3.73.100 $\int \frac{36+88x^2+80x^4+40x^6+10x^8+x^{10}+(80+160x^2+120x^4+40x^6+5x^8)\log (x)+(80+120x^2+60x^4+10x^6){\log }^2(x)+(40+40x^2+10x^4){\log }^3(x)+(10+5x^2){\log }^4(x)+{\log }^5(x)}{-98x-241x^3-240x^5-120x^7-30x^9-3x^{11}+(-209x-400x^3-280x^5-80x^7-5x^9+x^{11})\log (x)+(-160x-200x^3-60x^5+10x^7+5x^9){\log }^2(x)+(-40x+30x^5+10x^7){\log }^3(x)+(10x+25x^3+10x^5){\log }^4(x)+(7x+5x^3){\log }^5(x)+x{\log }^6(x)}dx$

Optimal. Leaf size=16 $$\log (3-\log (x)+\frac{1}{{(2+x^2+\log (x))}^4})$$

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Rubi [B]  time = 4.27, antiderivative size = 138, normalized size of antiderivative = 8.62, number of steps used = 4, number of rules used = 2, integrand size = 254, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.008, Rules used = {6742, 6684} $$\begin{array}{c}\log (3x^8+x^8(-\log (x))+24x^6-4x^6{\log }^2(x)+4x^6\log (x)+72x^4-6x^4{\log }^3(x)-6x^4{\log }^2(x)+48x^4\log (x)+96x^2-4x^2{\log }^4(x)-12x^2{\log }^3(x)+24x^2{\log }^2(x)+112x^2\log (x)-{\log }^5(x)-5{\log }^4(x)+40{\log }^2(x)+80\log (x)+49)-4\log (x^2+\log (x)+2)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(36 + 88*x^2 + 80*x^4 + 40*x^6 + 10*x^8 + x^10 + (80 + 160*x^2 + 120*x^4 + 40*x^6 + 5*x^8)*Log[x] + (80 +
120*x^2 + 60*x^4 + 10*x^6)*Log[x]^2 + (40 + 40*x^2 + 10*x^4)*Log[x]^3 + (10 + 5*x^2)*Log[x]^4 + Log[x]^5)/(-98
*x - 241*x^3 - 240*x^5 - 120*x^7 - 30*x^9 - 3*x^11 + (-209*x - 400*x^3 - 280*x^5 - 80*x^7 - 5*x^9 + x^11)*Log[
x] + (-160*x - 200*x^3 - 60*x^5 + 10*x^7 + 5*x^9)*Log[x]^2 + (-40*x + 30*x^5 + 10*x^7)*Log[x]^3 + (10*x + 25*x
^3 + 10*x^5)*Log[x]^4 + (7*x + 5*x^3)*Log[x]^5 + x*Log[x]^6),x]

[Out]

-4*Log[2 + x^2 + Log[x]] + Log[49 + 96*x^2 + 72*x^4 + 24*x^6 + 3*x^8 + 80*Log[x] + 112*x^2*Log[x] + 48*x^4*Log
[x] + 4*x^6*Log[x] - x^8*Log[x] + 40*Log[x]^2 + 24*x^2*Log[x]^2 - 6*x^4*Log[x]^2 - 4*x^6*Log[x]^2 - 12*x^2*Log
[x]^3 - 6*x^4*Log[x]^3 - 5*Log[x]^4 - 4*x^2*Log[x]^4 - Log[x]^5]

