\(\int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+(250 x^3+250 x^4+e^{3 x^2} (2 x^3+2 x^4)+e^{2 x^2} (30 x^3+30 x^4)+e^{x^2} (150 x^3+150 x^4)) \log (1+x)+(625+625 x-9650 x^2-9650 x^3+e^{3 x^2} (-85 x^2-85 x^3)+e^{x^2} (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5)+e^{2 x^2} (-1235 x^2-835 x^3+320 x^4-80 x^5)) \log ^2(1+x)}{(625 x^2+625 x^3+e^{3 x^2} (5 x^2+5 x^3)+e^{2 x^2} (75 x^2+75 x^3)+e^{x^2} (375 x^2+375 x^3)) \log ^2(1+x)} \, dx\) [5262]

Optimal result

Integrand size = 287, antiderivative size = 38 \[ \int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^3+250 x^4+e^{3 x^2} \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (30 x^3+30 x^4\right )+e^{x^2} \left (150 x^3+150 x^4\right )\right ) \log (1+x)+\left (625+625 x-9650 x^2-9650 x^3+e^{3 x^2} \left (-85 x^2-85 x^3\right )+e^{x^2} \left (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5\right )+e^{2 x^2} \left (-1235 x^2-835 x^3+320 x^4-80 x^5\right )\right ) \log ^2(1+x)}{\left (625 x^2+625 x^3+e^{3 x^2} \left (5 x^2+5 x^3\right )+e^{2 x^2} \left (75 x^2+75 x^3\right )+e^{x^2} \left (375 x^2+375 x^3\right )\right ) \log ^2(1+x)} \, dx=x \left (-1-\left (4-\frac {-5+x}{\left (5+e^{x^2}\right ) x}\right )^2+\frac {x}{5 \log (1+x)}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^3+250 x^4+e^{3 x^2} \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (30 x^3+30 x^4\right )+e^{x^2} \left (150 x^3+150 x^4\right )\right ) \log (1+x)+\left (625+625 x-9650 x^2-9650 x^3+e^{3 x^2} \left (-85 x^2-85 x^3\right )+e^{x^2} \left (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5\right )+e^{2 x^2} \left (-1235 x^2-835 x^3+320 x^4-80 x^5\right )\right ) \log ^2(1+x)}{\left (625 x^2+625 x^3+e^{3 x^2} \left (5 x^2+5 x^3\right )+e^{2 x^2} \left (75 x^2+75 x^3\right )+e^{x^2} \left (375 x^2+375 x^3\right )\right ) \log ^2(1+x)} \, dx=\int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^3+250 x^4+e^{3 x^2} \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (30 x^3+30 x^4\right )+e^{x^2} \left (150 x^3+150 x^4\right )\right ) \log (1+x)+\left (625+625 x-9650 x^2-9650 x^3+e^{3 x^2} \left (-85 x^2-85 x^3\right )+e^{x^2} \left (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5\right )+e^{2 x^2} \left (-1235 x^2-835 x^3+320 x^4-80 x^5\right )\right ) \log ^2(1+x)}{\left (625 x^2+625 x^3+e^{3 x^2} \left (5 x^2+5 x^3\right )+e^{2 x^2} \left (75 x^2+75 x^3\right )+e^{x^2} \left (375 x^2+375 x^3\right )\right ) \log ^2(1+x)} \, dx \]

[In]

Int[(-125*x^4 - 75*E^x^2*x^4 - 15*E^(2*x^2)*x^4 - E^(3*x^2)*x^4 + (250*x^3 + 250*x^4 + E^(3*x^2)*(2*x^3 + 2*x^
4) + E^(2*x^2)*(30*x^3 + 30*x^4) + E^x^2*(150*x^3 + 150*x^4))*Log[1 + x] + (625 + 625*x - 9650*x^2 - 9650*x^3
+ E^(3*x^2)*(-85*x^2 - 85*x^3) + E^x^2*(125 + 125*x - 5480*x^2 - 3680*x^3 + 1420*x^4 - 380*x^5) + E^(2*x^2)*(-
1235*x^2 - 835*x^3 + 320*x^4 - 80*x^5))*Log[1 + x]^2)/((625*x^2 + 625*x^3 + E^(3*x^2)*(5*x^2 + 5*x^3) + E^(2*x
^2)*(75*x^2 + 75*x^3) + E^x^2*(375*x^2 + 375*x^3))*Log[1 + x]^2),x]

