3.25.53 \(\int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} (80-240 x+240 x^2)+e^{2 x} (-120+240 x-280 x^2+240 x^3)+e^x (80-80 x+80 x^4)+(20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} (-80-80 x+240 x^2)+e^{2 x} (120-160 x-40 x^2+240 x^3)+e^x (-80+240 x-320 x^2+160 x^3+80 x^4)) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx\)

Optimal. Leaf size=23 \[ 5+\frac {20 x \left (x+\left (-1+e^x+x\right )^2\right )^2}{(1+\log (x))^2} \]

________________________________________________________________________________________

Rubi [F]  time = 5.77, 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 {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} \left (80-240 x+240 x^2\right )+e^{2 x} \left (-120+240 x-280 x^2+240 x^3\right )+e^x \left (80-80 x+80 x^4\right )+\left (20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} \left (-80-80 x+240 x^2\right )+e^{2 x} \left (120-160 x-40 x^2+240 x^3\right )+e^x \left (-80+240 x-320 x^2+160 x^3+80 x^4\right )\right ) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-20 + 60*x^2 - 80*x^3 + 60*x^4 + E^(4*x)*(-20 + 80*x) + E^(3*x)*(80 - 240*x + 240*x^2) + E^(2*x)*(-120 +
240*x - 280*x^2 + 240*x^3) + E^x*(80 - 80*x + 80*x^4) + (20 - 80*x + 180*x^2 - 160*x^3 + 100*x^4 + E^(4*x)*(20
+ 80*x) + E^(3*x)*(-80 - 80*x + 240*x^2) + E^(2*x)*(120 - 160*x - 40*x^2 + 240*x^3) + E^x*(-80 + 240*x - 320*
x^2 + 160*x^3 + 80*x^4))*Log[x])/(1 + 3*Log[x] + 3*Log[x]^2 + Log[x]^3),x]

[Out]

