\(\int \frac {(-100 x+50 e^5 x) \log (\frac {1}{3} (9+2 x-e^5 x))+(450+100 x-50 e^5 x) \log ^2(\frac {1}{3} (9+2 x-e^5 x))+e^{10 e^x} ((-4 x+2 e^5 x) \log (\frac {1}{3} (9+2 x-e^5 x))+(18+4 x-2 e^5 x+e^x (-90 x-20 x^2+10 e^5 x^2)) \log ^2(\frac {1}{3} (9+2 x-e^5 x)))+e^{5 e^x} ((-40 x+20 e^5 x) \log (\frac {1}{3} (9+2 x-e^5 x))+(180+40 x-20 e^5 x+e^x (-450 x-100 x^2+50 e^5 x^2)) \log ^2(\frac {1}{3} (9+2 x-e^5 x)))}{-9 x^3-2 x^4+e^5 x^4} \, dx\) [9359]

Optimal result

Integrand size = 253, antiderivative size = 32 \[ \int \frac {\left (-100 x+50 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (450+100 x-50 e^5 x\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+e^{10 e^x} \left (\left (-4 x+2 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (18+4 x-2 e^5 x+e^x \left (-90 x-20 x^2+10 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )+e^{5 e^x} \left (\left (-40 x+20 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (180+40 x-20 e^5 x+e^x \left (-450 x-100 x^2+50 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )}{-9 x^3-2 x^4+e^5 x^4} \, dx=\frac {\left (5+e^{5 e^x}\right )^2 \log ^2\left (3+\frac {1}{3} \left (2-e^5\right ) x\right )}{x^2} \]

[Out]

(exp(5*exp(x))+5)^2/x^2*ln(3+1/3*(2-exp(5))*x)^2

Rubi [F]

\[ \int \frac {\left (-100 x+50 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (450+100 x-50 e^5 x\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+e^{10 e^x} \left (\left (-4 x+2 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (18+4 x-2 e^5 x+e^x \left (-90 x-20 x^2+10 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )+e^{5 e^x} \left (\left (-40 x+20 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (180+40 x-20 e^5 x+e^x \left (-450 x-100 x^2+50 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )}{-9 x^3-2 x^4+e^5 x^4} \, dx=\int \frac {\left (-100 x+50 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (450+100 x-50 e^5 x\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+e^{10 e^x} \left (\left (-4 x+2 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (18+4 x-2 e^5 x+e^x \left (-90 x-20 x^2+10 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )+e^{5 e^x} \left (\left (-40 x+20 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (180+40 x-20 e^5 x+e^x \left (-450 x-100 x^2+50 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )}{-9 x^3-2 x^4+e^5 x^4} \, dx \]

[In]

Int[((-100*x + 50*E^5*x)*Log[(9 + 2*x - E^5*x)/3] + (450 + 100*x - 50*E^5*x)*Log[(9 + 2*x - E^5*x)/3]^2 + E^(1
0*E^x)*((-4*x + 2*E^5*x)*Log[(9 + 2*x - E^5*x)/3] + (18 + 4*x - 2*E^5*x + E^x*(-90*x - 20*x^2 + 10*E^5*x^2))*L
og[(9 + 2*x - E^5*x)/3]^2) + E^(5*E^x)*((-40*x + 20*E^5*x)*Log[(9 + 2*x - E^5*x)/3] + (180 + 40*x - 20*E^5*x +
E^x*(-450*x - 100*x^2 + 50*E^5*x^2))*Log[(9 + 2*x - E^5*x)/3]^2))/(-9*x^3 - 2*x^4 + E^5*x^4),x]

[Out]

(25*Log[3 + ((2 - E^5)*x)/3]^2)/x^2 + (20*(2 - E^5)*Log[3 + ((2 - E^5)*x)/3]*Defer[Int][E^(5*E^x)/x^2, x])/9 +
(2*(2 - E^5)*Log[3 + ((2 - E^5)*x)/3]*Defer[Int][E^(10*E^x)/x^2, x])/9 - (20*(2 - E^5)^2*Log[3 + ((2 - E^5)*x
)/3]*Defer[Int][E^(5*E^x)/x, x])/81 - (2*(2 - E^5)^2*Log[3 + ((2 - E^5)*x)/3]*Defer[Int][E^(10*E^x)/x, x])/81
+ (20*(2 - E^5)^3*Log[3 + ((2 - E^5)*x)/3]*Defer[Int][E^(5*E^x)/(9 + (2 - E^5)*x), x])/81 + (2*(2 - E^5)^3*Log
[3 + ((2 - E^5)*x)/3]*Defer[Int][E^(10*E^x)/(9 + (2 - E^5)*x), x])/81 - 20*Defer[Int][(E^(5*E^x)*Log[3 - ((-2
+ E^5)*x)/3]^2)/x^3, x] - 2*Defer[Int][(E^(10*E^x)*Log[3 - ((-2 + E^5)*x)/3]^2)/x^3, x] + 50*Defer[Int][(E^(5*
E^x + x)*Log[3 - ((-2 + E^5)*x)/3]^2)/x^2, x] + 10*Defer[Int][(E^(10*E^x + x)*Log[3 - ((-2 + E^5)*x)/3]^2)/x^2
, x] - (20*(2 - E^5)^2*Defer[Int][Defer[Int][E^(5*E^x)/x^2, x]/(9 + (2 - E^5)*x), x])/9 - (2*(2 - E^5)^2*Defer
[Int][Defer[Int][E^(10*E^x)/x^2, x]/(9 + (2 - E^5)*x), x])/9 + (20*(2 - E^5)^3*Defer[Int][Defer[Int][E^(5*E^x)
/x, x]/(9 + (2 - E^5)*x), x])/81 + (2*(2 - E^5)^3*Defer[Int][Defer[Int][E^(10*E^x)/x, x]/(9 + (2 - E^5)*x), x]
)/81 - (2*(2 - E^5)^4*Defer[Int][Defer[Int][E^(10*E^x)/(9 + (2 - E^5)*x), x]/(9 + (2 - E^5)*x), x])/81 - (20*(
2 - E^5)^4*Defer[Int][Defer[Int][E^(5*E^x)/(9 - (-2 + E^5)*x), x]/(9 + (2 - E^5)*x), x])/81

