3.27.99 \(\int \frac {500-200 x-455 x^2+200 x^3-20 x^4+e^4 (-190 x^2+80 x^3-8 x^4)+(-500-275 x+180 x^2-20 x^3+e^4 (-380 x+160 x^2-16 x^3)) \log (x)+e^4 (-190+80 x-8 x^2) \log ^2(x)}{125 x^2-50 x^3+5 x^4+e^4 (50 x^2-20 x^3+2 x^4)+(125 x-50 x^2+5 x^3+e^4 (100 x-40 x^2+4 x^3)) \log (x)+e^4 (50-20 x+2 x^2) \log ^2(x)} \, dx\)

Optimal. Leaf size=29 \[ \frac {x}{5-x}-4 \left (x+\log \left (2+\frac {5 x}{e^4 (x+\log (x))}\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 1.05, antiderivative size = 40, normalized size of antiderivative = 1.38, number of steps used = 7, number of rules used = 4, integrand size = 186, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6741, 6742, 683, 6684} \begin {gather*} -4 x+\frac {5}{5-x}+4 \log (x+\log (x))-4 \log \left (\left (5+2 e^4\right ) x+2 e^4 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(500 - 200*x - 455*x^2 + 200*x^3 - 20*x^4 + E^4*(-190*x^2 + 80*x^3 - 8*x^4) + (-500 - 275*x + 180*x^2 - 20
*x^3 + E^4*(-380*x + 160*x^2 - 16*x^3))*Log[x] + E^4*(-190 + 80*x - 8*x^2)*Log[x]^2)/(125*x^2 - 50*x^3 + 5*x^4
+ E^4*(50*x^2 - 20*x^3 + 2*x^4) + (125*x - 50*x^2 + 5*x^3 + E^4*(100*x - 40*x^2 + 4*x^3))*Log[x] + E^4*(50 -
20*x + 2*x^2)*Log[x]^2),x]

[Out]

5/(5 - x) - 4*x + 4*Log[x + Log[x]] - 4*Log[(5 + 2*E^4)*x + 2*E^4*Log[x]]

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {500-200 x-455 x^2+200 x^3-20 x^4+e^4 \left (-190 x^2+80 x^3-8 x^4\right )+\left (-500-275 x+180 x^2-20 x^3+e^4 \left (-380 x+160 x^2-16 x^3\right )\right ) \log (x)+e^4 \left (-190+80 x-8 x^2\right ) \log ^2(x)}{(5-x)^2 \left (5 \left (1+\frac {2 e^4}{5}\right ) x^2+5 \left (1+\frac {4 e^4}{5}\right ) x \log (x)+2 e^4 \log ^2(x)\right )} \, dx\\ &=\int \left (\frac {-95+40 x-4 x^2}{(-5+x)^2}+\frac {4 (1+x)}{x (x+\log (x))}+\frac {4 \left (-2 e^4-\left (5+2 e^4\right ) x\right )}{x \left (5 \left (1+\frac {2 e^4}{5}\right ) x+2 e^4 \log (x)\right )}\right ) \, dx\\ &=4 \int \frac {1+x}{x (x+\log (x))} \, dx+4 \int \frac {-2 e^4-\left (5+2 e^4\right ) x}{x \left (5 \left (1+\frac {2 e^4}{5}\right ) x+2 e^4 \log (x)\right )} \, dx+\int \frac {-95+40 x-4 x^2}{(-5+x)^2} \, dx\\ &=4 \log (x+\log (x))-4 \log \left (\left (5+2 e^4\right ) x+2 e^4 \log (x)\right )+\int \left (-4+\frac {5}{(-5+x)^2}\right ) \, dx\\ &=\frac {5}{5-x}-4 x+4 \log (x+\log (x))-4 \log \left (\left (5+2 e^4\right ) x+2 e^4 \log (x)\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 38, normalized size = 1.31 \begin {gather*} -\frac {5}{-5+x}-4 x+4 \log (x+\log (x))-4 \log \left (5 x+2 e^4 x+2 e^4 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(500 - 200*x - 455*x^2 + 200*x^3 - 20*x^4 + E^4*(-190*x^2 + 80*x^3 - 8*x^4) + (-500 - 275*x + 180*x^
2 - 20*x^3 + E^4*(-380*x + 160*x^2 - 16*x^3))*Log[x] + E^4*(-190 + 80*x - 8*x^2)*Log[x]^2)/(125*x^2 - 50*x^3 +
5*x^4 + E^4*(50*x^2 - 20*x^3 + 2*x^4) + (125*x - 50*x^2 + 5*x^3 + E^4*(100*x - 40*x^2 + 4*x^3))*Log[x] + E^4*
(50 - 20*x + 2*x^2)*Log[x]^2),x]

