Integrand size = 237, antiderivative size = 31 \[ \int \frac {e^{\frac {1}{5} \log (1+x) \log (25-x+\log (5))} \left (-125-120 x+5 x^2+(-5-5 x) \log (5)+\left (-8-6 x+2 x^2+\left (4+3 x-x^2\right ) \log \left (\frac {4-x}{2}\right )\right ) \log (1+x)+\left (200-58 x+2 x^2+(8-2 x) \log (5)+\left (-100+29 x-x^2+(-4+x) \log (5)\right ) \log \left (\frac {4-x}{2}\right )\right ) \log (25-x+\log (5))\right )}{-2000-1420 x+560 x^2-20 x^3+\left (-80-60 x+20 x^2\right ) \log (5)+\left (2000+1420 x-560 x^2+20 x^3+\left (80+60 x-20 x^2\right ) \log (5)\right ) \log \left (\frac {4-x}{2}\right )+\left (-500-355 x+140 x^2-5 x^3+\left (-20-15 x+5 x^2\right ) \log (5)\right ) \log ^2\left (\frac {4-x}{2}\right )} \, dx=\frac {e^{\frac {1}{5} \log (1+x) \log (25-x+\log (5))}}{-2+\log \left (2-\frac {x}{2}\right )} \]
\[ \int \frac {e^{\frac {1}{5} \log (1+x) \log (25-x+\log (5))} \left (-125-120 x+5 x^2+(-5-5 x) \log (5)+\left (-8-6 x+2 x^2+\left (4+3 x-x^2\right ) \log \left (\frac {4-x}{2}\right )\right ) \log (1+x)+\left (200-58 x+2 x^2+(8-2 x) \log (5)+\left (-100+29 x-x^2+(-4+x) \log (5)\right ) \log \left (\frac {4-x}{2}\right )\right ) \log (25-x+\log (5))\right )}{-2000-1420 x+560 x^2-20 x^3+\left (-80-60 x+20 x^2\right ) \log (5)+\left (2000+1420 x-560 x^2+20 x^3+\left (80+60 x-20 x^2\right ) \log (5)\right ) \log \left (\frac {4-x}{2}\right )+\left (-500-355 x+140 x^2-5 x^3+\left (-20-15 x+5 x^2\right ) \log (5)\right ) \log ^2\left (\frac {4-x}{2}\right )} \, dx=\int \frac {e^{\frac {1}{5} \log (1+x) \log (25-x+\log (5))} \left (-125-120 x+5 x^2+(-5-5 x) \log (5)+\left (-8-6 x+2 x^2+\left (4+3 x-x^2\right ) \log \left (\frac {4-x}{2}\right )\right ) \log (1+x)+\left (200-58 x+2 x^2+(8-2 x) \log (5)+\left (-100+29 x-x^2+(-4+x) \log (5)\right ) \log \left (\frac {4-x}{2}\right )\right ) \log (25-x+\log (5))\right )}{-2000-1420 x+560 x^2-20 x^3+\left (-80-60 x+20 x^2\right ) \log (5)+\left (2000+1420 x-560 x^2+20 x^3+\left (80+60 x-20 x^2\right ) \log (5)\right ) \log \left (\frac {4-x}{2}\right )+\left (-500-355 x+140 x^2-5 x^3+\left (-20-15 x+5 x^2\right ) \log (5)\right ) \log ^2\left (\frac {4-x}{2}\right )} \, dx \]
Integrate[(E^((Log[1 + x]*Log[25 - x + Log[5]])/5)*(-125 - 120*x + 5*x^2 + (-5 - 5*x)*Log[5] + (-8 - 6*x + 2*x^2 + (4 + 3*x - x^2)*Log[(4 - x)/2])*L og[1 + x] + (200 - 58*x + 2*x^2 + (8 - 2*x)*Log[5] + (-100 + 29*x - x^2 + (-4 + x)*Log[5])*Log[(4 - x)/2])*Log[25 - x + Log[5]]))/(-2000 - 1420*x + 560*x^2 - 20*x^3 + (-80 - 60*x + 20*x^2)*Log[5] + (2000 + 1420*x - 560*x^2 + 20*x^3 + (80 + 60*x - 20*x^2)*Log[5])*Log[(4 - x)/2] + (-500 - 355*x + 140*x^2 - 5*x^3 + (-20 - 15*x + 5*x^2)*Log[5])*Log[(4 - x)/2]^2),x]
