Integrand size = 545, antiderivative size = 28 \begin {dmath*} \int \frac {-6 x+56 x^2-2 e^{3 x/4} x^2-54 x^3+18 x^4-2 x^5+e^{x/2} \left (18 x^2-6 x^3\right )+e^{x/4} \left (2 x-54 x^2+36 x^3-6 x^4\right )+\left (-2 x+54 x^2+6 e^{x/2} x^2-36 x^3+6 x^4+e^{x/4} \left (-36 x^2+12 x^3\right )\right ) \log (x)+\left (18 x^2-6 e^{x/4} x^2-6 x^3\right ) \log ^2(x)+2 x^2 \log ^3(x)+\left (-10+528 x-434 x^2+72 x^3+16 x^4-4 x^5+e^{3 x/4} \left (-20 x-4 x^2\right )+e^{x/2} \left (180 x-24 x^2-12 x^3\right )+e^{x/4} \left (10-543 x+251 x^2+12 x^3-12 x^4\right )+\left (-10+538 x-252 x^2-12 x^3+12 x^4+e^{x/2} \left (60 x+12 x^2\right )+e^{x/4} \left (-360 x+48 x^2+24 x^3\right )\right ) \log (x)+\left (180 x-24 x^2-12 x^3+e^{x/4} \left (-60 x-12 x^2\right )\right ) \log ^2(x)+\left (20 x+4 x^2\right ) \log ^3(x)\right ) \log (5+x)}{270+e^{3 x/4} (-10-2 x)-216 x+36 x^2+8 x^3-2 x^4+e^{x/2} \left (90-12 x-6 x^2\right )+e^{x/4} \left (-270+126 x+6 x^2-6 x^3\right )+\left (270-126 x-6 x^2+6 x^3+e^{x/2} (30+6 x)+e^{x/4} \left (-180+24 x+12 x^2\right )\right ) \log (x)+\left (90+e^{x/4} (-30-6 x)-12 x-6 x^2\right ) \log ^2(x)+(10+2 x) \log ^3(x)} \, dx=x \left (x-\frac {1}{\left (3-e^{x/4}-x+\log (x)\right )^2}\right ) \log (5+x) \end {dmath*}
Leaf count is larger than twice the leaf count of optimal. \(75\) vs. \(2(28)=56\).
Time = 0.42 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.68 \begin {dmath*} \int \frac {-6 x+56 x^2-2 e^{3 x/4} x^2-54 x^3+18 x^4-2 x^5+e^{x/2} \left (18 x^2-6 x^3\right )+e^{x/4} \left (2 x-54 x^2+36 x^3-6 x^4\right )+\left (-2 x+54 x^2+6 e^{x/2} x^2-36 x^3+6 x^4+e^{x/4} \left (-36 x^2+12 x^3\right )\right ) \log (x)+\left (18 x^2-6 e^{x/4} x^2-6 x^3\right ) \log ^2(x)+2 x^2 \log ^3(x)+\left (-10+528 x-434 x^2+72 x^3+16 x^4-4 x^5+e^{3 x/4} \left (-20 x-4 x^2\right )+e^{x/2} \left (180 x-24 x^2-12 x^3\right )+e^{x/4} \left (10-543 x+251 x^2+12 x^3-12 x^4\right )+\left (-10+538 x-252 x^2-12 x^3+12 x^4+e^{x/2} \left (60 x+12 x^2\right )+e^{x/4} \left (-360 x+48 x^2+24 x^3\right )\right ) \log (x)+\left (180 x-24 x^2-12 x^3+e^{x/4} \left (-60 x-12 x^2\right )\right ) \log ^2(x)+\left (20 x+4 x^2\right ) \log ^3(x)\right ) \log (5+x)}{270+e^{3 x/4} (-10-2 x)-216 x+36 x^2+8 x^3-2 x^4+e^{x/2} \left (90-12 x-6 x^2\right )+e^{x/4} \left (-270+126 x+6 x^2-6 x^3\right )+\left (270-126 x-6 x^2+6 x^3+e^{x/2} (30+6 x)+e^{x/4} \left (-180+24 x+12 x^2\right )\right ) \log (x)+\left (90+e^{x/4} (-30-6 x)-12 x-6 x^2\right ) \log ^2(x)+(10+2 x) \log ^3(x)} \, dx=\frac {x \left (-1+\left (-3+e^{x/4}\right )^2 x+2 \left (-3+e^{x/4}\right ) x^2+x^3-2 x \left (-3+e^{x/4}+x\right ) \log (x)+x \log ^2(x)\right ) \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^2} \end {dmath*}
Integrate[(-6*x + 56*x^2 - 2*E^((3*x)/4)*x^2 - 54*x^3 + 18*x^4 - 2*x^5 + E ^(x/2)*(18*x^2 - 6*x^3) + E^(x/4)*(2*x - 54*x^2 + 36*x^3 - 6*x^4) + (-2*x + 54*x^2 + 6*E^(x/2)*x^2 - 36*x^3 + 6*x^4 + E^(x/4)*(-36*x^2 + 12*x^3))*Lo g[x] + (18*x^2 - 6*E^(x/4)*x^2 - 6*x^3)*Log[x]^2 + 2*x^2*Log[x]^3 + (-10 + 528*x - 434*x^2 + 72*x^3 + 16*x^4 - 4*x^5 + E^((3*x)/4)*(-20*x - 4*x^2) + E^(x/2)*(180*x - 24*x^2 - 12*x^3) + E^(x/4)*(10 - 543*x + 251*x^2 + 12*x^ 3 - 12*x^4) + (-10 + 538*x - 252*x^2 - 12*x^3 + 12*x^4 + E^(x/2)*(60*x + 1 2*x^2) + E^(x/4)*(-360*x + 48*x^2 + 24*x^3))*Log[x] + (180*x - 24*x^2 - 12 *x^3 + E^(x/4)*(-60*x - 12*x^2))*Log[x]^2 + (20*x + 4*x^2)*Log[x]^3)*Log[5 + x])/(270 + E^((3*x)/4)*(-10 - 2*x) - 216*x + 36*x^2 + 8*x^3 - 2*x^4 + E ^(x/2)*(90 - 12*x - 6*x^2) + E^(x/4)*(-270 + 126*x + 6*x^2 - 6*x^3) + (270 - 126*x - 6*x^2 + 6*x^3 + E^(x/2)*(30 + 6*x) + E^(x/4)*(-180 + 24*x + 12* x^2))*Log[x] + (90 + E^(x/4)*(-30 - 6*x) - 12*x - 6*x^2)*Log[x]^2 + (10 + 2*x)*Log[x]^3),x]
(x*(-1 + (-3 + E^(x/4))^2*x + 2*(-3 + E^(x/4))*x^2 + x^3 - 2*x*(-3 + E^(x/ 4) + x)*Log[x] + x*Log[x]^2)*Log[5 + x])/(-3 + E^(x/4) + x - Log[x])^2
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.
