Optimal. Leaf size=22 \[ \left (\left (-2+e^{4 e^{2 x}}-\frac {4}{x}\right ) x\right )^{\frac {1}{x}} \]
________________________________________________________________________________________
Rubi [F] time = 6.59, 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 {\left (-4-2 x+e^{4 e^{2 x}} x\right )^{\frac {1}{x}} \left (-2 x+e^{4 e^{2 x}} \left (x+8 e^{2 x} x^2\right )+\left (4+2 x-e^{4 e^{2 x}} x\right ) \log \left (-4-2 x+e^{4 e^{2 x}} x\right )\right )}{-4 x^2-2 x^3+e^{4 e^{2 x}} x^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (8 e^{2 \left (2 e^{2 x}+x\right )} \left (-4-2 x+e^{4 e^{2 x}} x\right )^{-1+\frac {1}{x}}-\frac {\left (-4-2 x+e^{4 e^{2 x}} x\right )^{-1+\frac {1}{x}} \left (2 x-e^{4 e^{2 x}} x-4 \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )-2 x \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )+e^{4 e^{2 x}} x \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )\right )}{x^2}\right ) \, dx\\ &=8 \int e^{2 \left (2 e^{2 x}+x\right )} \left (-4-2 x+e^{4 e^{2 x}} x\right )^{-1+\frac {1}{x}} \, dx-\int \frac {\left (-4-2 x+e^{4 e^{2 x}} x\right )^{-1+\frac {1}{x}} \left (2 x-e^{4 e^{2 x}} x-4 \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )-2 x \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )+e^{4 e^{2 x}} x \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )\right )}{x^2} \, dx\\ &=8 \int e^{2 \left (2 e^{2 x}+x\right )} \left (-4-2 x+e^{4 e^{2 x}} x\right )^{-1+\frac {1}{x}} \, dx-\int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}} \left (-\left (\left (-2+e^{4 e^{2 x}}\right ) x\right )+\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right ) \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )\right )}{x^2} \, dx\\ &=8 \int e^{2 \left (2 e^{2 x}+x\right )} \left (-4-2 x+e^{4 e^{2 x}} x\right )^{-1+\frac {1}{x}} \, dx-\int \left (\frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}} \left (-1+\log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )\right )}{x}-\frac {2 \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}} \left (-x+2 \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )+x \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )\right )}{x^2}\right ) \, dx\\ &=2 \int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}} \left (-x+2 \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )+x \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )\right )}{x^2} \, dx+8 \int e^{2 \left (2 e^{2 x}+x\right )} \left (-4-2 x+e^{4 e^{2 x}} x\right )^{-1+\frac {1}{x}} \, dx-\int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}} \left (-1+\log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )\right )}{x} \, dx\\ &=2 \int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}} \left (-x+(2+x) \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )\right )}{x^2} \, dx+8 \int e^{2 \left (2 e^{2 x}+x\right )} \left (-4-2 x+e^{4 e^{2 x}} x\right )^{-1+\frac {1}{x}} \, dx-\int \left (-\frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x}+\frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}} \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )}{x}\right ) \, dx\\ &=2 \int \left (-\frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x}+\frac {(2+x) \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}} \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )}{x^2}\right ) \, dx+8 \int e^{2 \left (2 e^{2 x}+x\right )} \left (-4-2 x+e^{4 e^{2 x}} x\right )^{-1+\frac {1}{x}} \, dx+\int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx-\int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}} \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )}{x} \, dx\\ &=-\left (2 \int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx\right )+2 \int \frac {(2+x) \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}} \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )}{x^2} \, dx+8 \int e^{2 \left (2 e^{2 x}+x\right )} \left (-4-2 x+e^{4 e^{2 x}} x\right )^{-1+\frac {1}{x}} \, dx-\log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right ) \int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx+\int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx+\int \frac {\left (-2+e^{4 e^{2 x}}+8 e^{4 e^{2 x}+2 x} x\right ) \int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx}{-4+\left (-2+e^{4 e^{2 x}}\right ) x} \, dx\\ &=-\left (2 \int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx\right )-2 \int \frac {\left (-2+e^{4 e^{2 x}}+8 e^{4 e^{2 x}+2 x} x\right ) \left (2 \int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x^2} \, dx+\int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx\right )}{-4+\left (-2+e^{4 e^{2 x}}\right ) x} \, dx+8 \int e^{2 \left (2 e^{2 x}+x\right )} \left (-4-2 x+e^{4 e^{2 x}} x\right )^{-1+\frac {1}{x}} \, dx-\log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right ) \int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx+\left (2 \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )\right ) \int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx+\left (4 \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )\right ) \int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x^2} \, dx+\int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx+\int \left (\frac {\left (-2+e^{4 e^{2 x}}\right ) \int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx}{-4-2 x+e^{4 e^{2 x}} x}+\frac {8 e^{2 \left (2 e^{2 x}+x\right )} x \int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx}{-4-2 x+e^{4 e^{2 x}} x}\right ) \, dx\\ &=-\left (2 \int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx\right )-2 \int \left (\frac {8 e^{2 \left (2 e^{2 x}+x\right )} x \left (-2 \int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x^2} \, dx-\int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx\right )}{4+2 x-e^{4 e^{2 x}} x}+\frac {\left (-2+e^{4 e^{2 x}}\right ) \left (2 \int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x^2} \, dx+\int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx\right )}{-4-2 x+e^{4 e^{2 x}} x}\right ) \, dx+8 \int e^{2 \left (2 e^{2 x}+x\right )} \left (-4-2 x+e^{4 e^{2 x}} x\right )^{-1+\frac {1}{x}} \, dx+8 \int \frac {e^{2 \left (2 e^{2 x}+x\right )} x \int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx}{-4-2 x+e^{4 e^{2 x}} x} \, dx-\log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right ) \int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx+\left (2 \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )\right ) \int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx+\left (4 \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )\right ) \int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x^2} \, dx+\int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx+\int \frac {\left (-2+e^{4 e^{2 x}}\right ) \int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx}{-4-2 x+e^{4 e^{2 x}} x} \, dx\\ &=-\left (2 \int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx\right )-2 \int \frac {\left (-2+e^{4 e^{2 x}}\right ) \left (2 \int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x^2} \, dx+\int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx\right )}{-4-2 x+e^{4 e^{2 x}} x} \, dx+8 \int e^{2 \left (2 e^{2 x}+x\right )} \left (-4-2 x+e^{4 e^{2 x}} x\right )^{-1+\frac {1}{x}} \, dx+8 \int \frac {e^{2 \left (2 e^{2 x}+x\right )} x \int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx}{-4-2 x+e^{4 e^{2 x}} x} \, dx-16 \int \frac {e^{2 \left (2 e^{2 x}+x\right )} x \left (-2 \int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x^2} \, dx-\int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx\right )}{4+2 x-e^{4 e^{2 x}} x} \, dx-\log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right ) \int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx+\left (2 \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )\right ) \int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx+\left (4 \log \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )\right ) \int \frac {\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x^2} \, dx+\int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx+\int \left (\frac {\int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx}{x}+\frac {4 \int \frac {e^{4 e^{2 x}} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{-1+\frac {1}{x}}}{x} \, dx}{x \left (-4-2 x+e^{4 e^{2 x}} x\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 19, normalized size = 0.86 \begin {gather*} \left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{\frac {1}{x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.93, size = 18, normalized size = 0.82 \begin {gather*} {\left (x e^{\left (4 \, e^{\left (2 \, x\right )}\right )} - 2 \, x - 4\right )}^{\left (\frac {1}{x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (8 \, x^{2} e^{\left (2 \, x\right )} + x\right )} e^{\left (4 \, e^{\left (2 \, x\right )}\right )} - {\left (x e^{\left (4 \, e^{\left (2 \, x\right )}\right )} - 2 \, x - 4\right )} \log \left (x e^{\left (4 \, e^{\left (2 \, x\right )}\right )} - 2 \, x - 4\right ) - 2 \, x\right )} {\left (x e^{\left (4 \, e^{\left (2 \, x\right )}\right )} - 2 \, x - 4\right )}^{\left (\frac {1}{x}\right )}}{x^{3} e^{\left (4 \, e^{\left (2 \, x\right )}\right )} - 2 \, x^{3} - 4 \, x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 19, normalized size = 0.86
method | result | size |
risch | \(\left (x \,{\mathrm e}^{4 \,{\mathrm e}^{2 x}}-2 x -4\right )^{\frac {1}{x}}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.46, size = 18, normalized size = 0.82 \begin {gather*} {\left (x e^{\left (4 \, e^{\left (2 \, x\right )}\right )} - 2 \, x - 4\right )}^{\left (\frac {1}{x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 8.24, size = 18, normalized size = 0.82 \begin {gather*} {\left (x\,{\mathrm {e}}^{4\,{\mathrm {e}}^{2\,x}}-2\,x-4\right )}^{1/x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [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]
________________________________________________________________________________________