3.38.40 \(\int \frac {-1125+(-675+2700 x) \log (3)+(-140+1080 x-2160 x^2) \log ^2(3)+(-10+104 x-432 x^2+576 x^3) \log ^3(3)+(675+(270-1080 x) \log (3)+(28-216 x+432 x^2) \log ^2(3)) \log (4)+(-135+(-27+108 x) \log (3)) \log ^2(4)+9 \log ^3(4)}{-1125+(-675+2700 x) \log (3)+(-135+1080 x-2160 x^2) \log ^2(3)+(-9+108 x-432 x^2+576 x^3) \log ^3(3)+(675+(270-1080 x) \log (3)+(27-216 x+432 x^2) \log ^2(3)) \log (4)+(-135+(-27+108 x) \log (3)) \log ^2(4)+9 \log ^3(4)} \, dx\)

Optimal. Leaf size=23 \[ x+\frac {x}{9 \left (-1+4 x+\frac {-5+\log (4)}{\log (3)}\right )^2} \]

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Rubi [B]  time = 0.12, antiderivative size = 64, normalized size of antiderivative = 2.78, number of steps used = 2, number of rules used = 1, integrand size = 191, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.005, Rules used = {2074} \begin {gather*} x-\frac {\log ^2(3)}{9 \log (81) \left (x (-\log (81))+5-\log \left (\frac {4}{3}\right )\right )}+\frac {\left (5-\log \left (\frac {4}{3}\right )\right ) \log ^2(3)}{9 \log (81) \left (x (-\log (81))+5-\log \left (\frac {4}{3}\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1125 + (-675 + 2700*x)*Log[3] + (-140 + 1080*x - 2160*x^2)*Log[3]^2 + (-10 + 104*x - 432*x^2 + 576*x^3)*
Log[3]^3 + (675 + (270 - 1080*x)*Log[3] + (28 - 216*x + 432*x^2)*Log[3]^2)*Log[4] + (-135 + (-27 + 108*x)*Log[
3])*Log[4]^2 + 9*Log[4]^3)/(-1125 + (-675 + 2700*x)*Log[3] + (-135 + 1080*x - 2160*x^2)*Log[3]^2 + (-9 + 108*x
- 432*x^2 + 576*x^3)*Log[3]^3 + (675 + (270 - 1080*x)*Log[3] + (27 - 216*x + 432*x^2)*Log[3]^2)*Log[4] + (-13
5 + (-27 + 108*x)*Log[3])*Log[4]^2 + 9*Log[4]^3),x]

[Out]

x + ((5 - Log[4/3])*Log[3]^2)/(9*Log[81]*(5 - Log[4/3] - x*Log[81])^2) - Log[3]^2/(9*Log[81]*(5 - Log[4/3] - x
*Log[81]))

