\(\int \frac {e^{x^4 \log ^2(\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2})} ((-10 x^5+20 x^6-10 x^7+\sqrt [5]{\frac {1}{x}} (12 x^3-22 x^4)) \log (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2})+(-20 x^5+40 x^6-20 x^7+\sqrt [5]{\frac {1}{x}} (-20 x^3+20 x^4)) \log ^2(\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}))}{-5 x^2+10 x^3-5 x^4+\sqrt [5]{\frac {1}{x}} (-5+5 x)} \, dx\) [6741]

Optimal result

Integrand size = 196, antiderivative size = 30 \[ \int \frac {e^{x^4 \log ^2\left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )} \left (\left (-10 x^5+20 x^6-10 x^7+\sqrt [5]{\frac {1}{x}} \left (12 x^3-22 x^4\right )\right ) \log \left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )+\left (-20 x^5+40 x^6-20 x^7+\sqrt [5]{\frac {1}{x}} \left (-20 x^3+20 x^4\right )\right ) \log ^2\left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )\right )}{-5 x^2+10 x^3-5 x^4+\sqrt [5]{\frac {1}{x}} (-5+5 x)} \, dx=e^{x^4 \log ^2\left (-x+\frac {\sqrt [5]{\frac {1}{x}}}{-x+x^2}\right )} \]

[Out]

exp(x^4*ln((1/x)^(1/5)/(x^2-x)-x)^2)

Rubi [F(-1)]

Timed out. \[ \int \frac {e^{x^4 \log ^2\left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )} \left (\left (-10 x^5+20 x^6-10 x^7+\sqrt [5]{\frac {1}{x}} \left (12 x^3-22 x^4\right )\right ) \log \left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )+\left (-20 x^5+40 x^6-20 x^7+\sqrt [5]{\frac {1}{x}} \left (-20 x^3+20 x^4\right )\right ) \log ^2\left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )\right )}{-5 x^2+10 x^3-5 x^4+\sqrt [5]{\frac {1}{x}} (-5+5 x)} \, dx=\text {\$Aborted} \]

[In]

Int[(E^(x^4*Log[((x^(-1))^(1/5) + x^2 - x^3)/(-x + x^2)]^2)*((-10*x^5 + 20*x^6 - 10*x^7 + (x^(-1))^(1/5)*(12*x
^3 - 22*x^4))*Log[((x^(-1))^(1/5) + x^2 - x^3)/(-x + x^2)] + (-20*x^5 + 40*x^6 - 20*x^7 + (x^(-1))^(1/5)*(-20*
x^3 + 20*x^4))*Log[((x^(-1))^(1/5) + x^2 - x^3)/(-x + x^2)]^2))/(-5*x^2 + 10*x^3 - 5*x^4 + (x^(-1))^(1/5)*(-5
+ 5*x)),x]

[Out]

$Aborted

Rubi steps Aborted

Mathematica [F]

\[ \int \frac {e^{x^4 \log ^2\left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )} \left (\left (-10 x^5+20 x^6-10 x^7+\sqrt [5]{\frac {1}{x}} \left (12 x^3-22 x^4\right )\right ) \log \left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )+\left (-20 x^5+40 x^6-20 x^7+\sqrt [5]{\frac {1}{x}} \left (-20 x^3+20 x^4\right )\right ) \log ^2\left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )\right )}{-5 x^2+10 x^3-5 x^4+\sqrt [5]{\frac {1}{x}} (-5+5 x)} \, dx=\int \frac {e^{x^4 \log ^2\left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )} \left (\left (-10 x^5+20 x^6-10 x^7+\sqrt [5]{\frac {1}{x}} \left (12 x^3-22 x^4\right )\right ) \log \left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )+\left (-20 x^5+40 x^6-20 x^7+\sqrt [5]{\frac {1}{x}} \left (-20 x^3+20 x^4\right )\right ) \log ^2\left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )\right )}{-5 x^2+10 x^3-5 x^4+\sqrt [5]{\frac {1}{x}} (-5+5 x)} \, dx \]

[In]