| \(\displaystyle \int \frac {-2 x^5+18 x^4-54 x^3-2 e^{3 x/4} x^2+56 x^2+2 x^2 \log ^3(x)+e^{x/2} \left (18 x^2-6 x^3\right )+\left (-6 x^3-6 e^{x/4} x^2+18 x^2\right ) \log ^2(x)+e^{x/4} \left (-6 x^4+36 x^3-54 x^2+2 x\right )+\left (6 x^4-36 x^3+6 e^{x/2} x^2+54 x^2+e^{x/4} \left (12 x^3-36 x^2\right )-2 x\right ) \log (x)+\left (-4 x^5+16 x^4+72 x^3-434 x^2+e^{3 x/4} \left (-4 x^2-20 x\right )+\left (4 x^2+20 x\right ) \log ^3(x)+e^{x/2} \left (-12 x^3-24 x^2+180 x\right )+\left (-12 x^3-24 x^2+e^{x/4} \left (-12 x^2-60 x\right )+180 x\right ) \log ^2(x)+e^{x/4} \left (-12 x^4+12 x^3+251 x^2-543 x+10\right )+\left (12 x^4-12 x^3-252 x^2+e^{x/2} \left (12 x^2+60 x\right )+e^{x/4} \left (24 x^3+48 x^2-360 x\right )+538 x-10\right ) \log (x)+528 x-10\right ) \log (x+5)-6 x}{-2 x^4+8 x^3+36 x^2+e^{x/2} \left (-6 x^2-12 x+90\right )+\left (-6 x^2-12 x+e^{x/4} (-6 x-30)+90\right ) \log ^2(x)+e^{x/4} \left (-6 x^3+6 x^2+126 x-270\right )+\left (6 x^3-6 x^2+e^{x/4} \left (12 x^2+24 x-180\right )-126 x+e^{x/2} (6 x+30)+270\right ) \log (x)-216 x+e^{3 x/4} (-2 x-10)+(2 x+10) \log ^3(x)+270} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (x \left (3 x^3+6 \left (e^{x/4}-3\right ) x^2+3 \left (e^{x/4}-3\right )^2 x-1\right )+(x+5) \left (6 x^3+12 \left (e^{x/4}-3\right ) x^2+6 \left (e^{x/4}-3\right )^2 x-1\right ) \log (x+5)\right ) \log (x)-2 x \left (x^4-9 x^3+27 x^2+e^{x/4} \left (3 x^3-18 x^2+27 x-1\right )+e^{3 x/4} x+3 e^{x/2} (x-3) x-28 x+3\right )-(x+5) \left (e^{x/4} \left (12 x^3-72 x^2+109 x-2\right )+2 \left (2 x^4-18 x^3+54 x^2-53 x+1\right )+4 e^{3 x/4} x+12 e^{x/2} (x-3) x\right ) \log (x+5)+2 x (x+2 (x+5) \log (x+5)) \log ^3(x)-6 x \left (x+e^{x/4}-3\right ) (x+2 (x+5) \log (x+5)) \log ^2(x)}{2 (x+5) \left (-x-e^{x/4}+\log (x)+3\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int -\frac {-2 x (x+2 (x+5) \log (x+5)) \log ^3(x)-6 \left (-x-e^{x/4}+3\right ) x (x+2 (x+5) \log (x+5)) \log ^2(x)+2 \left (x \left (-3 x^3+6 \left (3-e^{x/4}\right ) x^2-3 \left (3-e^{x/4}\right )^2 x+1\right )+(x+5) \left (-6 x^3+12 \left (3-e^{x/4}\right ) x^2-6 \left (3-e^{x/4}\right )^2 x+1\right ) \log (x+5)\right ) \log (x)+2 x \left (x^4-9 x^3+27 x^2+e^{3 x/4} x-3 e^{x/2} (3-x) x-28 x-e^{x/4} \left (-3 x^3+18 x^2-27 x+1\right )+3\right )+(x+5) \left (4 e^{3 x/4} x-12 e^{x/2} (3-x) x-e^{x/4} \left (-12 x^3+72 x^2-109 x+2\right )+2 \left (2 x^4-18 x^3+54 x^2-53 x+1\right )\right ) \log (x+5)}{(x+5) \left (-x-e^{x/4}+\log (x)+3\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{2} \int \frac {-2 x (x+2 (x+5) \log (x+5)) \log ^3(x)-6 \left (-x-e^{x/4}+3\right ) x (x+2 (x+5) \log (x+5)) \log ^2(x)+2 \left (x \left (-3 x^3+6 \left (3-e^{x/4}\right ) x^2-3 \left (3-e^{x/4}\right )^2 x+1\right )+(x+5) \left (-6 x^3+12 \left (3-e^{x/4}\right ) x^2-6 \left (3-e^{x/4}\right )^2 x+1\right ) \log (x+5)\right ) \log (x)+2 x \left (x^4-9 x^3+27 x^2+e^{3 x/4} x-3 e^{x/2} (3-x) x-28 x-e^{x/4} \left (-3 x^3+18 x^2-27 x+1\right )+3\right )+(x+5) \left (4 e^{3 x/4} x-12 e^{x/2} (3-x) x-e^{x/4} \left (-12 x^3+72 x^2-109 x+2\right )+2 \left (2 x^4-18 x^3+54 x^2-53 x+1\right )\right ) \log (x+5)}{(x+5) \left (-x-e^{x/4}+\log (x)+3\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{2} \int \left (\frac {\left (x^2-\log (x) x-7 x+4\right ) \log (x+5)}{\left (x+e^{x/4}-\log (x)-3\right )^3}-\frac {2 x (2 \log (x+5) x+x+10 \log (x+5))}{x+5}-\frac {\log (x+5) x^2+3 \log (x+5) x-2 x-10 \log (x+5)}{(x+5) \left (x+e^{x/4}-\log (x)-3\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-8 \text {Subst}\left (\int \frac {1}{\left (4 x+e^x-\log (4 x)-3\right )^2}dx,x,\frac {x}{4}\right )+40 \text {Subst}\left (\int \frac {1}{(4 x+5) \left (4 x+e^x-\log (4 x)-3\right )^2}dx,x,\frac {x}{4}\right )-\int \frac {x^2 \log (x+5)}{\left (x+e^{x/4}-\log (x)-3\right )^3}dx-4 \int \frac {\log (x+5)}{\left (x+e^{x/4}-\log (x)-3\right )^3}dx+7 \int \frac {x \log (x+5)}{\left (x+e^{x/4}-\log (x)-3\right )^3}dx-2 \int \frac {\log (x+5)}{\left (x+e^{x/4}-\log (x)-3\right )^2}dx+\int \frac {x \log (x+5)}{\left (x+e^{x/4}-\log (x)-3\right )^2}dx+\int \frac {x \log (x) \log (x+5)}{\left (x+e^{x/4}-\log (x)-3\right )^3}dx+2 x^2 \log (x+5)\right )\) |
Int[(-6*x + 56*x^2 - 2*E^((3*x)/4)*x^2 - 54*x^3 + 18*x^4 - 2*x^5 + E^(x/2) *(18*x^2 - 6*x^3) + E^(x/4)*(2*x - 54*x^2 + 36*x^3 - 6*x^4) + (-2*x + 54*x ^2 + 6*E^(x/2)*x^2 - 36*x^3 + 6*x^4 + E^(x/4)*(-36*x^2 + 12*x^3))*Log[x] + (18*x^2 - 6*E^(x/4)*x^2 - 6*x^3)*Log[x]^2 + 2*x^2*Log[x]^3 + (-10 + 528*x - 434*x^2 + 72*x^3 + 16*x^4 - 4*x^5 + E^((3*x)/4)*(-20*x - 4*x^2) + E^(x/ 2)*(180*x - 24*x^2 - 12*x^3) + E^(x/4)*(10 - 543*x + 251*x^2 + 12*x^3 - 12 *x^4) + (-10 + 538*x - 252*x^2 - 12*x^3 + 12*x^4 + E^(x/2)*(60*x + 12*x^2) + E^(x/4)*(-360*x + 48*x^2 + 24*x^3))*Log[x] + (180*x - 24*x^2 - 12*x^3 + E^(x/4)*(-60*x - 12*x^2))*Log[x]^2 + (20*x + 4*x^2)*Log[x]^3)*Log[5 + x]) /(270 + E^((3*x)/4)*(-10 - 2*x) - 216*x + 36*x^2 + 8*x^3 - 2*x^4 + E^(x/2) *(90 - 12*x - 6*x^2) + E^(x/4)*(-270 + 126*x + 6*x^2 - 6*x^3) + (270 - 126 *x - 6*x^2 + 6*x^3 + E^(x/2)*(30 + 6*x) + E^(x/4)*(-180 + 24*x + 12*x^2))* Log[x] + (90 + E^(x/4)*(-30 - 6*x) - 12*x - 6*x^2)*Log[x]^2 + (10 + 2*x)*L og[x]^3),x]
3.13.59.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(81\) vs. \(2(25)=50\).
Time = 14.81 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.93
| method | result | size |
| risch | \(\frac {\left (x^{3}+2 x^{2} {\mathrm e}^{\frac {x}{4}}-2 x^{2} \ln \left (x \right )+x \,{\mathrm e}^{\frac {x}{2}}-2 \ln \left (x \right ) {\mathrm e}^{\frac {x}{4}} x +x \ln \left (x \right )^{2}-6 x^{2}-6 x \,{\mathrm e}^{\frac {x}{4}}+6 x \ln \left (x \right )+9 x -1\right ) x \ln \left (5+x \right )}{\left (x +{\mathrm e}^{\frac {x}{4}}-\ln \left (x \right )-3\right )^{2}}\) | \(82\) |
| parallelrisch | \(\frac {-8 \ln \left (x \right ) {\mathrm e}^{\frac {x}{4}} \ln \left (5+x \right ) x^{2}+4 \ln \left (5+x \right ) x^{4}-24 \ln \left (5+x \right ) x^{3}+36 \ln \left (5+x \right ) x^{2}-4 x \ln \left (5+x \right )+4 \ln \left (x \right )^{2} \ln \left (5+x \right ) x^{2}-8 \ln \left (x \right ) \ln \left (5+x \right ) x^{3}+4 \ln \left (5+x \right ) {\mathrm e}^{\frac {x}{2}} x^{2}+8 \,{\mathrm e}^{\frac {x}{4}} \ln \left (5+x \right ) x^{3}+24 \ln \left (x \right ) \ln \left (5+x \right ) x^{2}-24 \,{\mathrm e}^{\frac {x}{4}} \ln \left (5+x \right ) x^{2}}{4 x^{2}-8 x \ln \left (x \right )+8 x \,{\mathrm e}^{\frac {x}{4}}+4 \ln \left (x \right )^{2}-8 \,{\mathrm e}^{\frac {x}{4}} \ln \left (x \right )+4 \,{\mathrm e}^{\frac {x}{2}}-24 x +24 \ln \left (x \right )-24 \,{\mathrm e}^{\frac {x}{4}}+36}\) | \(179\) |
int((((4*x^2+20*x)*ln(x)^3+((-12*x^2-60*x)*exp(1/4*x)-12*x^3-24*x^2+180*x) *ln(x)^2+((12*x^2+60*x)*exp(1/4*x)^2+(24*x^3+48*x^2-360*x)*exp(1/4*x)+12*x ^4-12*x^3-252*x^2+538*x-10)*ln(x)+(-4*x^2-20*x)*exp(1/4*x)^3+(-12*x^3-24*x ^2+180*x)*exp(1/4*x)^2+(-12*x^4+12*x^3+251*x^2-543*x+10)*exp(1/4*x)-4*x^5+ 16*x^4+72*x^3-434*x^2+528*x-10)*ln(5+x)+2*x^2*ln(x)^3+(-6*x^2*exp(1/4*x)-6 *x^3+18*x^2)*ln(x)^2+(6*x^2*exp(1/4*x)^2+(12*x^3-36*x^2)*exp(1/4*x)+6*x^4- 36*x^3+54*x^2-2*x)*ln(x)-2*x^2*exp(1/4*x)^3+(-6*x^3+18*x^2)*exp(1/4*x)^2+( -6*x^4+36*x^3-54*x^2+2*x)*exp(1/4*x)-2*x^5+18*x^4-54*x^3+56*x^2-6*x)/((2*x +10)*ln(x)^3+((-6*x-30)*exp(1/4*x)-6*x^2-12*x+90)*ln(x)^2+((6*x+30)*exp(1/ 4*x)^2+(12*x^2+24*x-180)*exp(1/4*x)+6*x^3-6*x^2-126*x+270)*ln(x)+(-2*x-10) *exp(1/4*x)^3+(-6*x^2-12*x+90)*exp(1/4*x)^2+(-6*x^3+6*x^2+126*x-270)*exp(1 /4*x)-2*x^4+8*x^3+36*x^2-216*x+270),x,method=_RETURNVERBOSE)