[Out]

10/(5 + E^x^2)^2 - 40/(5 + E^x^2) - 17*x + 1/(5*Log[1 + x]) - (2*(1 + x))/(5*Log[1 + x]) + (1 + x)^2/(5*Log[1
+ x]) - 500*Defer[Int][(5 + E^x^2)^(-3), x] + 99*Defer[Int][(5 + E^x^2)^(-2), x] + 8*Defer[Int][(5 + E^x^2)^(-
1), x] + 25*Defer[Int][1/((5 + E^x^2)^2*x^2), x] - 20*Defer[Int][x^2/(5 + E^x^2)^3, x] + 84*Defer[Int][x^2/(5
+ E^x^2)^2, x] - 16*Defer[Int][x^2/(5 + E^x^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {125-1930 x^2-17 e^{3 x^2} x^2-e^{2 x^2} x^2 \left (247-80 x+16 x^2\right )+e^{x^2} \left (25-1096 x^2+360 x^3-76 x^4\right )}{\left (5+e^{x^2}\right )^3 x^2}-\frac {x^2}{5 (1+x) \log ^2(1+x)}+\frac {2 x}{5 \log (1+x)}\right ) \, dx \\ & = -\left (\frac {1}{5} \int \frac {x^2}{(1+x) \log ^2(1+x)} \, dx\right )+\frac {2}{5} \int \frac {x}{\log (1+x)} \, dx+\int \frac {125-1930 x^2-17 e^{3 x^2} x^2-e^{2 x^2} x^2 \left (247-80 x+16 x^2\right )+e^{x^2} \left (25-1096 x^2+360 x^3-76 x^4\right )}{\left (5+e^{x^2}\right )^3 x^2} \, dx \\ & = -\left (\frac {1}{5} \text {Subst}\left (\int \frac {(-1+x)^2}{x \log ^2(x)} \, dx,x,1+x\right )\right )+\frac {2}{5} \int \left (-\frac {1}{\log (1+x)}+\frac {1+x}{\log (1+x)}\right ) \, dx+\int \left (-17-\frac {20 (-5+x)^2}{\left (5+e^{x^2}\right )^3}-\frac {8 \left (-1-10 x+2 x^2\right )}{5+e^{x^2}}+\frac {25+99 x^2-440 x^3+84 x^4}{\left (5+e^{x^2}\right )^2 x^2}\right ) \, dx \\ & = -17 x-\frac {1}{5} \text {Subst}\left (\int \left (-\frac {2}{\log ^2(x)}+\frac {1}{x \log ^2(x)}+\frac {x}{\log ^2(x)}\right ) \, dx,x,1+x\right )-\frac {2}{5} \int \frac {1}{\log (1+x)} \, dx+\frac {2}{5} \int \frac {1+x}{\log (1+x)} \, dx-8 \int \frac {-1-10 x+2 x^2}{5+e^{x^2}} \, dx-20 \int \frac {(-5+x)^2}{\left (5+e^{x^2}\right )^3} \, dx+\int \frac {25+99 x^2-440 x^3+84 x^4}{\left (5+e^{x^2}\right )^2 x^2} \, dx \\ & = -17 x-\frac {1}{5} \text {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,1+x\right )-\frac {1}{5} \text {Subst}\left (\int \frac {x}{\log ^2(x)} \, dx,x,1+x\right )+\frac {2}{5} \text {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,1+x\right )-\frac {2}{5} \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,1+x\right )+\frac {2}{5} \text {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,1+x\right )-8 \int \left (-\frac {1}{5+e^{x^2}}-\frac {10 x}{5+e^{x^2}}+\frac {2 x^2}{5+e^{x^2}}\right ) \, dx-20 \int \left (\frac {25}{\left (5+e^{x^2}\right )^3}-\frac {10 x}{\left (5+e^{x^2}\right )^3}+\frac {x^2}{\left (5+e^{x^2}\right )^3}\right ) \, dx+\int \left (\frac {99}{\left (5+e^{x^2}\right )^2}+\frac {25}{\left (5+e^{x^2}\right )^2 x^2}-\frac {440 x}{\left (5+e^{x^2}\right )^2}+\frac {84 x^2}{\left (5+e^{x^2}\right )^2}\right ) \, dx \\ & = -17 x-\frac {2 (1+x)}{5 \log (1+x)}+\frac {(1+x)^2}{5 \log (1+x)}-\frac {2 \text {li}(1+x)}{5}-\frac {1}{5} \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (1+x)\right )+\frac {2}{5} \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (1+x)\right )+\frac {2}{5} \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,1+x\right )-\frac {2}{5} \text {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,1+x\right )+8 \int \frac {1}{5+e^{x^2}} \, dx-16 \int \frac {x^2}{5+e^{x^2}} \, dx-20 \int \frac {x^2}{\left (5+e^{x^2}\right )^3} \, dx+25 \int \frac {1}{\left (5+e^{x^2}\right )^2 x^2} \, dx+80 \int \frac {x}{5+e^{x^2}} \, dx+84 \int \frac {x^2}{\left (5+e^{x^2}\right )^2} \, dx+99 \int \frac {1}{\left (5+e^{x^2}\right )^2} \, dx+200 \int \frac {x}{\left (5+e^{x^2}\right )^3} \, dx-440 \int \frac {x}{\left (5+e^{x^2}\right )^2} \, dx-500 \int \frac {1}{\left (5+e^{x^2}\right )^3} \, dx \\ & = -17 x+\frac {2}{5} \text {Ei}(2 \log (1+x))+\frac {1}{5 \log (1+x)}-\frac {2 (1+x)}{5 \log (1+x)}+\frac {(1+x)^2}{5 \log (1+x)}-\frac {2}{5} \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (1+x)\right )+8 \int \frac {1}{5+e^{x^2}} \, dx-16 \int \frac {x^2}{5+e^{x^2}} \, dx-20 \int \frac {x^2}{\left (5+e^{x^2}\right )^3} \, dx+25 \int \frac {1}{\left (5+e^{x^2}\right )^2 x^2} \, dx+40 \text {Subst}\left (\int \frac {1}{5+e^x} \, dx,x,x^2\right )+84 \int \frac {x^2}{\left (5+e^{x^2}\right )^2} \, dx+99 \int \frac {1}{\left (5+e^{x^2}\right )^2} \, dx+100 \text {Subst}\left (\int \frac {1}{\left (5+e^x\right )^3} \, dx,x,x^2\right )-220 \text {Subst}\left (\int \frac {1}{\left (5+e^x\right )^2} \, dx,x,x^2\right )-500 \int \frac {1}{\left (5+e^{x^2}\right )^3} \, dx \\ & = -17 x+\frac {1}{5 \log (1+x)}-\frac {2 (1+x)}{5 \log (1+x)}+\frac {(1+x)^2}{5 \log (1+x)}+8 \int \frac {1}{5+e^{x^2}} \, dx-16 \int \frac {x^2}{5+e^{x^2}} \, dx-20 \int \frac {x^2}{\left (5+e^{x^2}\right )^3} \, dx+25 \int \frac {1}{\left (5+e^{x^2}\right )^2 x^2} \, dx+40 \text {Subst}\left (\int \frac {1}{x (5+x)} \, dx,x,e^{x^2}\right )+84 \int \frac {x^2}{\left (5+e^{x^2}\right )^2} \, dx+99 \int \frac {1}{\left (5+e^{x^2}\right )^2} \, dx+100 \text {Subst}\left (\int \frac {1}{x (5+x)^3} \, dx,x,e^{x^2}\right )-220 \text {Subst}\left (\int \frac {1}{x (5+x)^2} \, dx,x,e^{x^2}\right )-500 \int \frac {1}{\left (5+e^{x^2}\right )^3} \, dx \\ & = -17 x+\frac {1}{5 \log (1+x)}-\frac {2 (1+x)}{5 \log (1+x)}+\frac {(1+x)^2}{5 \log (1+x)}+8 \int \frac {1}{5+e^{x^2}} \, dx+8 \text {Subst}\left (\int \frac {1}{x} \, dx,x,e^{x^2}\right )-8 \text {Subst}\left (\int \frac {1}{5+x} \, dx,x,e^{x^2}\right )-16 \int \frac {x^2}{5+e^{x^2}} \, dx-20 \int \frac {x^2}{\left (5+e^{x^2}\right )^3} \, dx+25 \int \frac {1}{\left (5+e^{x^2}\right )^2 x^2} \, dx+84 \int \frac {x^2}{\left (5+e^{x^2}\right )^2} \, dx+99 \int \frac {1}{\left (5+e^{x^2}\right )^2} \, dx+100 \text {Subst}\left (\int \left (\frac {1}{125 x}-\frac {1}{5 (5+x)^3}-\frac {1}{25 (5+x)^2}-\frac {1}{125 (5+x)}\right ) \, dx,x,e^{x^2}\right )-220 \text {Subst}\left (\int \left (\frac {1}{25 x}-\frac {1}{5 (5+x)^2}-\frac {1}{25 (5+x)}\right ) \, dx,x,e^{x^2}\right )-500 \int \frac {1}{\left (5+e^{x^2}\right )^3} \, dx \\ & = \frac {10}{\left (5+e^{x^2}\right )^2}-\frac {40}{5+e^{x^2}}-17 x+\frac {1}{5 \log (1+x)}-\frac {2 (1+x)}{5 \log (1+x)}+\frac {(1+x)^2}{5 \log (1+x)}+8 \int \frac {1}{5+e^{x^2}} \, dx-16 \int \frac {x^2}{5+e^{x^2}} \, dx-20 \int \frac {x^2}{\left (5+e^{x^2}\right )^3} \, dx+25 \int \frac {1}{\left (5+e^{x^2}\right )^2 x^2} \, dx+84 \int \frac {x^2}{\left (5+e^{x^2}\right )^2} \, dx+99 \int \frac {1}{\left (5+e^{x^2}\right )^2} \, dx-500 \int \frac {1}{\left (5+e^{x^2}\right )^3} \, dx \\ \end{align*}