Integrate[(E^((Log[1 + x]*Log[25 - x + Log[5]])/5)*(-125 - 120*x + 5*x^2 + (-5 - 5*x)*Log[5] + (-8 - 6*x + 2*x^2 + (4 + 3*x - x^2)*Log[(4 - x)/2])*L og[1 + x] + (200 - 58*x + 2*x^2 + (8 - 2*x)*Log[5] + (-100 + 29*x - x^2 + (-4 + x)*Log[5])*Log[(4 - x)/2])*Log[25 - x + Log[5]]))/(-2000 - 1420*x + 560*x^2 - 20*x^3 + (-80 - 60*x + 20*x^2)*Log[5] + (2000 + 1420*x - 560*x^2 + 20*x^3 + (80 + 60*x - 20*x^2)*Log[5])*Log[(4 - x)/2] + (-500 - 355*x + 140*x^2 - 5*x^3 + (-20 - 15*x + 5*x^2)*Log[5])*Log[(4 - x)/2]^2), x]
Leaf count is larger than twice the leaf count of optimal. \(255\) vs. \(2(31)=62\).
Time = 0.77 (sec) , antiderivative size = 255, normalized size of antiderivative = 8.23, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.004, Rules used = {2726}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^{\frac {1}{5} \log (x+1) \log (-x+25+\log (5))} \left (5 x^2+\left (2 x^2+\left (-x^2+3 x+4\right ) \log \left (\frac {4-x}{2}\right )-6 x-8\right ) \log (x+1)+\left (2 x^2+\left (-x^2+29 x+(x-4) \log (5)-100\right ) \log \left (\frac {4-x}{2}\right )-58 x+(8-2 x) \log (5)+200\right ) \log (-x+25+\log (5))-120 x+(-5 x-5) \log (5)-125\right )}{-20 x^3+560 x^2+\left (20 x^2-60 x-80\right ) \log (5)+\left (-5 x^3+140 x^2+\left (5 x^2-15 x-20\right ) \log (5)-355 x-500\right ) \log ^2\left (\frac {4-x}{2}\right )+\left (20 x^3-560 x^2+\left (-20 x^2+60 x+80\right ) \log (5)+1420 x+2000\right ) \log \left (\frac {4-x}{2}\right )-1420 x-2000} \, dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle -\frac {e^{\frac {1}{5} \log (x+1) \log (-x+25+\log (5))} \left (\left (-2 x^2-\left (-x^2+3 x+4\right ) \log \left (\frac {4-x}{2}\right )+6 x+8\right ) \log (x+1)-\left (2 x^2-\left (x^2-29 x+(4-x) \log (5)+100\right ) \log \left (\frac {4-x}{2}\right )-58 x+2 (4-x) \log (5)+200\right ) \log (-x+25+\log (5))\right )}{\left (4 x^3-112 x^2+4 \left (-x^2+3 x+4\right ) \log (5)+\left (x^3-28 x^2+\left (-x^2+3 x+4\right ) \log (5)+71 x+100\right ) \log ^2\left (\frac {4-x}{2}\right )-4 \left (x^3-28 x^2+\left (-x^2+3 x+4\right ) \log (5)+71 x+100\right ) \log \left (\frac {4-x}{2}\right )+284 x+400\right ) \left (\frac {\log (x+1)}{-x+25+\log (5)}-\frac {\log (-x+25+\log (5))}{x+1}\right )}\) |
