\(\int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+(-20 e^x x^3+20 e^{2 x} x^3) \log (\frac {5}{\log (625)})+(-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4) \log ^2(\frac {5}{\log (625)})}{(-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3) \log ^2(\frac {5}{\log (625)})} \, dx\) [6658]

Optimal result

Integrand size = 135, antiderivative size = 29 \[ \int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )+\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{\left (-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3\right ) \log ^2\left (\frac {5}{\log (625)}\right )} \, dx=\left (-x+\frac {10}{\left (x-e^{-x} x\right ) \log \left (\frac {5}{\log (625)}\right )}\right )^2 \]

[Out]

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

Rubi [F]

\[ \int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )+\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{\left (-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3\right ) \log ^2\left (\frac {5}{\log (625)}\right )} \, dx=\int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )+\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{\left (-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3\right ) \log ^2\left (\frac {5}{\log (625)}\right )} \, dx \]

[In]

Int[(-200*E^(3*x) + E^(2*x)*(200 - 200*x) + (-20*E^x*x^3 + 20*E^(2*x)*x^3)*Log[5/Log[625]] + (-2*x^4 + 6*E^x*x
^4 - 6*E^(2*x)*x^4 + 2*E^(3*x)*x^4)*Log[5/Log[625]]^2)/((-x^3 + 3*E^x*x^3 - 3*E^(2*x)*x^3 + E^(3*x)*x^3)*Log[5
/Log[625]]^2),x]

[Out]

x^2 + 100/(x^2*Log[5/Log[625]]^2) + 20/((1 - E^x)*Log[5/Log[625]]) - (200*Defer[Int][1/((-1 + E^x)^2*x^3), x])
/Log[5/Log[625]]^2 - (400*Defer[Int][1/((-1 + E^x)*x^3), x])/Log[5/Log[625]]^2 - (200*Defer[Int][1/((-1 + E^x)
^3*x^2), x])/Log[5/Log[625]]^2 - (400*Defer[Int][1/((-1 + E^x)^2*x^2), x])/Log[5/Log[625]]^2 - (200*Defer[Int]
[1/((-1 + E^x)*x^2), x])/Log[5/Log[625]]^2