(x^3+2*x^2*exp(1/4*x)-2*x^2*ln(x)+x*exp(1/2*x)-2*ln(x)*exp(1/4*x)*x+x*ln(x )^2-6*x^2-6*x*exp(1/4*x)+6*x*ln(x)+9*x-1)*x/(x+exp(1/4*x)-ln(x)-3)^2*ln(5+ x)
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (23) = 46\).
Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.00 \begin {dmath*} \int \frac {-6 x+56 x^2-2 e^{3 x/4} x^2-54 x^3+18 x^4-2 x^5+e^{x/2} \left (18 x^2-6 x^3\right )+e^{x/4} \left (2 x-54 x^2+36 x^3-6 x^4\right )+\left (-2 x+54 x^2+6 e^{x/2} x^2-36 x^3+6 x^4+e^{x/4} \left (-36 x^2+12 x^3\right )\right ) \log (x)+\left (18 x^2-6 e^{x/4} x^2-6 x^3\right ) \log ^2(x)+2 x^2 \log ^3(x)+\left (-10+528 x-434 x^2+72 x^3+16 x^4-4 x^5+e^{3 x/4} \left (-20 x-4 x^2\right )+e^{x/2} \left (180 x-24 x^2-12 x^3\right )+e^{x/4} \left (10-543 x+251 x^2+12 x^3-12 x^4\right )+\left (-10+538 x-252 x^2-12 x^3+12 x^4+e^{x/2} \left (60 x+12 x^2\right )+e^{x/4} \left (-360 x+48 x^2+24 x^3\right )\right ) \log (x)+\left (180 x-24 x^2-12 x^3+e^{x/4} \left (-60 x-12 x^2\right )\right ) \log ^2(x)+\left (20 x+4 x^2\right ) \log ^3(x)\right ) \log (5+x)}{270+e^{3 x/4} (-10-2 x)-216 x+36 x^2+8 x^3-2 x^4+e^{x/2} \left (90-12 x-6 x^2\right )+e^{x/4} \left (-270+126 x+6 x^2-6 x^3\right )+\left (270-126 x-6 x^2+6 x^3+e^{x/2} (30+6 x)+e^{x/4} \left (-180+24 x+12 x^2\right )\right ) \log (x)+\left (90+e^{x/4} (-30-6 x)-12 x-6 x^2\right ) \log ^2(x)+(10+2 x) \log ^3(x)} \, dx=\frac {{\left (x^{4} + x^{2} \log \left (x\right )^{2} - 6 \, x^{3} + x^{2} e^{\left (\frac {1}{2} \, x\right )} + 9 \, x^{2} + 2 \, {\left (x^{3} - 3 \, x^{2}\right )} e^{\left (\frac {1}{4} \, x\right )} - 2 \, {\left (x^{3} + x^{2} e^{\left (\frac {1}{4} \, x\right )} - 3 \, x^{2}\right )} \log \left (x\right ) - x\right )} \log \left (x + 5\right )}{x^{2} + 2 \, {\left (x - 3\right )} e^{\left (\frac {1}{4} \, x\right )} - 2 \, {\left (x + e^{\left (\frac {1}{4} \, x\right )} - 3\right )} \log \left (x\right ) + \log \left (x\right )^{2} - 6 \, x + e^{\left (\frac {1}{2} \, x\right )} + 9} \end {dmath*}
integrate((((4*x^2+20*x)*log(x)^3+((-12*x^2-60*x)*exp(1/4*x)-12*x^3-24*x^2 +180*x)*log(x)^2+((12*x^2+60*x)*exp(1/4*x)^2+(24*x^3+48*x^2-360*x)*exp(1/4 *x)+12*x^4-12*x^3-252*x^2+538*x-10)*log(x)+(-4*x^2-20*x)*exp(1/4*x)^3+(-12 *x^3-24*x^2+180*x)*exp(1/4*x)^2+(-12*x^4+12*x^3+251*x^2-543*x+10)*exp(1/4* x)-4*x^5+16*x^4+72*x^3-434*x^2+528*x-10)*log(5+x)+2*x^2*log(x)^3+(-6*x^2*e xp(1/4*x)-6*x^3+18*x^2)*log(x)^2+(6*x^2*exp(1/4*x)^2+(12*x^3-36*x^2)*exp(1 /4*x)+6*x^4-36*x^3+54*x^2-2*x)*log(x)-2*x^2*exp(1/4*x)^3+(-6*x^3+18*x^2)*e xp(1/4*x)^2+(-6*x^4+36*x^3-54*x^2+2*x)*exp(1/4*x)-2*x^5+18*x^4-54*x^3+56*x ^2-6*x)/((2*x+10)*log(x)^3+((-6*x-30)*exp(1/4*x)-6*x^2-12*x+90)*log(x)^2+( (6*x+30)*exp(1/4*x)^2+(12*x^2+24*x-180)*exp(1/4*x)+6*x^3-6*x^2-126*x+270)* log(x)+(-2*x-10)*exp(1/4*x)^3+(-6*x^2-12*x+90)*exp(1/4*x)^2+(-6*x^3+6*x^2+ 126*x-270)*exp(1/4*x)-2*x^4+8*x^3+36*x^2-216*x+270),x, algorithm=\
(x^4 + x^2*log(x)^2 - 6*x^3 + x^2*e^(1/2*x) + 9*x^2 + 2*(x^3 - 3*x^2)*e^(1 /4*x) - 2*(x^3 + x^2*e^(1/4*x) - 3*x^2)*log(x) - x)*log(x + 5)/(x^2 + 2*(x - 3)*e^(1/4*x) - 2*(x + e^(1/4*x) - 3)*log(x) + log(x)^2 - 6*x + e^(1/2*x ) + 9)
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (22) = 44\).