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {2 \left (-5+\log \left (\frac {4}{3}\right )\right ) \log ^2(3)}{9 \left (-5+\log \left (\frac {4}{3}\right )+x \log (81)\right )^3}-\frac {\log ^2(3)}{9 \left (-5+\log \left (\frac {4}{3}\right )+x \log (81)\right )^2}\right ) \, dx\\ &=x+\frac {\left (5-\log \left (\frac {4}{3}\right )\right ) \log ^2(3)}{9 \log (81) \left (5-\log \left (\frac {4}{3}\right )-x \log (81)\right )^2}-\frac {\log ^2(3)}{9 \log (81) \left (5-\log \left (\frac {4}{3}\right )-x \log (81)\right )}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.15, size = 220, normalized size = 9.57 \begin {gather*} \frac {2304 x^2 \left (-5+\log \left (\frac {4}{3}\right )\right ) \log ^3(3) \log ^2(81)+27 \log (3) (-5+\log (4))^2 \left (20+\log \left (\frac {6561}{256}\right )\right ) \log ^2(81)+4 \log ^2(3) (-5+\log (4)) \left (-270+54 \log \left (\frac {4}{3}\right )-7 \log (81)\right ) \log ^2(81)+1152 x^3 \log ^3(3) \log ^3(81)-9 (-5+\log (4))^3 \log ^3(81)-8 x \log (3) \log (81) \left (-54 \log (3) (-5+\log (4)) \log ^2(81)+27 (-5+\log (4))^2 \log ^2(81)+\log ^2(3) \left (-14400+5760 \log \left (\frac {4}{3}\right )-576 \log ^2\left (\frac {4}{3}\right )+26 \log ^2(81)\right )\right )+2 \log ^3(3) \left (-17280 \log ^2\left (\frac {4}{3}\right )+1152 \log ^3\left (\frac {4}{3}\right )+\log \left (\frac {4}{3}\right ) \left (86400-52 \log ^2(81)\right )+5 \left (-28800+52 \log ^2(81)+\log ^3(81)\right )\right )}{18 \log ^4(81) \left (-5+\log \left (\frac {4}{3}\right )+x \log (81)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1125 + (-675 + 2700*x)*Log[3] + (-140 + 1080*x - 2160*x^2)*Log[3]^2 + (-10 + 104*x - 432*x^2 + 576
*x^3)*Log[3]^3 + (675 + (270 - 1080*x)*Log[3] + (28 - 216*x + 432*x^2)*Log[3]^2)*Log[4] + (-135 + (-27 + 108*x
)*Log[3])*Log[4]^2 + 9*Log[4]^3)/(-1125 + (-675 + 2700*x)*Log[3] + (-135 + 1080*x - 2160*x^2)*Log[3]^2 + (-9 +
108*x - 432*x^2 + 576*x^3)*Log[3]^3 + (675 + (270 - 1080*x)*Log[3] + (27 - 216*x + 432*x^2)*Log[3]^2)*Log[4]
+ (-135 + (-27 + 108*x)*Log[3])*Log[4]^2 + 9*Log[4]^3),x]

[Out]

(2304*x^2*(-5 + Log[4/3])*Log[3]^3*Log[81]^2 + 27*Log[3]*(-5 + Log[4])^2*(20 + Log[6561/256])*Log[81]^2 + 4*Lo
g[3]^2*(-5 + Log[4])*(-270 + 54*Log[4/3] - 7*Log[81])*Log[81]^2 + 1152*x^3*Log[3]^3*Log[81]^3 - 9*(-5 + Log[4]
)^3*Log[81]^3 - 8*x*Log[3]*Log[81]*(-54*Log[3]*(-5 + Log[4])*Log[81]^2 + 27*(-5 + Log[4])^2*Log[81]^2 + Log[3]
^2*(-14400 + 5760*Log[4/3] - 576*Log[4/3]^2 + 26*Log[81]^2)) + 2*Log[3]^3*(-17280*Log[4/3]^2 + 1152*Log[4/3]^3
+ Log[4/3]*(86400 - 52*Log[81]^2) + 5*(-28800 + 52*Log[81]^2 + Log[81]^3)))/(18*Log[81]^4*(-5 + Log[4/3] + x*
Log[81])^2)

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fricas [B]  time = 0.74, size = 111, normalized size = 4.83 \begin {gather*} \frac {2 \, {\left (72 \, x^{3} - 36 \, x^{2} + 5 \, x\right )} \log \relax (3)^{2} + 36 \, x \log \relax (2)^{2} - 18 \, {\left (20 \, x^{2} - 2 \, {\left (4 \, x^{2} - x\right )} \log \relax (2) - 5 \, x\right )} \log \relax (3) - 180 \, x \log \relax (2) + 225 \, x}{9 \, {\left ({\left (16 \, x^{2} - 8 \, x + 1\right )} \log \relax (3)^{2} + 2 \, {\left (2 \, {\left (4 \, x - 1\right )} \log \relax (2) - 20 \, x + 5\right )} \log \relax (3) + 4 \, \log \relax (2)^{2} - 20 \, \log \relax (2) + 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((72*log(2)^3+4*((108*x-27)*log(3)-135)*log(2)^2+2*((432*x^2-216*x+28)*log(3)^2+(-1080*x+270)*log(3)+
675)*log(2)+(576*x^3-432*x^2+104*x-10)*log(3)^3+(-2160*x^2+1080*x-140)*log(3)^2+(2700*x-675)*log(3)-1125)/(72*
log(2)^3+4*((108*x-27)*log(3)-135)*log(2)^2+2*((432*x^2-216*x+27)*log(3)^2+(-1080*x+270)*log(3)+675)*log(2)+(5
76*x^3-432*x^2+108*x-9)*log(3)^3+(-2160*x^2+1080*x-135)*log(3)^2+(2700*x-675)*log(3)-1125),x, algorithm="frica
s")