Int[(E^((Log[1 + x]*Log[25 - x + Log[5]])/5)*(-125 - 120*x + 5*x^2 + (-5 - 5*x)*Log[5] + (-8 - 6*x + 2*x^2 + (4 + 3*x - x^2)*Log[(4 - x)/2])*Log[1 + x] + (200 - 58*x + 2*x^2 + (8 - 2*x)*Log[5] + (-100 + 29*x - x^2 + (-4 + x)*Log[5])*Log[(4 - x)/2])*Log[25 - x + Log[5]]))/(-2000 - 1420*x + 560*x^ 2 - 20*x^3 + (-80 - 60*x + 20*x^2)*Log[5] + (2000 + 1420*x - 560*x^2 + 20* x^3 + (80 + 60*x - 20*x^2)*Log[5])*Log[(4 - x)/2] + (-500 - 355*x + 140*x^ 2 - 5*x^3 + (-20 - 15*x + 5*x^2)*Log[5])*Log[(4 - x)/2]^2),x]
-((E^((Log[1 + x]*Log[25 - x + Log[5]])/5)*((8 + 6*x - 2*x^2 - (4 + 3*x - x^2)*Log[(4 - x)/2])*Log[1 + x] - (200 - 58*x + 2*x^2 + 2*(4 - x)*Log[5] - (100 - 29*x + x^2 + (4 - x)*Log[5])*Log[(4 - x)/2])*Log[25 - x + Log[5]]) )/((400 + 284*x - 112*x^2 + 4*x^3 + 4*(4 + 3*x - x^2)*Log[5] - 4*(100 + 71 *x - 28*x^2 + x^3 + (4 + 3*x - x^2)*Log[5])*Log[(4 - x)/2] + (100 + 71*x - 28*x^2 + x^3 + (4 + 3*x - x^2)*Log[5])*Log[(4 - x)/2]^2)*(Log[1 + x]/(25 - x + Log[5]) - Log[25 - x + Log[5]]/(1 + x))))
3.28.30.3.1 Defintions of rubi rules used
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Time = 280.41 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84
method | result | size |
risch | \(\frac {\left (\ln \left (5\right )-x +25\right )^{\frac {\ln \left (1+x \right )}{5}}}{\ln \left (2-\frac {x}{2}\right )-2}\) | \(26\) |
parallelrisch | \(\frac {{\mathrm e}^{\frac {\ln \left (1+x \right ) \ln \left (\ln \left (5\right )-x +25\right )}{5}}}{\ln \left (2-\frac {x}{2}\right )-2}\) | \(27\) |
int(((((x-4)*ln(5)-x^2+29*x-100)*ln(2-1/2*x)+(-2*x+8)*ln(5)+2*x^2-58*x+200 )*ln(ln(5)-x+25)+((-x^2+3*x+4)*ln(2-1/2*x)+2*x^2-6*x-8)*ln(1+x)+(-5*x-5)*l n(5)+5*x^2-120*x-125)*exp(1/5*ln(1+x)*ln(ln(5)-x+25))/(((5*x^2-15*x-20)*ln (5)-5*x^3+140*x^2-355*x-500)*ln(2-1/2*x)^2+((-20*x^2+60*x+80)*ln(5)+20*x^3 -560*x^2+1420*x+2000)*ln(2-1/2*x)+(20*x^2-60*x-80)*ln(5)-20*x^3+560*x^2-14 20*x-2000),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {e^{\frac {1}{5} \log (1+x) \log (25-x+\log (5))} \left (-125-120 x+5 x^2+(-5-5 x) \log (5)+\left (-8-6 x+2 x^2+\left (4+3 x-x^2\right ) \log \left (\frac {4-x}{2}\right )\right ) \log (1+x)+\left (200-58 x+2 x^2+(8-2 x) \log (5)+\left (-100+29 x-x^2+(-4+x) \log (5)\right ) \log \left (\frac {4-x}{2}\right )\right ) \log (25-x+\log (5))\right )}{-2000-1420 x+560 x^2-20 x^3+\left (-80-60 x+20 x^2\right ) \log (5)+\left (2000+1420 x-560 x^2+20 x^3+\left (80+60 x-20 x^2\right ) \log (5)\right ) \log \left (\frac {4-x}{2}\right )+\left (-500-355 x+140 x^2-5 x^3+\left (-20-15 x+5 x^2\right ) \log (5)\right ) \log ^2\left (\frac {4-x}{2}\right )} \, dx=\frac {e^{\left (\frac {1}{5} \, \log \left (x + 1\right ) \log \left (-x + \log \left (5\right ) + 25\right )\right )}}{\log \left (-\frac {1}{2} \, x + 2\right ) - 2} \]