Integrate[(E^(x^4*Log[((x^(-1))^(1/5) + x^2 - x^3)/(-x + x^2)]^2)*((-10*x^5 + 20*x^6 - 10*x^7 + (x^(-1))^(1/5)
*(12*x^3 - 22*x^4))*Log[((x^(-1))^(1/5) + x^2 - x^3)/(-x + x^2)] + (-20*x^5 + 40*x^6 - 20*x^7 + (x^(-1))^(1/5)
*(-20*x^3 + 20*x^4))*Log[((x^(-1))^(1/5) + x^2 - x^3)/(-x + x^2)]^2))/(-5*x^2 + 10*x^3 - 5*x^4 + (x^(-1))^(1/5
)*(-5 + 5*x)),x]

[Out]

Integrate[(E^(x^4*Log[((x^(-1))^(1/5) + x^2 - x^3)/(-x + x^2)]^2)*((-10*x^5 + 20*x^6 - 10*x^7 + (x^(-1))^(1/5)
*(12*x^3 - 22*x^4))*Log[((x^(-1))^(1/5) + x^2 - x^3)/(-x + x^2)] + (-20*x^5 + 40*x^6 - 20*x^7 + (x^(-1))^(1/5)
*(-20*x^3 + 20*x^4))*Log[((x^(-1))^(1/5) + x^2 - x^3)/(-x + x^2)]^2))/(-5*x^2 + 10*x^3 - 5*x^4 + (x^(-1))^(1/5
)*(-5 + 5*x)), x]

Maple [F]

\[\int \frac {\left (\left (\left (20 x^{4}-20 x^{3}\right ) \left (\frac {1}{x}\right )^{\frac {1}{5}}-20 x^{7}+40 x^{6}-20 x^{5}\right ) \ln \left (\frac {\left (\frac {1}{x}\right )^{\frac {1}{5}}-x^{3}+x^{2}}{x^{2}-x}\right )^{2}+\left (\left (-22 x^{4}+12 x^{3}\right ) \left (\frac {1}{x}\right )^{\frac {1}{5}}-10 x^{7}+20 x^{6}-10 x^{5}\right ) \ln \left (\frac {\left (\frac {1}{x}\right )^{\frac {1}{5}}-x^{3}+x^{2}}{x^{2}-x}\right )\right ) {\mathrm e}^{x^{4} \ln \left (\frac {\left (\frac {1}{x}\right )^{\frac {1}{5}}-x^{3}+x^{2}}{x^{2}-x}\right )^{2}}}{\left (5 x -5\right ) \left (\frac {1}{x}\right )^{\frac {1}{5}}-5 x^{4}+10 x^{3}-5 x^{2}}d x\]

[In]

int((((20*x^4-20*x^3)*(1/x)^(1/5)-20*x^7+40*x^6-20*x^5)*ln(((1/x)^(1/5)-x^3+x^2)/(x^2-x))^2+((-22*x^4+12*x^3)*
(1/x)^(1/5)-10*x^7+20*x^6-10*x^5)*ln(((1/x)^(1/5)-x^3+x^2)/(x^2-x)))*exp(x^4*ln(((1/x)^(1/5)-x^3+x^2)/(x^2-x))
^2)/((5*x-5)*(1/x)^(1/5)-5*x^4+10*x^3-5*x^2),x)

[Out]

int((((20*x^4-20*x^3)*(1/x)^(1/5)-20*x^7+40*x^6-20*x^5)*ln(((1/x)^(1/5)-x^3+x^2)/(x^2-x))^2+((-22*x^4+12*x^3)*
(1/x)^(1/5)-10*x^7+20*x^6-10*x^5)*ln(((1/x)^(1/5)-x^3+x^2)/(x^2-x)))*exp(x^4*ln(((1/x)^(1/5)-x^3+x^2)/(x^2-x))
^2)/((5*x-5)*(1/x)^(1/5)-5*x^4+10*x^3-5*x^2),x)

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {e^{x^4 \log ^2\left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )} \left (\left (-10 x^5+20 x^6-10 x^7+\sqrt [5]{\frac {1}{x}} \left (12 x^3-22 x^4\right )\right ) \log \left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )+\left (-20 x^5+40 x^6-20 x^7+\sqrt [5]{\frac {1}{x}} \left (-20 x^3+20 x^4\right )\right ) \log ^2\left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )\right )}{-5 x^2+10 x^3-5 x^4+\sqrt [5]{\frac {1}{x}} (-5+5 x)} \, dx=e^{\left (x^{4} \log \left (-\frac {x^{4} - x^{3} - x^{\frac {4}{5}}}{x^{3} - x^{2}}\right )^{2}\right )} \]