[Out]

1/9*(2*(72*x^3 - 36*x^2 + 5*x)*log(3)^2 + 36*x*log(2)^2 - 18*(20*x^2 - 2*(4*x^2 - x)*log(2) - 5*x)*log(3) - 18
0*x*log(2) + 225*x)/((16*x^2 - 8*x + 1)*log(3)^2 + 2*(2*(4*x - 1)*log(2) - 20*x + 5)*log(3) + 4*log(2)^2 - 20*
log(2) + 25)

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giac [A]  time = 0.29, size = 26, normalized size = 1.13 \begin {gather*} x + \frac {x \log \relax (3)^{2}}{9 \, {\left (4 \, x \log \relax (3) - \log \relax (3) + 2 \, \log \relax (2) - 5\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((72*log(2)^3+4*((108*x-27)*log(3)-135)*log(2)^2+2*((432*x^2-216*x+28)*log(3)^2+(-1080*x+270)*log(3)+
675)*log(2)+(576*x^3-432*x^2+104*x-10)*log(3)^3+(-2160*x^2+1080*x-140)*log(3)^2+(2700*x-675)*log(3)-1125)/(72*
log(2)^3+4*((108*x-27)*log(3)-135)*log(2)^2+2*((432*x^2-216*x+27)*log(3)^2+(-1080*x+270)*log(3)+675)*log(2)+(5
76*x^3-432*x^2+108*x-9)*log(3)^3+(-2160*x^2+1080*x-135)*log(3)^2+(2700*x-675)*log(3)-1125),x, algorithm="giac"
)

[Out]

x + 1/9*x*log(3)^2/(4*x*log(3) - log(3) + 2*log(2) - 5)^2

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maple [A]  time = 0.18, size = 53, normalized size = 2.30