Time = 0.64 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.61 \begin {dmath*} \int \frac {-6 x+56 x^2-2 e^{3 x/4} x^2-54 x^3+18 x^4-2 x^5+e^{x/2} \left (18 x^2-6 x^3\right )+e^{x/4} \left (2 x-54 x^2+36 x^3-6 x^4\right )+\left (-2 x+54 x^2+6 e^{x/2} x^2-36 x^3+6 x^4+e^{x/4} \left (-36 x^2+12 x^3\right )\right ) \log (x)+\left (18 x^2-6 e^{x/4} x^2-6 x^3\right ) \log ^2(x)+2 x^2 \log ^3(x)+\left (-10+528 x-434 x^2+72 x^3+16 x^4-4 x^5+e^{3 x/4} \left (-20 x-4 x^2\right )+e^{x/2} \left (180 x-24 x^2-12 x^3\right )+e^{x/4} \left (10-543 x+251 x^2+12 x^3-12 x^4\right )+\left (-10+538 x-252 x^2-12 x^3+12 x^4+e^{x/2} \left (60 x+12 x^2\right )+e^{x/4} \left (-360 x+48 x^2+24 x^3\right )\right ) \log (x)+\left (180 x-24 x^2-12 x^3+e^{x/4} \left (-60 x-12 x^2\right )\right ) \log ^2(x)+\left (20 x+4 x^2\right ) \log ^3(x)\right ) \log (5+x)}{270+e^{3 x/4} (-10-2 x)-216 x+36 x^2+8 x^3-2 x^4+e^{x/2} \left (90-12 x-6 x^2\right )+e^{x/4} \left (-270+126 x+6 x^2-6 x^3\right )+\left (270-126 x-6 x^2+6 x^3+e^{x/2} (30+6 x)+e^{x/4} \left (-180+24 x+12 x^2\right )\right ) \log (x)+\left (90+e^{x/4} (-30-6 x)-12 x-6 x^2\right ) \log ^2(x)+(10+2 x) \log ^3(x)} \, dx=- \frac {x \log {\left (x + 5 \right )}}{x^{2} - 2 x \log {\left (x \right )} - 6 x + \left (2 x - 2 \log {\left (x \right )} - 6\right ) e^{\frac {x}{4}} + e^{\frac {x}{2}} + \log {\left (x \right )}^{2} + 6 \log {\left (x \right )} + 9} + \left (x^{2} - \frac {25}{3}\right ) \log {\left (x + 5 \right )} + \frac {25 \log {\left (3 x + 15 \right )}}{3} \end {dmath*}
integrate((((4*x**2+20*x)*ln(x)**3+((-12*x**2-60*x)*exp(1/4*x)-12*x**3-24* x**2+180*x)*ln(x)**2+((12*x**2+60*x)*exp(1/4*x)**2+(24*x**3+48*x**2-360*x) *exp(1/4*x)+12*x**4-12*x**3-252*x**2+538*x-10)*ln(x)+(-4*x**2-20*x)*exp(1/ 4*x)**3+(-12*x**3-24*x**2+180*x)*exp(1/4*x)**2+(-12*x**4+12*x**3+251*x**2- 543*x+10)*exp(1/4*x)-4*x**5+16*x**4+72*x**3-434*x**2+528*x-10)*ln(5+x)+2*x **2*ln(x)**3+(-6*x**2*exp(1/4*x)-6*x**3+18*x**2)*ln(x)**2+(6*x**2*exp(1/4* x)**2+(12*x**3-36*x**2)*exp(1/4*x)+6*x**4-36*x**3+54*x**2-2*x)*ln(x)-2*x** 2*exp(1/4*x)**3+(-6*x**3+18*x**2)*exp(1/4*x)**2+(-6*x**4+36*x**3-54*x**2+2 *x)*exp(1/4*x)-2*x**5+18*x**4-54*x**3+56*x**2-6*x)/((2*x+10)*ln(x)**3+((-6 *x-30)*exp(1/4*x)-6*x**2-12*x+90)*ln(x)**2+((6*x+30)*exp(1/4*x)**2+(12*x** 2+24*x-180)*exp(1/4*x)+6*x**3-6*x**2-126*x+270)*ln(x)+(-2*x-10)*exp(1/4*x) **3+(-6*x**2-12*x+90)*exp(1/4*x)**2+(-6*x**3+6*x**2+126*x-270)*exp(1/4*x)- 2*x**4+8*x**3+36*x**2-216*x+270),x)
-x*log(x + 5)/(x**2 - 2*x*log(x) - 6*x + (2*x - 2*log(x) - 6)*exp(x/4) + e xp(x/2) + log(x)**2 + 6*log(x) + 9) + (x**2 - 25/3)*log(x + 5) + 25*log(3* x + 15)/3
Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (23) = 46\).