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )+\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )} \\ & = \frac {\int \frac {200 e^{3 x}-e^{2 x} (200-200 x)-\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )-\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{\left (1-e^x\right )^3 x^3} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )} \\ & = \frac {\int \frac {200 e^{3 x}+200 e^{2 x} (-1+x)-20 e^x \left (-1+e^x\right ) x^3 \log \left (\frac {5}{\log (625)}\right )-2 \left (-1+e^x\right )^3 x^4 \log ^2\left (\frac {5}{\log (625)}\right )}{\left (x-e^x x\right )^3} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )} \\ & = \frac {\int \left (-\frac {200}{\left (-1+e^x\right )^3 x^2}+\frac {20 \left (-10-20 x+x^3 \log \left (\frac {5}{\log (625)}\right )\right )}{\left (-1+e^x\right )^2 x^3}+\frac {20 \left (-20-10 x+x^3 \log \left (\frac {5}{\log (625)}\right )\right )}{\left (-1+e^x\right ) x^3}+\frac {2 \left (-100+x^4 \log ^2\left (\frac {5}{\log (625)}\right )\right )}{x^3}\right ) \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )} \\ & = \frac {2 \int \frac {-100+x^4 \log ^2\left (\frac {5}{\log (625)}\right )}{x^3} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}+\frac {20 \int \frac {-10-20 x+x^3 \log \left (\frac {5}{\log (625)}\right )}{\left (-1+e^x\right )^2 x^3} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}+\frac {20 \int \frac {-20-10 x+x^3 \log \left (\frac {5}{\log (625)}\right )}{\left (-1+e^x\right ) x^3} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}-\frac {200 \int \frac {1}{\left (-1+e^x\right )^3 x^2} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )} \\ & = \frac {2 \int \left (-\frac {100}{x^3}+x \log ^2\left (\frac {5}{\log (625)}\right )\right ) \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}+\frac {20 \int \left (-\frac {10}{\left (-1+e^x\right )^2 x^3}-\frac {20}{\left (-1+e^x\right )^2 x^2}+\frac {\log \left (\frac {5}{\log (625)}\right )}{\left (-1+e^x\right )^2}\right ) \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}+\frac {20 \int \left (-\frac {20}{\left (-1+e^x\right ) x^3}-\frac {10}{\left (-1+e^x\right ) x^2}+\frac {\log \left (\frac {5}{\log (625)}\right )}{-1+e^x}\right ) \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}-\frac {200 \int \frac {1}{\left (-1+e^x\right )^3 x^2} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )} \\ & = x^2+\frac {100}{x^2 \log ^2\left (\frac {5}{\log (625)}\right )}-\frac {200 \int \frac {1}{\left (-1+e^x\right )^2 x^3} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}-\frac {200 \int \frac {1}{\left (-1+e^x\right )^3 x^2} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}-\frac {200 \int \frac {1}{\left (-1+e^x\right ) x^2} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}-\frac {400 \int \frac {1}{\left (-1+e^x\right ) x^3} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}-\frac {400 \int \frac {1}{\left (-1+e^x\right )^2 x^2} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}+\frac {20 \int \frac {1}{\left (-1+e^x\right )^2} \, dx}{\log \left (\frac {5}{\log (625)}\right )}+\frac {20 \int \frac {1}{-1+e^x} \, dx}{\log \left (\frac {5}{\log (625)}\right )} \\ & = x^2+\frac {100}{x^2 \log ^2\left (\frac {5}{\log (625)}\right )}-\frac {200 \int \frac {1}{\left (-1+e^x\right )^2 x^3} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}-\frac {200 \int \frac {1}{\left (-1+e^x\right )^3 x^2} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}-\frac {200 \int \frac {1}{\left (-1+e^x\right ) x^2} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}-\frac {400 \int \frac {1}{\left (-1+e^x\right ) x^3} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}-\frac {400 \int \frac {1}{\left (-1+e^x\right )^2 x^2} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}+\frac {20 \text {Subst}\left (\int \frac {1}{(-1+x)^2 x} \, dx,x,e^x\right )}{\log \left (\frac {5}{\log (625)}\right )}+\frac {20 \text {Subst}\left (\int \frac {1}{(-1+x) x} \, dx,x,e^x\right )}{\log \left (\frac {5}{\log (625)}\right )} \\ & = x^2+\frac {100}{x^2 \log ^2\left (\frac {5}{\log (625)}\right )}-\frac {200 \int \frac {1}{\left (-1+e^x\right )^2 x^3} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}-\frac {200 \int \frac {1}{\left (-1+e^x\right )^3 x^2} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}-\frac {200 \int \frac {1}{\left (-1+e^x\right ) x^2} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}-\frac {400 \int \frac {1}{\left (-1+e^x\right ) x^3} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}-\frac {400 \int \frac {1}{\left (-1+e^x\right )^2 x^2} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}+\frac {20 \text {Subst}\left (\int \left (\frac {1}{1-x}+\frac {1}{(-1+x)^2}+\frac {1}{x}\right ) \, dx,x,e^x\right )}{\log \left (\frac {5}{\log (625)}\right )}+\frac {20 \text {Subst}\left (\int \frac {1}{-1+x} \, dx,x,e^x\right )}{\log \left (\frac {5}{\log (625)}\right )}-\frac {20 \text {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )}{\log \left (\frac {5}{\log (625)}\right )} \\ & = x^2+\frac {100}{x^2 \log ^2\left (\frac {5}{\log (625)}\right )}+\frac {20}{\left (1-e^x\right ) \log \left (\frac {5}{\log (625)}\right )}-\frac {200 \int \frac {1}{\left (-1+e^x\right )^2 x^3} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}-\frac {200 \int \frac {1}{\left (-1+e^x\right )^3 x^2} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}-\frac {200 \int \frac {1}{\left (-1+e^x\right ) x^2} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}-\frac {400 \int \frac {1}{\left (-1+e^x\right ) x^3} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )}-\frac {400 \int \frac {1}{\left (-1+e^x\right )^2 x^2} \, dx}{\log ^2\left (\frac {5}{\log (625)}\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.97 \[ \int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )+\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{\left (-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3\right ) \log ^2\left (\frac {5}{\log (625)}\right )} \, dx=\frac {\frac {100 e^{2 x}}{\left (-1+e^x\right )^2 x^2}+\log \left (\frac {5}{\log (625)}\right ) \left (-\frac {20}{-1+e^x}+x^2 \log \left (\frac {5}{\log (625)}\right )\right )}{\log ^2\left (\frac {5}{\log (625)}\right )} \]