Mathematica [C] (verified)

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

Time = 2.84 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.76 \[ \int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^3+250 x^4+e^{3 x^2} \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (30 x^3+30 x^4\right )+e^{x^2} \left (150 x^3+150 x^4\right )\right ) \log (1+x)+\left (625+625 x-9650 x^2-9650 x^3+e^{3 x^2} \left (-85 x^2-85 x^3\right )+e^{x^2} \left (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5\right )+e^{2 x^2} \left (-1235 x^2-835 x^3+320 x^4-80 x^5\right )\right ) \log ^2(1+x)}{\left (625 x^2+625 x^3+e^{3 x^2} \left (5 x^2+5 x^3\right )+e^{2 x^2} \left (75 x^2+75 x^3\right )+e^{x^2} \left (375 x^2+375 x^3\right )\right ) \log ^2(1+x)} \, dx=\frac {8 (-5+x)}{5+e^{x^2}}-\frac {(-5+x)^2}{\left (5+e^{x^2}\right )^2 x}-17 x-\frac {2}{5} \operatorname {ExpIntegralEi}(\log (1+x))+\frac {x^2}{5 \log (1+x)}+\frac {2 \operatorname {LogIntegral}(1+x)}{5} \]

[In]

Integrate[(-125*x^4 - 75*E^x^2*x^4 - 15*E^(2*x^2)*x^4 - E^(3*x^2)*x^4 + (250*x^3 + 250*x^4 + E^(3*x^2)*(2*x^3
+ 2*x^4) + E^(2*x^2)*(30*x^3 + 30*x^4) + E^x^2*(150*x^3 + 150*x^4))*Log[1 + x] + (625 + 625*x - 9650*x^2 - 965
0*x^3 + E^(3*x^2)*(-85*x^2 - 85*x^3) + E^x^2*(125 + 125*x - 5480*x^2 - 3680*x^3 + 1420*x^4 - 380*x^5) + E^(2*x
^2)*(-1235*x^2 - 835*x^3 + 320*x^4 - 80*x^5))*Log[1 + x]^2)/((625*x^2 + 625*x^3 + E^(3*x^2)*(5*x^2 + 5*x^3) +
E^(2*x^2)*(75*x^2 + 75*x^3) + E^x^2*(375*x^2 + 375*x^3))*Log[1 + x]^2),x]

[Out]

(8*(-5 + x))/(5 + E^x^2) - (-5 + x)^2/((5 + E^x^2)^2*x) - 17*x - (2*ExpIntegralEi[Log[1 + x]])/5 + x^2/(5*Log[
1 + x]) + (2*LogIntegral[1 + x])/5