method	result	size
default	\(x +\frac {\left (\ln \relax (3)-2 \ln \relax (2)+5\right ) \ln \relax (3)}{36 \left (4 x \ln \relax (3)-\ln \relax (3)+2 \ln \relax (2)-5\right )^{2}}+\frac {\ln \relax (3)}{144 x \ln \relax (3)-36 \ln \relax (3)+72 \ln \relax (2)-180}\)	\(53\)
risch	\(x +\frac {x \ln \relax (3)^{2}}{144 x^{2} \ln \relax (3)^{2}-72 x \ln \relax (3)^{2}+144 x \ln \relax (2) \ln \relax (3)+9 \ln \relax (3)^{2}-36 \ln \relax (2) \ln \relax (3)-360 x \ln \relax (3)+36 \ln \relax (2)^{2}+90 \ln \relax (3)-180 \ln \relax (2)+225}\)	\(66\)
norman	\(\frac {\left (-\frac {26 \ln \relax (3)^{2}}{9}+12 \ln \relax (2) \ln \relax (3)-12 \ln \relax (2)^{2}-30 \ln \relax (3)+60 \ln \relax (2)-75\right ) x +16 x^{3} \ln \relax (3)^{2}+\frac {\ln \relax (3)^{3}-6 \ln \relax (2) \ln \relax (3)^{2}+12 \ln \relax (2)^{2} \ln \relax (3)-8 \ln \relax (2)^{3}+15 \ln \relax (3)^{2}-60 \ln \relax (2) \ln \relax (3)+60 \ln \relax (2)^{2}+75 \ln \relax (3)-150 \ln \relax (2)+125}{2 \ln \relax (3)}}{\left (4 x \ln \relax (3)-\ln \relax (3)+2 \ln \relax (2)-5\right )^{2}}\)	\(119\)
gosper	\(\frac {288 x^{3} \ln \relax (3)^{3}-52 x \ln \relax (3)^{3}+216 x \ln \relax (2) \ln \relax (3)^{2}-216 \ln \relax (3) \ln \relax (2)^{2} x +9 \ln \relax (3)^{3}-54 \ln \relax (2) \ln \relax (3)^{2}-540 x \ln \relax (3)^{2}+108 \ln \relax (2)^{2} \ln \relax (3)+1080 x \ln \relax (2) \ln \relax (3)-72 \ln \relax (2)^{3}+135 \ln \relax (3)^{2}-540 \ln \relax (2) \ln \relax (3)-1350 x \ln \relax (3)+540 \ln \relax (2)^{2}+675 \ln \relax (3)-1350 \ln \relax (2)+1125}{18 \ln \relax (3) \left (16 x^{2} \ln \relax (3)^{2}-8 x \ln \relax (3)^{2}+16 x \ln \relax (2) \ln \relax (3)+\ln \relax (3)^{2}-4 \ln \relax (2) \ln \relax (3)-40 x \ln \relax (3)+4 \ln \relax (2)^{2}+10 \ln \relax (3)-20 \ln \relax (2)+25\right )}\)	\(172\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((72*ln(2)^3+4*((108*x-27)*ln(3)-135)*ln(2)^2+2*((432*x^2-216*x+28)*ln(3)^2+(-1080*x+270)*ln(3)+675)*ln(2)+
(576*x^3-432*x^2+104*x-10)*ln(3)^3+(-2160*x^2+1080*x-140)*ln(3)^2+(2700*x-675)*ln(3)-1125)/(72*ln(2)^3+4*((108
*x-27)*ln(3)-135)*ln(2)^2+2*((432*x^2-216*x+27)*ln(3)^2+(-1080*x+270)*ln(3)+675)*ln(2)+(576*x^3-432*x^2+108*x-
9)*ln(3)^3+(-2160*x^2+1080*x-135)*ln(3)^2+(2700*x-675)*ln(3)-1125),x,method=_RETURNVERBOSE)

[Out]

x+1/36*(ln(3)-2*ln(2)+5)*ln(3)/(4*x*ln(3)-ln(3)+2*ln(2)-5)^2+1/36*ln(3)/(4*x*ln(3)-ln(3)+2*ln(2)-5)

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maxima [B]  time = 0.36, size = 65, normalized size = 2.83 \begin {gather*} \frac {x \log \relax (3)^{2}}{9 \, {\left (16 \, x^{2} \log \relax (3)^{2} + 8 \, {\left ({\left (2 \, \log \relax (2) - 5\right )} \log \relax (3) - \log \relax (3)^{2}\right )} x - 2 \, {\left (2 \, \log \relax (2) - 5\right )} \log \relax (3) + \log \relax (3)^{2} + 4 \, \log \relax (2)^{2} - 20 \, \log \relax (2) + 25\right )}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((72*log(2)^3+4*((108*x-27)*log(3)-135)*log(2)^2+2*((432*x^2-216*x+28)*log(3)^2+(-1080*x+270)*log(3)+
675)*log(2)+(576*x^3-432*x^2+104*x-10)*log(3)^3+(-2160*x^2+1080*x-140)*log(3)^2+(2700*x-675)*log(3)-1125)/(72*
log(2)^3+4*((108*x-27)*log(3)-135)*log(2)^2+2*((432*x^2-216*x+27)*log(3)^2+(-1080*x+270)*log(3)+675)*log(2)+(5
76*x^3-432*x^2+108*x-9)*log(3)^3+(-2160*x^2+1080*x-135)*log(3)^2+(2700*x-675)*log(3)-1125),x, algorithm="maxim
a")

