Optimal. Leaf size=23 \[ \log \left (\sqrt [9]{e} \left (3+3 e^{\frac {20 e^2}{x}}\right )+x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 0.62, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {-60 e^{\frac {19}{9}+\frac {20 e^2}{x}}+x^2}{3 \sqrt [9]{e} x^2+3 e^{\frac {1}{9}+\frac {20 e^2}{x}} x^2+x^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {20 e^2}{x^2}+\frac {60 e^{19/9}+20 e^2 x+x^2}{x^2 \left (3 \sqrt [9]{e}+3 e^{\frac {1}{9}+\frac {20 e^2}{x}}+x\right )}\right ) \, dx\\ &=\frac {20 e^2}{x}+\int \frac {60 e^{19/9}+20 e^2 x+x^2}{x^2 \left (3 \sqrt [9]{e}+3 e^{\frac {1}{9}+\frac {20 e^2}{x}}+x\right )} \, dx\\ &=\frac {20 e^2}{x}+\int \left (\frac {1}{3 \sqrt [9]{e}+3 e^{\frac {1}{9}+\frac {20 e^2}{x}}+x}+\frac {60 e^{19/9}}{x^2 \left (3 \sqrt [9]{e}+3 e^{\frac {1}{9}+\frac {20 e^2}{x}}+x\right )}+\frac {20 e^2}{x \left (3 \sqrt [9]{e}+3 e^{\frac {1}{9}+\frac {20 e^2}{x}}+x\right )}\right ) \, dx\\ &=\frac {20 e^2}{x}+\left (20 e^2\right ) \int \frac {1}{x \left (3 \sqrt [9]{e}+3 e^{\frac {1}{9}+\frac {20 e^2}{x}}+x\right )} \, dx+\left (60 e^{19/9}\right ) \int \frac {1}{x^2 \left (3 \sqrt [9]{e}+3 e^{\frac {1}{9}+\frac {20 e^2}{x}}+x\right )} \, dx+\int \frac {1}{3 \sqrt [9]{e}+3 e^{\frac {1}{9}+\frac {20 e^2}{x}}+x} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.12, size = 26, normalized size = 1.13 \begin {gather*} \log \left (3 \sqrt [9]{e}+3 e^{\frac {1}{9}+\frac {20 e^2}{x}}+x\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 4.77, size = 20, normalized size = 0.87
method | result | size |
norman | \(\ln \left (3 \,{\mathrm e}^{\frac {1}{9}} {\mathrm e}^{\frac {20 \,{\mathrm e}^{2}}{x}}+3 \,{\mathrm e}^{\frac {1}{9}}+x \right )\) | \(20\) |
risch | \(\ln \left ({\mathrm e}^{\frac {20 \,{\mathrm e}^{2}}{x}}+\frac {\left (3 \,{\mathrm e}^{\frac {1}{9}}+x \right ) {\mathrm e}^{-\frac {1}{9}}}{3}\right )\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.31, size = 23, normalized size = 1.00 \begin {gather*} \log \left (\frac {1}{3} \, {\left (x + 3 \, e^{\frac {1}{9}} + 3 \, e^{\left (\frac {20 \, e^{2}}{x} + \frac {1}{9}\right )}\right )} e^{\left (-\frac {1}{9}\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 26, normalized size = 1.13 \begin {gather*} \log \left (x e^{2} + 3 \, e^{\frac {19}{9}} + 3 \, e^{\left (\frac {19 \, x + 180 \, e^{2}}{9 \, x}\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.10, size = 24, normalized size = 1.04 \begin {gather*} \log {\left (\frac {x + 3 e^{\frac {1}{9}}}{3 e^{\frac {1}{9}}} + e^{\frac {20 e^{2}}{x}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs.
\(2 (17) = 34\).
time = 0.44, size = 44, normalized size = 1.91 \begin {gather*} {\left (e^{2} \log \left (\frac {3 \, e^{2}}{x} + \frac {3 \, e^{\left (\frac {20 \, e^{2}}{x} + 2\right )}}{x} + e^{\frac {17}{9}}\right ) - e^{2} \log \left (\frac {e^{2}}{x}\right )\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.69, size = 30, normalized size = 1.30 \begin {gather*} \ln \left (\frac {x+3\,{\mathrm {e}}^{1/9}+3\,{\mathrm {e}}^{\frac {20\,{\mathrm {e}}^2}{x}}\,{\mathrm {e}}^{1/9}}{x}\right )-\ln \left (\frac {1}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________