3.84.53 \(\int \frac {e^{-5+x} (2+e^x) (80+14 x-16 x^2+2 x^3+e^x (80+14 x-16 x^2+2 x^3))+(-92+40 x-4 x^2+e^x (-46+20 x-2 x^2)+e^{-5+x} (2+e^x) (46-20 x+2 x^2+e^x (23-10 x+x^2))) \log (-2+e^{-5+x} (2+e^x))}{-100+40 x-4 x^2+e^x (-50+20 x-2 x^2)+e^{-5+x} (2+e^x) (50-20 x+2 x^2+e^x (25-10 x+x^2))} \, dx\)

Optimal. Leaf size=25 \[ \left (2+\frac {2}{-5+x}+x\right ) \log \left (-2+e^{-5+x} \left (2+e^x\right )\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 8.85, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{-5+x} \left (2+e^x\right ) \left (80+14 x-16 x^2+2 x^3+e^x \left (80+14 x-16 x^2+2 x^3\right )\right )+\left (-92+40 x-4 x^2+e^x \left (-46+20 x-2 x^2\right )+e^{-5+x} \left (2+e^x\right ) \left (46-20 x+2 x^2+e^x \left (23-10 x+x^2\right )\right )\right ) \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )}{-100+40 x-4 x^2+e^x \left (-50+20 x-2 x^2\right )+e^{-5+x} \left (2+e^x\right ) \left (50-20 x+2 x^2+e^x \left (25-10 x+x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-5 + x)*(2 + E^x)*(80 + 14*x - 16*x^2 + 2*x^3 + E^x*(80 + 14*x - 16*x^2 + 2*x^3)) + (-92 + 40*x - 4*x^
2 + E^x*(-46 + 20*x - 2*x^2) + E^(-5 + x)*(2 + E^x)*(46 - 20*x + 2*x^2 + E^x*(23 - 10*x + x^2)))*Log[-2 + E^(-
5 + x)*(2 + E^x)])/(-100 + 40*x - 4*x^2 + E^x*(-50 + 20*x - 2*x^2) + E^(-5 + x)*(2 + E^x)*(50 - 20*x + 2*x^2 +
E^x*(25 - 10*x + x^2))),x]

[Out]