-160*x^2 + 540*x^3 - 640*x^4 + 500*x^5 - (80*ExpIntegralEi[2*(1 + Log[x])])/E^2 + (540*ExpIntegralEi[3*(1 + Lo
g[x])])/E^3 - (960*ExpIntegralEi[4*(1 + Log[x])])/E^4 + (1000*ExpIntegralEi[5*(1 + Log[x])])/E^5 - (160*ExpInt
egralEi[2*(1 + Log[x])]*Log[x])/E^2 + (810*ExpIntegralEi[3*(1 + Log[x])]*Log[x])/E^3 - (1280*ExpIntegralEi[4*(
1 + Log[x])]*Log[x])/E^4 + (1250*ExpIntegralEi[5*(1 + Log[x])]*Log[x])/E^5 + (20*x)/(1 + Log[x])^2 - (30*x^3)/
(1 + Log[x])^2 + (40*x^4)/(1 + Log[x])^2 - (30*x^5)/(1 + Log[x])^2 + (40*x^2*Log[x])/(1 + Log[x])^2 - (90*x^3*
Log[x])/(1 + Log[x])^2 + (80*x^4*Log[x])/(1 + Log[x])^2 - (50*x^5*Log[x])/(1 + Log[x])^2 - (90*x^3)/(1 + Log[x
]) + (160*x^4)/(1 + Log[x]) - (150*x^5)/(1 + Log[x]) + (80*x^2*Log[x])/(1 + Log[x]) - (270*x^3*Log[x])/(1 + Lo
g[x]) + (320*x^4*Log[x])/(1 + Log[x]) - (250*x^5*Log[x])/(1 + Log[x]) + (320*ExpIntegralEi[2*(1 + Log[x])]*(1
+ Log[x]))/E^2 - (1620*ExpIntegralEi[3*(1 + Log[x])]*(1 + Log[x]))/E^3 + (2560*ExpIntegralEi[4*(1 + Log[x])]*(
1 + Log[x]))/E^4 - (2500*ExpIntegralEi[5*(1 + Log[x])]*(1 + Log[x]))/E^5 - (80*ExpIntegralEi[2*(1 + Log[x])]*(
3 + 2*Log[x]))/E^2 + (40*x^2*(3 + 2*Log[x]))/(1 + Log[x]) + (270*ExpIntegralEi[3*(1 + Log[x])]*(4 + 3*Log[x]))
/E^3 - (90*x^3*(4 + 3*Log[x]))/(1 + Log[x]) - (320*ExpIntegralEi[4*(1 + Log[x])]*(5 + 4*Log[x]))/E^4 + (80*x^4
*(5 + 4*Log[x]))/(1 + Log[x]) + (250*ExpIntegralEi[5*(1 + Log[x])]*(6 + 5*Log[x]))/E^5 - (50*x^5*(6 + 5*Log[x]
))/(1 + Log[x]) + (20*E^(4*x)*(x + x*Log[x]))/(1 + Log[x])^3 + 160*Defer[Int][E^x/(1 + Log[x])^3, x] - 240*Def
er[Int][E^(2*x)/(1 + Log[x])^3, x] + 160*Defer[Int][E^(3*x)/(1 + Log[x])^3, x] - 320*Defer[Int][(E^x*x)/(1 + L
og[x])^3, x] + 400*Defer[Int][(E^(2*x)*x)/(1 + Log[x])^3, x] - 160*Defer[Int][(E^(3*x)*x)/(1 + Log[x])^3, x] +
320*Defer[Int][(E^x*x^2)/(1 + Log[x])^3, x] - 240*Defer[Int][(E^(2*x)*x^2)/(1 + Log[x])^3, x] - 160*Defer[Int
][(E^x*x^3)/(1 + Log[x])^3, x] - 80*Defer[Int][E^x/(1 + Log[x])^2, x] + 120*Defer[Int][E^(2*x)/(1 + Log[x])^2,
x] - 80*Defer[Int][E^(3*x)/(1 + Log[x])^2, x] + 240*Defer[Int][(E^x*x)/(1 + Log[x])^2, x] - 160*Defer[Int][(E
^(2*x)*x)/(1 + Log[x])^2, x] - 80*Defer[Int][(E^(3*x)*x)/(1 + Log[x])^2, x] - 320*Defer[Int][(E^x*x^2)/(1 + Lo
g[x])^2, x] - 40*Defer[Int][(E^(2*x)*x^2)/(1 + Log[x])^2, x] + 240*Defer[Int][(E^(3*x)*x^2)/(1 + Log[x])^2, x]
+ 160*Defer[Int][(E^x*x^3)/(1 + Log[x])^2, x] + 240*Defer[Int][(E^(2*x)*x^3)/(1 + Log[x])^2, x] + 80*Defer[In
t][(E^x*x^4)/(1 + Log[x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {20 \left (1+e^{2 x}+2 e^x (-1+x)-x+x^2\right ) \left (-1-x+3 x^2+e^{2 x} (-1+4 x)+e^x \left (2-2 x+4 x^2\right )+\left (1-3 x+5 x^2+e^{2 x} (1+4 x)+2 e^x \left (-1+x+2 x^2\right )\right ) \log (x)\right )}{(1+\log (x))^3} \, dx\\ &=20 \int \frac {\left (1+e^{2 x}+2 e^x (-1+x)-x+x^2\right ) \left (-1-x+3 x^2+e^{2 x} (-1+4 x)+e^x \left (2-2 x+4 x^2\right )+\left (1-3 x+5 x^2+e^{2 x} (1+4 x)+2 e^x \left (-1+x+2 x^2\right )\right ) \log (x)\right )}{(1+\log (x))^3} \, dx\\ &=20 \int \left (-\frac {1}{(1+\log (x))^3}+\frac {3 x^2}{(1+\log (x))^3}-\frac {4 x^3}{(1+\log (x))^3}+\frac {3 x^4}{(1+\log (x))^3}+\frac {\log (x)}{(1+\log (x))^3}-\frac {4 x \log (x)}{(1+\log (x))^3}+\frac {9 x^2 \log (x)}{(1+\log (x))^3}-\frac {8 x^3 \log (x)}{(1+\log (x))^3}+\frac {5 x^4 \log (x)}{(1+\log (x))^3}+\frac {e^{4 x} (-1+4 x+\log (x)+4 x \log (x))}{(1+\log (x))^3}+\frac {4 e^{3 x} \left (1-3 x+3 x^2-\log (x)-x \log (x)+3 x^2 \log (x)\right )}{(1+\log (x))^3}+\frac {2 e^{2 x} \left (-3+6 x-7 x^2+6 x^3+3 \log (x)-4 x \log (x)-x^2 \log (x)+6 x^3 \log (x)\right )}{(1+\log (x))^3}+\frac {4 e^x \left (1-x+x^4-\log (x)+3 x \log (x)-4 x^2 \log (x)+2 x^3 \log (x)+x^4 \log (x)\right )}{(1+\log (x))^3}\right ) \, dx\\ &=-\left (20 \int \frac {1}{(1+\log (x))^3} \, dx\right )+20 \int \frac {\log (x)}{(1+\log (x))^3} \, dx+20 \int \frac {e^{4 x} (-1+4 x+\log (x)+4 x \log (x))}{(1+\log (x))^3} \, dx+40 \int \frac {e^{2 x} \left (-3+6 x-7 x^2+6 x^3+3 \log (x)-4 x \log (x)-x^2 \log (x)+6 x^3 \log (x)\right )}{(1+\log (x))^3} \, dx+60 \int \frac {x^2}{(1+\log (x))^3} \, dx+60 \int \frac {x^4}{(1+\log (x))^3} \, dx-80 \int \frac {x^3}{(1+\log (x))^3} \, dx-80 \int \frac {x \log (x)}{(1+\log (x))^3} \, dx+80 \int \frac {e^{3 x} \left (1-3 x+3 x^2-\log (x)-x \log (x)+3 x^2 \log (x)\right )}{(1+\log (x))^3} \, dx+80 \int \frac {e^x \left (1-x+x^4-\log (x)+3 x \log (x)-4 x^2 \log (x)+2 x^3 \log (x)+x^4 \log (x)\right )}{(1+\log (x))^3} \, dx+100 \int \frac {x^4 \log (x)}{(1+\log (x))^3} \, dx-160 \int \frac {x^3 \log (x)}{(1+\log (x))^3} \, dx+180 \int \frac {x^2 \log (x)}{(1+\log (x))^3} \, dx\\ &=-\frac {160 \text {Ei}(2 (1+\log (x))) \log (x)}{e^2}+\frac {810 \text {Ei}(3 (1+\log (x))) \log (x)}{e^3}-\frac {1280 \text {Ei}(4 (1+\log (x))) \log (x)}{e^4}+\frac {1250 \text {Ei}(5 (1+\log (x))) \log (x)}{e^5}+\frac {10 x}{(1+\log (x))^2}-\frac {30 x^3}{(1+\log (x))^2}+\frac {40 x^4}{(1+\log (x))^2}-\frac {30 x^5}{(1+\log (x))^2}+\frac {40 x^2 \log (x)}{(1+\log (x))^2}-\frac {90 x^3 \log (x)}{(1+\log (x))^2}+\frac {80 x^4 \log (x)}{(1+\log (x))^2}-\frac {50 x^5 \log (x)}{(1+\log (x))^2}+\frac {80 x^2 \log (x)}{1+\log (x)}-\frac {270 x^3 \log (x)}{1+\log (x)}+\frac {320 x^4 \log (x)}{1+\log (x)}-\frac {250 x^5 \log (x)}{1+\log (x)}+\frac {20 e^{4 x} (x+x \log (x))}{(1+\log (x))^3}-10 \int \frac {1}{(1+\log (x))^2} \, dx+20 \int \left (-\frac {1}{(1+\log (x))^3}+\frac {1}{(1+\log (x))^2}\right ) \, dx+40 \int \left (-\frac {2 e^{2 x} \left (3-5 x+3 x^2\right )}{(1+\log (x))^3}+\frac {e^{2 x} \left (3-4 x-x^2+6 x^3\right )}{(1+\log (x))^2}\right ) \, dx+80 \int \left (-\frac {2 e^{3 x} (-1+x)}{(1+\log (x))^3}+\frac {e^{3 x} \left (-1-x+3 x^2\right )}{(1+\log (x))^2}\right ) \, dx+80 \int \left (-\frac {2 e^x \left (-1+2 x-2 x^2+x^3\right )}{(1+\log (x))^3}+\frac {e^x \left (-1+3 x-4 x^2+2 x^3+x^4\right )}{(1+\log (x))^2}\right ) \, dx+80 \int \left (\frac {2 \text {Ei}(2 (1+\log (x)))}{e^2 x}-\frac {x (3+2 \log (x))}{2 (1+\log (x))^2}\right ) \, dx+90 \int \frac {x^2}{(1+\log (x))^2} \, dx-100 \int \left (\frac {25 \text {Ei}(5 (1+\log (x)))}{2 e^5 x}-\frac {x^4 (6+5 \log (x))}{2 (1+\log (x))^2}\right ) \, dx+150 \int \frac {x^4}{(1+\log (x))^2} \, dx-160 \int \frac {x^3}{(1+\log (x))^2} \, dx+160 \int \left (\frac {8 \text {Ei}(4 (1+\log (x)))}{e^4 x}-\frac {x^3 (5+4 \log (x))}{2 (1+\log (x))^2}\right ) \, dx-180 \int \left (\frac {9 \text {Ei}(3 (1+\log (x)))}{2 e^3 x}-\frac {x^2 (4+3 \log (x))}{2 (1+\log (x))^2}\right ) \, dx\\ &=-\frac {160 \text {Ei}(2 (1+\log (x))) \log (x)}{e^2}+\frac {810 \text {Ei}(3 (1+\log (x))) \log (x)}{e^3}-\frac {1280 \text {Ei}(4 (1+\log (x))) \log (x)}{e^4}+\frac {1250 \text {Ei}(5 (1+\log (x))) \log (x)}{e^5}+\frac {10 x}{(1+\log (x))^2}-\frac {30 x^3}{(1+\log (x))^2}+\frac {40 x^4}{(1+\log (x))^2}-\frac {30 x^5}{(1+\log (x))^2}+\frac {40 x^2 \log (x)}{(1+\log (x))^2}-\frac {90 x^3 \log (x)}{(1+\log (x))^2}+\frac {80 x^4 \log (x)}{(1+\log (x))^2}-\frac {50 x^5 \log (x)}{(1+\log (x))^2}+\frac {10 x}{1+\log (x)}-\frac {90 x^3}{1+\log (x)}+\frac {160 x^4}{1+\log (x)}-\frac {150 x^5}{1+\log (x)}+\frac {80 x^2 \log (x)}{1+\log (x)}-\frac {270 x^3 \log (x)}{1+\log (x)}+\frac {320 x^4 \log (x)}{1+\log (x)}-\frac {250 x^5 \log (x)}{1+\log (x)}+\frac {20 e^{4 x} (x+x \log (x))}{(1+\log (x))^3}-10 \int \frac {1}{1+\log (x)} \, dx-20 \int \frac {1}{(1+\log (x))^3} \, dx+20 \int \frac {1}{(1+\log (x))^2} \, dx+40 \int \frac {e^{2 x} \left (3-4 x-x^2+6 x^3\right )}{(1+\log (x))^2} \, dx-40 \int \frac {x (3+2 \log (x))}{(1+\log (x))^2} \, dx+50 \int \frac {x^4 (6+5 \log (x))}{(1+\log (x))^2} \, dx-80 \int \frac {e^{2 x} \left (3-5 x+3 x^2\right )}{(1+\log (x))^3} \, dx+80 \int \frac {e^{3 x} \left (-1-x+3 x^2\right )}{(1+\log (x))^2} \, dx+80 \int \frac {e^x \left (-1+3 x-4 x^2+2 x^3+x^4\right )}{(1+\log (x))^2} \, dx-80 \int \frac {x^3 (5+4 \log (x))}{(1+\log (x))^2} \, dx+90 \int \frac {x^2 (4+3 \log (x))}{(1+\log (x))^2} \, dx-160 \int \frac {e^{3 x} (-1+x)}{(1+\log (x))^3} \, dx-160 \int \frac {e^x \left (-1+2 x-2 x^2+x^3\right )}{(1+\log (x))^3} \, dx+270 \int \frac {x^2}{1+\log (x)} \, dx-640 \int \frac {x^3}{1+\log (x)} \, dx+750 \int \frac {x^4}{1+\log (x)} \, dx-\frac {1250 \int \frac {\text {Ei}(5 (1+\log (x)))}{x} \, dx}{e^5}+\frac {1280 \int \frac {\text {Ei}(4 (1+\log (x)))}{x} \, dx}{e^4}-\frac {810 \int \frac {\text {Ei}(3 (1+\log (x)))}{x} \, dx}{e^3}+\frac {160 \int \frac {\text {Ei}(2 (1+\log (x)))}{x} \, dx}{e^2}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.11, size = 32, normalized size = 1.39 \begin {gather*} \frac {20 x \left (1+e^{2 x}+2 e^x (-1+x)-x+x^2\right )^2}{(1+\log (x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-20 + 60*x^2 - 80*x^3 + 60*x^4 + E^(4*x)*(-20 + 80*x) + E^(3*x)*(80 - 240*x + 240*x^2) + E^(2*x)*(-
120 + 240*x - 280*x^2 + 240*x^3) + E^x*(80 - 80*x + 80*x^4) + (20 - 80*x + 180*x^2 - 160*x^3 + 100*x^4 + E^(4*
x)*(20 + 80*x) + E^(3*x)*(-80 - 80*x + 240*x^2) + E^(2*x)*(120 - 160*x - 40*x^2 + 240*x^3) + E^x*(-80 + 240*x
- 320*x^2 + 160*x^3 + 80*x^4))*Log[x])/(1 + 3*Log[x] + 3*Log[x]^2 + Log[x]^3),x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.64, size = 94, normalized size = 4.09 \begin {gather*} \frac {20 \, {\left (x^{5} - 2 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} + x e^{\left (4 \, x\right )} + 4 \, {\left (x^{2} - x\right )} e^{\left (3 \, x\right )} + 2 \, {\left (3 \, x^{3} - 5 \, x^{2} + 3 \, x\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{4} - 2 \, x^{3} + 2 \, x^{2} - x\right )} e^{x} + x\right )}}{\log \relax (x)^{2} + 2 \, \log \relax (x) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((80*x+20)*exp(x)^4+(240*x^2-80*x-80)*exp(x)^3+(240*x^3-40*x^2-160*x+120)*exp(x)^2+(80*x^4+160*x^3-
320*x^2+240*x-80)*exp(x)+100*x^4-160*x^3+180*x^2-80*x+20)*log(x)+(80*x-20)*exp(x)^4+(240*x^2-240*x+80)*exp(x)^
3+(240*x^3-280*x^2+240*x-120)*exp(x)^2+(80*x^4-80*x+80)*exp(x)+60*x^4-80*x^3+60*x^2-20)/(log(x)^3+3*log(x)^2+3
*log(x)+1),x, algorithm="fricas")

