Optimal. Leaf size=20 \[ \frac {-e^{2 x}-3 x}{-625+e+2 x} \]
________________________________________________________________________________________
Rubi [A] time = 0.27, antiderivative size = 36, normalized size of antiderivative = 1.80, number of steps used = 5, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6741, 27, 6742, 2197} \begin {gather*} \frac {e^{2 x}}{-2 x-e+625}+\frac {3 (625-e)}{2 (-2 x-e+625)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 2197
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1875 \left (1-\frac {e}{625}\right )+e^{2 x} (1252-2 e-4 x)}{(-625+e)^2-4 (625-e) x+4 x^2} \, dx\\ &=\int \frac {1875 \left (1-\frac {e}{625}\right )+e^{2 x} (1252-2 e-4 x)}{(-625+e+2 x)^2} \, dx\\ &=\int \left (-\frac {3 (-625+e)}{(-625+e+2 x)^2}-\frac {2 e^{2 x} (-626+e+2 x)}{(-625+e+2 x)^2}\right ) \, dx\\ &=\frac {3 (625-e)}{2 (625-e-2 x)}-2 \int \frac {e^{2 x} (-626+e+2 x)}{(-625+e+2 x)^2} \, dx\\ &=\frac {3 (625-e)}{2 (625-e-2 x)}+\frac {e^{2 x}}{625-e-2 x}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 23, normalized size = 1.15 \begin {gather*} \frac {1875-3 e+2 e^{2 x}}{1250-2 e-4 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.25, size = 23, normalized size = 1.15 \begin {gather*} \frac {3 \, e - 2 \, e^{\left (2 \, x\right )} - 1875}{2 \, {\left (2 \, x + e - 625\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.58, size = 23, normalized size = 1.15 \begin {gather*} \frac {3 \, e - 2 \, e^{\left (2 \, x\right )} - 1875}{2 \, {\left (2 \, x + e - 625\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.32, size = 23, normalized size = 1.15
method | result | size |
norman | \(\frac {-{\mathrm e}^{2 x}-\frac {1875}{2}+\frac {3 \,{\mathrm e}}{2}}{{\mathrm e}-625+2 x}\) | \(23\) |
risch | \(-\frac {1875}{2 \left ({\mathrm e}-625+2 x \right )}+\frac {3 \,{\mathrm e}}{2 \left ({\mathrm e}-625+2 x \right )}-\frac {{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}\) | \(41\) |
derivativedivides | \(-\frac {1875}{2 \left ({\mathrm e}-625+2 x \right )}+\frac {3 \,{\mathrm e}}{2 \left ({\mathrm e}-625+2 x \right )}-\frac {626 \,{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-626 \,{\mathrm e}^{-{\mathrm e}+625} \expIntegralEi \left (1, 625-{\mathrm e}-2 x \right )-\frac {{\mathrm e}^{2 x} \left ({\mathrm e}-625\right )}{{\mathrm e}-625+2 x}+\left (-{\mathrm e}+626\right ) {\mathrm e}^{-{\mathrm e}+625} \expIntegralEi \left (1, 625-{\mathrm e}-2 x \right )-{\mathrm e} \left (-\frac {{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-{\mathrm e}^{-{\mathrm e}+625} \expIntegralEi \left (1, 625-{\mathrm e}-2 x \right )\right )\) | \(145\) |
default | \(-\frac {1875}{2 \left ({\mathrm e}-625+2 x \right )}+\frac {3 \,{\mathrm e}}{2 \left ({\mathrm e}-625+2 x \right )}-\frac {626 \,{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-626 \,{\mathrm e}^{-{\mathrm e}+625} \expIntegralEi \left (1, 625-{\mathrm e}-2 x \right )-\frac {{\mathrm e}^{2 x} \left ({\mathrm e}-625\right )}{{\mathrm e}-625+2 x}+\left (-{\mathrm e}+626\right ) {\mathrm e}^{-{\mathrm e}+625} \expIntegralEi \left (1, 625-{\mathrm e}-2 x \right )-{\mathrm e} \left (-\frac {{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-{\mathrm e}^{-{\mathrm e}+625} \expIntegralEi \left (1, 625-{\mathrm e}-2 x \right )\right )\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {2 \, x e^{\left (2 \, x\right )}}{4 \, x^{2} + 4 \, x {\left (e - 625\right )} + e^{2} - 1250 \, e + 390625} - \frac {626 \, e^{\left (-e + 625\right )} E_{2}\left (-2 \, x - e + 625\right )}{2 \, x + e - 625} + \frac {3 \, e}{2 \, {\left (2 \, x + e - 625\right )}} - \frac {1875}{2 \, {\left (2 \, x + e - 625\right )}} - \int \frac {2 \, {\left (2 \, x {\left (e + 1\right )} + e^{2} - 626 \, e + 625\right )} e^{\left (2 \, x\right )}}{8 \, x^{3} + 12 \, x^{2} {\left (e - 625\right )} + 6 \, x {\left (e^{2} - 1250 \, e + 390625\right )} + e^{3} - 1875 \, e^{2} + 1171875 \, e - 244140625}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.04, size = 19, normalized size = 0.95 \begin {gather*} -\frac {3\,x+{\mathrm {e}}^{2\,x}}{2\,x+\mathrm {e}-625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.23, size = 31, normalized size = 1.55 \begin {gather*} - \frac {1875 - 3 e}{4 x - 1250 + 2 e} - \frac {e^{2 x}}{2 x - 625 + e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________