[In]

integrate((((20*x^4-20*x^3)*(1/x)^(1/5)-20*x^7+40*x^6-20*x^5)*log(((1/x)^(1/5)-x^3+x^2)/(x^2-x))^2+((-22*x^4+1
2*x^3)*(1/x)^(1/5)-10*x^7+20*x^6-10*x^5)*log(((1/x)^(1/5)-x^3+x^2)/(x^2-x)))*exp(x^4*log(((1/x)^(1/5)-x^3+x^2)
/(x^2-x))^2)/((5*x-5)*(1/x)^(1/5)-5*x^4+10*x^3-5*x^2),x, algorithm="fricas")

[Out]

e^(x^4*log(-(x^4 - x^3 - x^(4/5))/(x^3 - x^2))^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{x^4 \log ^2\left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )} \left (\left (-10 x^5+20 x^6-10 x^7+\sqrt [5]{\frac {1}{x}} \left (12 x^3-22 x^4\right )\right ) \log \left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )+\left (-20 x^5+40 x^6-20 x^7+\sqrt [5]{\frac {1}{x}} \left (-20 x^3+20 x^4\right )\right ) \log ^2\left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )\right )}{-5 x^2+10 x^3-5 x^4+\sqrt [5]{\frac {1}{x}} (-5+5 x)} \, dx=\text {Timed out} \]

[In]

integrate((((20*x**4-20*x**3)*(1/x)**(1/5)-20*x**7+40*x**6-20*x**5)*ln(((1/x)**(1/5)-x**3+x**2)/(x**2-x))**2+(
(-22*x**4+12*x**3)*(1/x)**(1/5)-10*x**7+20*x**6-10*x**5)*ln(((1/x)**(1/5)-x**3+x**2)/(x**2-x)))*exp(x**4*ln(((
1/x)**(1/5)-x**3+x**2)/(x**2-x))**2)/((5*x-5)*(1/x)**(1/5)-5*x**4+10*x**3-5*x**2),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (25) = 50\).

Time = 0.74 (sec) , antiderivative size = 193, normalized size of antiderivative = 6.43 \[ \int \frac {e^{x^4 \log ^2\left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )} \left (\left (-10 x^5+20 x^6-10 x^7+\sqrt [5]{\frac {1}{x}} \left (12 x^3-22 x^4\right )\right ) \log \left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )+\left (-20 x^5+40 x^6-20 x^7+\sqrt [5]{\frac {1}{x}} \left (-20 x^3+20 x^4\right )\right ) \log ^2\left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )\right )}{-5 x^2+10 x^3-5 x^4+\sqrt [5]{\frac {1}{x}} (-5+5 x)} \, dx=e^{\left (x^{4} \log \left (-x^{\frac {16}{5}} + x^{\frac {11}{5}} + 1\right )^{2} - \frac {12}{5} \, x^{4} \log \left (-x^{\frac {16}{5}} + x^{\frac {11}{5}} + 1\right ) \log \left (x\right ) + \frac {36}{25} \, x^{4} \log \left (x\right )^{2} - 2 \, x^{4} \log \left (-x^{\frac {16}{5}} + x^{\frac {11}{5}} + 1\right ) \log \left (x^{\frac {4}{5}} + x^{\frac {3}{5}} + x^{\frac {2}{5}} + x^{\frac {1}{5}} + 1\right ) + \frac {12}{5} \, x^{4} \log \left (x\right ) \log \left (x^{\frac {4}{5}} + x^{\frac {3}{5}} + x^{\frac {2}{5}} + x^{\frac {1}{5}} + 1\right ) + x^{4} \log \left (x^{\frac {4}{5}} + x^{\frac {3}{5}} + x^{\frac {2}{5}} + x^{\frac {1}{5}} + 1\right )^{2} - 2 \, x^{4} \log \left (-x^{\frac {16}{5}} + x^{\frac {11}{5}} + 1\right ) \log \left (x^{\frac {1}{5}} - 1\right ) + \frac {12}{5} \, x^{4} \log \left (x\right ) \log \left (x^{\frac {1}{5}} - 1\right ) + 2 \, x^{4} \log \left (x^{\frac {4}{5}} + x^{\frac {3}{5}} + x^{\frac {2}{5}} + x^{\frac {1}{5}} + 1\right ) \log \left (x^{\frac {1}{5}} - 1\right ) + x^{4} \log \left (x^{\frac {1}{5}} - 1\right )^{2}\right )} \]