2*Log[2*E^5 - 2*E^x - E^(2*x)] - (2*Log[-2 + E^(-5 + x)*(2 + E^x)])/(5 - x) + x*Log[-2 + E^(-5 + x)*(2 + E^x)]
+ x*Log[1 + E^x/(1 - Sqrt[1 + 2*E^5])] - (x*Log[1 + E^x/(1 - Sqrt[1 + 2*E^5])])/Sqrt[1 + 2*E^5] + x*Log[1 + E
^x/(1 + Sqrt[1 + 2*E^5])] + (x*Log[1 + E^x/(1 + Sqrt[1 + 2*E^5])])/Sqrt[1 + 2*E^5] + 4*Log[5 - x] + PolyLog[2,
-(E^x/(1 - Sqrt[1 + 2*E^5]))] - PolyLog[2, -(E^x/(1 - Sqrt[1 + 2*E^5]))]/Sqrt[1 + 2*E^5] + PolyLog[2, -(E^x/(
1 + Sqrt[1 + 2*E^5]))] + PolyLog[2, -(E^x/(1 + Sqrt[1 + 2*E^5]))]/Sqrt[1 + 2*E^5] - 8*E^5*Defer[Int][1/((2*E^5
- 2*E^x - E^(2*x))*(-5 + x)), x] - 8*Defer[Int][E^x/((-2*E^5 + 2*E^x + E^(2*x))*(-5 + x)), x] - 4*Defer[Int][
E^(2*x)/((-2*E^5 + 2*E^x + E^(2*x))*(-5 + x)), x] - 2*Defer[Int][(E^(2*x)*x)/(-2*E^5 + 2*E^x + E^(2*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\frac {2 e^x \left (1+e^x\right ) \left (40+7 x-8 x^2+x^3\right )}{-2 e^5+2 e^x+e^{2 x}}+\left (23-10 x+x^2\right ) \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )}{(5-x)^2} \, dx\\ &=\int \left (-\frac {2 \left (-2 e^5+e^x\right ) \left (-8-3 x+x^2\right )}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)}+\frac {80+14 x-16 x^2+2 x^3+23 \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )-10 x \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )+x^2 \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )}{(-5+x)^2}\right ) \, dx\\ &=-\left (2 \int \frac {\left (-2 e^5+e^x\right ) \left (-8-3 x+x^2\right )}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)} \, dx\right )+\int \frac {80+14 x-16 x^2+2 x^3+23 \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )-10 x \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )+x^2 \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )}{(-5+x)^2} \, dx\\ &=-\left (2 \int \left (\frac {2 \left (-2 e^5+e^x\right )}{-2 e^5+2 e^x+e^{2 x}}+\frac {2 \left (-2 e^5+e^x\right )}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)}+\frac {\left (-2 e^5+e^x\right ) x}{-2 e^5+2 e^x+e^{2 x}}\right ) \, dx\right )+\int \frac {2 \left (40+7 x-8 x^2+x^3\right )+\left (23-10 x+x^2\right ) \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )}{(5-x)^2} \, dx\\ &=-\left (2 \int \frac {\left (-2 e^5+e^x\right ) x}{-2 e^5+2 e^x+e^{2 x}} \, dx\right )-4 \int \frac {-2 e^5+e^x}{-2 e^5+2 e^x+e^{2 x}} \, dx-4 \int \frac {-2 e^5+e^x}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)} \, dx+\int \left (\frac {2 \left (-8-3 x+x^2\right )}{-5+x}+\frac {\left (23-10 x+x^2\right ) \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )}{(-5+x)^2}\right ) \, dx\\ &=2 \int \frac {-8-3 x+x^2}{-5+x} \, dx-4 \int \left (\frac {2 e^5}{\left (2 e^5-2 e^x-e^{2 x}\right ) (-5+x)}+\frac {e^x}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)}\right ) \, dx-4 \operatorname {Subst}\left (\int \frac {-2 e^5+x}{x \left (-2 e^5+2 x+x^2\right )} \, dx,x,e^x\right )-\left (2 \left (1-\sqrt {1+2 e^5}\right )\right ) \int \frac {x}{2+2 e^x-2 \sqrt {1+2 e^5}} \, dx-\left (2 \left (1+\sqrt {1+2 e^5}\right )\right ) \int \frac {x}{2+2 e^x+2 \sqrt {1+2 e^5}} \, dx+\int \frac {\left (23-10 x+x^2\right ) \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )}{(-5+x)^2} \, dx\\ &=-x^2-\frac {2 \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )}{5-x}+x \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )+2 \int \frac {e^x x}{2+2 e^x-2 \sqrt {1+2 e^5}} \, dx+2 \int \frac {e^x x}{2+2 e^x+2 \sqrt {1+2 e^5}} \, dx+2 \int \left (2+\frac {2}{-5+x}+x\right ) \, dx-4 \int \frac {e^x}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)} \, dx-4 \operatorname {Subst}\left (\int \left (\frac {1}{x}+\frac {-1-x}{-2 e^5+2 x+x^2}\right ) \, dx,x,e^x\right )-\left (8 e^5\right ) \int \frac {1}{\left (2 e^5-2 e^x-e^{2 x}\right ) (-5+x)} \, dx-\int \frac {2 e^x \left (1+e^x\right ) \left (2-5 x+x^2\right )}{\left (2 e^5-2 e^x-e^{2 x}\right ) (5-x)} \, dx\\ &=-\frac {2 \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )}{5-x}+x \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )+x \log \left (1+\frac {e^x}{1-\sqrt {1+2 e^5}}\right )+x \log \left (1+\frac {e^x}{1+\sqrt {1+2 e^5}}\right )+4 \log (5-x)-2 \int \frac {e^x \left (1+e^x\right ) \left (2-5 x+x^2\right )}{\left (2 e^5-2 e^x-e^{2 x}\right ) (5-x)} \, dx-4 \int \frac {e^x}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)} \, dx-4 \operatorname {Subst}\left (\int \frac {-1-x}{-2 e^5+2 x+x^2} \, dx,x,e^x\right )-\left (8 e^5\right ) \int \frac {1}{\left (2 e^5-2 e^x-e^{2 x}\right ) (-5+x)} \, dx-\int \log \left (1+\frac {2 e^x}{2-2 \sqrt {1+2 e^5}}\right ) \, dx-\int \log \left (1+\frac {2 e^x}{2+2 \sqrt {1+2 e^5}}\right ) \, dx\\ &=2 \log \left (2 e^5-2 e^x-e^{2 x}\right )-\frac {2 \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )}{5-x}+x \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )+x \log \left (1+\frac {e^x}{1-\sqrt {1+2 e^5}}\right )+x \log \left (1+\frac {e^x}{1+\sqrt {1+2 e^5}}\right )+4 \log (5-x)-2 \int \left (\frac {2 e^x \left (1+e^x\right )}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)}+\frac {e^x \left (1+e^x\right ) x}{-2 e^5+2 e^x+e^{2 x}}\right ) \, dx-4 \int \frac {e^x}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)} \, dx-\left (8 e^5\right ) \int \frac {1}{\left (2 e^5-2 e^x-e^{2 x}\right ) (-5+x)} \, dx-\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{2-2 \sqrt {1+2 e^5}}\right )}{x} \, dx,x,e^x\right )-\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{2+2 \sqrt {1+2 e^5}}\right )}{x} \, dx,x,e^x\right )\\ &=2 \log \left (2 e^5-2 e^x-e^{2 x}\right )-\frac {2 \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )}{5-x}+x \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )+x \log \left (1+\frac {e^x}{1-\sqrt {1+2 e^5}}\right )+x \log \left (1+\frac {e^x}{1+\sqrt {1+2 e^5}}\right )+4 \log (5-x)+\text {Li}_2\left (-\frac {e^x}{1-\sqrt {1+2 e^5}}\right )+\text {Li}_2\left (-\frac {e^x}{1+\sqrt {1+2 e^5}}\right )-2 \int \frac {e^x \left (1+e^x\right ) x}{-2 e^5+2 e^x+e^{2 x}} \, dx-4 \int \frac {e^x}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)} \, dx-4 \int \frac {e^x \left (1+e^x\right )}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)} \, dx-\left (8 e^5\right ) \int \frac {1}{\left (2 e^5-2 e^x-e^{2 x}\right ) (-5+x)} \, dx\\ &=2 \log \left (2 e^5-2 e^x-e^{2 x}\right )-\frac {2 \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )}{5-x}+x \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )+x \log \left (1+\frac {e^x}{1-\sqrt {1+2 e^5}}\right )+x \log \left (1+\frac {e^x}{1+\sqrt {1+2 e^5}}\right )+4 \log (5-x)+\text {Li}_2\left (-\frac {e^x}{1-\sqrt {1+2 e^5}}\right )+\text {Li}_2\left (-\frac {e^x}{1+\sqrt {1+2 e^5}}\right )-2 \int \left (\frac {e^x x}{-2 e^5+2 e^x+e^{2 x}}+\frac {e^{2 x} x}{-2 e^5+2 e^x+e^{2 x}}\right ) \, dx-4 \int \left (\frac {e^x}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)}+\frac {e^{2 x}}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)}\right ) \, dx-4 \int \frac {e^x}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)} \, dx-\left (8 e^5\right ) \int \frac {1}{\left (2 e^5-2 e^x-e^{2 x}\right ) (-5+x)} \, dx\\ &=2 \log \left (2 e^5-2 e^x-e^{2 x}\right )-\frac {2 \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )}{5-x}+x \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )+x \log \left (1+\frac {e^x}{1-\sqrt {1+2 e^5}}\right )+x \log \left (1+\frac {e^x}{1+\sqrt {1+2 e^5}}\right )+4 \log (5-x)+\text {Li}_2\left (-\frac {e^x}{1-\sqrt {1+2 e^5}}\right )+\text {Li}_2\left (-\frac {e^x}{1+\sqrt {1+2 e^5}}\right )-2 \int \frac {e^x x}{-2 e^5+2 e^x+e^{2 x}} \, dx-2 \int \frac {e^{2 x} x}{-2 e^5+2 e^x+e^{2 x}} \, dx-2 \left (4 \int \frac {e^x}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)} \, dx\right )-4 \int \frac {e^{2 x}}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)} \, dx-\left (8 e^5\right ) \int \frac {1}{\left (2 e^5-2 e^x-e^{2 x}\right ) (-5+x)} \, dx\\ &=2 \log \left (2 e^5-2 e^x-e^{2 x}\right )-\frac {2 \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )}{5-x}+x \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )+x \log \left (1+\frac {e^x}{1-\sqrt {1+2 e^5}}\right )+x \log \left (1+\frac {e^x}{1+\sqrt {1+2 e^5}}\right )+4 \log (5-x)+\text {Li}_2\left (-\frac {e^x}{1-\sqrt {1+2 e^5}}\right )+\text {Li}_2\left (-\frac {e^x}{1+\sqrt {1+2 e^5}}\right )-2 \int \frac {e^{2 x} x}{-2 e^5+2 e^x+e^{2 x}} \, dx-2 \left (4 \int \frac {e^x}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)} \, dx\right )-4 \int \frac {e^{2 x}}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)} \, dx-\left (8 e^5\right ) \int \frac {1}{\left (2 e^5-2 e^x-e^{2 x}\right ) (-5+x)} \, dx-\frac {2 \int \frac {e^x x}{2+2 e^x-2 \sqrt {1+2 e^5}} \, dx}{\sqrt {1+2 e^5}}+\frac {2 \int \frac {e^x x}{2+2 e^x+2 \sqrt {1+2 e^5}} \, dx}{\sqrt {1+2 e^5}}\\ &=2 \log \left (2 e^5-2 e^x-e^{2 x}\right )-\frac {2 \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )}{5-x}+x \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )+x \log \left (1+\frac {e^x}{1-\sqrt {1+2 e^5}}\right )-\frac {x \log \left (1+\frac {e^x}{1-\sqrt {1+2 e^5}}\right )}{\sqrt {1+2 e^5}}+x \log \left (1+\frac {e^x}{1+\sqrt {1+2 e^5}}\right )+\frac {x \log \left (1+\frac {e^x}{1+\sqrt {1+2 e^5}}\right )}{\sqrt {1+2 e^5}}+4 \log (5-x)+\text {Li}_2\left (-\frac {e^x}{1-\sqrt {1+2 e^5}}\right )+\text {Li}_2\left (-\frac {e^x}{1+\sqrt {1+2 e^5}}\right )-2 \int \frac {e^{2 x} x}{-2 e^5+2 e^x+e^{2 x}} \, dx-2 \left (4 \int \frac {e^x}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)} \, dx\right )-4 \int \frac {e^{2 