Rule 6684

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

Rule 6742

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

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (-\frac{4(1+2x^2)}{x(2+x^2+\log (x))}+\frac{{(2+x^2+\log (x))}^3(-10-23x^2+5\log (x)+8x^2\log (x))}{x(-49-96x^2-72x^4-24x^6-3x^8-80\log (x)-112x^2\log (x)-48x^4\log (x)-4x^6\log (x)+x^8\log (x)-40{\log }^2(x)-24x^2{\log }^2(x)+6x^4{\log }^2(x)+4x^6{\log }^2(x)+12x^2{\log }^3(x)+6x^4{\log }^3(x)+5{\log }^4(x)+4x^2{\log }^4(x)+{\log }^5(x))})dx\\ & =-(4\int \frac{1+2x^2}{x(2+x^2+\log (x))}dx)+\int \frac{{(2+x^2+\log (x))}^3(-10-23x^2+5\log (x)+8x^2\log (x))}{x(-49-96x^2-72x^4-24x^6-3x^8-80\log (x)-112x^2\log (x)-48x^4\log (x)-4x^6\log (x)+x^8\log (x)-40{\log }^2(x)-24x^2{\log }^2(x)+6x^4{\log }^2(x)+4x^6{\log }^2(x)+12x^2{\log }^3(x)+6x^4{\log }^3(x)+5{\log }^4(x)+4x^2{\log }^4(x)+{\log }^5(x))}dx\\ & =-4\log (2+x^2+\log (x))+\log (49+96x^2+72x^4+24x^6+3x^8+80\log (x)+112x^2\log (x)+48x^4\log (x)+4x^6\log (x)-x^8\log (x)+40{\log }^2(x)+24x^2{\log }^2(x)-6x^4{\log }^2(x)-4x^6{\log }^2(x)-12x^2{\log }^3(x)-6x^4{\log }^3(x)-5{\log }^4(x)-4x^2{\log }^4(x)-{\log }^5(x))\end{array}\end{array}$$

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Mathematica [B]  time = 0.56, size = 138, normalized size = 8.62 $$\begin{array}{c}-4\log (2+x^2+\log (x))+\log (49+96x^2+72x^4+24x^6+3x^8+80\log (x)+112x^2\log (x)+48x^4\log (x)+4x^6\log (x)-x^8\log (x)+40{\log }^2(x)+24x^2{\log }^2(x)-6x^4{\log }^2(x)-4x^6{\log }^2(x)-12x^2{\log }^3(x)-6x^4{\log }^3(x)-5{\log }^4(x)-4x^2{\log }^4(x)-{\log }^5(x))\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(36 + 88*x^2 + 80*x^4 + 40*x^6 + 10*x^8 + x^10 + (80 + 160*x^2 + 120*x^4 + 40*x^6 + 5*x^8)*Log[x] +
(80 + 120*x^2 + 60*x^4 + 10*x^6)*Log[x]^2 + (40 + 40*x^2 + 10*x^4)*Log[x]^3 + (10 + 5*x^2)*Log[x]^4 + Log[x]^5
)/(-98*x - 241*x^3 - 240*x^5 - 120*x^7 - 30*x^9 - 3*x^11 + (-209*x - 400*x^3 - 280*x^5 - 80*x^7 - 5*x^9 + x^11
)*Log[x] + (-160*x - 200*x^3 - 60*x^5 + 10*x^7 + 5*x^9)*Log[x]^2 + (-40*x + 30*x^5 + 10*x^7)*Log[x]^3 + (10*x
+ 25*x^3 + 10*x^5)*Log[x]^4 + (7*x + 5*x^3)*Log[x]^5 + x*Log[x]^6),x]

[Out]

-4*Log[2 + x^2 + Log[x]] + Log[49 + 96*x^2 + 72*x^4 + 24*x^6 + 3*x^8 + 80*Log[x] + 112*x^2*Log[x] + 48*x^4*Log
[x] + 4*x^6*Log[x] - x^8*Log[x] + 40*Log[x]^2 + 24*x^2*Log[x]^2 - 6*x^4*Log[x]^2 - 4*x^6*Log[x]^2 - 12*x^2*Log
[x]^3 - 6*x^4*Log[x]^3 - 5*Log[x]^4 - 4*x^2*Log[x]^4 - Log[x]^5]