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (-100 x+50 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (450+100 x-50 e^5 x\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+e^{10 e^x} \left (\left (-4 x+2 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (18+4 x-2 e^5 x+e^x \left (-90 x-20 x^2+10 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )+e^{5 e^x} \left (\left (-40 x+20 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (180+40 x-20 e^5 x+e^x \left (-450 x-100 x^2+50 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )}{-9 x^3+\left (-2+e^5\right ) x^4} \, dx \\ & = \int \frac {\left (-100 x+50 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (450+100 x-50 e^5 x\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+e^{10 e^x} \left (\left (-4 x+2 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (18+4 x-2 e^5 x+e^x \left (-90 x-20 x^2+10 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )+e^{5 e^x} \left (\left (-40 x+20 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (180+40 x-20 e^5 x+e^x \left (-450 x-100 x^2+50 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )}{x^3 \left (-9+\left (-2+e^5\right ) x\right )} \, dx \\ & = \int \frac {2 \left (5+e^{5 e^x}\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right ) \left (-\left (\left (-2+e^5\right ) \left (5+e^{5 e^x}\right ) x\right )-\left (-5-e^{5 e^x}+5 e^{5 e^x+x} x\right ) \left (-9+\left (-2+e^5\right ) x\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right )}{x^3 \left (9-\left (-2+e^5\right ) x\right )} \, dx \\ & = 2 \int \frac {\left (5+e^{5 e^x}\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right ) \left (-\left (\left (-2+e^5\right ) \left (5+e^{5 e^x}\right ) x\right )-\left (-5-e^{5 e^x}+5 e^{5 e^x+x} x\right ) \left (-9+\left (-2+e^5\right ) x\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right )}{x^3 \left (9-\left (-2+e^5\right ) x\right )} \, dx \\ & = 2 \int \left (\frac {5 e^{5 e^x+x} \left (5+e^{5 e^x}\right ) \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2}+\frac {\left (5+e^{5 e^x}\right )^2 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right ) \left (2 \left (1-\frac {e^5}{2}\right ) x-9 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )-2 \left (1-\frac {e^5}{2}\right ) x \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right )}{x^3 \left (9+\left (2-e^5\right ) x\right )}\right ) \, dx \\ & = 2 \int \frac {\left (5+e^{5 e^x}\right )^2 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right ) \left (2 \left (1-\frac {e^5}{2}\right ) x-9 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )-2 \left (1-\frac {e^5}{2}\right ) x \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right )}{x^3 \left (9+\left (2-e^5\right ) x\right )} \, dx+10 \int \frac {e^{5 e^x+x} \left (5+e^{5 e^x}\right ) \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx \\ & = 2 \int \frac {\left (5+e^{5 e^x}\right )^2 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right ) \left (-\left (\left (-2+e^5\right ) x\right )+\left (-9+\left (-2+e^5\right ) x\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right )}{x^3 \left (9-\left (-2+e^5\right ) x\right )} \, dx+10 \int \left (\frac {5 e^{5 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2}+\frac {e^{10 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2}\right ) \, dx \\ & = 2 \int \left (\frac {25 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right ) \left (2 \left (1-\frac {e^5}{2}\right ) x-9 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )-2 \left (1-\frac {e^5}{2}\right ) x \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right )}{x^3 \left (9+\left (2-e^5\right ) x\right )}+\frac {10 e^{5 e^x} \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right ) \left (2 \left (1-\frac {e^5}{2}\right ) x-9 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )-2 \left (1-\frac {e^5}{2}\right ) x \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right )}{x^3 \left (9+\left (2-e^5\right ) x\right )}+\frac {e^{10 e^x} \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right ) \left (2 \left (1-\frac {e^5}{2}\right ) x-9 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )-2 \left (1-\frac {e^5}{2}\right ) x \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right )}{x^3 \left (9+\left (2-e^5\right ) x\right )}\right ) \, dx+10 \int \frac {e^{10 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx+50 \int \frac {e^{5 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx \\ & = 2 \int \frac {e^{10 e^x} \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right ) \left (2 \left (1-\frac {e^5}{2}\right ) x-9 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )-2 \left (1-\frac {e^5}{2}\right ) x \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right )}{x^3 \left (9+\left (2-e^5\right ) x\right )} \, dx+10 \int \frac {e^{10 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx+20 \int \frac {e^{5 e^x} \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right ) \left (2 \left (1-\frac {e^5}{2}\right ) x-9 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )-2 \left (1-\frac {e^5}{2}\right ) x \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right )}{x^3 \left (9+\left (2-e^5\right ) x\right )} \, dx+50 \int \frac {e^{5 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx+50 \int \frac {\log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right ) \left (2 \left (1-\frac {e^5}{2}\right ) x-9 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )-2 \left (1-\frac {e^5}{2}\right ) x \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right )}{x^3 \left (9+\left (2-e^5\right ) x\right )} \, dx \\ & = 2 \int \frac {e^{10 e^x} \left (\frac {\left (-2+e^5\right ) x}{-9+\left (-2+e^5\right ) x}-\log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3} \, dx+10 \int \frac {e^{10 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx+20 \int \frac {e^{5 e^x} \left (\frac {\left (-2+e^5\right ) x}{-9+\left (-2+e^5\right ) x}-\log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3} \, dx+50 \int \frac {e^{5 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx+50 \int \frac {\log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right ) \left (-\left (\left (-2+e^5\right ) x\right )-\left (9-\left (-2+e^5\right ) x\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right )}{x^3 \left (9-\left (-2+e^5\right ) x\right )} \, dx \\ & = 2 \int \left (\frac {e^{10 e^x} \left (2-e^5\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2 \left (9+\left (2-e^5\right ) x\right )}-\frac {e^{10 e^x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3}\right ) \, dx+10 \int \frac {e^{10 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx+20 \int \left (\frac {e^{5 e^x} \left (2-e^5\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2 \left (9+\left (2-e^5\right ) x\right )}-\frac {e^{5 e^x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3}\right ) \, dx+50 \int \frac {e^{5 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx+50 \int \left (\frac {\left (2-e^5\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2 \left (9+\left (2-e^5\right ) x\right )}-\frac {\log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3}\right ) \, dx \\ & = -\left (2 \int \frac {e^{10 e^x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3} \, dx\right )+10 \int \frac {e^{10 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx-20 \int \frac {e^{5 e^x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3} \, dx-50 \int \frac {\log ^2\left (3+\frac {1}{3} \left (2-e^5\right ) x\right )}{x^3} \, dx+50 \int \frac {e^{5 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx+\left (2 \left (2-e^5\right )\right ) \int \frac {e^{10 e^x} \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2 \left (9+\left (2-e^5\right ) x\right )} \, dx+\left (20 \left (2-e^5\right )\right ) \int \frac {e^{5 e^x} \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2 \left (9+\left (2-e^5\right ) x\right )} \, dx+\left (50 \left (2-e^5\right )\right ) \int \frac {\log \left (3+\frac {1}{3} \left (2-e^5\right ) x\right )}{x^2 \left (9+\left (2-e^5\right ) x\right )} \, dx \\ & = \frac {25 \log ^2\left (3+\frac {1}{3} \left (2-e^5\right ) x\right )}{x^2}-2 \int \frac {e^{10 e^x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3} \, dx+10 \int \frac {e^{10 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx-20 \int \frac {e^{5 e^x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3} \, dx+50 \int \frac {e^{5 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx+150 \text {Subst}\left (\int \frac {\log (x)}{3 x \left (-\frac {9}{2-e^5}+\frac {3 x}{2-e^5}\right )^2} \, dx,x,3+\frac {1}{3} \left (2-e^5\right ) x\right )-\left (2 \left (2-e^5\right )\right ) \int \frac {\left (2-e^5\right ) \left (9 \int \frac {e^{10 e^x}}{x^2} \, dx+\left (-2+e^5\right ) \left (\int \frac {e^{10 e^x}}{x} \, dx+\left (-2+e^5\right ) \int \frac {e^{10 e^x}}{9+2 x-e^5 x} \, dx\right )\right )}{81 \left (9-\left (-2+e^5\right ) x\right )} \, dx-\frac {1}{3} \left (50 \left (2-e^5\right )\right ) \int \frac {\log \left (3+\frac {1}{3} \left (2-e^5\right ) x\right )}{x^2 \left (3+\frac {1}{3} \left (2-e^5\right ) x\right )} \, dx-\left (20 \left (2-e^5\right )\right ) \int \frac {\left (2-e^5\right ) \left (9 \int \frac {e^{5 e^x}}{x^2} \, dx+\left (-2+e^5\right ) \left (\int \frac {e^{5 e^x}}{x} \, dx+\left (-2+e^5\right ) \int \frac {e^{5 e^x}}{9+2 x-e^5 x} \, dx\right )\right )}{81 \left (9-\left (-2+e^5\right ) x\right )} \, dx+\frac {1}{9} \left (2 \left (2-e^5\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{10 e^x}}{x^2} \, dx+\frac {1}{9} \left (20 \left (2-e^5\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{5 e^x}}{x^2} \, dx-\frac {1}{81} \left (2 \left (2-e^5\right )^2 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{10 e^x}}{x} \, dx-\frac {1}{81} \left (20 \left (2-e^5\right )^2 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{5 e^x}}{x} \, dx+\frac {1}{81} \left (2 \left (2-e^5\right )^3 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{10 e^x}}{9+\left (2-e^5\right ) x} \, dx+\frac {1}{81} \left (20 \left (2-e^5\right )^3 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{5 e^x}}{9+\left (2-e^5\right ) x} \, dx \\ & = \frac {25 \log ^2\left (3+\frac {1}{3} \left (2-e^5\right ) x\right )}{x^2}-2 \int \frac {e^{10 e^x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3} \, dx+10 \int \frac {e^{10 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx-20 \int \frac {e^{5 e^x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3} \, dx+50 \int \frac {e^{5 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx-\frac {1}{81} \left (2 \left (2-e^5\right )^2\right ) \int \frac {9 \int \frac {e^{10 e^x}}{x^2} \, dx+\left (-2+e^5\right ) \left (\int \frac {e^{10 e^x}}{x} \, dx+\left (-2+e^5\right ) \int \frac {e^{10 e^x}}{9+\left (2-e^5\right ) x} \, dx\right )}{9-\left (-2+e^5\right ) x} \, dx-\frac {1}{81} \left (20 \left (2-e^5\right )^2\right ) \int \frac {9 \int \frac {e^{5 e^x}}{x^2} \, dx+\left (-2+e^5\right ) \left (\int \frac {e^{5 e^x}}{x} \, dx+\left (-2+e^5\right ) \int \frac {e^{5 e^x}}{9+\left (2-e^5\right ) x} \, dx\right )}{9-\left (-2+e^5\right ) x} \, dx+\frac {1}{9} \left (2 \left (2-e^5\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{10 e^x}}{x^2} \, dx+\frac {1}{9} \left (20 \left (2-e^5\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{5 e^x}}{x^2} \, dx-\frac {1}{81} \left (2 \left (2-e^5\right )^2 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{10 e^x}}{x} \, dx-\frac {1}{81} \left (20 \left (2-e^5\right )^2 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{5 e^x}}{x} \, dx+\frac {1}{81} \left (2 \left (2-e^5\right )^3 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{10 e^x}}{9+\left (2-e^5\right ) x} \, dx+\frac {1}{81} \left (20 \left (2-e^5\right )^3 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{5 e^x}}{9+\left (2-e^5\right ) x} \, dx \\ & = \frac {25 \log ^2\left (3+\frac {1}{3} \left (2-e^5\right ) x\right )}{x^2}-2 \int \frac {e^{10 e^x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3} \, dx+10 \int \frac {e^{10 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx-20 \int \frac {e^{5 e^x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3} \, dx+50 \int \frac {e^{5 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx-\frac {1}{81} \left (2 \left (2-e^5\right )^2\right ) \int \left (\frac {9 \int \frac {e^{10 e^x}}{x^2} \, dx-2 \left (1-\frac {e^5}{2}\right ) \int \frac {e^{10 e^x}}{x} \, dx}{9+\left (2-e^5\right ) x}+\frac {\left (2-e^5\right )^2 \int \frac {e^{10 e^x}}{9+\left (2-e^5\right ) x} \, dx}{9+\left (2-e^5\right ) x}\right ) \, dx-\frac {1}{81} \left (20 \left (2-e^5\right )^2\right ) \int \left (\frac {9 \int \frac {e^{5 e^x}}{x^2} \, dx-2 \left (1-\frac {e^5}{2}\right ) \int \frac {e^{5 e^x}}{x} \, dx}{9+\left (2-e^5\right ) x}+\frac {\left (2-e^5\right )^2 \int \frac {e^{5 e^x}}{9+\left (2-e^5\right ) x} \, dx}{9+\left (2-e^5\right ) x}\right ) \, dx+\frac {1}{9} \left (2 \left (2-e^5\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{10 e^x}}{x^2} \, dx+\frac {1}{9} \left (20 \left (2-e^5\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{5 e^x}}{x^2} \, dx-\frac {1}{81} \left (2 \left (2-e^5\right )^2 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{10 e^x}}{x} \, dx-\frac {1}{81} \left (20 \left (2-e^5\right )^2 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{5 e^x}}{x} \, dx+\frac {1}{81} \left (2 \left (2-e^5\right )^3 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{10 e^x}}{9+\left (2-e^5\right ) x} \, dx+\frac {1}{81} \left (20 \left (2-e^5\right )^3 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{5 e^x}}{9+\left (2-e^5\right ) x} \, dx \\ & = \frac {25 \log ^2\left (3+\frac {1}{3} \left (2-e^5\right ) x\right )}{x^2}-2 \int \frac {e^{10 e^x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3} \, dx+10 \int \frac {e^{10 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx-20 \int \frac {e^{5 e^x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3} \, dx+50 \int \frac {e^{5 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx-\frac {1}{81} \left (2 \left (2-e^5\right )^2\right ) \int \frac {9 \int \frac {e^{10 e^x}}{x^2} \, dx-2 \left (1-\frac {e^5}{2}\right ) \int \frac {e^{10 e^x}}{x} \, dx}{9+\left (2-e^5\right ) x} \, dx-\frac {1}{81} \left (20 \left (2-e^5\right )^2\right ) \int \frac {9 \int \frac {e^{5 e^x}}{x^2} \, dx-2 \left (1-\frac {e^5}{2}\right ) \int \frac {e^{5 e^x}}{x} \, dx}{9+\left (2-e^5\right ) x} \, dx-\frac {1}{81} \left (2 \left (2-e^5\right )^4\right ) \int \frac {\int \frac {e^{10 e^x}}{9+\left (2-e^5\right ) x} \, dx}{9+\left (2-e^5\right ) x} \, dx-\frac {1}{81} \left (20 \left (2-e^5\right )^4\right ) \int \frac {\int \frac {e^{5 e^x}}{9+\left (2-e^5\right ) x} \, dx}{9+\left (2-e^5\right ) x} \, dx+\frac {1}{9} \left (2 \left (2-e^5\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{10 e^x}}{x^2} \, dx+\frac {1}{9} \left (20 \left (2-e^5\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{5 e^x}}{x^2} \, dx-\frac {1}{81} \left (2 \left (2-e^5\right )^2 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{10 e^x}}{x} \, dx-\frac {1}{81} \left (20 \left (2-e^5\right )^2 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{5 e^x}}{x} \, dx+\frac {1}{81} \left (2 \left (2-e^5\right )^3 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{10 e^x}}{9+\left (2-e^5\right ) x} \, dx+\frac {1}{81} \left (20 \left (2-e^5\right )^3 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{5 e^x}}{9+\left (2-e^5\right ) x} \, dx \\ & = \frac {25 \log ^2\left (3+\frac {1}{3} \left (2-e^5\right ) x\right )}{x^2}-2 \int \frac {e^{10 e^x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3} \, dx+10 \int \frac {e^{10 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx-20 \int \frac {e^{5 e^x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3} \, dx+50 \int \frac {e^{5 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx-\frac {1}{81} \left (2 \left (2-e^5\right )^2\right ) \int \left (\frac {9 \int \frac {e^{10 e^x}}{x^2} \, dx}{9+\left (2-e^5\right ) x}+\frac {\left (-2+e^5\right ) \int \frac {e^{10 e^x}}{x} \, dx}{9+\left (2-e^5\right ) x}\right ) \, dx-\frac {1}{81} \left (20 \left (2-e^5\right )^2\right ) \int \left (\frac {9 \int \frac {e^{5 e^x}}{x^2} \, dx}{9+\left (2-e^5\right ) x}+\frac {\left (-2+e^5\right ) \int \frac {e^{5 e^x}}{x} \, dx}{9+\left (2-e^5\right ) x}\right ) \, dx-\frac {1}{81} \left (2 \left (2-e^5\right )^4\right ) \int \frac {\int \frac {e^{10 e^x}}{9+\left (2-e^5\right ) x} \, dx}{9+\left (2-e^5\right ) x} \, dx-\frac {1}{81} \left (20 \left (2-e^5\right )^4\right ) \int \frac {\int \frac {e^{5 e^x}}{9+\left (2-e^5\right ) x} \, dx}{9+\left (2-e^5\right ) x} \, dx+\frac {1}{9} \left (2 \left (2-e^5\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{10 e^x}}{x^2} \, dx+\frac {1}{9} \left (20 \left (2-e^5\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{5 e^x}}{x^2} \, dx-\frac {1}{81} \left (2 \left (2-e^5\right )^2 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{10 e^x}}{x} \, dx-\frac {1}{81} \left (20 \left (2-e^5\right )^2 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{5 e^x}}{x} \, dx+\frac {1}{81} \left (2 \left (2-e^5\right )^3 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{10 e^x}}{9+\left (2-e^5\right ) x} \, dx+\frac {1}{81} \left (20 \left (2-e^5\right )^3 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{5 e^x}}{9+\left (2-e^5\right ) x} \, dx \\ & = \frac {25 \log ^2\left (3+\frac {1}{3} \left (2-e^5\right ) x\right )}{x^2}-2 \int \frac {e^{10 e^x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3} \, dx+10 \int \frac {e^{10 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx-20 \int \frac {e^{5 e^x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^3} \, dx+50 \int \frac {e^{5 e^x+x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2} \, dx-\frac {1}{9} \left (2 \left (2-e^5\right )^2\right ) \int \frac {\int \frac {e^{10 e^x}}{x^2} \, dx}{9+\left (2-e^5\right ) x} \, dx-\frac {1}{9} \left (20 \left (2-e^5\right )^2\right ) \int \frac {\int \frac {e^{5 e^x}}{x^2} \, dx}{9+\left (2-e^5\right ) x} \, dx+\frac {1}{81} \left (2 \left (2-e^5\right )^3\right ) \int \frac {\int \frac {e^{10 e^x}}{x} \, dx}{9+\left (2-e^5\right ) x} \, dx+\frac {1}{81} \left (20 \left (2-e^5\right )^3\right ) \int \frac {\int \frac {e^{5 e^x}}{x} \, dx}{9+\left (2-e^5\right ) x} \, dx-\frac {1}{81} \left (2 \left (2-e^5\right )^4\right ) \int \frac {\int \frac {e^{10 e^x}}{9+\left (2-e^5\right ) x} \, dx}{9+\left (2-e^5\right ) x} \, dx-\frac {1}{81} \left (20 \left (2-e^5\right )^4\right ) \int \frac {\int \frac {e^{5 e^x}}{9+\left (2-e^5\right ) x} \, dx}{9+\left (2-e^5\right ) x} \, dx+\frac {1}{9} \left (2 \left (2-e^5\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{10 e^x}}{x^2} \, dx+\frac {1}{9} \left (20 \left (2-e^5\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{5 e^x}}{x^2} \, dx-\frac {1}{81} \left (2 \left (2-e^5\right )^2 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{10 e^x}}{x} \, dx-\frac {1}{81} \left (20 \left (2-e^5\right )^2 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{5 e^x}}{x} \, dx+\frac {1}{81} \left (2 \left (2-e^5\right )^3 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{10 e^x}}{9+\left (2-e^5\right ) x} \, dx+\frac {1}{81} \left (20 \left (2-e^5\right )^3 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right ) \int \frac {e^{5 e^x}}{9+\left (2-e^5\right ) x} \, dx \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.63 (sec) , antiderivative size = 277, normalized size of antiderivative = 8.66 \[ \int \frac {\left (-100 x+50 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (450+100 x-50 e^5 x\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+e^{10 e^x} \left (\left (-4 x+2 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (18+4 x-2 e^5 x+e^x \left (-90 x-20 x^2+10 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )+e^{5 e^x} \left (\left (-40 x+20 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (180+40 x-20 e^5 x+e^x \left (-450 x-100 x^2+50 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )}{-9 x^3-2 x^4+e^5 x^4} \, dx=\frac {1}{81} \left (\frac {810 e^{5 e^x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2}+\frac {81 e^{10 e^x} \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x^2}+25 \left (-2+e^5\right ) \left (2 \left (-2+e^5\right ) \left (\log (x)-\log \left (9+2 x-e^5 x\right )\right )+\frac {18 \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )}{x}+\left (-2+e^5\right ) \log ^2\left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )-2 \left (-2+e^5\right ) \left (\log (3) \log (x)-\operatorname {PolyLog}\left (2,\frac {1}{9} \left (-2+e^5\right ) x\right )\right )\right )+25 \left (2 \left (-2+e^5\right )^2 \log \left (\frac {1}{9} \left (-2+e^5\right ) x\right ) \left (-1+\log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right )-\frac {\left (-9+\left (-2+e^5\right ) x\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right ) \left (-2 \left (-2+e^5\right ) x+\left (9+\left (-2+e^5\right ) x\right ) \log \left (3-\frac {1}{3} \left (-2+e^5\right ) x\right )\right )}{x^2}+2 \left (-2+e^5\right )^2 \operatorname {PolyLog}\left (2,1-\frac {1}{9} \left (-2+e^5\right ) x\right )\right )\right ) \]