[Out]

-5/(-5 + x) - 4*x + 4*Log[x + Log[x]] - 4*Log[5*x + 2*E^4*x + 2*E^4*Log[x]]

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 48, normalized size = 1.66 \begin {gather*} -\frac {4 \, x^{2} + 4 \, {\left (x - 5\right )} \log \left (2 \, x e^{4} + 2 \, e^{4} \log \relax (x) + 5 \, x\right ) - 4 \, {\left (x - 5\right )} \log \left (x + \log \relax (x)\right ) - 20 \, x + 5}{x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2+80*x-190)*exp(4)*log(x)^2+((-16*x^3+160*x^2-380*x)*exp(4)-20*x^3+180*x^2-275*x-500)*log(x)+
(-8*x^4+80*x^3-190*x^2)*exp(4)-20*x^4+200*x^3-455*x^2-200*x+500)/((2*x^2-20*x+50)*exp(4)*log(x)^2+((4*x^3-40*x
^2+100*x)*exp(4)+5*x^3-50*x^2+125*x)*log(x)+(2*x^4-20*x^3+50*x^2)*exp(4)+5*x^4-50*x^3+125*x^2),x, algorithm="f
ricas")

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.37, size = 77, normalized size = 2.66 \begin {gather*} -\frac {4 \, x^{2} + 4 \, x \log \left (2 \, x e^{4} + 2 \, e^{4} \log \relax (x) + 5 \, x\right ) - 4 \, x \log \left (-x - \log \relax (x)\right ) - 20 \, x - 20 \, \log \left (2 \, x e^{4} + 2 \, e^{4} \log \relax (x) + 5 \, x\right ) + 20 \, \log \left (-x - \log \relax (x)\right ) + 5}{x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2+80*x-190)*exp(4)*log(x)^2+((-16*x^3+160*x^2-380*x)*exp(4)-20*x^3+180*x^2-275*x-500)*log(x)+
(-8*x^4+80*x^3-190*x^2)*exp(4)-20*x^4+200*x^3-455*x^2-200*x+500)/((2*x^2-20*x+50)*exp(4)*log(x)^2+((4*x^3-40*x
^2+100*x)*exp(4)+5*x^3-50*x^2+125*x)*log(x)+(2*x^4-20*x^3+50*x^2)*exp(4)+5*x^4-50*x^3+125*x^2),x, algorithm="g
iac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.23, size = 40, normalized size = 1.38

method	result	size
norman	\(\frac {-4 x^{2}+95}{x -5}+4 \ln \left (x +\ln \relax (x )\right )-4 \ln \left (2 x \,{\mathrm e}^{4}+2 \,{\mathrm e}^{4} \ln \relax (x )+5 x \right )\)	\(40\)
risch	\(-\frac {4 x^{2}-20 x +5}{x -5}+4 \ln \left (x +\ln \relax (x )\right )-4 \ln \left (\ln \relax (x )+\frac {\left (2 \,{\mathrm e}^{4}+5\right ) x \,{\mathrm e}^{-4}}{2}\right )\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x^2+80*x-190)*exp(4)*ln(x)^2+((-16*x^3+160*x^2-380*x)*exp(4)-20*x^3+180*x^2-275*x-500)*ln(x)+(-8*x^4+
80*x^3-190*x^2)*exp(4)-20*x^4+200*x^3-455*x^2-200*x+500)/((2*x^2-20*x+50)*exp(4)*ln(x)^2+((4*x^3-40*x^2+100*x)
*exp(4)+5*x^3-50*x^2+125*x)*ln(x)+(2*x^4-20*x^3+50*x^2)*exp(4)+5*x^4-50*x^3+125*x^2),x,method=_RETURNVERBOSE)

[Out]

(-4*x^2+95)/(x-5)+4*ln(x+ln(x))-4*ln(2*x*exp(4)+2*exp(4)*ln(x)+5*x)

