 \, {\left (x^{4} - x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.32, size = 25, normalized size = 0.96 \begin {gather*} -\frac {{\left ({\left (x - 1\right )} e^{2} - 6 \, x\right )} e^{\left (-2\right )}}{25 \, {\left (x^{4} - x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.27, size = 27, normalized size = 1.04 \begin {gather*} - \frac {x \left (-6 + e^{2}\right ) - e^{2}}{25 x^{4} e^{2} - 25 x^{3} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.40, size = 28, normalized size = 1.08 \begin {gather*} \frac {1}{25} \, {\left (\frac {6}{x - 1} - \frac {6 \, x^{2} + 6 \, x + e^{2}}{x^{3}}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.13, size = 29, normalized size = 1.12 \begin {gather*} -\frac {{\mathrm {e}}^2-x\,\left ({\mathrm {e}}^2-6\right )}{25\,x^3\,{\mathrm {e}}^2-25\,x^4\,{\mathrm {e}}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________