[In]

Integrate[((-100*x + 50*E^5*x)*Log[(9 + 2*x - E^5*x)/3] + (450 + 100*x - 50*E^5*x)*Log[(9 + 2*x - E^5*x)/3]^2
+ E^(10*E^x)*((-4*x + 2*E^5*x)*Log[(9 + 2*x - E^5*x)/3] + (18 + 4*x - 2*E^5*x + E^x*(-90*x - 20*x^2 + 10*E^5*x
^2))*Log[(9 + 2*x - E^5*x)/3]^2) + E^(5*E^x)*((-40*x + 20*E^5*x)*Log[(9 + 2*x - E^5*x)/3] + (180 + 40*x - 20*E
^5*x + E^x*(-450*x - 100*x^2 + 50*E^5*x^2))*Log[(9 + 2*x - E^5*x)/3]^2))/(-9*x^3 - 2*x^4 + E^5*x^4),x]

[Out]

((810*E^(5*E^x)*Log[3 - ((-2 + E^5)*x)/3]^2)/x^2 + (81*E^(10*E^x)*Log[3 - ((-2 + E^5)*x)/3]^2)/x^2 + 25*(-2 +
E^5)*(2*(-2 + E^5)*(Log[x] - Log[9 + 2*x - E^5*x]) + (18*Log[3 - ((-2 + E^5)*x)/3])/x + (-2 + E^5)*Log[3 - ((-
2 + E^5)*x)/3]^2 - 2*(-2 + E^5)*(Log[3]*Log[x] - PolyLog[2, ((-2 + E^5)*x)/9])) + 25*(2*(-2 + E^5)^2*Log[((-2
+ E^5)*x)/9]*(-1 + Log[3 - ((-2 + E^5)*x)/3]) - ((-9 + (-2 + E^5)*x)*Log[3 - ((-2 + E^5)*x)/3]*(-2*(-2 + E^5)*
x + (9 + (-2 + E^5)*x)*Log[3 - ((-2 + E^5)*x)/3]))/x^2 + 2*(-2 + E^5)^2*PolyLog[2, 1 - ((-2 + E^5)*x)/9]))/81

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs. \(2(27)=54\).

Time = 10.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.00

method	result	size
parallelrisch	\(-\frac {-81 \ln \left (-\frac {x \,{\mathrm e}^{5}}{3}+\frac {2 x}{3}+3\right )^{2} {\mathrm e}^{10 \,{\mathrm e}^{x}}-810 \ln \left (-\frac {x \,{\mathrm e}^{5}}{3}+\frac {2 x}{3}+3\right )^{2} {\mathrm e}^{5 \,{\mathrm e}^{x}}-2025 \ln \left (-\frac {x \,{\mathrm e}^{5}}{3}+\frac {2 x}{3}+3\right )^{2}}{81 x^{2}}\)	\(64\)
risch	\(\frac {25 \ln \left (-\frac {x \,{\mathrm e}^{5}}{3}+\frac {2 x}{3}+3\right )^{2}}{x^{2}}+\frac {\ln \left (-\frac {x \,{\mathrm e}^{5}}{3}+\frac {2 x}{3}+3\right )^{2} {\mathrm e}^{10 \,{\mathrm e}^{x}}}{x^{2}}+\frac {10 \ln \left (-\frac {x \,{\mathrm e}^{5}}{3}+\frac {2 x}{3}+3\right )^{2} {\mathrm e}^{5 \,{\mathrm e}^{x}}}{x^{2}}\)	\(67\)

[In]

int(((((10*x^2*exp(5)-20*x^2-90*x)*exp(x)-2*x*exp(5)+4*x+18)*ln(-1/3*x*exp(5)+2/3*x+3)^2+(2*x*exp(5)-4*x)*ln(-
1/3*x*exp(5)+2/3*x+3))*exp(5*exp(x))^2+(((50*x^2*exp(5)-100*x^2-450*x)*exp(x)-20*x*exp(5)+40*x+180)*ln(-1/3*x*
exp(5)+2/3*x+3)^2+(20*x*exp(5)-40*x)*ln(-1/3*x*exp(5)+2/3*x+3))*exp(5*exp(x))+(-50*x*exp(5)+100*x+450)*ln(-1/3
*x*exp(5)+2/3*x+3)^2+(50*x*exp(5)-100*x)*ln(-1/3*x*exp(5)+2/3*x+3))/(x^4*exp(5)-2*x^4-9*x^3),x,method=_RETURNV
ERBOSE)

[Out]

-1/81*(-81*ln(-1/3*x*exp(5)+2/3*x+3)^2*exp(5*exp(x))^2-810*ln(-1/3*x*exp(5)+2/3*x+3)^2*exp(5*exp(x))-2025*ln(-
1/3*x*exp(5)+2/3*x+3)^2)/x^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (25) = 50\).

Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.84 \[ \int \frac {\left (-100 x+50 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (450+100 x-50 e^5 x\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+e^{10 e^x} \left (\left (-4 x+2 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (18+4 x-2 e^5 x+e^x \left (-90 x-20 x^2+10 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )+e^{5 e^x} \left (\left (-40 x+20 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (180+40 x-20 e^5 x+e^x \left (-450 x-100 x^2+50 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )}{-9 x^3-2 x^4+e^5 x^4} \, dx=\frac {e^{\left (10 \, e^{x}\right )} \log \left (-\frac {1}{3} \, x e^{5} + \frac {2}{3} \, x + 3\right )^{2} + 10 \, e^{\left (5 \, e^{x}\right )} \log \left (-\frac {1}{3} \, x e^{5} + \frac {2}{3} \, x + 3\right )^{2} + 25 \, \log \left (-\frac {1}{3} \, x e^{5} + \frac {2}{3} \, x + 3\right )^{2}}{x^{2}} \]