[Out]

1/9*x*log(3)^2/(16*x^2*log(3)^2 + 8*((2*log(2) - 5)*log(3) - log(3)^2)*x - 2*(2*log(2) - 5)*log(3) + log(3)^2
+ 4*log(2)^2 - 20*log(2) + 25) + x

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mupad [B]  time = 4.02, size = 923, normalized size = 40.13 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3)*(2700*x - 675) + 4*log(2)^2*(log(3)*(108*x - 27) - 135) - log(3)^2*(2160*x^2 - 1080*x + 140) + 2*l
og(2)*(log(3)^2*(432*x^2 - 216*x + 28) - log(3)*(1080*x - 270) + 675) + 72*log(2)^3 + log(3)^3*(104*x - 432*x^
2 + 576*x^3 - 10) - 1125)/(log(3)*(2700*x - 675) + 4*log(2)^2*(log(3)*(108*x - 27) - 135) - log(3)^2*(2160*x^2
- 1080*x + 135) + 2*log(2)*(log(3)^2*(432*x^2 - 216*x + 27) - log(3)*(1080*x - 270) + 675) + 72*log(2)^3 + lo
g(3)^3*(108*x - 432*x^2 + 576*x^3 - 9) - 1125),x)

[Out]

x + symsum(log((51200*root(480*log(2)^2*log(3)^6 + 96*log(2)^2*log(3)^7 - 64*log(2)^3*log(3)^6 - 1200*log(2)*l
og(3)^6 - 480*log(2)*log(3)^7 - 48*log(2)*log(3)^8 + 1000*log(3)^6 + 600*log(3)^7 + 120*log(3)^8 + 8*log(3)^9,
z, k)*log(3)^7)/3 + (20480*root(480*log(2)^2*log(3)^6 + 96*log(2)^2*log(3)^7 - 64*log(2)^3*log(3)^6 - 1200*lo
g(2)*log(3)^6 - 480*log(2)*log(3)^7 - 48*log(2)*log(3)^8 + 1000*log(3)^6 + 600*log(3)^7 + 120*log(3)^8 + 8*log
(3)^9, z, k)*log(3)^8)/3 + (2048*root(480*log(2)^2*log(3)^6 + 96*log(2)^2*log(3)^7 - 64*log(2)^3*log(3)^6 - 12
00*log(2)*log(3)^6 - 480*log(2)*log(3)^7 - 48*log(2)*log(3)^8 + 1000*log(3)^6 + 600*log(3)^7 + 120*log(3)^8 +
8*log(3)^9, z, k)*log(3)^9)/3 - (512*log(2)*log(3)^8)/81 + (1024*x*log(3)^9)/81 + (1280*log(3)^8)/81 + (256*lo
g(3)^9)/81 + (8192*root(480*log(2)^2*log(3)^6 + 96*log(2)^2*log(3)^7 - 64*log(2)^3*log(3)^6 - 1200*log(2)*log(
3)^6 - 480*log(2)*log(3)^7 - 48*log(2)*log(3)^8 + 1000*log(3)^6 + 600*log(3)^7 + 120*log(3)^8 + 8*log(3)^9, z,
k)*log(2)^2*log(3)^7)/3 - (40960*root(480*log(2)^2*log(3)^6 + 96*log(2)^2*log(3)^7 - 64*log(2)^3*log(3)^6 - 1
200*log(2)*log(3)^6 - 480*log(2)*log(3)^7 - 48*log(2)*log(3)^8 + 1000*log(3)^6 + 600*log(3)^7 + 120*log(3)^8 +
8*log(3)^9, z, k)*log(2)*log(3)^7)/3 - (8192*root(480*log(2)^2*log(3)^6 + 96*log(2)^2*log(3)^7 - 64*log(2)^3*
log(3)^6 - 1200*log(2)*log(3)^6 - 480*log(2)*log(3)^7 - 48*log(2)*log(3)^8 + 1000*log(3)^6 + 600*log(3)^7 + 12
0*log(3)^8 + 8*log(3)^9, z, k)*log(2)*log(3)^8)/3 - (40960*root(480*log(2)^2*log(3)^6 + 96*log(2)^2*log(3)^7 -
64*log(2)^3*log(3)^6 - 1200*log(2)*log(3)^6 - 480*log(2)*log(3)^7 - 48*log(2)*log(3)^8 + 1000*log(3)^6 + 600*
log(3)^7 + 120*log(3)^8 + 8*log(3)^9, z, k)*x*log(3)^8)/3 - (8192*root(480*log(2)^2*log(3)^6 + 96*log(2)^2*log
(3)^7 - 64*log(2)^3*log(3)^6 - 1200*log(2)*log(3)^6 - 480*log(2)*log(3)^7 - 48*log(2)*log(3)^8 + 1000*log(3)^6
+ 600*log(3)^7 + 120*log(3)^8 + 8*log(3)^9, z, k)*x*log(3)^9)/3 + (16384*root(480*log(2)^2*log(3)^6 + 96*log(
2)^2*log(3)^7 - 64*log(2)^3*log(3)^6 - 1200*log(2)*log(3)^6 - 480*log(2)*log(3)^7 - 48*log(2)*log(3)^8 + 1000*
log(3)^6 + 600*log(3)^7 + 120*log(3)^8 + 8*log(3)^9, z, k)*x*log(2)*log(3)^8)/3)*root(480*log(2)^2*log(3)^6 +
96*log(2)^2*log(3)^7 - 64*log(2)^3*log(3)^6 - 1200*log(2)*log(3)^6 - 480*log(2)*log(3)^7 - 48*log(2)*log(3)^8
+ 1000*log(3)^6 + 600*log(3)^7 + 120*log(3)^8 + 8*log(3)^9, z, k), k, 1, 3)