[In]

integrate((((20*x^4-20*x^3)*(1/x)^(1/5)-20*x^7+40*x^6-20*x^5)*log(((1/x)^(1/5)-x^3+x^2)/(x^2-x))^2+((-22*x^4+1
2*x^3)*(1/x)^(1/5)-10*x^7+20*x^6-10*x^5)*log(((1/x)^(1/5)-x^3+x^2)/(x^2-x)))*exp(x^4*log(((1/x)^(1/5)-x^3+x^2)
/(x^2-x))^2)/((5*x-5)*(1/x)^(1/5)-5*x^4+10*x^3-5*x^2),x, algorithm="maxima")

[Out]

e^(x^4*log(-x^(16/5) + x^(11/5) + 1)^2 - 12/5*x^4*log(-x^(16/5) + x^(11/5) + 1)*log(x) + 36/25*x^4*log(x)^2 -
2*x^4*log(-x^(16/5) + x^(11/5) + 1)*log(x^(4/5) + x^(3/5) + x^(2/5) + x^(1/5) + 1) + 12/5*x^4*log(x)*log(x^(4/
5) + x^(3/5) + x^(2/5) + x^(1/5) + 1) + x^4*log(x^(4/5) + x^(3/5) + x^(2/5) + x^(1/5) + 1)^2 - 2*x^4*log(-x^(1
6/5) + x^(11/5) + 1)*log(x^(1/5) - 1) + 12/5*x^4*log(x)*log(x^(1/5) - 1) + 2*x^4*log(x^(4/5) + x^(3/5) + x^(2/
5) + x^(1/5) + 1)*log(x^(1/5) - 1) + x^4*log(x^(1/5) - 1)^2)

Giac [F]

\[ \int \frac {e^{x^4 \log ^2\left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )} \left (\left (-10 x^5+20 x^6-10 x^7+\sqrt [5]{\frac {1}{x}} \left (12 x^3-22 x^4\right )\right ) \log \left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )+\left (-20 x^5+40 x^6-20 x^7+\sqrt [5]{\frac {1}{x}} \left (-20 x^3+20 x^4\right )\right ) \log ^2\left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )\right )}{-5 x^2+10 x^3-5 x^4+\sqrt [5]{\frac {1}{x}} (-5+5 x)} \, dx=\int { \frac {2 \, {\left (10 \, {\left (x^{7} - 2 \, x^{6} + x^{5} - \frac {x^{4} - x^{3}}{x^{\frac {1}{5}}}\right )} \log \left (-\frac {x^{3} - x^{2} - \frac {1}{x^{\frac {1}{5}}}}{x^{2} - x}\right )^{2} + {\left (5 \, x^{7} - 10 \, x^{6} + 5 \, x^{5} + \frac {11 \, x^{4} - 6 \, x^{3}}{x^{\frac {1}{5}}}\right )} \log \left (-\frac {x^{3} - x^{2} - \frac {1}{x^{\frac {1}{5}}}}{x^{2} - x}\right )\right )} e^{\left (x^{4} \log \left (-\frac {x^{3} - x^{2} - \frac {1}{x^{\frac {1}{5}}}}{x^{2} - x}\right )^{2}\right )}}{5 \, {\left (x^{4} - 2 \, x^{3} + x^{2} - \frac {x - 1}{x^{\frac {1}{5}}}\right )}} \,d x } \]