[Out]

20*(x^5 - 2*x^4 + 3*x^3 - 2*x^2 + x*e^(4*x) + 4*(x^2 - x)*e^(3*x) + 2*(3*x^3 - 5*x^2 + 3*x)*e^(2*x) + 4*(x^4 -
2*x^3 + 2*x^2 - x)*e^x + x)/(log(x)^2 + 2*log(x) + 1)

________________________________________________________________________________________

giac [B]  time = 0.83, size = 107, normalized size = 4.65 \begin {gather*} \frac {20 \, {\left (x^{5} + 4 \, x^{4} e^{x} - 2 \, x^{4} + 6 \, x^{3} e^{\left (2 \, x\right )} - 8 \, x^{3} e^{x} + 3 \, x^{3} + 4 \, x^{2} e^{\left (3 \, x\right )} - 10 \, x^{2} e^{\left (2 \, x\right )} + 8 \, x^{2} e^{x} - 2 \, x^{2} + x e^{\left (4 \, x\right )} - 4 \, x e^{\left (3 \, x\right )} + 6 \, x e^{\left (2 \, x\right )} - 4 \, x e^{x} + x\right )}}{\log \relax (x)^{2} + 2 \, \log \relax (x) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((80*x+20)*exp(x)^4+(240*x^2-80*x-80)*exp(x)^3+(240*x^3-40*x^2-160*x+120)*exp(x)^2+(80*x^4+160*x^3-
320*x^2+240*x-80)*exp(x)+100*x^4-160*x^3+180*x^2-80*x+20)*log(x)+(80*x-20)*exp(x)^4+(240*x^2-240*x+80)*exp(x)^
3+(240*x^3-280*x^2+240*x-120)*exp(x)^2+(80*x^4-80*x+80)*exp(x)+60*x^4-80*x^3+60*x^2-20)/(log(x)^3+3*log(x)^2+3
*log(x)+1),x, algorithm="giac")

[Out]