integrate(((((x-4)*log(5)-x^2+29*x-100)*log(2-1/2*x)+(-2*x+8)*log(5)+2*x^2 -58*x+200)*log(log(5)-x+25)+((-x^2+3*x+4)*log(2-1/2*x)+2*x^2-6*x-8)*log(1+ x)+(-5*x-5)*log(5)+5*x^2-120*x-125)*exp(1/5*log(1+x)*log(log(5)-x+25))/((( 5*x^2-15*x-20)*log(5)-5*x^3+140*x^2-355*x-500)*log(2-1/2*x)^2+((-20*x^2+60 *x+80)*log(5)+20*x^3-560*x^2+1420*x+2000)*log(2-1/2*x)+(20*x^2-60*x-80)*lo g(5)-20*x^3+560*x^2-1420*x-2000),x, algorithm=\
Time = 1.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {e^{\frac {1}{5} \log (1+x) \log (25-x+\log (5))} \left (-125-120 x+5 x^2+(-5-5 x) \log (5)+\left (-8-6 x+2 x^2+\left (4+3 x-x^2\right ) \log \left (\frac {4-x}{2}\right )\right ) \log (1+x)+\left (200-58 x+2 x^2+(8-2 x) \log (5)+\left (-100+29 x-x^2+(-4+x) \log (5)\right ) \log \left (\frac {4-x}{2}\right )\right ) \log (25-x+\log (5))\right )}{-2000-1420 x+560 x^2-20 x^3+\left (-80-60 x+20 x^2\right ) \log (5)+\left (2000+1420 x-560 x^2+20 x^3+\left (80+60 x-20 x^2\right ) \log (5)\right ) \log \left (\frac {4-x}{2}\right )+\left (-500-355 x+140 x^2-5 x^3+\left (-20-15 x+5 x^2\right ) \log (5)\right ) \log ^2\left (\frac {4-x}{2}\right )} \, dx=\frac {e^{\frac {\log {\left (x + 1 \right )} \log {\left (- x + \log {\left (5 \right )} + 25 \right )}}{5}}}{\log {\left (2 - \frac {x}{2} \right )} - 2} \]
integrate(((((x-4)*ln(5)-x**2+29*x-100)*ln(2-1/2*x)+(-2*x+8)*ln(5)+2*x**2- 58*x+200)*ln(ln(5)-x+25)+((-x**2+3*x+4)*ln(2-1/2*x)+2*x**2-6*x-8)*ln(1+x)+ (-5*x-5)*ln(5)+5*x**2-120*x-125)*exp(1/5*ln(1+x)*ln(ln(5)-x+25))/(((5*x**2 -15*x-20)*ln(5)-5*x**3+140*x**2-355*x-500)*ln(2-1/2*x)**2+((-20*x**2+60*x+ 80)*ln(5)+20*x**3-560*x**2+1420*x+2000)*ln(2-1/2*x)+(20*x**2-60*x-80)*ln(5 )-20*x**3+560*x**2-1420*x-2000),x)
Time = 0.37 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {1}{5} \log (1+x) \log (25-x+\log (5))} \left (-125-120 x+5 x^2+(-5-5 x) \log (5)+\left (-8-6 x+2 x^2+\left (4+3 x-x^2\right ) \log \left (\frac {4-x}{2}\right )\right ) \log (1+x)+\left (200-58 x+2 x^2+(8-2 x) \log (5)+\left (-100+29 x-x^2+(-4+x) \log (5)\right ) \log \left (\frac {4-x}{2}\right )\right ) \log (25-x+\log (5))\right )}{-2000-1420 x+560 x^2-20 x^3+\left (-80-60 x+20 