Maple [A] (verified)

Time = 1.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.66

method	result	size
risch	\(-\frac {17 \,{\mathrm e}^{2 x^{2}} x^{2}+162 x^{2} {\mathrm e}^{x^{2}}+386 x^{2}+40 \,{\mathrm e}^{x^{2}} x +190 x +25}{x \left (5+{\mathrm e}^{x^{2}}\right )^{2}}+\frac {x^{2}}{5 \ln \left (1+x \right )}\)	\(63\)
parallelrisch	\(-\frac {-20 \,{\mathrm e}^{2 x^{2}} x^{3}+1700 \ln \left (1+x \right ) {\mathrm e}^{2 x^{2}} x^{2}-200 x^{3} {\mathrm e}^{x^{2}}+16200 \ln \left (1+x \right ) {\mathrm e}^{x^{2}} x^{2}-3640 \ln \left (1+x \right ) x \,{\mathrm e}^{2 x^{2}}-500 x^{3}+38600 x^{2} \ln \left (1+x \right )-32400 \ln \left (1+x \right ) {\mathrm e}^{x^{2}} x -72000 x \ln \left (1+x \right )+2500 \ln \left (1+x \right )}{100 \ln \left (1+x \right ) \left ({\mathrm e}^{2 x^{2}}+10 \,{\mathrm e}^{x^{2}}+25\right ) x}\)	\(128\)

[In]

int((((-85*x^3-85*x^2)*exp(x^2)^3+(-80*x^5+320*x^4-835*x^3-1235*x^2)*exp(x^2)^2+(-380*x^5+1420*x^4-3680*x^3-54
80*x^2+125*x+125)*exp(x^2)-9650*x^3-9650*x^2+625*x+625)*ln(1+x)^2+((2*x^4+2*x^3)*exp(x^2)^3+(30*x^4+30*x^3)*ex
p(x^2)^2+(150*x^4+150*x^3)*exp(x^2)+250*x^4+250*x^3)*ln(1+x)-x^4*exp(x^2)^3-15*x^4*exp(x^2)^2-75*x^4*exp(x^2)-
125*x^4)/((5*x^3+5*x^2)*exp(x^2)^3+(75*x^3+75*x^2)*exp(x^2)^2+(375*x^3+375*x^2)*exp(x^2)+625*x^3+625*x^2)/ln(1
+x)^2,x,method=_RETURNVERBOSE)

[Out]

-(17*x^2*exp(x^2)^2+162*x^2*exp(x^2)+386*x^2+40*exp(x^2)*x+190*x+25)/x/(5+exp(x^2))^2+1/5*x^2/ln(1+x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (35) = 70\).

Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.53 \[ \int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^3+250 x^4+e^{3 x^2} \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (30 x^3+30 x^4\right )+e^{x^2} \left (150 x^3+150 x^4\right )\right ) \log (1+x)+\left (625+625 x-9650 x^2-9650 x^3+e^{3 x^2} \left (-85 x^2-85 x^3\right )+e^{x^2} \left (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5\right )+e^{2 x^2} \left (-1235 x^2-835 x^3+320 x^4-80 x^5\right )\right ) \log ^2(1+x)}{\left (625 x^2+625 x^3+e^{3 x^2} \left (5 x^2+5 x^3\right )+e^{2 x^2} \left (75 x^2+75 x^3\right )+e^{x^2} \left (375 x^2+375 x^3\right )\right ) \log ^2(1+x)} \, dx=\frac {x^{3} e^{\left (2 \, x^{2}\right )} + 10 \, x^{3} e^{\left (x^{2}\right )} + 25 \, x^{3} - 5 \, {\left (17 \, x^{2} e^{\left (2 \, x^{2}\right )} + 386 \, x^{2} + 2 \, {\left (81 \, x^{2} + 20 \, x\right )} e^{\left (x^{2}\right )} + 190 \, x + 25\right )} \log \left (x + 1\right )}{5 \, {\left (x e^{\left (2 \, x^{2}\right )} + 10 \, x e^{\left (x^{2}\right )} + 25 \, x\right )} \log \left (x + 1\right )} \]