20*(x^5 + 4*x^4*e^x - 2*x^4 + 6*x^3*e^(2*x) - 8*x^3*e^x + 3*x^3 + 4*x^2*e^(3*x) - 10*x^2*e^(2*x) + 8*x^2*e^x -
2*x^2 + x*e^(4*x) - 4*x*e^(3*x) + 6*x*e^(2*x) - 4*x*e^x + x)/(log(x)^2 + 2*log(x) + 1)

________________________________________________________________________________________

maple [B]  time = 0.11, size = 90, normalized size = 3.91

method	result	size
risch	\(\frac {20 x \left (x^{4}+4 \,{\mathrm e}^{x} x^{3}+6 \,{\mathrm e}^{2 x} x^{2}+4 x \,{\mathrm e}^{3 x}+{\mathrm e}^{4 x}-2 x^{3}-8 \,{\mathrm e}^{x} x^{2}-10 x \,{\mathrm e}^{2 x}-4 \,{\mathrm e}^{3 x}+3 x^{2}+8 \,{\mathrm e}^{x} x +6 \,{\mathrm e}^{2 x}-2 x -4 \,{\mathrm e}^{x}+1\right )}{\left (\ln \relax (x )+1\right )^{2}}\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((80*x+20)*exp(x)^4+(240*x^2-80*x-80)*exp(x)^3+(240*x^3-40*x^2-160*x+120)*exp(x)^2+(80*x^4+160*x^3-320*x^
2+240*x-80)*exp(x)+100*x^4-160*x^3+180*x^2-80*x+20)*ln(x)+(80*x-20)*exp(x)^4+(240*x^2-240*x+80)*exp(x)^3+(240*
x^3-280*x^2+240*x-120)*exp(x)^2+(80*x^4-80*x+80)*exp(x)+60*x^4-80*x^3+60*x^2-20)/(ln(x)^3+3*ln(x)^2+3*ln(x)+1)
,x,method=_RETURNVERBOSE)

[Out]

20*x*(x^4+4*exp(x)*x^3+6*exp(2*x)*x^2+4*x*exp(3*x)+exp(4*x)-2*x^3-8*exp(x)*x^2-10*x*exp(2*x)-4*exp(3*x)+3*x^2+
8*exp(x)*x+6*exp(2*x)-2*x-4*exp(x)+1)/(ln(x)+1)^2

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {10 \, {\left (20 \, x^{5} - 24 \, x^{4} + 18 \, x^{3} - 4 \, x^{2} + 2 \, x e^{\left (4 \, x\right )} + 8 \, {\left (x^{2} - x\right )} e^{\left (3 \, x\right )} + 4 \, {\left (3 \, x^{3} - 5 \, x^{2} + 3 \, x\right )} e^{\left (2 \, x\right )} + 8 \, {\left (x^{4} - 2 \, x^{3} + 2 \, x^{2} - x\right )} e^{x} + {\left (15 \, x^{5} - 16 \, x^{4} + 9 \, x^{3} - x\right )} \log \relax (x)\right )}}{\log \relax (x)^{2} + 2 \, \log \relax (x) + 1} + \frac {20 \, e^{\left (-1\right )} E_{3}\left (-\log \relax (x) - 1\right )}{{\left (\log \relax (x) + 1\right )}^{2}} - \frac {60 \, e^{\left (-3\right )} E_{3}\left (-3 \, \log \relax (x) - 3\right )}{{\left (\log \relax (x) + 1\right )}^{2}} + \frac {80 \, e^{\left (-4\right )} E_{3}\left (-4 \, \log \relax (x) - 4\right )}{{\left (\log \relax (x) + 1\right )}^{2}} - \frac {60 \, e^{\left (-5\right )} E_{3}\left (-5 \, \log \relax (x) - 5\right )}{{\left (\log \relax (x) + 1\right )}^{2}} - 20 \, \int \frac {75 \, x^{4} - 64 \, x^{3} + 27 \, x^{2} - 1}{2 \, {\left (\log \relax (x) + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((80*x+20)*exp(x)^4+(240*x^2-80*x-80)*exp(x)^3+(240*x^3-40*x^2-160*x+120)*exp(x)^2+(80*x^4+160*x^3-
320*x^2+240*x-80)*exp(x)+100*x^4-160*x^3+180*x^2-80*x+20)*log(x)+(80*x-20)*exp(x)^4+(240*x^2-240*x+80)*exp(x)^
3+(240*x^3-280*x^2+240*x-120)*exp(x)^2+(80*x^4-80*x+80)*exp(x)+60*x^4-80*x^3+60*x^2-20)/(log(x)^3+3*log(x)^2+3
*log(x)+1),x, algorithm="maxima")

[Out]

10*(20*x^5 - 24*x^4 + 18*x^3 - 4*x^2 + 2*x*e^(4*x) + 8*(x^2 - x)*e^(3*x) + 4*(3*x^3 - 5*x^2 + 3*x)*e^(2*x) + 8
*(x^4 - 2*x^3 + 2*x^2 - x)*e^x + (15*x^5 - 16*x^4 + 9*x^3 - x)*log(x))/(log(x)^2 + 2*log(x) + 1) + 20*e^(-1)*e
xp_integral_e(3, -log(x) - 1)/(log(x) + 1)^2 - 60*e^(-3)*exp_integral_e(3, -3*log(x) - 3)/(log(x) + 1)^2 + 80*
e^(-4)*exp_integral_e(3, -4*log(x) - 4)/(log(x) + 1)^2 - 60*e^(-5)*exp_integral_e(3, -5*log(x) - 5)/(log(x) +
1)^2 - 20*integrate(1/2*(75*x^4 - 64*x^3 + 27*x^2 - 1)/(log(x) + 1), x)