Time = 0.46 (sec) , antiderivative size = 121, normalized size of antiderivative = 4.32 \begin {dmath*} \int \frac {-6 x+56 x^2-2 e^{3 x/4} x^2-54 x^3+18 x^4-2 x^5+e^{x/2} \left (18 x^2-6 x^3\right )+e^{x/4} \left (2 x-54 x^2+36 x^3-6 x^4\right )+\left (-2 x+54 x^2+6 e^{x/2} x^2-36 x^3+6 x^4+e^{x/4} \left (-36 x^2+12 x^3\right )\right ) \log (x)+\left (18 x^2-6 e^{x/4} x^2-6 x^3\right ) \log ^2(x)+2 x^2 \log ^3(x)+\left (-10+528 x-434 x^2+72 x^3+16 x^4-4 x^5+e^{3 x/4} \left (-20 x-4 x^2\right )+e^{x/2} \left (180 x-24 x^2-12 x^3\right )+e^{x/4} \left (10-543 x+251 x^2+12 x^3-12 x^4\right )+\left (-10+538 x-252 x^2-12 x^3+12 x^4+e^{x/2} \left (60 x+12 x^2\right )+e^{x/4} \left (-360 x+48 x^2+24 x^3\right )\right ) \log (x)+\left (180 x-24 x^2-12 x^3+e^{x/4} \left (-60 x-12 x^2\right )\right ) \log ^2(x)+\left (20 x+4 x^2\right ) \log ^3(x)\right ) \log (5+x)}{270+e^{3 x/4} (-10-2 x)-216 x+36 x^2+8 x^3-2 x^4+e^{x/2} \left (90-12 x-6 x^2\right )+e^{x/4} \left (-270+126 x+6 x^2-6 x^3\right )+\left (270-126 x-6 x^2+6 x^3+e^{x/2} (30+6 x)+e^{x/4} \left (-180+24 x+12 x^2\right )\right ) \log (x)+\left (90+e^{x/4} (-30-6 x)-12 x-6 x^2\right ) \log ^2(x)+(10+2 x) \log ^3(x)} \, dx=\frac {x^{2} e^{\left (\frac {1}{2} \, x\right )} \log \left (x + 5\right ) + 2 \, {\left (x^{3} - x^{2} \log \left (x\right ) - 3 \, x^{2}\right )} e^{\left (\frac {1}{4} \, x\right )} \log \left (x + 5\right ) + {\left (x^{4} + x^{2} \log \left (x\right )^{2} - 6 \, x^{3} + 9 \, x^{2} - 2 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (x\right ) - x\right )} \log \left (x + 5\right )}{x^{2} + 2 \, {\left (x - \log \left (x\right ) - 3\right )} e^{\left (\frac {1}{4} \, x\right )} - 2 \, {\left (x - 3\right )} \log \left (x\right ) + \log \left (x\right )^{2} - 6 \, x + e^{\left (\frac {1}{2} \, x\right )} + 9} \end {dmath*}
integrate((((4*x^2+20*x)*log(x)^3+((-12*x^2-60*x)*exp(1/4*x)-12*x^3-24*x^2 +180*x)*log(x)^2+((12*x^2+60*x)*exp(1/4*x)^2+(24*x^3+48*x^2-360*x)*exp(1/4 *x)+12*x^4-12*x^3-252*x^2+538*x-10)*log(x)+(-4*x^2-20*x)*exp(1/4*x)^3+(-12 *x^3-24*x^2+180*x)*exp(1/4*x)^2+(-12*x^4+12*x^3+251*x^2-543*x+10)*exp(1/4* x)-4*x^5+16*x^4+72*x^3-434*x^2+528*x-10)*log(5+x)+2*x^2*log(x)^3+(-6*x^2*e xp(1/4*x)-6*x^3+18*x^2)*log(x)^2+(6*x^2*exp(1/4*x)^2+(12*x^3-36*x^2)*exp(1 /4*x)+6*x^4-36*x^3+54*x^2-2*x)*log(x)-2*x^2*exp(1/4*x)^3+(-6*x^3+18*x^2)*e xp(1/4*x)^2+(-6*x^4+36*x^3-54*x^2+2*x)*exp(1/4*x)-2*x^5+18*x^4-54*x^3+56*x ^2-6*x)/((2*x+10)*log(x)^3+((-6*x-30)*exp(1/4*x)-6*x^2-12*x+90)*log(x)^2+( (6*x+30)*exp(1/4*x)^2+(12*x^2+24*x-180)*exp(1/4*x)+6*x^3-6*x^2-126*x+270)* log(x)+(-2*x-10)*exp(1/4*x)^3+(-6*x^2-12*x+90)*exp(1/4*x)^2+(-6*x^3+6*x^2+ 126*x-270)*exp(1/4*x)-2*x^4+8*x^3+36*x^2-216*x+270),x, algorithm=\
(x^2*e^(1/2*x)*log(x + 5) + 2*(x^3 - x^2*log(x) - 3*x^2)*e^(1/4*x)*log(x + 5) + (x^4 + x^2*log(x)^2 - 6*x^3 + 9*x^2 - 2*(x^3 - 3*x^2)*log(x) - x)*lo g(x + 5))/(x^2 + 2*(x - log(x) - 3)*e^(1/4*x) - 2*(x - 3)*log(x) + log(x)^ 2 - 6*x + e^(1/2*x) + 9)
Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (23) = 46\).