[In]

integrate((((-85*x^3-85*x^2)*exp(x^2)^3+(-80*x^5+320*x^4-835*x^3-1235*x^2)*exp(x^2)^2+(-380*x^5+1420*x^4-3680*
x^3-5480*x^2+125*x+125)*exp(x^2)-9650*x^3-9650*x^2+625*x+625)*log(1+x)^2+((2*x^4+2*x^3)*exp(x^2)^3+(30*x^4+30*
x^3)*exp(x^2)^2+(150*x^4+150*x^3)*exp(x^2)+250*x^4+250*x^3)*log(1+x)-x^4*exp(x^2)^3-15*x^4*exp(x^2)^2-75*x^4*e
xp(x^2)-125*x^4)/((5*x^3+5*x^2)*exp(x^2)^3+(75*x^3+75*x^2)*exp(x^2)^2+(375*x^3+375*x^2)*exp(x^2)+625*x^3+625*x
^2)/log(1+x)^2,x, algorithm="fricas")

[Out]

1/5*(x^3*e^(2*x^2) + 10*x^3*e^(x^2) + 25*x^3 - 5*(17*x^2*e^(2*x^2) + 386*x^2 + 2*(81*x^2 + 20*x)*e^(x^2) + 190
*x + 25)*log(x + 1))/((x*e^(2*x^2) + 10*x*e^(x^2) + 25*x)*log(x + 1))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).

Time = 0.15 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.47 \[ \int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^3+250 x^4+e^{3 x^2} \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (30 x^3+30 x^4\right )+e^{x^2} \left (150 x^3+150 x^4\right )\right ) \log (1+x)+\left (625+625 x-9650 x^2-9650 x^3+e^{3 x^2} \left (-85 x^2-85 x^3\right )+e^{x^2} \left (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5\right )+e^{2 x^2} \left (-1235 x^2-835 x^3+320 x^4-80 x^5\right )\right ) \log ^2(1+x)}{\left (625 x^2+625 x^3+e^{3 x^2} \left (5 x^2+5 x^3\right )+e^{2 x^2} \left (75 x^2+75 x^3\right )+e^{x^2} \left (375 x^2+375 x^3\right )\right ) \log ^2(1+x)} \, dx=\frac {x^{2}}{5 \log {\left (x + 1 \right )}} - 17 x + \frac {39 x^{2} - 190 x + \left (8 x^{2} - 40 x\right ) e^{x^{2}} - 25}{x e^{2 x^{2}} + 10 x e^{x^{2}} + 25 x} \]

[In]

integrate((((-85*x**3-85*x**2)*exp(x**2)**3+(-80*x**5+320*x**4-835*x**3-1235*x**2)*exp(x**2)**2+(-380*x**5+142
0*x**4-3680*x**3-5480*x**2+125*x+125)*exp(x**2)-9650*x**3-9650*x**2+625*x+625)*ln(1+x)**2+((2*x**4+2*x**3)*exp
(x**2)**3+(30*x**4+30*x**3)*exp(x**2)**2+(150*x**4+150*x**3)*exp(x**2)+250*x**4+250*x**3)*ln(1+x)-x**4*exp(x**
2)**3-15*x**4*exp(x**2)**2-75*x**4*exp(x**2)-125*x**4)/((5*x**3+5*x**2)*exp(x**2)**3+(75*x**3+75*x**2)*exp(x**
2)**2+(375*x**3+375*x**2)*exp(x**2)+625*x**3+625*x**2)/ln(1+x)**2,x)

[Out]

x**2/(5*log(x + 1)) - 17*x + (39*x**2 - 190*x + (8*x**2 - 40*x)*exp(x**2) - 25)/(x*exp(2*x**2) + 10*x*exp(x**2
) + 25*x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (35) = 70\).

Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.68 \[ \int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^3+250 x^4+e^{3 x^2} \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (30 x^3+30 x^4\right )+e^{x^2} \left (150 x^3+150 x^4\right )\right ) \log (1+x)+\left (625+625 x-9650 x^2-9650 x^3+e^{3 x^2} \left (-85 x^2-85 x^3\right )+e^{x^2} \left (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5\right )+e^{2 x^2} \left (-1235 x^2-835 x^3+320 x^4-80 x^5\right )\right ) \log ^2(1+x)}{\left (625 x^2+625 x^3+e^{3 x^2} \left (5 x^2+5 x^3\right )+e^{2 x^2} \left (75 x^2+75 x^3\right )+e^{x^2} \left (375 x^2+375 x^3\right )\right ) \log ^2(1+x)} \, dx=\frac {25 \, x^{3} + {\left (x^{3} - 85 \, x^{2} \log \left (x + 1\right )\right )} e^{\left (2 \, x^{2}\right )} + 10 \, {\left (x^{3} - {\left (81 \, x^{2} + 20 \, x\right )} \log \left (x + 1\right )\right )} e^{\left (x^{2}\right )} - 5 \, {\left (386 \, x^{2} + 190 \, x + 25\right )} \log \left (x + 1\right )}{5 \, {\left (x e^{\left (2 \, x^{2}\right )} \log \left (x + 1\right ) + 10 \, x e^{\left (x^{2}\right )} \log \left (x + 1\right ) + 25 \, x \log \left (x + 1\right )\right )}} \]

[In]

integrate((((-85*x^3-85*x^2)*exp(x^2)^3+(-80*x^5+320*x^4-835*x^3-1235*x^2)*exp(x^2)^2+(-380*x^5+1420*x^4-3680*
x^3-5480*x^2+125*x+125)*exp(x^2)-9650*x^3-9650*x^2+625*x+625)*log(1+x)^2+((2*x^4+2*x^3)*exp(x^2)^3+(30*x^4+30*
x^3)*exp(x^2)^2+(150*x^4+150*x^3)*exp(x^2)+250*x^4+250*x^3)*log(1+x)-x^4*exp(x^2)^3-15*x^4*exp(x^2)^2-75*x^4*e
xp(x^2)-125*x^4)/((5*x^3+5*x^2)*exp(x^2)^3+(75*x^3+75*x^2)*exp(x^2)^2+(375*x^3+375*x^2)*exp(x^2)+625*x^3+625*x
^2)/log(1+x)^2,x, algorithm="maxima")

[Out]

1/5*(25*x^3 + (x^3 - 85*x^2*log(x + 1))*e^(2*x^2) + 10*(x^3 - (81*x^2 + 20*x)*log(x + 1))*e^(x^2) - 5*(386*x^2
+ 190*x + 25)*log(x + 1))/(x*e^(2*x^2)*log(x + 1) + 10*x*e^(x^2)*log(x + 1) + 25*x*log(x + 1))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (35) = 70\).

Time = 0.38 (sec) , antiderivative size = 121, normalized size of antiderivative = 3.18 \[ \int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^3+250 x^4+e^{3 x^2} \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (30 x^3+30 x^4\right )+e^{x^2} \left (150 x^3+150 x^4\right )\right ) \log (1+x)+\left (625+625 x-9650 x^2-9650 x^3+e^{3 x^2} \left (-85 x^2-85 x^3\right )+e^{x^2} \left (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5\right )+e^{2 x^2} \left (-1235 x^2-835 x^3+320 x^4-80 x^5\right )\right ) \log ^2(1+x)}{\left (625 x^2+625 x^3+e^{3 x^2} \left (5 x^2+5 x^3\right )+e^{2 x^2} \left (75 x^2+75 x^3\right )+e^{x^2} \left (375 x^2+375 x^3\right )\right ) \log ^2(1+x)} \, dx=\frac {x^{3} e^{\left (2 \, x^{2}\right )} + 10 \, x^{3} e^{\left (x^{2}\right )} - 85 \, x^{2} e^{\left (2 \, x^{2}\right )} \log \left (x + 1\right ) - 810 \, x^{2} e^{\left (x^{2}\right )} \log \left (x + 1\right ) + 25 \, x^{3} - 1930 \, x^{2} \log \left (x + 1\right ) - 200 \, x e^{\left (x^{2}\right )} \log \left (x + 1\right ) - 950 \, x \log \left (x + 1\right ) - 125 \, \log \left (x + 1\right )}{5 \, {\left (x e^{\left (2 \, x^{2}\right )} \log \left (x + 1\right ) + 10 \, x e^{\left (x^{2}\right )} \log \left (x + 1\right ) + 25 \, x \log \left (x + 1\right )\right )}} \]

