3.72.68 \(\int \frac {(-35+14 x+x^2) \log (3)+28 \log ^2(3)}{25 x^2-10 x^3+x^4+(-40 x^2+8 x^3) \log (3)+16 x^2 \log ^2(3)} \, dx\)

Optimal. Leaf size=27 \[ 1-\frac {1+\frac {7}{x}}{4+\frac {-5+x}{\log (3)}}+5 \log (64) \]

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Rubi [A]  time = 0.10, antiderivative size = 36, normalized size of antiderivative = 1.33, number of steps used = 5, number of rules used = 5, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {6, 1680, 12, 1814, 8} \begin {gather*} -\frac {4 (x+7) \log (3)}{4 \left (x+\frac {1}{4} (8 \log (3)-10)\right )^2-(5-\log (81))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-35 + 14*x + x^2)*Log[3] + 28*Log[3]^2)/(25*x^2 - 10*x^3 + x^4 + (-40*x^2 + 8*x^3)*Log[3] + 16*x^2*Log[3
]^2),x]

[Out]

(-4*(7 + x)*Log[3])/(4*(x + (-10 + 8*Log[3])/4)^2 - (5 - Log[81])^2)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-35+14 x+x^2\right ) \log (3)+28 \log ^2(3)}{-10 x^3+x^4+\left (-40 x^2+8 x^3\right ) \log (3)+x^2 \left (25+16 \log ^2(3)\right )} \, dx\\ &=\operatorname {Subst}\left (\int \frac {4 \log (3) \left (4 x^2+4 x (19-4 \log (3))+(-5+\log (81))^2\right )}{\left (4 x^2-(-5+\log (81))^2\right )^2} \, dx,x,x+\frac {1}{4} (-10+8 \log (3))\right )\\ &=(4 \log (3)) \operatorname {Subst}\left (\int \frac {4 x^2+4 x (19-4 \log (3))+(-5+\log (81))^2}{\left (4 x^2-(-5+\log (81))^2\right )^2} \, dx,x,x+\frac {1}{4} (-10+8 \log (3))\right )\\ &=-\frac {4 (7+x) \log (3)}{(-5+2 x+2 \log (9))^2-(5-\log (81))^2}+\frac {(2 \log (3)) \operatorname {Subst}\left (\int 0 \, dx,x,x+\frac {1}{4} (-10+8 \log (3))\right )}{(5-\log (81))^2}\\ &=-\frac {4 (7+x) \log (3)}{(-5+2 x+2 \log (9))^2-(5-\log (81))^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 17, normalized size = 0.63 \begin {gather*} -\frac {(7+x) \log (3)}{x (-5+x+\log (81))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-35 + 14*x + x^2)*Log[3] + 28*Log[3]^2)/(25*x^2 - 10*x^3 + x^4 + (-40*x^2 + 8*x^3)*Log[3] + 16*x^2
*Log[3]^2),x]

[Out]

-(((7 + x)*Log[3])/(x*(-5 + x + Log[81])))

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fricas [A]  time = 1.17, size = 21, normalized size = 0.78 \begin {gather*} -\frac {{\left (x + 7\right )} \log \relax (3)}{x^{2} + 4 \, x \log \relax (3) - 5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((28*log(3)^2+(x^2+14*x-35)*log(3))/(16*x^2*log(3)^2+(8*x^3-40*x^2)*log(3)+x^4-10*x^3+25*x^2),x, algo
rithm="fricas")

[Out]

-(x + 7)*log(3)/(x^2 + 4*x*log(3) - 5*x)

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giac [A]  time = 0.21, size = 25, normalized size = 0.93 \begin {gather*} -\frac {x \log \relax (3) + 7 \, \log \relax (3)}{x^{2} + 4 \, x \log \relax (3) - 5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((28*log(3)^2+(x^2+14*x-35)*log(3))/(16*x^2*log(3)^2+(8*x^3-40*x^2)*log(3)+x^4-10*x^3+25*x^2),x, algo
rithm="giac")

[Out]

-(x*log(3) + 7*log(3))/(x^2 + 4*x*log(3) - 5*x)

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maple [A]  time = 0.07, size = 20, normalized size = 0.74

method	result	size
gosper	\(-\frac {\left (x +7\right ) \ln \relax (3)}{x \left (-5+4 \ln \relax (3)+x \right )}\)	\(20\)
norman	\(\frac {-x \ln \relax (3)-7 \ln \relax (3)}{x \left (-5+4 \ln \relax (3)+x \right )}\)	\(24\)
risch	\(\frac {-x \ln \relax (3)-7 \ln \relax (3)}{x \left (-5+4 \ln \relax (3)+x \right )}\)	\(25\)
default	\(\ln \relax (3) \left (-\frac {7}{\left (-5+4 \ln \relax (3)\right ) x}-\frac {4 \ln \relax (3)-12}{\left (-5+4 \ln \relax (3)\right ) \left (-5+4 \ln \relax (3)+x \right )}\right )\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((28*ln(3)^2+(x^2+14*x-35)*ln(3))/(16*x^2*ln(3)^2+(8*x^3-40*x^2)*ln(3)+x^4-10*x^3+25*x^2),x,method=_RETURNV
ERBOSE)

[Out]

-1/x*(x+7)*ln(3)/(-5+4*ln(3)+x)

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maxima [A]  time = 0.35, size = 25, normalized size = 0.93 \begin {gather*} -\frac {x \log \relax (3) + 7 \, \log \relax (3)}{x^{2} + x {\left (4 \, \log \relax (3) - 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((28*log(3)^2+(x^2+14*x-35)*log(3))/(16*x^2*log(3)^2+(8*x^3-40*x^2)*log(3)+x^4-10*x^3+25*x^2),x, algo
rithm="maxima")