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fricas [B]  time = 0.84, size = 111, normalized size = 6.94 $$\begin{array}{c}\log (-3x^8-24x^6+(4x^2+5)\log (x{)}^4+\log (x{)}^5-72x^4+6(x^4+2x^2)\log (x{)}^3+2(2x^6+3x^4-12x^2-20)\log (x{)}^2-96x^2+(x^8-4x^6-48x^4-112x^2-80)\log (x)-49)-4\log (x^2+\log (x)+2)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)^5+(5*x^2+10)*log(x)^4+(10*x^4+40*x^2+40)*log(x)^3+(10*x^6+60*x^4+120*x^2+80)*log(x)^2+(5*x^8
+40*x^6+120*x^4+160*x^2+80)*log(x)+x^10+10*x^8+40*x^6+80*x^4+88*x^2+36)/(x*log(x)^6+(5*x^3+7*x)*log(x)^5+(10*x
^5+25*x^3+10*x)*log(x)^4+(10*x^7+30*x^5-40*x)*log(x)^3+(5*x^9+10*x^7-60*x^5-200*x^3-160*x)*log(x)^2+(x^11-5*x^
9-80*x^7-280*x^5-400*x^3-209*x)*log(x)-3*x^11-30*x^9-120*x^7-240*x^5-241*x^3-98*x),x, algorithm="fricas")

[Out]

log(-3*x^8 - 24*x^6 + (4*x^2 + 5)*log(x)^4 + log(x)^5 - 72*x^4 + 6*(x^4 + 2*x^2)*log(x)^3 + 2*(2*x^6 + 3*x^4 -
12*x^2 - 20)*log(x)^2 - 96*x^2 + (x^8 - 4*x^6 - 48*x^4 - 112*x^2 - 80)*log(x) - 49) - 4*log(x^2 + log(x) + 2)

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giac [B]  time = 1.48, size = 135, normalized size = 8.44 $$\begin{array}{c}\log (x^8\log (x)-3x^8+4x^6\log (x{)}^2-4x^6\log (x)+6x^4\log (x{)}^3-24x^6+6x^4\log (x{)}^2+4x^2\log (x{)}^4-48x^4\log (x)+12x^2\log (x{)}^3+\log (x{)}^5-72x^4-24x^2\log (x{)}^2+5\log (x{)}^4-112x^2\log (x)-96x^2-40\log (x{)}^2-80\log (x)-49)-4\log (x^2+\log (x)+2)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)^5+(5*x^2+10)*log(x)^4+(10*x^4+40*x^2+40)*log(x)^3+(10*x^6+60*x^4+120*x^2+80)*log(x)^2+(5*x^8
+40*x^6+120*x^4+160*x^2+80)*log(x)+x^10+10*x^8+40*x^6+80*x^4+88*x^2+36)/(x*log(x)^6+(5*x^3+7*x)*log(x)^5+(10*x
^5+25*x^3+10*x)*log(x)^4+(10*x^7+30*x^5-40*x)*log(x)^3+(5*x^9+10*x^7-60*x^5-200*x^3-160*x)*log(x)^2+(x^11-5*x^
9-80*x^7-280*x^5-400*x^3-209*x)*log(x)-3*x^11-30*x^9-120*x^7-240*x^5-241*x^3-98*x),x, algorithm="giac")

[Out]

log(x^8*log(x) - 3*x^8 + 4*x^6*log(x)^2 - 4*x^6*log(x) + 6*x^4*log(x)^3 - 24*x^6 + 6*x^4*log(x)^2 + 4*x^2*log(
x)^4 - 48*x^4*log(x) + 12*x^2*log(x)^3 + log(x)^5 - 72*x^4 - 24*x^2*log(x)^2 + 5*log(x)^4 - 112*x^2*log(x) - 9
6*x^2 - 40*log(x)^2 - 80*log(x) - 49) - 4*log(x^2 + log(x) + 2)