[In]

integrate((((-85*x^3-85*x^2)*exp(x^2)^3+(-80*x^5+320*x^4-835*x^3-1235*x^2)*exp(x^2)^2+(-380*x^5+1420*x^4-3680*
x^3-5480*x^2+125*x+125)*exp(x^2)-9650*x^3-9650*x^2+625*x+625)*log(1+x)^2+((2*x^4+2*x^3)*exp(x^2)^3+(30*x^4+30*
x^3)*exp(x^2)^2+(150*x^4+150*x^3)*exp(x^2)+250*x^4+250*x^3)*log(1+x)-x^4*exp(x^2)^3-15*x^4*exp(x^2)^2-75*x^4*e
xp(x^2)-125*x^4)/((5*x^3+5*x^2)*exp(x^2)^3+(75*x^3+75*x^2)*exp(x^2)^2+(375*x^3+375*x^2)*exp(x^2)+625*x^3+625*x
^2)/log(1+x)^2,x, algorithm="giac")

[Out]

1/5*(x^3*e^(2*x^2) + 10*x^3*e^(x^2) - 85*x^2*e^(2*x^2)*log(x + 1) - 810*x^2*e^(x^2)*log(x + 1) + 25*x^3 - 1930
*x^2*log(x + 1) - 200*x*e^(x^2)*log(x + 1) - 950*x*log(x + 1) - 125*log(x + 1))/(x*e^(2*x^2)*log(x + 1) + 10*x
*e^(x^2)*log(x + 1) + 25*x*log(x + 1))

Mupad [B] (verification not implemented)

Time = 14.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.34 \[ \int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^3+250 x^4+e^{3 x^2} \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (30 x^3+30 x^4\right )+e^{x^2} \left (150 x^3+150 x^4\right )\right ) \log (1+x)+\left (625+625 x-9650 x^2-9650 x^3+e^{3 x^2} \left (-85 x^2-85 x^3\right )+e^{x^2} \left (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5\right )+e^{2 x^2} \left (-1235 x^2-835 x^3+320 x^4-80 x^5\right )\right ) \log ^2(1+x)}{\left (625 x^2+625 x^3+e^{3 x^2} \left (5 x^2+5 x^3\right )+e^{2 x^2} \left (75 x^2+75 x^3\right )+e^{x^2} \left (375 x^2+375 x^3\right )\right ) \log ^2(1+x)} \, dx=\frac {\frac {x^2}{5}-\frac {2\,x\,\ln \left (x+1\right )\,\left (x+1\right )}{5}}{\ln \left (x+1\right )}-\frac {83\,x}{5}+\frac {2\,x^2}{5}-\frac {8\,\left (5\,x-x^2\right )}{x\,\left ({\mathrm {e}}^{x^2}+5\right )}-\frac {x^4-10\,x^3+25\,x^2}{x^3\,\left (10\,{\mathrm {e}}^{x^2}+{\mathrm {e}}^{2\,x^2}+25\right )} \]

[In]

int(-(log(x + 1)^2*(exp(2*x^2)*(1235*x^2 + 835*x^3 - 320*x^4 + 80*x^5) - 625*x - exp(x^2)*(125*x - 5480*x^2 -
3680*x^3 + 1420*x^4 - 380*x^5 + 125) + exp(3*x^2)*(85*x^2 + 85*x^3) + 9650*x^2 + 9650*x^3 - 625) - log(x + 1)*
(exp(x^2)*(150*x^3 + 150*x^4) + exp(3*x^2)*(2*x^3 + 2*x^4) + exp(2*x^2)*(30*x^3 + 30*x^4) + 250*x^3 + 250*x^4)
+ 75*x^4*exp(x^2) + 15*x^4*exp(2*x^2) + x^4*exp(3*x^2) + 125*x^4)/(log(x + 1)^2*(exp(x^2)*(375*x^2 + 375*x^3)
+ exp(3*x^2)*(5*x^2 + 5*x^3) + exp(2*x^2)*(75*x^2 + 75*x^3) + 625*x^2 + 625*x^3)),x)

[Out]

(x^2/5 - (2*x*log(x + 1)*(x + 1))/5)/log(x + 1) - (83*x)/5 + (2*x^2)/5 - (8*(5*x - x^2))/(x*(exp(x^2) + 5)) -
(25*x^2 - 10*x^3 + x^4)/(x^3*(10*exp(x^2) + exp(2*x^2) + 25))