[In]

Integrate[(-200*E^(3*x) + E^(2*x)*(200 - 200*x) + (-20*E^x*x^3 + 20*E^(2*x)*x^3)*Log[5/Log[625]] + (-2*x^4 + 6
*E^x*x^4 - 6*E^(2*x)*x^4 + 2*E^(3*x)*x^4)*Log[5/Log[625]]^2)/((-x^3 + 3*E^x*x^3 - 3*E^(2*x)*x^3 + E^(3*x)*x^3)
*Log[5/Log[625]]^2),x]

[Out]

((100*E^(2*x))/((-1 + E^x)^2*x^2) + Log[5/Log[625]]*(-20/(-1 + E^x) + x^2*Log[5/Log[625]]))/Log[5/Log[625]]^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(104\) vs. \(2(30)=60\).

Time = 6.61 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.62

method	result	size
parallelrisch	\(\frac {{\mathrm e}^{2 x} \ln \left (\frac {5}{4 \ln \left (5\right )}\right )^{2} x^{4}-2 \,{\mathrm e}^{x} \ln \left (\frac {5}{4 \ln \left (5\right )}\right )^{2} x^{4}+\ln \left (\frac {5}{4 \ln \left (5\right )}\right )^{2} x^{4}-20 \,{\mathrm e}^{x} x^{2} \ln \left (\frac {5}{4 \ln \left (5\right )}\right )+20 x^{2} \ln \left (\frac {5}{4 \ln \left (5\right )}\right )+100 \,{\mathrm e}^{2 x}}{\ln \left (\frac {5}{4 \ln \left (5\right )}\right )^{2} \left ({\mathrm e}^{2 x}-2 \,{\mathrm e}^{x}+1\right ) x^{2}}\)	\(105\)
norman	\(\frac {20 x^{2}+\left (\ln \left (5\right )-2 \ln \left (2\right )-\ln \left (\ln \left (5\right )\right )\right ) x^{4}-20 \,{\mathrm e}^{x} x^{2}+\left (\ln \left (5\right )-2 \ln \left (2\right )-\ln \left (\ln \left (5\right )\right )\right ) x^{4} {\mathrm e}^{2 x}+\left (2 \ln \left (\ln \left (5\right )\right )-2 \ln \left (5\right )+4 \ln \left (2\right )\right ) x^{4} {\mathrm e}^{x}-\frac {100 \,{\mathrm e}^{2 x}}{\ln \left (\ln \left (5\right )\right )-\ln \left (5\right )+2 \ln \left (2\right )}}{x^{2} \left ({\mathrm e}^{x}-1\right )^{2} \ln \left (\frac {5}{4 \ln \left (5\right )}\right )}\)	\(109\)
risch	\(\frac {x^{2} \ln \left (\ln \left (5\right )\right )^{2}}{\left (\ln \left (5\right )-2 \ln \left (2\right )-\ln \left (\ln \left (5\right )\right )\right )^{2}}-\frac {2 x^{2} \ln \left (5\right ) \ln \left (\ln \left (5\right )\right )}{\left (\ln \left (5\right )-2 \ln \left (2\right )-\ln \left (\ln \left (5\right )\right )\right )^{2}}+\frac {4 x^{2} \ln \left (2\right ) \ln \left (\ln \left (5\right )\right )}{\left (\ln \left (5\right )-2 \ln \left (2\right )-\ln \left (\ln \left (5\right )\right )\right )^{2}}+\frac {x^{2} \ln \left (5\right )^{2}}{\left (\ln \left (5\right )-2 \ln \left (2\right )-\ln \left (\ln \left (5\right )\right )\right )^{2}}-\frac {4 x^{2} \ln \left (2\right ) \ln \left (5\right )}{\left (\ln \left (5\right )-2 \ln \left (2\right )-\ln \left (\ln \left (5\right )\right )\right )^{2}}+\frac {4 x^{2} \ln \left (2\right )^{2}}{\left (\ln \left (5\right )-2 \ln \left (2\right )-\ln \left (\ln \left (5\right )\right )\right )^{2}}+\frac {100}{\left (\ln \left (5\right )-2 \ln \left (2\right )-\ln \left (\ln \left (5\right )\right )\right )^{2} x^{2}}+\frac {20 \ln \left (\ln \left (5\right )\right ) x^{2} {\mathrm e}^{x}-20 x^{2} \ln \left (5\right ) {\mathrm e}^{x}+40 x^{2} \ln \left (2\right ) {\mathrm e}^{x}-20 \ln \left (\ln \left (5\right )\right ) x^{2}+20 x^{2} \ln \left (5\right )-40 x^{2} \ln \left (2\right )+200 \,{\mathrm e}^{x}-100}{\left (\ln \left (5\right )-2 \ln \left (2\right )-\ln \left (\ln \left (5\right )\right )\right )^{2} x^{2} \left ({\mathrm e}^{x}-1\right )^{2}}\)	\(239\)