________________________________________________________________________________________

mupad [B]  time = 1.77, size = 657, normalized size = 28.57 \begin {gather*} 10\,x-\frac {10\,x\,\left (4\,{\mathrm {e}}^{3\,x}-6\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+12\,x\,{\mathrm {e}}^{2\,x}-12\,x\,{\mathrm {e}}^{3\,x}+4\,x\,{\mathrm {e}}^{4\,x}+4\,x^4\,{\mathrm {e}}^x-14\,x^2\,{\mathrm {e}}^{2\,x}+12\,x^2\,{\mathrm {e}}^{3\,x}+12\,x^3\,{\mathrm {e}}^{2\,x}-4\,x\,{\mathrm {e}}^x+3\,x^2-4\,x^3+3\,x^4-1\right )+10\,x\,\ln \relax (x)\,\left (6\,{\mathrm {e}}^{2\,x}-4\,x-4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^x-8\,x\,{\mathrm {e}}^{2\,x}-4\,x\,{\mathrm {e}}^{3\,x}+4\,x\,{\mathrm {e}}^{4\,x}-16\,x^2\,{\mathrm {e}}^x+8\,x^3\,{\mathrm {e}}^x+4\,x^4\,{\mathrm {e}}^x-2\,x^2\,{\mathrm {e}}^{2\,x}+12\,x^2\,{\mathrm {e}}^{3\,x}+12\,x^3\,{\mathrm {e}}^{2\,x}+12\,x\,{\mathrm {e}}^x+9\,x^2-8\,x^3+5\,x^4+1\right )}{{\ln \relax (x)}^2+2\,\ln \relax (x)+1}+{\mathrm {e}}^{4\,x}\,\left (160\,x^3+120\,x^2+10\,x\right )-\frac {20\,x\,\left (2\,x\,{\mathrm {e}}^{2\,x}-2\,x-8\,x\,{\mathrm {e}}^{3\,x}+4\,x\,{\mathrm {e}}^{4\,x}-10\,x^2\,{\mathrm {e}}^x+4\,x^3\,{\mathrm {e}}^x+12\,x^4\,{\mathrm {e}}^x+2\,x^5\,{\mathrm {e}}^x-10\,x^2\,{\mathrm {e}}^{2\,x}+6\,x^2\,{\mathrm {e}}^{3\,x}+16\,x^3\,{\mathrm {e}}^{2\,x}+8\,x^2\,{\mathrm {e}}^{4\,x}+18\,x^3\,{\mathrm {e}}^{3\,x}+12\,x^4\,{\mathrm {e}}^{2\,x}+4\,x\,{\mathrm {e}}^x+9\,x^2-12\,x^3+10\,x^4\right )+10\,x\,\ln \relax (x)\,\left (6\,{\mathrm {e}}^{2\,x}-8\,x-4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^x-4\,x\,{\mathrm {e}}^{2\,x}-20\,x\,{\mathrm {e}}^{3\,x}+12\,x\,{\mathrm {e}}^{4\,x}-36\,x^2\,{\mathrm {e}}^x+16\,x^3\,{\mathrm {e}}^x+28\,x^4\,{\mathrm {e}}^x+4\,x^5\,{\mathrm {e}}^x-22\,x^2\,{\mathrm {e}}^{2\,x}+24\,x^2\,{\mathrm {e}}^{3\,x}+44\,x^3\,{\mathrm {e}}^{2\,x}+16\,x^2\,{\mathrm {e}}^{4\,x}+36\,x^3\,{\mathrm {e}}^{3\,x}+24\,x^4\,{\mathrm {e}}^{2\,x}+20\,x\,{\mathrm {e}}^x+27\,x^2-32\,x^3+25\,x^4+1\right )}{\ln \relax (x)+1}-{\mathrm {e}}^{3\,x}\,\left (-360\,x^4-240\,x^3+200\,x^2+40\,x\right )+{\mathrm {e}}^x\,\left (40\,x^6+280\,x^5+160\,x^4-360\,x^3+200\,x^2-40\,x\right )+{\mathrm {e}}^{2\,x}\,\left (240\,x^5+440\,x^4-220\,x^3-40\,x^2+60\,x\right )-80\,x^2+270\,x^3-320\,x^4+250\,x^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3*x)*(240*x^2 - 240*x + 80) + exp(2*x)*(240*x - 280*x^2 + 240*x^3 - 120) + log(x)*(exp(x)*(240*x - 32
0*x^2 + 160*x^3 + 80*x^4 - 80) - exp(3*x)*(80*x - 240*x^2 + 80) - 80*x - exp(2*x)*(160*x + 40*x^2 - 240*x^3 -
120) + exp(4*x)*(80*x + 20) + 180*x^2 - 160*x^3 + 100*x^4 + 20) + exp(x)*(80*x^4 - 80*x + 80) + exp(4*x)*(80*x
- 20) + 60*x^2 - 80*x^3 + 60*x^4 - 20)/(3*log(x) + 3*log(x)^2 + log(x)^3 + 1),x)