x^2\right ) \log (5)+\left (2000+1420 x-560 x^2+20 x^3+\left (80+60 x-20 x^2\right ) \log (5)\right ) \log \left (\frac {4-x}{2}\right )+\left (-500-355 x+140 x^2-5 x^3+\left (-20-15 x+5 x^2\right ) \log (5)\right ) \log ^2\left (\frac {4-x}{2}\right )} \, dx=-\frac {e^{\left (\frac {1}{5} \, \log \left (x + 1\right ) \log \left (-x + \log \left (5\right ) + 25\right )\right )}}{\log \left (2\right ) - \log \left (-x + 4\right ) + 2} \]
integrate(((((x-4)*log(5)-x^2+29*x-100)*log(2-1/2*x)+(-2*x+8)*log(5)+2*x^2 -58*x+200)*log(log(5)-x+25)+((-x^2+3*x+4)*log(2-1/2*x)+2*x^2-6*x-8)*log(1+ x)+(-5*x-5)*log(5)+5*x^2-120*x-125)*exp(1/5*log(1+x)*log(log(5)-x+25))/((( 5*x^2-15*x-20)*log(5)-5*x^3+140*x^2-355*x-500)*log(2-1/2*x)^2+((-20*x^2+60 *x+80)*log(5)+20*x^3-560*x^2+1420*x+2000)*log(2-1/2*x)+(20*x^2-60*x-80)*lo g(5)-20*x^3+560*x^2-1420*x-2000),x, algorithm=\
\[ \int \frac {e^{\frac {1}{5} \log (1+x) \log (25-x+\log (5))} \left (-125-120 x+5 x^2+(-5-5 x) \log (5)+\left (-8-6 x+2 x^2+\left (4+3 x-x^2\right ) \log \left (\frac {4-x}{2}\right )\right ) \log (1+x)+\left (200-58 x+2 x^2+(8-2 x) \log (5)+\left (-100+29 x-x^2+(-4+x) \log (5)\right ) \log \left (\frac {4-x}{2}\right )\right ) \log (25-x+\log (5))\right )}{-2000-1420 x+560 x^2-20 x^3+\left (-80-60 x+20 x^2\right ) \log (5)+\left (2000+1420 x-560 x^2+20 x^3+\left (80+60 x-20 x^2\right ) \log (5)\right ) \log \left (\frac {4-x}{2}\right )+\left (-500-355 x+140 x^2-5 x^3+\left (-20-15 x+5 x^2\right ) \log (5)\right ) \log ^2\left (\frac {4-x}{2}\right )} \, dx=\int { -\frac {{\left (5 \, x^{2} - 5 \, {\left (x + 1\right )} \log \left (5\right ) + {\left (2 \, x^{2} - {\left (x^{2} - 3 \, x - 4\right )} \log \left (-\frac {1}{2} \, x + 2\right ) - 6 \, x - 8\right )} \log \left (x + 1\right ) + {\left (2 \, x^{2} - 2 \, {\left (x - 4\right )} \log \left (5\right ) - {\left (x^{2} - {\left (x - 4\right )} \log \left (5\right ) - 29 \, x + 100\right )} \log \left (-\frac {1}{2} \, x + 2\right ) - 58 \, x + 200\right )} \log \left (-x + \log \left (5\right ) + 25\right ) - 120 \, x - 125\right )} e^{\left (\frac {1}{5} \, \log \left (x + 1\right ) \log \left (-x + \log \left (5\right ) + 25\right )\right )}}{5 \, {\left (4 \, x^{3} + {\left (x^{3} - 28 \, x^{2} - {\left (x^{2} - 3 \, x - 4\right )} \log \left (5\right ) + 71 \, x + 100\right )} \log \left (-\frac {1}{2} \, x + 2\right )^{2} - 112 \, x^{2} - 4 \, {\left (x^{2} - 3 \, x - 4\right )} \log \left (5\right ) - 4 \, {\left (x^{3} - 28 \, x^{2} - {\left (x^{2} - 3 \, x - 4\right )} \log \left (5\right ) + 71 \, x + 100\right )} \log \left (-\frac {1}{2} \, x + 2\right ) + 284 \, x + 400\right )}} \,d x } \]