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maple [B]  time = 0.07, size = 112, normalized size = 7.00

method	result	size
risch	$-4\ln (2+x^2+\ln (x))+\ln (\ln (x{)}^5+(4x^2+5)\ln (x{)}^4+(6x^4+12x^2)\ln (x{)}^3+(4x^6+6x^4-24x^2-40)\ln (x{)}^2+(x^8-4x^6-48x^4-112x^2-80)\ln (x)-3x^8-24x^6-72x^4-96x^2-49)$	$112$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(x)^5+(5*x^2+10)*ln(x)^4+(10*x^4+40*x^2+40)*ln(x)^3+(10*x^6+60*x^4+120*x^2+80)*ln(x)^2+(5*x^8+40*x^6+12
0*x^4+160*x^2+80)*ln(x)+x^10+10*x^8+40*x^6+80*x^4+88*x^2+36)/(x*ln(x)^6+(5*x^3+7*x)*ln(x)^5+(10*x^5+25*x^3+10*
x)*ln(x)^4+(10*x^7+30*x^5-40*x)*ln(x)^3+(5*x^9+10*x^7-60*x^5-200*x^3-160*x)*ln(x)^2+(x^11-5*x^9-80*x^7-280*x^5
-400*x^3-209*x)*ln(x)-3*x^11-30*x^9-120*x^7-240*x^5-241*x^3-98*x),x,method=_RETURNVERBOSE)

[Out]

-4*ln(2+x^2+ln(x))+ln(ln(x)^5+(4*x^2+5)*ln(x)^4+(6*x^4+12*x^2)*ln(x)^3+(4*x^6+6*x^4-24*x^2-40)*ln(x)^2+(x^8-4*
x^6-48*x^4-112*x^2-80)*ln(x)-3*x^8-24*x^6-72*x^4-96*x^2-49)

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maxima [B]  time = 0.41, size = 111, normalized size = 6.94 $$\begin{array}{c}\log (-3x^8-24x^6+(4x^2+5)\log (x{)}^4+\log (x{)}^5-72x^4+6(x^4+2x^2)\log (x{)}^3+2(2x^6+3x^4-12x^2-20)\log (x{)}^2-96x^2+(x^8-4x^6-48x^4-112x^2-80)\log (x)-49)-4\log (x^2+\log (x)+2)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)^5+(5*x^2+10)*log(x)^4+(10*x^4+40*x^2+40)*log(x)^3+(10*x^6+60*x^4+120*x^2+80)*log(x)^2+(5*x^8
+40*x^6+120*x^4+160*x^2+80)*log(x)+x^10+10*x^8+40*x^6+80*x^4+88*x^2+36)/(x*log(x)^6+(5*x^3+7*x)*log(x)^5+(10*x
^5+25*x^3+10*x)*log(x)^4+(10*x^7+30*x^5-40*x)*log(x)^3+(5*x^9+10*x^7-60*x^5-200*x^3-160*x)*log(x)^2+(x^11-5*x^
9-80*x^7-280*x^5-400*x^3-209*x)*log(x)-3*x^11-30*x^9-120*x^7-240*x^5-241*x^3-98*x),x, algorithm="maxima")

[Out]

log(-3*x^8 - 24*x^6 + (4*x^2 + 5)*log(x)^4 + log(x)^5 - 72*x^4 + 6*(x^4 + 2*x^2)*log(x)^3 + 2*(2*x^6 + 3*x^4 -
12*x^2 - 20)*log(x)^2 - 96*x^2 + (x^8 - 4*x^6 - 48*x^4 - 112*x^2 - 80)*log(x) - 49) - 4*log(x^2 + log(x) + 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 $$\begin{array}{c}\int -\frac{{\ln (x)}^4(5x^2+10)+{\ln (x)}^5+{\ln (x)}^3(10x^4+40x^2+40)+\ln (x)(5x^8+40x^6+120x^4+160x^2+80)+{\ln (x)}^2(10x^6+60x^4+120x^2+80)+88x^2+80x^4+40x^6+10x^8+x^{10}+36}{98x-{\ln (x)}^5(5x^3+7x)-x{\ln (x)}^6-{\ln (x)}^4(10x^5+25x^3+10x)-{\ln (x)}^3(10x^7+30x^5-40x)+\ln (x)(-x^{11}+5x^9+80x^7+280x^5+400x^3+209x)+{\ln (x)}^2(-5x^9-10x^7+60x^5+200x^3+160x)+241x^3+240x^5+120x^7+30x^9+3x^{11}}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)^4*(5*x^2 + 10) + log(x)^5 + log(x)^3*(40*x^2 + 10*x^4 + 40) + log(x)*(160*x^2 + 120*x^4 + 40*x^6
+ 5*x^8 + 80) + log(x)^2*(120*x^2 + 60*x^4 + 10*x^6 + 80) + 88*x^2 + 80*x^4 + 40*x^6 + 10*x^8 + x^10 + 36)/(98
*x - log(x)^5*(7*x + 5*x^3) - x*log(x)^6 - log(x)^4*(10*x + 25*x^3 + 10*x^5) - log(x)^3*(30*x^5 - 40*x + 10*x^
7) + log(x)*(209*x + 400*x^3 + 280*x^5 + 80*x^7 + 5*x^9 - x^11) + log(x)^2*(160*x + 200*x^3 + 60*x^5 - 10*x^7
- 5*x^9) + 241*x^3 + 240*x^5 + 120*x^7 + 30*x^9 + 3*x^11),x)