[In]

integrate((((20*x^4-20*x^3)*(1/x)^(1/5)-20*x^7+40*x^6-20*x^5)*log(((1/x)^(1/5)-x^3+x^2)/(x^2-x))^2+((-22*x^4+1
2*x^3)*(1/x)^(1/5)-10*x^7+20*x^6-10*x^5)*log(((1/x)^(1/5)-x^3+x^2)/(x^2-x)))*exp(x^4*log(((1/x)^(1/5)-x^3+x^2)
/(x^2-x))^2)/((5*x-5)*(1/x)^(1/5)-5*x^4+10*x^3-5*x^2),x, algorithm="giac")

[Out]

integrate(2/5*(10*(x^7 - 2*x^6 + x^5 - (x^4 - x^3)/x^(1/5))*log(-(x^3 - x^2 - 1/x^(1/5))/(x^2 - x))^2 + (5*x^7
- 10*x^6 + 5*x^5 + (11*x^4 - 6*x^3)/x^(1/5))*log(-(x^3 - x^2 - 1/x^(1/5))/(x^2 - x)))*e^(x^4*log(-(x^3 - x^2
- 1/x^(1/5))/(x^2 - x))^2)/(x^4 - 2*x^3 + x^2 - (x - 1)/x^(1/5)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{x^4 \log ^2\left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )} \left (\left (-10 x^5+20 x^6-10 x^7+\sqrt [5]{\frac {1}{x}} \left (12 x^3-22 x^4\right )\right ) \log \left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )+\left (-20 x^5+40 x^6-20 x^7+\sqrt [5]{\frac {1}{x}} \left (-20 x^3+20 x^4\right )\right ) \log ^2\left (\frac {\sqrt [5]{\frac {1}{x}}+x^2-x^3}{-x+x^2}\right )\right )}{-5 x^2+10 x^3-5 x^4+\sqrt [5]{\frac {1}{x}} (-5+5 x)} \, dx=\int -\frac {{\mathrm {e}}^{x^4\,{\ln \left (-\frac {{\left (\frac {1}{x}\right )}^{1/5}+x^2-x^3}{x-x^2}\right )}^2}\,\left ({\ln \left (-\frac {{\left (\frac {1}{x}\right )}^{1/5}+x^2-x^3}{x-x^2}\right )}^2\,\left (\left (20\,x^3-20\,x^4\right )\,{\left (\frac {1}{x}\right )}^{1/5}+20\,x^5-40\,x^6+20\,x^7\right )-\ln \left (-\frac {{\left (\frac {1}{x}\right )}^{1/5}+x^2-x^3}{x-x^2}\right )\,\left (\left (12\,x^3-22\,x^4\right )\,{\left (\frac {1}{x}\right )}^{1/5}-10\,x^5+20\,x^6-10\,x^7\right )\right )}{\left (5\,x-5\right )\,{\left (\frac {1}{x}\right )}^{1/5}-5\,x^2+10\,x^3-5\,x^4} \,d x \]

[In]

int(-(exp(x^4*log(-((1/x)^(1/5) + x^2 - x^3)/(x - x^2))^2)*(log(-((1/x)^(1/5) + x^2 - x^3)/(x - x^2))^2*((20*x
^3 - 20*x^4)*(1/x)^(1/5) + 20*x^5 - 40*x^6 + 20*x^7) - log(-((1/x)^(1/5) + x^2 - x^3)/(x - x^2))*((12*x^3 - 22
*x^4)*(1/x)^(1/5) - 10*x^5 + 20*x^6 - 10*x^7)))/((5*x - 5)*(1/x)^(1/5) - 5*x^2 + 10*x^3 - 5*x^4),x)

[Out]

int(-(exp(x^4*log(-((1/x)^(1/5) + x^2 - x^3)/(x - x^2))^2)*(log(-((1/x)^(1/5) + x^2 - x^3)/(x - x^2))^2*((20*x
^3 - 20*x^4)*(1/x)^(1/5) + 20*x^5 - 40*x^6 + 20*x^7) - log(-((1/x)^(1/5) + x^2 - x^3)/(x - x^2))*((12*x^3 - 22
*x^4)*(1/x)^(1/5) - 10*x^5 + 20*x^6 - 10*x^7)))/((5*x - 5)*(1/x)^(1/5) - 5*x^2 + 10*x^3 - 5*x^4), x)