[In]

int(((2*x^4*exp(x)^3-6*exp(x)^2*x^4+6*exp(x)*x^4-2*x^4)*ln(5/4/ln(5))^2+(20*exp(x)^2*x^3-20*exp(x)*x^3)*ln(5/4
/ln(5))-200*exp(x)^3+(-200*x+200)*exp(x)^2)/(x^3*exp(x)^3-3*exp(x)^2*x^3+3*exp(x)*x^3-x^3)/ln(5/4/ln(5))^2,x,m
ethod=_RETURNVERBOSE)

[Out]

1/ln(5/4/ln(5))^2*(exp(x)^2*ln(5/4/ln(5))^2*x^4-2*exp(x)*ln(5/4/ln(5))^2*x^4+ln(5/4/ln(5))^2*x^4-20*exp(x)*x^2
*ln(5/4/ln(5))+20*x^2*ln(5/4/ln(5))+100*exp(x)^2)/(exp(x)^2-2*exp(x)+1)/x^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (27) = 54\).

Time = 0.43 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.03 \[ \int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )+\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{\left (-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3\right ) \log ^2\left (\frac {5}{\log (625)}\right )} \, dx=\frac {{\left (x^{4} e^{\left (2 \, x\right )} - 2 \, x^{4} e^{x} + x^{4}\right )} \log \left (\frac {5}{4 \, \log \left (5\right )}\right )^{2} - 20 \, {\left (x^{2} e^{x} - x^{2}\right )} \log \left (\frac {5}{4 \, \log \left (5\right )}\right ) + 100 \, e^{\left (2 \, x\right )}}{{\left (x^{2} e^{\left (2 \, x\right )} - 2 \, x^{2} e^{x} + x^{2}\right )} \log \left (\frac {5}{4 \, \log \left (5\right )}\right )^{2}} \]

[In]

integrate(((2*x^4*exp(x)^3-6*exp(x)^2*x^4+6*exp(x)*x^4-2*x^4)*log(5/4/log(5))^2+(20*exp(x)^2*x^3-20*exp(x)*x^3
)*log(5/4/log(5))-200*exp(x)^3+(-200*x+200)*exp(x)^2)/(x^3*exp(x)^3-3*exp(x)^2*x^3+3*exp(x)*x^3-x^3)/log(5/4/l
og(5))^2,x, algorithm="fricas")

[Out]

((x^4*e^(2*x) - 2*x^4*e^x + x^4)*log(5/4/log(5))^2 - 20*(x^2*e^x - x^2)*log(5/4/log(5)) + 100*e^(2*x))/((x^2*e
^(2*x) - 2*x^2*e^x + x^2)*log(5/4/log(5))^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (22) = 44\).