x}}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)} \, dx-\left (8 e^5\right ) \int \frac {1}{\left (2 e^5-2 e^x-e^{2 x}\right ) (-5+x)} \, dx+\frac {\int \log \left (1+\frac {2 e^x}{2-2 \sqrt {1+2 e^5}}\right ) \, dx}{\sqrt {1+2 e^5}}-\frac {\int \log \left (1+\frac {2 e^x}{2+2 \sqrt {1+2 e^5}}\right ) \, dx}{\sqrt {1+2 e^5}}\\ &=2 \log \left (2 e^5-2 e^x-e^{2 x}\right )-\frac {2 \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )}{5-x}+x \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )+x \log \left (1+\frac {e^x}{1-\sqrt {1+2 e^5}}\right )-\frac {x \log \left (1+\frac {e^x}{1-\sqrt {1+2 e^5}}\right )}{\sqrt {1+2 e^5}}+x \log \left (1+\frac {e^x}{1+\sqrt {1+2 e^5}}\right )+\frac {x \log \left (1+\frac {e^x}{1+\sqrt {1+2 e^5}}\right )}{\sqrt {1+2 e^5}}+4 \log (5-x)+\text {Li}_2\left (-\frac {e^x}{1-\sqrt {1+2 e^5}}\right )+\text {Li}_2\left (-\frac {e^x}{1+\sqrt {1+2 e^5}}\right )-2 \int \frac {e^{2 x} x}{-2 e^5+2 e^x+e^{2 x}} \, dx-2 \left (4 \int \frac {e^x}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)} \, dx\right )-4 \int \frac {e^{2 x}}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)} \, dx-\left (8 e^5\right ) \int \frac {1}{\left (2 e^5-2 e^x-e^{2 x}\right ) (-5+x)} \, dx+\frac {\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{2-2 \sqrt {1+2 e^5}}\right )}{x} \, dx,x,e^x\right )}{\sqrt {1+2 e^5}}-\frac {\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{2+2 \sqrt {1+2 e^5}}\right )}{x} \, dx,x,e^x\right )}{\sqrt {1+2 e^5}}\\ &=2 \log \left (2 e^5-2 e^x-e^{2 x}\right )-\frac {2 \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )}{5-x}+x \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )+x \log \left (1+\frac {e^x}{1-\sqrt {1+2 e^5}}\right )-\frac {x \log \left (1+\frac {e^x}{1-\sqrt {1+2 e^5}}\right )}{\sqrt {1+2 e^5}}+x \log \left (1+\frac {e^x}{1+\sqrt {1+2 e^5}}\right )+\frac {x \log \left (1+\frac {e^x}{1+\sqrt {1+2 e^5}}\right )}{\sqrt {1+2 e^5}}+4 \log (5-x)+\text {Li}_2\left (-\frac {e^x}{1-\sqrt {1+2 e^5}}\right )-\frac {\text {Li}_2\left (-\frac {e^x}{1-\sqrt {1+2 e^5}}\right )}{\sqrt {1+2 e^5}}+\text {Li}_2\left (-\frac {e^x}{1+\sqrt {1+2 e^5}}\right )+\frac {\text {Li}_2\left (-\frac {e^x}{1+\sqrt {1+2 e^5}}\right )}{\sqrt {1+2 e^5}}-2 \int \frac {e^{2 x} x}{-2 e^5+2 e^x+e^{2 x}} \, dx-2 \left (4 \int \frac {e^x}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)} \, dx\right )-4 \int \frac {e^{2 x}}{\left (-2 e^5+2 e^x+e^{2 x}\right ) (-5+x)} \, dx-\left (8 e^5\right ) \int \frac {1}{\left (2 e^5-2 e^x-e^{2 x}\right ) (-5+x)} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.70, size = 48, normalized size = 1.92 \begin {gather*} 2 \log \left (-2 e^5+2 e^x+e^{2 x}\right )+\frac {\left (2-5 x+x^2\right ) \log \left (-2+e^{-5+x} \left (2+e^x\right )\right )}{-5+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-5 + x)*(2 + E^x)*(80 + 14*x - 16*x^2 + 2*x^3 + E^x*(80 + 14*x - 16*x^2 + 2*x^3)) + (-92 + 40*x
- 4*x^2 + E^x*(-46 + 20*x - 2*x^2) + E^(-5 + x)*(2 + E^x)*(46 - 20*x + 2*x^2 + E^x*(23 - 10*x + x^2)))*Log[-2
+ E^(-5 + x)*(2 + E^x)])/(-100 + 40*x - 4*x^2 + E^x*(-50 + 20*x - 2*x^2) + E^(-5 + x)*(2 + E^x)*(50 - 20*x + 2
*x^2 + E^x*(25 - 10*x + x^2))),x]