[Out]

10*x - (10*x*(4*exp(3*x) - 6*exp(2*x) - exp(4*x) + 4*exp(x) + 12*x*exp(2*x) - 12*x*exp(3*x) + 4*x*exp(4*x) + 4
*x^4*exp(x) - 14*x^2*exp(2*x) + 12*x^2*exp(3*x) + 12*x^3*exp(2*x) - 4*x*exp(x) + 3*x^2 - 4*x^3 + 3*x^4 - 1) +
10*x*log(x)*(6*exp(2*x) - 4*x - 4*exp(3*x) + exp(4*x) - 4*exp(x) - 8*x*exp(2*x) - 4*x*exp(3*x) + 4*x*exp(4*x)
- 16*x^2*exp(x) + 8*x^3*exp(x) + 4*x^4*exp(x) - 2*x^2*exp(2*x) + 12*x^2*exp(3*x) + 12*x^3*exp(2*x) + 12*x*exp(
x) + 9*x^2 - 8*x^3 + 5*x^4 + 1))/(2*log(x) + log(x)^2 + 1) + exp(4*x)*(10*x + 120*x^2 + 160*x^3) - (20*x*(2*x*
exp(2*x) - 2*x - 8*x*exp(3*x) + 4*x*exp(4*x) - 10*x^2*exp(x) + 4*x^3*exp(x) + 12*x^4*exp(x) + 2*x^5*exp(x) - 1
0*x^2*exp(2*x) + 6*x^2*exp(3*x) + 16*x^3*exp(2*x) + 8*x^2*exp(4*x) + 18*x^3*exp(3*x) + 12*x^4*exp(2*x) + 4*x*e
xp(x) + 9*x^2 - 12*x^3 + 10*x^4) + 10*x*log(x)*(6*exp(2*x) - 8*x - 4*exp(3*x) + exp(4*x) - 4*exp(x) - 4*x*exp(
2*x) - 20*x*exp(3*x) + 12*x*exp(4*x) - 36*x^2*exp(x) + 16*x^3*exp(x) + 28*x^4*exp(x) + 4*x^5*exp(x) - 22*x^2*e
xp(2*x) + 24*x^2*exp(3*x) + 44*x^3*exp(2*x) + 16*x^2*exp(4*x) + 36*x^3*exp(3*x) + 24*x^4*exp(2*x) + 20*x*exp(x
) + 27*x^2 - 32*x^3 + 25*x^4 + 1))/(log(x) + 1) - exp(3*x)*(40*x + 200*x^2 - 240*x^3 - 360*x^4) + exp(x)*(200*
x^2 - 40*x - 360*x^3 + 160*x^4 + 280*x^5 + 40*x^6) + exp(2*x)*(60*x - 40*x^2 - 220*x^3 + 440*x^4 + 240*x^5) -
80*x^2 + 270*x^3 - 320*x^4 + 250*x^5