[In]

integrate(((((10*x^2*exp(5)-20*x^2-90*x)*exp(x)-2*x*exp(5)+4*x+18)*log(-1/3*x*exp(5)+2/3*x+3)^2+(2*x*exp(5)-4*
x)*log(-1/3*x*exp(5)+2/3*x+3))*exp(5*exp(x))^2+(((50*x^2*exp(5)-100*x^2-450*x)*exp(x)-20*x*exp(5)+40*x+180)*lo
g(-1/3*x*exp(5)+2/3*x+3)^2+(20*x*exp(5)-40*x)*log(-1/3*x*exp(5)+2/3*x+3))*exp(5*exp(x))+(-50*x*exp(5)+100*x+45
0)*log(-1/3*x*exp(5)+2/3*x+3)^2+(50*x*exp(5)-100*x)*log(-1/3*x*exp(5)+2/3*x+3))/(x^4*exp(5)-2*x^4-9*x^3),x, al
gorithm="fricas")

[Out]

(e^(10*e^x)*log(-1/3*x*e^5 + 2/3*x + 3)^2 + 10*e^(5*e^x)*log(-1/3*x*e^5 + 2/3*x + 3)^2 + 25*log(-1/3*x*e^5 + 2
/3*x + 3)^2)/x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (27) = 54\).

Time = 0.29 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.50 \[ \int \frac {\left (-100 x+50 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (450+100 x-50 e^5 x\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+e^{10 e^x} \left (\left (-4 x+2 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (18+4 x-2 e^5 x+e^x \left (-90 x-20 x^2+10 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )+e^{5 e^x} \left (\left (-40 x+20 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (180+40 x-20 e^5 x+e^x \left (-450 x-100 x^2+50 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )}{-9 x^3-2 x^4+e^5 x^4} \, dx=\frac {25 \log {\left (- \frac {x e^{5}}{3} + \frac {2 x}{3} + 3 \right )}^{2}}{x^{2}} + \frac {x^{2} e^{10 e^{x}} \log {\left (- \frac {x e^{5}}{3} + \frac {2 x}{3} + 3 \right )}^{2} + 10 x^{2} e^{5 e^{x}} \log {\left (- \frac {x e^{5}}{3} + \frac {2 x}{3} + 3 \right )}^{2}}{x^{4}} \]

[In]

integrate(((((10*x**2*exp(5)-20*x**2-90*x)*exp(x)-2*x*exp(5)+4*x+18)*ln(-1/3*x*exp(5)+2/3*x+3)**2+(2*x*exp(5)-
4*x)*ln(-1/3*x*exp(5)+2/3*x+3))*exp(5*exp(x))**2+(((50*x**2*exp(5)-100*x**2-450*x)*exp(x)-20*x*exp(5)+40*x+180
)*ln(-1/3*x*exp(5)+2/3*x+3)**2+(20*x*exp(5)-40*x)*ln(-1/3*x*exp(5)+2/3*x+3))*exp(5*exp(x))+(-50*x*exp(5)+100*x
+450)*ln(-1/3*x*exp(5)+2/3*x+3)**2+(50*x*exp(5)-100*x)*ln(-1/3*x*exp(5)+2/3*x+3))/(x**4*exp(5)-2*x**4-9*x**3),
x)

[Out]

25*log(-x*exp(5)/3 + 2*x/3 + 3)**2/x**2 + (x**2*exp(10*exp(x))*log(-x*exp(5)/3 + 2*x/3 + 3)**2 + 10*x**2*exp(5
*exp(x))*log(-x*exp(5)/3 + 2*x/3 + 3)**2)/x**4

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (25) = 50\).

Time = 0.38 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.91 \[ \int \frac {\left (-100 x+50 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (450+100 x-50 e^5 x\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+e^{10 e^x} \left (\left (-4 x+2 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (18+4 x-2 e^5 x+e^x \left (-90 x-20 x^2+10 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )+e^{5 e^x} \left (\left (-40 x+20 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (180+40 x-20 e^5 x+e^x \left (-450 x-100 x^2+50 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )}{-9 x^3-2 x^4+e^5 x^4} \, dx=\frac {e^{\left (10 \, e^{x}\right )} \log \left (3\right )^{2} + 10 \, e^{\left (5 \, e^{x}\right )} \log \left (3\right )^{2} + {\left (e^{\left (10 \, e^{x}\right )} + 10 \, e^{\left (5 \, e^{x}\right )} + 25\right )} \log \left (-x {\left (e^{5} - 2\right )} + 9\right )^{2} + 25 \, \log \left (3\right )^{2} - 2 \, {\left (e^{\left (10 \, e^{x}\right )} \log \left (3\right ) + 10 \, e^{\left (5 \, e^{x}\right )} \log \left (3\right ) + 25 \, \log \left (3\right )\right )} \log \left (-x {\left (e^{5} - 2\right )} + 9\right )}{x^{2}} \]

[In]

integrate(((((10*x^2*exp(5)-20*x^2-90*x)*exp(x)-2*x*exp(5)+4*x+18)*log(-1/3*x*exp(5)+2/3*x+3)^2+(2*x*exp(5)-4*
x)*log(-1/3*x*exp(5)+2/3*x+3))*exp(5*exp(x))^2+(((50*x^2*exp(5)-100*x^2-450*x)*exp(x)-20*x*exp(5)+40*x+180)*lo
g(-1/3*x*exp(5)+2/3*x+3)^2+(20*x*exp(5)-40*x)*log(-1/3*x*exp(5)+2/3*x+3))*exp(5*exp(x))+(-50*x*exp(5)+100*x+45
0)*log(-1/3*x*exp(5)+2/3*x+3)^2+(50*x*exp(5)-100*x)*log(-1/3*x*exp(5)+2/3*x+3))/(x^4*exp(5)-2*x^4-9*x^3),x, al
gorithm="maxima")

[Out]

(e^(10*e^x)*log(3)^2 + 10*e^(5*e^x)*log(3)^2 + (e^(10*e^x) + 10*e^(5*e^x) + 25)*log(-x*(e^5 - 2) + 9)^2 + 25*l
og(3)^2 - 2*(e^(10*e^x)*log(3) + 10*e^(5*e^x)*log(3) + 25*log(3))*log(-x*(e^5 - 2) + 9))/x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (25) = 50\).