Time = 0.45 (sec) , antiderivative size = 386, normalized size of antiderivative = 13.31 \[ \int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )+\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{\left (-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3\right ) \log ^2\left (\frac {5}{\log (625)}\right )} \, dx=\frac {x^{2} \left (- 4 \log {\left (2 \right )} \log {\left (5 \right )} - 2 \log {\left (5 \right )} \log {\left (\log {\left (5 \right )} \right )} + \log {\left (\log {\left (5 \right )} \right )}^{2} + 4 \log {\left (2 \right )} \log {\left (\log {\left (5 \right )} \right )} + 4 \log {\left (2 \right )}^{2} + \log {\left (5 \right )}^{2}\right ) + \frac {100}{x^{2}}}{- 4 \log {\left (2 \right )} \log {\left (5 \right )} - 2 \log {\left (5 \right )} \log {\left (\log {\left (5 \right )} \right )} + \log {\left (\log {\left (5 \right )} \right )}^{2} + 4 \log {\left (2 \right )} \log {\left (\log {\left (5 \right )} \right )} + 4 \log {\left (2 \right )}^{2} + \log {\left (5 \right )}^{2}} + \frac {- 640 x^{2} \log {\left (2 \right )} - 320 x^{2} \log {\left (\log {\left (5 \right )} \right )} + 320 x^{2} \log {\left (5 \right )} + \left (- 320 x^{2} \log {\left (5 \right )} + 320 x^{2} \log {\left (\log {\left (5 \right )} \right )} + 640 x^{2} \log {\left (2 \right )} + 3200\right ) e^{x} - 1600}{- 64 x^{2} \log {\left (2 \right )} \log {\left (5 \right )} - 32 x^{2} \log {\left (5 \right )} \log {\left (\log {\left (5 \right )} \right )} + 16 x^{2} \log {\left (\log {\left (5 \right )} \right )}^{2} + 64 x^{2} \log {\left (2 \right )} \log {\left (\log {\left (5 \right )} \right )} + 64 x^{2} \log {\left (2 \right )}^{2} + 16 x^{2} \log {\left (5 \right )}^{2} + \left (- 32 x^{2} \log {\left (5 \right )}^{2} - 128 x^{2} \log {\left (2 \right )}^{2} - 128 x^{2} \log {\left (2 \right )} \log {\left (\log {\left (5 \right )} \right )} - 32 x^{2} \log {\left (\log {\left (5 \right )} \right )}^{2} + 64 x^{2} \log {\left (5 \right )} \log {\left (\log {\left (5 \right )} \right )} + 128 x^{2} \log {\left (2 \right )} \log {\left (5 \right )}\right ) e^{x} + \left (- 64 x^{2} \log {\left (2 \right )} \log {\left (5 \right )} - 32 x^{2} \log {\left (5 \right )} \log {\left (\log {\left (5 \right )} \right )} + 16 x^{2} \log {\left (\log {\left (5 \right )} \right )}^{2} + 64 x^{2} \log {\left (2 \right )} \log {\left (\log {\left (5 \right )} \right )} + 64 x^{2} \log {\left (2 \right )}^{2} + 16 x^{2} \log {\left (5 \right )}^{2}\right ) e^{2 x}} \]

[In]

integrate(((2*x**4*exp(x)**3-6*exp(x)**2*x**4+6*exp(x)*x**4-2*x**4)*ln(5/4/ln(5))**2+(20*exp(x)**2*x**3-20*exp
(x)*x**3)*ln(5/4/ln(5))-200*exp(x)**3+(-200*x+200)*exp(x)**2)/(x**3*exp(x)**3-3*exp(x)**2*x**3+3*exp(x)*x**3-x
**3)/ln(5/4/ln(5))**2,x)

[Out]

(x**2*(-4*log(2)*log(5) - 2*log(5)*log(log(5)) + log(log(5))**2 + 4*log(2)*log(log(5)) + 4*log(2)**2 + log(5)*
*2) + 100/x**2)/(-4*log(2)*log(5) - 2*log(5)*log(log(5)) + log(log(5))**2 + 4*log(2)*log(log(5)) + 4*log(2)**2
+ log(5)**2) + (-640*x**2*log(2) - 320*x**2*log(log(5)) + 320*x**2*log(5) + (-320*x**2*log(5) + 320*x**2*log(
log(5)) + 640*x**2*log(2) + 3200)*exp(x) - 1600)/(-64*x**2*log(2)*log(5) - 32*x**2*log(5)*log(log(5)) + 16*x**
2*log(log(5))**2 + 64*x**2*log(2)*log(log(5)) + 64*x**2*log(2)**2 + 16*x**2*log(5)**2 + (-32*x**2*log(5)**2 -
128*x**2*log(2)**2 - 128*x**2*log(2)*log(log(5)) - 32*x**2*log(log(5))**2 + 64*x**2*log(5)*log(log(5)) + 128*x
**2*log(2)*log(5))*exp(x) + (-64*x**2*log(2)*log(5) - 32*x**2*log(5)*log(log(5)) + 16*x**2*log(log(5))**2 + 64
*x**2*log(2)*log(log(5)) + 64*x**2*log(2)**2 + 16*x**2*log(5)**2)*exp(2*x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (27) = 54\).