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sympy [B]  time = 0.88, size = 73, normalized size = 3.17 \begin {gather*} x + \frac {x \log {\relax (3 )}^{2}}{144 x^{2} \log {\relax (3 )}^{2} + x \left (- 360 \log {\relax (3 )} - 72 \log {\relax (3 )}^{2} + 144 \log {\relax (2 )} \log {\relax (3 )}\right ) - 180 \log {\relax (2 )} - 36 \log {\relax (2 )} \log {\relax (3 )} + 9 \log {\relax (3 )}^{2} + 36 \log {\relax (2 )}^{2} + 90 \log {\relax (3 )} + 225} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((72*ln(2)**3+4*((108*x-27)*ln(3)-135)*ln(2)**2+2*((432*x**2-216*x+28)*ln(3)**2+(-1080*x+270)*ln(3)+6
75)*ln(2)+(576*x**3-432*x**2+104*x-10)*ln(3)**3+(-2160*x**2+1080*x-140)*ln(3)**2+(2700*x-675)*ln(3)-1125)/(72*
ln(2)**3+4*((108*x-27)*ln(3)-135)*ln(2)**2+2*((432*x**2-216*x+27)*ln(3)**2+(-1080*x+270)*ln(3)+675)*ln(2)+(576
*x**3-432*x**2+108*x-9)*ln(3)**3+(-2160*x**2+1080*x-135)*ln(3)**2+(2700*x-675)*ln(3)-1125),x)

[Out]

x + x*log(3)**2/(144*x**2*log(3)**2 + x*(-360*log(3) - 72*log(3)**2 + 144*log(2)*log(3)) - 180*log(2) - 36*log
(2)*log(3) + 9*log(3)**2 + 36*log(2)**2 + 90*log(3) + 225)

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