________________________________________________________________________________________

maxima [A]  time = 0.84, size = 47, normalized size = 1.62 \begin {gather*} -\frac {4 \, x^{2} - 20 \, x + 5}{x - 5} - 4 \, \log \left (\frac {1}{2} \, {\left (x {\left (2 \, e^{4} + 5\right )} + 2 \, e^{4} \log \relax (x)\right )} e^{\left (-4\right )}\right ) + 4 \, \log \left (x + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2+80*x-190)*exp(4)*log(x)^2+((-16*x^3+160*x^2-380*x)*exp(4)-20*x^3+180*x^2-275*x-500)*log(x)+
(-8*x^4+80*x^3-190*x^2)*exp(4)-20*x^4+200*x^3-455*x^2-200*x+500)/((2*x^2-20*x+50)*exp(4)*log(x)^2+((4*x^3-40*x
^2+100*x)*exp(4)+5*x^3-50*x^2+125*x)*log(x)+(2*x^4-20*x^3+50*x^2)*exp(4)+5*x^4-50*x^3+125*x^2),x, algorithm="m
axima")

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {200\,x+\ln \relax (x)\,\left (275\,x+{\mathrm {e}}^4\,\left (16\,x^3-160\,x^2+380\,x\right )-180\,x^2+20\,x^3+500\right )+{\mathrm {e}}^4\,\left (8\,x^4-80\,x^3+190\,x^2\right )+455\,x^2-200\,x^3+20\,x^4+{\mathrm {e}}^4\,{\ln \relax (x)}^2\,\left (8\,x^2-80\,x+190\right )-500}{\ln \relax (x)\,\left (125\,x+{\mathrm {e}}^4\,\left (4\,x^3-40\,x^2+100\,x\right )-50\,x^2+5\,x^3\right )+{\mathrm {e}}^4\,\left (2\,x^4-20\,x^3+50\,x^2\right )+125\,x^2-50\,x^3+5\,x^4+{\mathrm {e}}^4\,{\ln \relax (x)}^2\,\left (2\,x^2-20\,x+50\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(200*x + log(x)*(275*x + exp(4)*(380*x - 160*x^2 + 16*x^3) - 180*x^2 + 20*x^3 + 500) + exp(4)*(190*x^2 -
80*x^3 + 8*x^4) + 455*x^2 - 200*x^3 + 20*x^4 + exp(4)*log(x)^2*(8*x^2 - 80*x + 190) - 500)/(log(x)*(125*x + ex
p(4)*(100*x - 40*x^2 + 4*x^3) - 50*x^2 + 5*x^3) + exp(4)*(50*x^2 - 20*x^3 + 2*x^4) + 125*x^2 - 50*x^3 + 5*x^4
+ exp(4)*log(x)^2*(2*x^2 - 20*x + 50)),x)

[Out]

int(-(200*x + log(x)*(275*x + exp(4)*(380*x - 160*x^2 + 16*x^3) - 180*x^2 + 20*x^3 + 500) + exp(4)*(190*x^2 -
80*x^3 + 8*x^4) + 455*x^2 - 200*x^3 + 20*x^4 + exp(4)*log(x)^2*(8*x^2 - 80*x + 190) - 500)/(log(x)*(125*x + ex
p(4)*(100*x - 40*x^2 + 4*x^3) - 50*x^2 + 5*x^3) + exp(4)*(50*x^2 - 20*x^3 + 2*x^4) + 125*x^2 - 50*x^3 + 5*x^4
+ exp(4)*log(x)^2*(2*x^2 - 20*x + 50)), x)

________________________________________________________________________________________

sympy [A]  time = 0.43, size = 37, normalized size = 1.28 \begin {gather*} - 4 x + 4 \log {\left (x + \log {\relax (x )} \right )} - 4 \log {\left (\frac {40 x + 16 x e^{4}}{16 e^{4}} + \log {\relax (x )} \right )} - \frac {5}{x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x**2+80*x-190)*exp(4)*ln(x)**2+((-16*x**3+160*x**2-380*x)*exp(4)-20*x**3+180*x**2-275*x-500)*ln
(x)+(-8*x**4+80*x**3-190*x**2)*exp(4)-20*x**4+200*x**3-455*x**2-200*x+500)/((2*x**2-20*x+50)*exp(4)*ln(x)**2+(
(4*x**3-40*x**2+100*x)*exp(4)+5*x**3-50*x**2+125*x)*ln(x)+(2*x**4-20*x**3+50*x**2)*exp(4)+5*x**4-50*x**3+125*x
**2),x)

[Out]

-4*x + 4*log(x + log(x)) - 4*log((40*x + 16*x*exp(4))*exp(-4)/16 + log(x)) - 5/(x - 5)

________________________________________________________________________________________