Time = 0.34 (sec) , antiderivative size = 195, normalized size of antiderivative = 6.72 \[ \int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )+\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{\left (-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3\right ) \log ^2\left (\frac {5}{\log (625)}\right )} \, dx=\frac {{\left (\log \left (5\right )^{2} - 4 \, {\left (\log \left (5\right ) - \log \left (\log \left (5\right )\right )\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 2 \, \log \left (5\right ) \log \left (\log \left (5\right )\right ) + \log \left (\log \left (5\right )\right )^{2}\right )} x^{4} + 20 \, x^{2} {\left (\log \left (5\right ) - 2 \, \log \left (2\right ) - \log \left (\log \left (5\right )\right )\right )} + {\left ({\left (\log \left (5\right )^{2} - 4 \, {\left (\log \left (5\right ) - \log \left (\log \left (5\right )\right )\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 2 \, \log \left (5\right ) \log \left (\log \left (5\right )\right ) + \log \left (\log \left (5\right )\right )^{2}\right )} x^{4} + 100\right )} e^{\left (2 \, x\right )} - 2 \, {\left ({\left (\log \left (5\right )^{2} - 4 \, {\left (\log \left (5\right ) - \log \left (\log \left (5\right )\right )\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 2 \, \log \left (5\right ) \log \left (\log \left (5\right )\right ) + \log \left (\log \left (5\right )\right )^{2}\right )} x^{4} + 10 \, x^{2} {\left (\log \left (5\right ) - 2 \, \log \left (2\right ) - \log \left (\log \left (5\right )\right )\right )}\right )} e^{x}}{{\left (x^{2} e^{\left (2 \, x\right )} - 2 \, x^{2} e^{x} + x^{2}\right )} \log \left (\frac {5}{4 \, \log \left (5\right )}\right )^{2}} \]

[In]

integrate(((2*x^4*exp(x)^3-6*exp(x)^2*x^4+6*exp(x)*x^4-2*x^4)*log(5/4/log(5))^2+(20*exp(x)^2*x^3-20*exp(x)*x^3
)*log(5/4/log(5))-200*exp(x)^3+(-200*x+200)*exp(x)^2)/(x^3*exp(x)^3-3*exp(x)^2*x^3+3*exp(x)*x^3-x^3)/log(5/4/l
og(5))^2,x, algorithm="maxima")

[Out]

((log(5)^2 - 4*(log(5) - log(log(5)))*log(2) + 4*log(2)^2 - 2*log(5)*log(log(5)) + log(log(5))^2)*x^4 + 20*x^2
*(log(5) - 2*log(2) - log(log(5))) + ((log(5)^2 - 4*(log(5) - log(log(5)))*log(2) + 4*log(2)^2 - 2*log(5)*log(
log(5)) + log(log(5))^2)*x^4 + 100)*e^(2*x) - 2*((log(5)^2 - 4*(log(5) - log(log(5)))*log(2) + 4*log(2)^2 - 2*
log(5)*log(log(5)) + log(log(5))^2)*x^4 + 10*x^2*(log(5) - 2*log(2) - log(log(5))))*e^x)/((x^2*e^(2*x) - 2*x^2
*e^x + x^2)*log(5/4/log(5))^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (27) = 54\).