[Out]

int(-(log(x)^4*(5*x^2 + 10) + log(x)^5 + log(x)^3*(40*x^2 + 10*x^4 + 40) + log(x)*(160*x^2 + 120*x^4 + 40*x^6
+ 5*x^8 + 80) + log(x)^2*(120*x^2 + 60*x^4 + 10*x^6 + 80) + 88*x^2 + 80*x^4 + 40*x^6 + 10*x^8 + x^10 + 36)/(98
*x - log(x)^5*(7*x + 5*x^3) - x*log(x)^6 - log(x)^4*(10*x + 25*x^3 + 10*x^5) - log(x)^3*(30*x^5 - 40*x + 10*x^
7) + log(x)*(209*x + 400*x^3 + 280*x^5 + 80*x^7 + 5*x^9 - x^11) + log(x)^2*(160*x + 200*x^3 + 60*x^5 - 10*x^7
- 5*x^9) + 241*x^3 + 240*x^5 + 120*x^7 + 30*x^9 + 3*x^11), x)

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sympy [B]  time = 1.47, size = 112, normalized size = 7.00 $$\begin{array}{c}-4\log (x^2+\log (x)+2)+\log (-3x^8-24x^6-72x^4-96x^2+(4x^2+5)\log {(x)}^4+(6x^4+12x^2)\log {(x)}^3+(4x^6+6x^4-24x^2-40)\log {(x)}^2+(x^8-4x^6-48x^4-112x^2-80)\log (x)+\log {(x)}^5-49)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(x)**5+(5*x**2+10)*ln(x)**4+(10*x**4+40*x**2+40)*ln(x)**3+(10*x**6+60*x**4+120*x**2+80)*ln(x)**2+
(5*x**8+40*x**6+120*x**4+160*x**2+80)*ln(x)+x**10+10*x**8+40*x**6+80*x**4+88*x**2+36)/(x*ln(x)**6+(5*x**3+7*x)
*ln(x)**5+(10*x**5+25*x**3+10*x)*ln(x)**4+(10*x**7+30*x**5-40*x)*ln(x)**3+(5*x**9+10*x**7-60*x**5-200*x**3-160
*x)*ln(x)**2+(x**11-5*x**9-80*x**7-280*x**5-400*x**3-209*x)*ln(x)-3*x**11-30*x**9-120*x**7-240*x**5-241*x**3-9
8*x),x)

[Out]

-4*log(x**2 + log(x) + 2) + log(-3*x**8 - 24*x**6 - 72*x**4 - 96*x**2 + (4*x**2 + 5)*log(x)**4 + (6*x**4 + 12*
x**2)*log(x)**3 + (4*x**6 + 6*x**4 - 24*x**2 - 40)*log(x)**2 + (x**8 - 4*x**6 - 48*x**4 - 112*x**2 - 80)*log(x
) + log(x)**5 - 49)

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