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

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

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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 {gather*} \log \left (3 x^8+x^8 (-\log (x))+24 x^6-4 x^6 \log ^2(x)+4 x^6 \log (x)+72 x^4-6 x^4 \log ^3(x)-6 x^4 \log ^2(x)+48 x^4 \log (x)+96 x^2-4 x^2 \log ^4(x)-12 x^2 \log ^3(x)+24 x^2 \log ^2(x)+112 x^2 \log (x)-\log ^5(x)-5 \log ^4(x)+40 \log ^2(x)+80 \log (x)+49\right )-4 \log \left (x^2+\log (x)+2\right ) \end {gather*}

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 {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {4 \left (1+2 x^2\right )}{x \left (2+x^2+\log (x)\right )}+\frac {\left (2+x^2+\log (x)\right )^3 \left (-10-23 x^2+5 \log (x)+8 x^2 \log (x)\right )}{x \left (-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 ^2(x)-24 x^2 \log ^2(x)+6 x^4 \log ^2(x)+4 x^6 \log ^2(x)+12 x^2 \log ^3(x)+6 x^4 \log ^3(x)+5 \log ^4(x)+4 x^2 \log ^4(x)+\log ^5(x)\right )}\right ) \, dx\\ &=-\left (4 \int \frac {1+2 x^2}{x \left (2+x^2+\log (x)\right )} \, dx\right )+\int \frac {\left (2+x^2+\log (x)\right )^3 \left (-10-23 x^2+5 \log (x)+8 x^2 \log (x)\right )}{x \left (-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 ^2(x)-24 x^2 \log ^2(x)+6 x^4 \log ^2(x)+4 x^6 \log ^2(x)+12 x^2 \log ^3(x)+6 x^4 \log ^3(x)+5 \log ^4(x)+4 x^2 \log ^4(x)+\log ^5(x)\right )} \, dx\\ &=-4 \log \left (2+x^2+\log (x)\right )+\log \left (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 ^2(x)+24 x^2 \log ^2(x)-6 x^4 \log ^2(x)-4 x^6 \log ^2(x)-12 x^2 \log ^3(x)-6 x^4 \log ^3(x)-5 \log ^4(x)-4 x^2 \log ^4(x)-\log ^5(x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.56, size = 138, normalized size = 8.62 \begin {gather*} -4 \log \left (2+x^2+\log (x)\right )+\log \left (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 ^2(x)+24 x^2 \log ^2(x)-6 x^4 \log ^2(x)-4 x^6 \log ^2(x)-12 x^2 \log ^3(x)-6 x^4 \log ^3(x)-5 \log ^4(x)-4 x^2 \log ^4(x)-\log ^5(x)\right ) \end {gather*}

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 {gather*} \log \left (-3 \, x^{8} - 24 \, x^{6} + {\left (4 \, x^{2} + 5\right )} \log \relax (x)^{4} + \log \relax (x)^{5} - 72 \, x^{4} + 6 \, {\left (x^{4} + 2 \, x^{2}\right )} \log \relax (x)^{3} + 2 \, {\left (2 \, x^{6} + 3 \, x^{4} - 12 \, x^{2} - 20\right )} \log \relax (x)^{2} - 96 \, x^{2} + {\left (x^{8} - 4 \, x^{6} - 48 \, x^{4} - 112 \, x^{2} - 80\right )} \log \relax (x) - 49\right ) - 4 \, \log \left (x^{2} + \log \relax (x) + 2\right ) \end {gather*}