Time = 0.31 (sec) , antiderivative size = 189, normalized size of antiderivative = 6.52 \[ \int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )+\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{\left (-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3\right ) \log ^2\left (\frac {5}{\log (625)}\right )} \, dx=\frac {x^{4} e^{\left (2 \, x\right )} \log \left (5\right )^{2} - 2 \, x^{4} e^{x} \log \left (5\right )^{2} - 2 \, x^{4} e^{\left (2 \, x\right )} \log \left (5\right ) \log \left (4 \, \log \left (5\right )\right ) + 4 \, x^{4} e^{x} \log \left (5\right ) \log \left (4 \, \log \left (5\right )\right ) + x^{4} e^{\left (2 \, x\right )} \log \left (4 \, \log \left (5\right )\right )^{2} - 2 \, x^{4} e^{x} \log \left (4 \, \log \left (5\right )\right )^{2} + x^{4} \log \left (5\right )^{2} - 2 \, x^{4} \log \left (5\right ) \log \left (4 \, \log \left (5\right )\right ) + x^{4} \log \left (4 \, \log \left (5\right )\right )^{2} - 20 \, x^{2} e^{x} \log \left (5\right ) + 20 \, x^{2} e^{x} \log \left (4 \, \log \left (5\right )\right ) + 20 \, x^{2} \log \left (5\right ) - 20 \, x^{2} \log \left (4 \, \log \left (5\right )\right ) + 100 \, e^{\left (2 \, x\right )}}{{\left (x^{2} e^{\left (2 \, x\right )} - 2 \, x^{2} e^{x} + x^{2}\right )} \log \left (\frac {5}{4 \, \log \left (5\right )}\right )^{2}} \]

[In]

integrate(((2*x^4*exp(x)^3-6*exp(x)^2*x^4+6*exp(x)*x^4-2*x^4)*log(5/4/log(5))^2+(20*exp(x)^2*x^3-20*exp(x)*x^3
)*log(5/4/log(5))-200*exp(x)^3+(-200*x+200)*exp(x)^2)/(x^3*exp(x)^3-3*exp(x)^2*x^3+3*exp(x)*x^3-x^3)/log(5/4/l
og(5))^2,x, algorithm="giac")

[Out]

(x^4*e^(2*x)*log(5)^2 - 2*x^4*e^x*log(5)^2 - 2*x^4*e^(2*x)*log(5)*log(4*log(5)) + 4*x^4*e^x*log(5)*log(4*log(5
)) + x^4*e^(2*x)*log(4*log(5))^2 - 2*x^4*e^x*log(4*log(5))^2 + x^4*log(5)^2 - 2*x^4*log(5)*log(4*log(5)) + x^4
*log(4*log(5))^2 - 20*x^2*e^x*log(5) + 20*x^2*e^x*log(4*log(5)) + 20*x^2*log(5) - 20*x^2*log(4*log(5)) + 100*e
^(2*x))/((x^2*e^(2*x) - 2*x^2*e^x + x^2)*log(5/4/log(5))^2)

Mupad [B] (verification not implemented)

Time = 12.87 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.69 \[ \int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )+\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{\left (-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3\right ) \log ^2\left (\frac {5}{\log (625)}\right )} \, dx=\frac {100}{x^2\,{\ln \left (\frac {5}{4\,\ln \left (5\right )}\right )}^2}+x^2-\frac {x^3\,\ln \left (\frac {1099511627776\,{\ln \left (5\right )}^{20}}{95367431640625}\right )-200\,x+20\,x^3\,\ln \left (\frac {5}{4\,\ln \left (5\right )}\right )}{2\,x^3\,{\ln \left (\frac {5}{4\,\ln \left (5\right )}\right )}^2\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {20\,\left (10\,x-x^3\,\ln \left (\frac {5}{4\,\ln \left (5\right )}\right )\right )}{x^3\,{\ln \left (\frac {5}{4\,\ln \left (5\right )}\right )}^2\,\left ({\mathrm {e}}^x-1\right )} \]

[In]

int(-(200*exp(3*x) + exp(2*x)*(200*x - 200) + log(5/(4*log(5)))*(20*x^3*exp(x) - 20*x^3*exp(2*x)) - log(5/(4*l
og(5)))^2*(6*x^4*exp(x) - 6*x^4*exp(2*x) + 2*x^4*exp(3*x) - 2*x^4))/(log(5/(4*log(5)))^2*(3*x^3*exp(x) - 3*x^3
*exp(2*x) + x^3*exp(3*x) - x^3)),x)

[Out]

100/(x^2*log(5/(4*log(5)))^2) + x^2 - (x^3*log((1099511627776*log(5)^20)/95367431640625) - 200*x + 20*x^3*log(
5/(4*log(5))))/(2*x^3*log(5/(4*log(5)))^2*(exp(2*x) - 2*exp(x) + 1)) + (20*(10*x - x^3*log(5/(4*log(5)))))/(x^
3*log(5/(4*log(5)))^2*(exp(x) - 1))