[Out]

2*Log[-2*E^5 + 2*E^x + E^(2*x)] + ((2 - 5*x + x^2)*Log[-2 + E^(-5 + x)*(2 + E^x)])/(-5 + x)

________________________________________________________________________________________

fricas [A]  time = 0.55, size = 34, normalized size = 1.36 \begin {gather*} \frac {{\left (x^{2} - 3 \, x - 8\right )} \log \left (-{\left (2 \, e^{5} - e^{\left (2 \, x\right )} - 2 \, e^{x}\right )} e^{\left (-5\right )}\right )}{x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x^2-10*x+23)*exp(x)+2*x^2-20*x+46)*exp(log(exp(x)+2)+x-5)+(-2*x^2+20*x-46)*exp(x)-4*x^2+40*x-92)
*log(exp(log(exp(x)+2)+x-5)-2)+((2*x^3-16*x^2+14*x+80)*exp(x)+2*x^3-16*x^2+14*x+80)*exp(log(exp(x)+2)+x-5))/((
(x^2-10*x+25)*exp(x)+2*x^2-20*x+50)*exp(log(exp(x)+2)+x-5)+(-2*x^2+20*x-50)*exp(x)-4*x^2+40*x-100),x, algorith
m="fricas")

[Out]

(x^2 - 3*x - 8)*log(-(2*e^5 - e^(2*x) - 2*e^x)*e^(-5))/(x - 5)

________________________________________________________________________________________

giac [B]  time = 8.29, size = 104, normalized size = 4.16 \begin {gather*} \frac {x^{2} \log \left (-2 \, e^{5} + e^{\left (2 \, x\right )} + 2 \, e^{x}\right ) - 5 \, x^{2} + 2 \, x \log \left (2 \, e^{5} - e^{\left (2 \, x\right )} - 2 \, e^{x}\right ) - 5 \, x \log \left (-2 \, e^{5} + e^{\left (2 \, x\right )} + 2 \, e^{x}\right ) + 25 \, x - 10 \, \log \left (2 \, e^{5} - e^{\left (2 \, x\right )} - 2 \, e^{x}\right ) + 2 \, \log \left (-2 \, e^{5} + e^{\left (2 \, x\right )} + 2 \, e^{x}\right ) - 10}{x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x^2-10*x+23)*exp(x)+2*x^2-20*x+46)*exp(log(exp(x)+2)+x-5)+(-2*x^2+20*x-46)*exp(x)-4*x^2+40*x-92)
*log(exp(log(exp(x)+2)+x-5)-2)+((2*x^3-16*x^2+14*x+80)*exp(x)+2*x^3-16*x^2+14*x+80)*exp(log(exp(x)+2)+x-5))/((
(x^2-10*x+25)*exp(x)+2*x^2-20*x+50)*exp(log(exp(x)+2)+x-5)+(-2*x^2+20*x-50)*exp(x)-4*x^2+40*x-100),x, algorith
m="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.08, size = 44, normalized size = 1.76

method	result	size
risch	\(\frac {\left (x^{2}-5 x +2\right ) \ln \left (\left ({\mathrm e}^{x}+2\right ) {\mathrm e}^{x -5}-2\right )}{x -5}+2 \ln \left ({\mathrm e}^{2 x}+2 \,{\mathrm e}^{x}-2 \,{\mathrm e}^{5}\right )\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((x^2-10*x+23)*exp(x)+2*x^2-20*x+46)*exp(ln(exp(x)+2)+x-5)+(-2*x^2+20*x-46)*exp(x)-4*x^2+40*x-92)*ln(exp
(ln(exp(x)+2)+x-5)-2)+((2*x^3-16*x^2+14*x+80)*exp(x)+2*x^3-16*x^2+14*x+80)*exp(ln(exp(x)+2)+x-5))/(((x^2-10*x+
25)*exp(x)+2*x^2-20*x+50)*exp(ln(exp(x)+2)+x-5)+(-2*x^2+20*x-50)*exp(x)-4*x^2+40*x-100),x,method=_RETURNVERBOS
E)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.46, size = 41, normalized size = 1.64 \begin {gather*} -\frac {5 \, x^{2} - {\left (x^{2} - 3 \, x - 8\right )} \log \left (-2 \, e^{5} + e^{\left (2 \, x\right )} + 2 \, e^{x}\right ) - 25 \, x + 10}{x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x^2-10*x+23)*exp(x)+2*x^2-20*x+46)*exp(log(exp(x)+2)+x-5)+(-2*x^2+20*x-46)*exp(x)-4*x^2+40*x-92)
*log(exp(log(exp(x)+2)+x-5)-2)+((2*x^3-16*x^2+14*x+80)*exp(x)+2*x^3-16*x^2+14*x+80)*exp(log(exp(x)+2)+x-5))/((
(x^2-10*x+25)*exp(x)+2*x^2-20*x+50)*exp(log(exp(x)+2)+x-5)+(-2*x^2+20*x-50)*exp(x)-4*x^2+40*x-100),x, algorith
m="maxima")