Time = 0.83 (sec) , antiderivative size = 282, normalized size of antiderivative = 10.07 \begin {dmath*} \int \frac {-6 x+56 x^2-2 e^{3 x/4} x^2-54 x^3+18 x^4-2 x^5+e^{x/2} \left (18 x^2-6 x^3\right )+e^{x/4} \left (2 x-54 x^2+36 x^3-6 x^4\right )+\left (-2 x+54 x^2+6 e^{x/2} x^2-36 x^3+6 x^4+e^{x/4} \left (-36 x^2+12 x^3\right )\right ) \log (x)+\left (18 x^2-6 e^{x/4} x^2-6 x^3\right ) \log ^2(x)+2 x^2 \log ^3(x)+\left (-10+528 x-434 x^2+72 x^3+16 x^4-4 x^5+e^{3 x/4} \left (-20 x-4 x^2\right )+e^{x/2} \left (180 x-24 x^2-12 x^3\right )+e^{x/4} \left (10-543 x+251 x^2+12 x^3-12 x^4\right )+\left (-10+538 x-252 x^2-12 x^3+12 x^4+e^{x/2} \left (60 x+12 x^2\right )+e^{x/4} \left (-360 x+48 x^2+24 x^3\right )\right ) \log (x)+\left (180 x-24 x^2-12 x^3+e^{x/4} \left (-60 x-12 x^2\right )\right ) \log ^2(x)+\left (20 x+4 x^2\right ) \log ^3(x)\right ) \log (5+x)}{270+e^{3 x/4} (-10-2 x)-216 x+36 x^2+8 x^3-2 x^4+e^{x/2} \left (90-12 x-6 x^2\right )+e^{x/4} \left (-270+126 x+6 x^2-6 x^3\right )+\left (270-126 x-6 x^2+6 x^3+e^{x/2} (30+6 x)+e^{x/4} \left (-180+24 x+12 x^2\right )\right ) \log (x)+\left (90+e^{x/4} (-30-6 x)-12 x-6 x^2\right ) \log ^2(x)+(10+2 x) \log ^3(x)} \, dx=\frac {x^{4} \log \left (x + 5\right ) + 2 \, x^{3} e^{\left (\frac {1}{4} \, x\right )} \log \left (x + 5\right ) - 4 \, x^{3} \log \left (2\right ) \log \left (x + 5\right ) - 4 \, x^{2} e^{\left (\frac {1}{4} \, x\right )} \log \left (2\right ) \log \left (x + 5\right ) + 4 \, x^{2} \log \left (2\right )^{2} \log \left (x + 5\right ) - 2 \, x^{3} \log \left (\frac {1}{4} \, x\right ) \log \left (x + 5\right ) - 2 \, x^{2} e^{\left (\frac {1}{4} \, x\right )} \log \left (\frac {1}{4} \, x\right ) \log \left (x + 5\right ) + 4 \, x^{2} \log \left (2\right ) \log \left (\frac {1}{4} \, x\right ) \log \left (x + 5\right ) + x^{2} \log \left (\frac {1}{4} \, x\right )^{2} \log \left (x + 5\right ) - 6 \, x^{3} \log \left (x + 5\right ) + x^{2} e^{\left (\frac {1}{2} \, x\right )} \log \left (x + 5\right ) - 6 \, x^{2} e^{\left (\frac {1}{4} \, x\right )} \log \left (x + 5\right ) + 12 \, x^{2} \log \left (2\right ) \log \left (x + 5\right ) + 6 \, x^{2} \log \left (\frac {1}{4} \, x\right ) \log \left (x + 5\right ) + 9 \, x^{2} \log \left (x + 5\right ) - x \log \left (x + 5\right )}{x^{2} + 2 \, x e^{\left (\frac {1}{4} \, x\right )} - 4 \, x \log \left (2\right ) - 4 \, e^{\left (\frac {1}{4} \, x\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 2 \, x \log \left (\frac {1}{4} \, x\right ) - 2 \, e^{\left (\frac {1}{4} \, x\right )} \log \left (\frac {1}{4} \, x\right ) + 4 \, \log \left (2\right ) \log \left (\frac {1}{4} \, x\right ) + \log \left (\frac {1}{4} \, x\right )^{2} - 6 \, x + e^{\left (\frac {1}{2} \, x\right )} - 6 \, e^{\left (\frac {1}{4} \, x\right )} + 12 \, \log \left (2\right ) + 6 \, \log \left (\frac {1}{4} \, x\right ) + 9} \end {dmath*}
integrate((((4*x^2+20*x)*log(x)^3+((-12*x^2-60*x)*exp(1/4*x)-12*x^3-24*x^2 +180*x)*log(x)^2+((12*x^2+60*x)*exp(1/4*x)^2+(24*x^3+48*x^2-360*x)*exp(1/4 *x)+12*x^4-12*x^3-252*x^2+538*x-10)*log(x)+(-4*x^2-20*x)*exp(1/4*x)^3+(-12 *x^3-24*x^2+180*x)*exp(1/4*x)^2+(-12*x^4+12*x^3+251*x^2-543*x+10)*exp(1/4* x)-4*x^5+16*x^4+72*x^3-434*x^2+528*x-10)*log(5+x)+2*x^2*log(x)^3+(-6*x^2*e xp(1/4*x)-6*x^3+18*x^2)*log(x)^2+(6*x^2*exp(1/4*x)^2+(12*x^3-36*x^2)*exp(1 /4*x)+6*x^4-36*x^3+54*x^2-2*x)*log(x)-2*x^2*exp(1/4*x)^3+(-6*x^3+18*x^2)*e xp(1/4*x)^2+(-6*x^4+36*x^3-54*x^2+2*x)*exp(1/4*x)-2*x^5+18*x^4-54*x^3+56*x ^2-6*x)/((2*x+10)*log(x)^3+((-6*x-30)*exp(1/4*x)-6*x^2-12*x+90)*log(x)^2+( (6*x+30)*exp(1/4*x)^2+(12*x^2+24*x-180)*exp(1/4*x)+6*x^3-6*x^2-126*x+270)* log(x)+(-2*x-10)*exp(1/4*x)^3+(-6*x^2-12*x+90)*exp(1/4*x)^2+(-6*x^3+6*x^2+ 126*x-270)*exp(1/4*x)-2*x^4+8*x^3+36*x^2-216*x+270),x, algorithm=\