________________________________________________________________________________________

sympy [B]  time = 0.76, size = 704, normalized size = 30.61 \begin {gather*} \frac {\left (20 x \log {\relax (x )}^{6} + 120 x \log {\relax (x )}^{5} + 300 x \log {\relax (x )}^{4} + 400 x \log {\relax (x )}^{3} + 300 x \log {\relax (x )}^{2} + 120 x \log {\relax (x )} + 20 x\right ) e^{4 x} + \left (80 x^{2} \log {\relax (x )}^{6} + 480 x^{2} \log {\relax (x )}^{5} + 1200 x^{2} \log {\relax (x )}^{4} + 1600 x^{2} \log {\relax (x )}^{3} + 1200 x^{2} \log {\relax (x )}^{2} + 480 x^{2} \log {\relax (x )} + 80 x^{2} - 80 x \log {\relax (x )}^{6} - 480 x \log {\relax (x )}^{5} - 1200 x \log {\relax (x )}^{4} - 1600 x \log {\relax (x )}^{3} - 1200 x \log {\relax (x )}^{2} - 480 x \log {\relax (x )} - 80 x\right ) e^{3 x} + \left (120 x^{3} \log {\relax (x )}^{6} + 720 x^{3} \log {\relax (x )}^{5} + 1800 x^{3} \log {\relax (x )}^{4} + 2400 x^{3} \log {\relax (x )}^{3} + 1800 x^{3} \log {\relax (x )}^{2} + 720 x^{3} \log {\relax (x )} + 120 x^{3} - 200 x^{2} \log {\relax (x )}^{6} - 1200 x^{2} \log {\relax (x )}^{5} - 3000 x^{2} \log {\relax (x )}^{4} - 4000 x^{2} \log {\relax (x )}^{3} - 3000 x^{2} \log {\relax (x )}^{2} - 1200 x^{2} \log {\relax (x )} - 200 x^{2} + 120 x \log {\relax (x )}^{6} + 720 x \log {\relax (x )}^{5} + 1800 x \log {\relax (x )}^{4} + 2400 x \log {\relax (x )}^{3} + 1800 x \log {\relax (x )}^{2} + 720 x \log {\relax (x )} + 120 x\right ) e^{2 x} + \left (80 x^{4} \log {\relax (x )}^{6} + 480 x^{4} \log {\relax (x )}^{5} + 1200 x^{4} \log {\relax (x )}^{4} + 1600 x^{4} \log {\relax (x )}^{3} + 1200 x^{4} \log {\relax (x )}^{2} + 480 x^{4} \log {\relax (x )} + 80 x^{4} - 160 x^{3} \log {\relax (x )}^{6} - 960 x^{3} \log {\relax (x )}^{5} - 2400 x^{3} \log {\relax (x )}^{4} - 3200 x^{3} \log {\relax (x )}^{3} - 2400 x^{3} \log {\relax (x )}^{2} - 960 x^{3} \log {\relax (x )} - 160 x^{3} + 160 x^{2} \log {\relax (x )}^{6} + 960 x^{2} \log {\relax (x )}^{5} + 2400 x^{2} \log {\relax (x )}^{4} + 3200 x^{2} \log {\relax (x )}^{3} + 2400 x^{2} \log {\relax (x )}^{2} + 960 x^{2} \log {\relax (x )} + 160 x^{2} - 80 x \log {\relax (x )}^{6} - 480 x \log {\relax (x )}^{5} - 1200 x \log {\relax (x )}^{4} - 1600 x \log {\relax (x )}^{3} - 1200 x \log {\relax (x )}^{2} - 480 x \log {\relax (x )} - 80 x\right ) e^{x}}{\log {\relax (x )}^{8} + 8 \log {\relax (x )}^{7} + 28 \log {\relax (x )}^{6} + 56 \log {\relax (x )}^{5} + 70 \log {\relax (x )}^{4} + 56 \log {\relax (x )}^{3} + 28 \log {\relax (x )}^{2} + 8 \log {\relax (x )} + 1} + \frac {20 x^{5} - 40 x^{4} + 60 x^{3} - 40 x^{2} + 20 x}{\log {\relax (x )}^{2} + 2 \log {\relax (x )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((80*x+20)*exp(x)**4+(240*x**2-80*x-80)*exp(x)**3+(240*x**3-40*x**2-160*x+120)*exp(x)**2+(80*x**4+1
60*x**3-320*x**2+240*x-80)*exp(x)+100*x**4-160*x**3+180*x**2-80*x+20)*ln(x)+(80*x-20)*exp(x)**4+(240*x**2-240*
x+80)*exp(x)**3+(240*x**3-280*x**2+240*x-120)*exp(x)**2+(80*x**4-80*x+80)*exp(x)+60*x**4-80*x**3+60*x**2-20)/(
ln(x)**3+3*ln(x)**2+3*ln(x)+1),x)

[Out]

((20*x*log(x)**6 + 120*x*log(x)**5 + 300*x*log(x)**4 + 400*x*log(x)**3 + 300*x*log(x)**2 + 120*x*log(x) + 20*x
)*exp(4*x) + (80*x**2*log(x)**6 + 480*x**2*log(x)**5 + 1200*x**2*log(x)**4 + 1600*x**2*log(x)**3 + 1200*x**2*l
og(x)**2 + 480*x**2*log(x) + 80*x**2 - 80*x*log(x)**6 - 480*x*log(x)**5 - 1200*x*log(x)**4 - 1600*x*log(x)**3
- 1200*x*log(x)**2 - 480*x*log(x) - 80*x)*exp(3*x) + (120*x**3*log(x)**6 + 720*x**3*log(x)**5 + 1800*x**3*log(
x)**4 + 2400*x**3*log(x)**3 + 1800*x**3*log(x)**2 + 720*x**3*log(x) + 120*x**3 - 200*x**2*log(x)**6 - 1200*x**
2*log(x)**5 - 3000*x**2*log(x)**4 - 4000*x**2*log(x)**3 - 3000*x**2*log(x)**2 - 1200*x**2*log(x) - 200*x**2 +
120*x*log(x)**6 + 720*x*log(x)**5 + 1800*x*log(x)**4 + 2400*x*log(x)**3 + 1800*x*log(x)**2 + 720*x*log(x) + 12
0*x)*exp(2*x) + (80*x**4*log(x)**6 + 480*x**4*log(x)**5 + 1200*x**4*log(x)**4 + 1600*x**4*log(x)**3 + 1200*x**
4*log(x)**2 + 480*x**4*log(x) + 80*x**4 - 160*x**3*log(x)**6 - 960*x**3*log(x)**5 - 2400*x**3*log(x)**4 - 3200
*x**3*log(x)**3 - 2400*x**3*log(x)**2 - 960*x**3*log(x) - 160*x**3 + 160*x**2*log(x)**6 + 960*x**2*log(x)**5 +
2400*x**2*log(x)**4 + 3200*x**2*log(x)**3 + 2400*x**2*log(x)**2 + 960*x**2*log(x) + 160*x**2 - 80*x*log(x)**6
- 480*x*log(x)**5 - 1200*x*log(x)**4 - 1600*x*log(x)**3 - 1200*x*log(x)**2 - 480*x*log(x) - 80*x)*exp(x))/(lo
g(x)**8 + 8*log(x)**7 + 28*log(x)**6 + 56*log(x)**5 + 70*log(x)**4 + 56*log(x)**3 + 28*log(x)**2 + 8*log(x) +
1) + (20*x**5 - 40*x**4 + 60*x**3 - 40*x**2 + 20*x)/(log(x)**2 + 2*log(x) + 1)

________________________________________________________________________________________