Time = 0.37 (sec) , antiderivative size = 141, normalized size of antiderivative = 4.41 \[ \int \frac {\left (-100 x+50 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (450+100 x-50 e^5 x\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+e^{10 e^x} \left (\left (-4 x+2 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (18+4 x-2 e^5 x+e^x \left (-90 x-20 x^2+10 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )+e^{5 e^x} \left (\left (-40 x+20 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (180+40 x-20 e^5 x+e^x \left (-450 x-100 x^2+50 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )}{-9 x^3-2 x^4+e^5 x^4} \, dx=\frac {e^{\left (10 \, e^{x}\right )} \log \left (3\right )^{2} + 10 \, e^{\left (5 \, e^{x}\right )} \log \left (3\right )^{2} - 2 \, e^{\left (10 \, e^{x}\right )} \log \left (3\right ) \log \left (-x e^{5} + 2 \, x + 9\right ) - 20 \, e^{\left (5 \, e^{x}\right )} \log \left (3\right ) \log \left (-x e^{5} + 2 \, x + 9\right ) + e^{\left (10 \, e^{x}\right )} \log \left (-x e^{5} + 2 \, x + 9\right )^{2} + 10 \, e^{\left (5 \, e^{x}\right )} \log \left (-x e^{5} + 2 \, x + 9\right )^{2} + 25 \, \log \left (3\right )^{2} - 50 \, \log \left (3\right ) \log \left (-x e^{5} + 2 \, x + 9\right ) + 25 \, \log \left (-x e^{5} + 2 \, x + 9\right )^{2}}{x^{2}} \]

[In]

integrate(((((10*x^2*exp(5)-20*x^2-90*x)*exp(x)-2*x*exp(5)+4*x+18)*log(-1/3*x*exp(5)+2/3*x+3)^2+(2*x*exp(5)-4*
x)*log(-1/3*x*exp(5)+2/3*x+3))*exp(5*exp(x))^2+(((50*x^2*exp(5)-100*x^2-450*x)*exp(x)-20*x*exp(5)+40*x+180)*lo
g(-1/3*x*exp(5)+2/3*x+3)^2+(20*x*exp(5)-40*x)*log(-1/3*x*exp(5)+2/3*x+3))*exp(5*exp(x))+(-50*x*exp(5)+100*x+45
0)*log(-1/3*x*exp(5)+2/3*x+3)^2+(50*x*exp(5)-100*x)*log(-1/3*x*exp(5)+2/3*x+3))/(x^4*exp(5)-2*x^4-9*x^3),x, al
gorithm="giac")

[Out]

(e^(10*e^x)*log(3)^2 + 10*e^(5*e^x)*log(3)^2 - 2*e^(10*e^x)*log(3)*log(-x*e^5 + 2*x + 9) - 20*e^(5*e^x)*log(3)
*log(-x*e^5 + 2*x + 9) + e^(10*e^x)*log(-x*e^5 + 2*x + 9)^2 + 10*e^(5*e^x)*log(-x*e^5 + 2*x + 9)^2 + 25*log(3)
^2 - 50*log(3)*log(-x*e^5 + 2*x + 9) + 25*log(-x*e^5 + 2*x + 9)^2)/x^2

Mupad [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {\left (-100 x+50 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (450+100 x-50 e^5 x\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+e^{10 e^x} \left (\left (-4 x+2 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (18+4 x-2 e^5 x+e^x \left (-90 x-20 x^2+10 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )+e^{5 e^x} \left (\left (-40 x+20 e^5 x\right ) \log \left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )+\left (180+40 x-20 e^5 x+e^x \left (-450 x-100 x^2+50 e^5 x^2\right )\right ) \log ^2\left (\frac {1}{3} \left (9+2 x-e^5 x\right )\right )\right )}{-9 x^3-2 x^4+e^5 x^4} \, dx={\ln \left (\frac {2\,x}{3}-\frac {x\,{\mathrm {e}}^5}{3}+3\right )}^2\,\left (\frac {25}{x^2}+\frac {10\,{\mathrm {e}}^{5\,{\mathrm {e}}^x}}{x^2}+\frac {{\mathrm {e}}^{10\,{\mathrm {e}}^x}}{x^2}\right ) \]

[In]

int(-(log((2*x)/3 - (x*exp(5))/3 + 3)^2*(100*x - 50*x*exp(5) + 450) + exp(10*exp(x))*(log((2*x)/3 - (x*exp(5))
/3 + 3)^2*(4*x - 2*x*exp(5) - exp(x)*(90*x - 10*x^2*exp(5) + 20*x^2) + 18) - log((2*x)/3 - (x*exp(5))/3 + 3)*(
4*x - 2*x*exp(5))) + exp(5*exp(x))*(log((2*x)/3 - (x*exp(5))/3 + 3)^2*(40*x - 20*x*exp(5) - exp(x)*(450*x - 50
*x^2*exp(5) + 100*x^2) + 180) - log((2*x)/3 - (x*exp(5))/3 + 3)*(40*x - 20*x*exp(5))) - log((2*x)/3 - (x*exp(5
))/3 + 3)*(100*x - 50*x*exp(5)))/(9*x^3 - x^4*exp(5) + 2*x^4),x)

[Out]

log((2*x)/3 - (x*exp(5))/3 + 3)^2*(25/x^2 + (10*exp(5*exp(x)))/x^2 + exp(10*exp(x))/x^2)