(x^4*log(x + 5) + 2*x^3*e^(1/4*x)*log(x + 5) - 4*x^3*log(2)*log(x + 5) - 4 *x^2*e^(1/4*x)*log(2)*log(x + 5) + 4*x^2*log(2)^2*log(x + 5) - 2*x^3*log(1 /4*x)*log(x + 5) - 2*x^2*e^(1/4*x)*log(1/4*x)*log(x + 5) + 4*x^2*log(2)*lo g(1/4*x)*log(x + 5) + x^2*log(1/4*x)^2*log(x + 5) - 6*x^3*log(x + 5) + x^2 *e^(1/2*x)*log(x + 5) - 6*x^2*e^(1/4*x)*log(x + 5) + 12*x^2*log(2)*log(x + 5) + 6*x^2*log(1/4*x)*log(x + 5) + 9*x^2*log(x + 5) - x*log(x + 5))/(x^2 + 2*x*e^(1/4*x) - 4*x*log(2) - 4*e^(1/4*x)*log(2) + 4*log(2)^2 - 2*x*log(1 /4*x) - 2*e^(1/4*x)*log(1/4*x) + 4*log(2)*log(1/4*x) + log(1/4*x)^2 - 6*x + e^(1/2*x) - 6*e^(1/4*x) + 12*log(2) + 6*log(1/4*x) + 9)
Time = 15.92 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.36 \begin {dmath*} \int \frac {-6 x+56 x^2-2 e^{3 x/4} x^2-54 x^3+18 x^4-2 x^5+e^{x/2} \left (18 x^2-6 x^3\right )+e^{x/4} \left (2 x-54 x^2+36 x^3-6 x^4\right )+\left (-2 x+54 x^2+6 e^{x/2} x^2-36 x^3+6 x^4+e^{x/4} \left (-36 x^2+12 x^3\right )\right ) \log (x)+\left (18 x^2-6 e^{x/4} x^2-6 x^3\right ) \log ^2(x)+2 x^2 \log ^3(x)+\left (-10+528 x-434 x^2+72 x^3+16 x^4-4 x^5+e^{3 x/4} \left (-20 x-4 x^2\right )+e^{x/2} \left (180 x-24 x^2-12 x^3\right )+e^{x/4} \left (10-543 x+251 x^2+12 x^3-12 x^4\right )+\left (-10+538 x-252 x^2-12 x^3+12 x^4+e^{x/2} \left (60 x+12 x^2\right )+e^{x/4} \left (-360 x+48 x^2+24 x^3\right )\right ) \log (x)+\left (180 x-24 x^2-12 x^3+e^{x/4} \left (-60 x-12 x^2\right )\right ) \log ^2(x)+\left (20 x+4 x^2\right ) \log ^3(x)\right ) \log (5+x)}{270+e^{3 x/4} (-10-2 x)-216 x+36 x^2+8 x^3-2 x^4+e^{x/2} \left (90-12 x-6 x^2\right )+e^{x/4} \left (-270+126 x+6 x^2-6 x^3\right )+\left (270-126 x-6 x^2+6 x^3+e^{x/2} (30+6 x)+e^{x/4} \left (-180+24 x+12 x^2\right )\right ) \log (x)+\left (90+e^{x/4} (-30-6 x)-12 x-6 x^2\right ) \log ^2(x)+(10+2 x) \log ^3(x)} \, dx=-\frac {\ln \left (x+5\right )\,\left (x-\frac {{\mathrm {e}}^{x/2}\,\left (x^3+5\,x^2\right )}{x+5}-\frac {\left (x^3+5\,x^2\right )\,{\left (\ln \left (x\right )-x+3\right )}^2}{x+5}+\frac {2\,{\mathrm {e}}^{x/4}\,\left (x^3+5\,x^2\right )\,\left (\ln \left (x\right )-x+3\right )}{x+5}\right )}{{\mathrm {e}}^{x/2}-6\,x+6\,\ln \left (x\right )-{\mathrm {e}}^{x/4}\,\left (2\,\ln \left (x\right )-2\,x+6\right )+{\ln \left (x\right )}^2-2\,x\,\ln \left (x\right )+x^2+9} \end {dmath*}
int(-(6*x - log(x + 5)*(528*x - exp((3*x)/4)*(20*x + 4*x^2) + log(x)^3*(20 *x + 4*x^2) - exp(x/2)*(24*x^2 - 180*x + 12*x^3) - log(x)^2*(exp(x/4)*(60* x + 12*x^2) - 180*x + 24*x^2 + 12*x^3) + log(x)*(538*x + exp(x/2)*(60*x + 12*x^2) + exp(x/4)*(48*x^2 - 360*x + 24*x^3) - 252*x^2 - 12*x^3 + 12*x^4 - 10) + exp(x/4)*(251*x^2 - 543*x + 12*x^3 - 12*x^4 + 10) - 434*x^2 + 72*x^ 3 + 16*x^4 - 4*x^5 - 10) - exp(x/2)*(18*x^2 - 6*x^3) + 2*x^2*exp((3*x)/4) - exp(x/4)*(2*x - 54*x^2 + 36*x^3 - 6*x^4) - 2*x^2*log(x)^3 - 56*x^2 + 54* x^3 - 18*x^4 + 2*x^5 + log(x)^2*(6*x^2*exp(x/4) - 18*x^2 + 6*x^3) + log(x) *(2*x + exp(x/4)*(36*x^2 - 12*x^3) - 6*x^2*exp(x/2) - 54*x^2 + 36*x^3 - 6* x^4))/(exp(x/4)*(126*x + 6*x^2 - 6*x^3 - 270) - exp(x/2)*(12*x + 6*x^2 - 9 0) - 216*x + log(x)*(exp(x/4)*(24*x + 12*x^2 - 180) - 126*x + exp(x/2)*(6* x + 30) - 6*x^2 + 6*x^3 + 270) - exp((3*x)/4)*(2*x + 10) + 36*x^2 + 8*x^3 - 2*x^4 + log(x)^3*(2*x + 10) - log(x)^2*(12*x + exp(x/4)*(6*x + 30) + 6*x ^2 - 90) + 270),x)