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 {gather*} \log \left (x^{8} \log \relax (x) - 3 \, x^{8} + 4 \, x^{6} \log \relax (x)^{2} - 4 \, x^{6} \log \relax (x) + 6 \, x^{4} \log \relax (x)^{3} - 24 \, x^{6} + 6 \, x^{4} \log \relax (x)^{2} + 4 \, x^{2} \log \relax (x)^{4} - 48 \, x^{4} \log \relax (x) + 12 \, x^{2} \log \relax (x)^{3} + \log \relax (x)^{5} - 72 \, x^{4} - 24 \, x^{2} \log \relax (x)^{2} + 5 \, \log \relax (x)^{4} - 112 \, x^{2} \log \relax (x) - 96 \, x^{2} - 40 \, \log \relax (x)^{2} - 80 \, \log \relax (x) - 49\right ) - 4 \, \log \left (x^{2} + \log \relax (x) + 2\right ) \end {gather*}

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	\(-4 \ln \left (2+x^{2}+\ln \relax (x )\right )+\ln \left (\ln \relax (x )^{5}+\left (4 x^{2}+5\right ) \ln \relax (x )^{4}+\left (6 x^{4}+12 x^{2}\right ) \ln \relax (x )^{3}+\left (4 x^{6}+6 x^{4}-24 x^{2}-40\right ) \ln \relax (x )^{2}+\left (x^{8}-4 x^{6}-48 x^{4}-112 x^{2}-80\right ) \ln \relax (x )-3 x^{8}-24 x^{6}-72 x^{4}-96 x^{2}-49\right )\)	\(112\)

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 {gather*} \log \left (-3 \, x^{8} - 24 \, x^{6} + {\left (4 \, x^{2} + 5\right )} \log \relax (x)^{4} + \log \relax (x)^{5} - 72 \, x^{4} + 6 \, {\left (x^{4} + 2 \, x^{2}\right )} \log \relax (x)^{3} + 2 \, {\left (2 \, x^{6} + 3 \, x^{4} - 12 \, x^{2} - 20\right )} \log \relax (x)^{2} - 96 \, x^{2} + {\left (x^{8} - 4 \, x^{6} - 48 \, x^{4} - 112 \, x^{2} - 80\right )} \log \relax (x) - 49\right ) - 4 \, \log \left (x^{2} + \log \relax (x) + 2\right ) \end {gather*}

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 {gather*} \int -\frac {{\ln \relax (x)}^4\,\left (5\,x^2+10\right )+{\ln \relax (x)}^5+{\ln \relax (x)}^3\,\left (10\,x^4+40\,x^2+40\right )+\ln \relax (x)\,\left (5\,x^8+40\,x^6+120\,x^4+160\,x^2+80\right )+{\ln \relax (x)}^2\,\left (10\,x^6+60\,x^4+120\,x^2+80\right )+88\,x^2+80\,x^4+40\,x^6+10\,x^8+x^{10}+36}{98\,x-{\ln \relax (x)}^5\,\left (5\,x^3+7\,x\right )-x\,{\ln \relax (x)}^6-{\ln \relax (x)}^4\,\left (10\,x^5+25\,x^3+10\,x\right )-{\ln \relax (x)}^3\,\left (10\,x^7+30\,x^5-40\,x\right )+\ln \relax (x)\,\left (-x^{11}+5\,x^9+80\,x^7+280\,x^5+400\,x^3+209\,x\right )+{\ln \relax (x)}^2\,\left (-5\,x^9-10\,x^7+60\,x^5+200\,x^3+160\,x\right )+241\,x^3+240\,x^5+120\,x^7+30\,x^9+3\,x^{11}} \,d x \end {gather*}

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 {gather*} - 4 \log {\left (x^{2} + \log {\relax (x )} + 2 \right )} + \log {\left (- 3 x^{8} - 24 x^{6} - 72 x^{4} - 96 x^{2} + \left (4 x^{2} + 5\right ) \log {\relax (x )}^{4} + \left (6 x^{4} + 12 x^{2}\right ) \log {\relax (x )}^{3} + \left (4 x^{6} + 6 x^{4} - 24 x^{2} - 40\right ) \log {\relax (x )}^{2} + \left (x^{8} - 4 x^{6} - 48 x^{4} - 112 x^{2} - 80\right ) \log {\relax (x )} + \log {\relax (x )}^{5} - 49 \right )} \end {gather*}

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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