[Out]

-(5*x^2 - (x^2 - 3*x - 8)*log(-2*e^5 + e^(2*x) + 2*e^x) - 25*x + 10)/(x - 5)

________________________________________________________________________________________

mupad [B]  time = 5.54, size = 61, normalized size = 2.44 \begin {gather*} -3\,\ln \left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^5+2\,{\mathrm {e}}^x\right )-\ln \left ({\mathrm {e}}^{-5}\,{\mathrm {e}}^x\,\left ({\mathrm {e}}^x+2\right )-2\right )\,\left (\frac {10\,x-2\,x^2}{x-5}+\frac {x^2-10\,x+23}{x-5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(exp(x + log(exp(x) + 2) - 5) - 2)*(exp(x)*(2*x^2 - 20*x + 46) - 40*x + 4*x^2 - exp(x + log(exp(x) + 2
) - 5)*(exp(x)*(x^2 - 10*x + 23) - 20*x + 2*x^2 + 46) + 92) - exp(x + log(exp(x) + 2) - 5)*(14*x - 16*x^2 + 2*
x^3 + exp(x)*(14*x - 16*x^2 + 2*x^3 + 80) + 80))/(exp(x)*(2*x^2 - 20*x + 50) - 40*x + 4*x^2 - exp(x + log(exp(
x) + 2) - 5)*(exp(x)*(x^2 - 10*x + 25) - 20*x + 2*x^2 + 50) + 100),x)

[Out]

- 3*log(exp(2*x) - 2*exp(5) + 2*exp(x)) - log(exp(-5)*exp(x)*(exp(x) + 2) - 2)*((10*x - 2*x^2)/(x - 5) + (x^2
- 10*x + 23)/(x - 5))

________________________________________________________________________________________

sympy [B]  time = 0.65, size = 44, normalized size = 1.76 \begin {gather*} 2 \log {\left (e^{2 x} + 2 e^{x} - 2 e^{5} \right )} + \frac {\left (x^{2} - 5 x + 2\right ) \log {\left (\frac {\left (e^{x} + 2\right ) e^{x}}{e^{5}} - 2 \right )}}{x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x**2-10*x+23)*exp(x)+2*x**2-20*x+46)*exp(ln(exp(x)+2)+x-5)+(-2*x**2+20*x-46)*exp(x)-4*x**2+40*x-
92)*ln(exp(ln(exp(x)+2)+x-5)-2)+((2*x**3-16*x**2+14*x+80)*exp(x)+2*x**3-16*x**2+14*x+80)*exp(ln(exp(x)+2)+x-5)
)/(((x**2-10*x+25)*exp(x)+2*x**2-20*x+50)*exp(ln(exp(x)+2)+x-5)+(-2*x**2+20*x-50)*exp(x)-4*x**2+40*x-100),x)

[Out]

2*log(exp(2*x) + 2*exp(x) - 2*exp(5)) + (x**2 - 5*x + 2)*log((exp(x) + 2)*exp(-5)*exp(x) - 2)/(x - 5)

________________________________________________________________________________________