integrate(((((x-4)*log(5)-x^2+29*x-100)*log(2-1/2*x)+(-2*x+8)*log(5)+2*x^2 -58*x+200)*log(log(5)-x+25)+((-x^2+3*x+4)*log(2-1/2*x)+2*x^2-6*x-8)*log(1+ x)+(-5*x-5)*log(5)+5*x^2-120*x-125)*exp(1/5*log(1+x)*log(log(5)-x+25))/((( 5*x^2-15*x-20)*log(5)-5*x^3+140*x^2-355*x-500)*log(2-1/2*x)^2+((-20*x^2+60 *x+80)*log(5)+20*x^3-560*x^2+1420*x+2000)*log(2-1/2*x)+(20*x^2-60*x-80)*lo g(5)-20*x^3+560*x^2-1420*x-2000),x, algorithm=\
integrate(-1/5*(5*x^2 - 5*(x + 1)*log(5) + (2*x^2 - (x^2 - 3*x - 4)*log(-1 /2*x + 2) - 6*x - 8)*log(x + 1) + (2*x^2 - 2*(x - 4)*log(5) - (x^2 - (x - 4)*log(5) - 29*x + 100)*log(-1/2*x + 2) - 58*x + 200)*log(-x + log(5) + 25 ) - 120*x - 125)*e^(1/5*log(x + 1)*log(-x + log(5) + 25))/(4*x^3 + (x^3 - 28*x^2 - (x^2 - 3*x - 4)*log(5) + 71*x + 100)*log(-1/2*x + 2)^2 - 112*x^2 - 4*(x^2 - 3*x - 4)*log(5) - 4*(x^3 - 28*x^2 - (x^2 - 3*x - 4)*log(5) + 71 *x + 100)*log(-1/2*x + 2) + 284*x + 400), x)
Time = 10.97 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {e^{\frac {1}{5} \log (1+x) \log (25-x+\log (5))} \left (-125-120 x+5 x^2+(-5-5 x) \log (5)+\left (-8-6 x+2 x^2+\left (4+3 x-x^2\right ) \log \left (\frac {4-x}{2}\right )\right ) \log (1+x)+\left (200-58 x+2 x^2+(8-2 x) \log (5)+\left (-100+29 x-x^2+(-4+x) \log (5)\right ) \log \left (\frac {4-x}{2}\right )\right ) \log (25-x+\log (5))\right )}{-2000-1420 x+560 x^2-20 x^3+\left (-80-60 x+20 x^2\right ) \log (5)+\left (2000+1420 x-560 x^2+20 x^3+\left (80+60 x-20 x^2\right ) \log (5)\right ) \log \left (\frac {4-x}{2}\right )+\left (-500-355 x+140 x^2-5 x^3+\left (-20-15 x+5 x^2\right ) \log (5)\right ) \log ^2\left (\frac {4-x}{2}\right )} \, dx=\frac {{\left (x+1\right )}^{\frac {\ln \left (\ln \left (5\right )-x+25\right )}{5}}}{\ln \left (2-\frac {x}{2}\right )-2} \]
int((exp((log(log(5) - x + 25)*log(x + 1))/5)*(120*x + log(5)*(5*x + 5) + log(x + 1)*(6*x - log(2 - x/2)*(3*x - x^2 + 4) - 2*x^2 + 8) - 5*x^2 - log( log(5) - x + 25)*(log(2 - x/2)*(29*x + log(5)*(x - 4) - x^2 - 100) - log(5 )*(2*x - 8) - 58*x + 2*x^2 + 200) + 125))/(1420*x + log(5)*(60*x - 20*x^2 + 80) + log(2 - x/2)^2*(355*x + log(5)*(15*x - 5*x^2 + 20) - 140*x^2 + 5*x ^3 + 500) - 560*x^2 + 20*x^3 - log(2 - x/2)*(1420*x + log(5)*(60*x - 20*x^ 2 + 80) - 560*x^2 + 20*x^3 + 2000) + 2000),x)