[Out]

-(x*log(3) + 7*log(3))/(x^2 + x*(4*log(3) - 5))

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mupad [B]  time = 5.06, size = 5106, normalized size = 189.11 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((28*log(3)^2 + log(3)*(14*x + x^2 - 35))/(16*x^2*log(3)^2 - log(3)*(40*x^2 - 8*x^3) + 25*x^2 - 10*x^3 + x^
4),x)

[Out]

log((2240*log(3)^4*log(81) - 1680*log(3)^3*log(81) - 8400*log(3)^2*log(81) + 33600*log(3)^3 - 29120*log(3)^4 +
6272*log(3)^5 + 2240*log(3)^2*log(81)^2 - 280*log(3)^2*log(81)^3 + 168*log(3)^3*log(81)^2 + 14*log(3)^2*log(8
1)^4 - 224*log(3)^4*log(81)^2)/(150*log(81)^2 - 500*log(81) - 20*log(81)^3 + log(81)^4 + 625) - ((14*log(3))/(
log(81) - 5)^2 - (14*log(3)*(4*log(3) - 5)^2)/(log(81) - 5)^4)*((37500*log(3) - 71250*log(3)*log(81) + 32125*l
og(3)*log(81)^2 - 14000*log(3)^2*log(81) - 6000*log(3)*log(81)^3 + 67200*log(3)^3*log(81) + 550*log(3)*log(81)
^4 - 17920*log(3)^4*log(81) - 30*log(3)*log(81)^5 + log(3)*log(81)^6 + 122500*log(3)^2 - 168000*log(3)^3 + 448
00*log(3)^4 - 12600*log(3)^2*log(81)^2 + 2800*log(3)^2*log(81)^3 - 6720*log(3)^3*log(81)^2 - 140*log(3)^2*log(
81)^4 + 1792*log(3)^4*log(81)^2)/(150*log(81)^2 - 500*log(81) - 20*log(81)^3 + log(81)^4 + 625) + ((125000*log
(3) + 187500*log(81) - 150000*log(3)*log(81) + 75000*log(3)*log(81)^2 - 20000*log(3)*log(81)^3 + 3000*log(3)*l
og(81)^4 - 240*log(3)*log(81)^5 + 8*log(3)*log(81)^6 - 93750*log(81)^2 + 25000*log(81)^3 - 3750*log(81)^4 + 30
0*log(81)^5 - 10*log(81)^6 - 156250)/(150*log(81)^2 - 500*log(81) - 20*log(81)^3 + log(81)^4 + 625) + (x*(1250
0*log(81) - 200000*log(3) + 160000*log(3)*log(81) - 48000*log(3)*log(81)^2 - 64000*log(3)^2*log(81) + 6400*log
(3)*log(81)^3 - 320*log(3)*log(81)^4 + 80000*log(3)^2 - 26250*log(81)^2 + 11000*log(81)^3 - 2050*log(81)^4 + 1
80*log(81)^5 - 6*log(81)^6 + 19200*log(3)^2*log(81)^2 - 2560*log(3)^2*log(81)^3 + 128*log(3)^2*log(81)^4 + 312
50))/(150*log(81)^2 - 500*log(81) - 20*log(81)^3 + log(81)^4 + 625))*((14*log(3))/(log(81) - 5)^2 - (14*log(3)
*(4*log(3) - 5)^2)/(log(81) - 5)^4) + (x*(19000*log(3)*log(81) - 15000*log(3) - 7100*log(3)*log(81)^2 + 7200*l
og(3)^2*log(81) + 1040*log(3)*log(81)^3 - 4480*log(3)^3*log(81) - 52*log(3)*log(81)^4 - 23000*log(3)^2 + 11200
*log(3)^3 + 80*log(3)^2*log(81)^2 - 160*log(3)^2*log(81)^3 + 448*log(3)^3*log(81)^2 + 8*log(3)^2*log(81)^4))/(
150*log(81)^2 - 500*log(81) - 20*log(81)^3 + log(81)^4 + 625)) + (x*(560*log(3)^3*log(81) - 1200*log(3)^2*log(
81) + 3600*log(3)^2 - 3360*log(3)^3 + 784*log(3)^4 + 220*log(3)^2*log(81)^2 - 20*log(3)^2*log(81)^3 - 56*log(3
)^3*log(81)^2 + log(3)^2*log(81)^4))/(150*log(81)^2 - 500*log(81) - 20*log(81)^3 + log(81)^4 + 625))*((14*log(
3))/(log(81) - 5)^2 - (14*log(3)*(4*log(3) - 5)^2)/(log(81) - 5)^4) + (7*(5*log(3) - 4*log(3)^2))/(x*(log(81)^
2 - 10*log(81) + 25)) - (log((2240*log(3)^4*log(81) - 1680*log(3)^3*log(81) - 8400*log(3)^2*log(81) + 33600*lo
g(3)^3 - 29120*log(3)^4 + 6272*log(3)^5 + 2240*log(3)^2*log(81)^2 - 280*log(3)^2*log(81)^3 + 168*log(3)^3*log(
81)^2 + 14*log(3)^2*log(81)^4 - 224*log(3)^4*log(81)^2)/(150*log(81)^2 - 500*log(81) - 20*log(81)^3 + log(81)^
4 + 625) + (x*(560*log(3)^3*log(81) - 1200*log(3)^2*log(81) + 3600*log(3)^2 - 3360*log(3)^3 + 784*log(3)^4 + 2
20*log(3)^2*log(81)^2 - 20*log(3)^2*log(81)^3 - 56*log(3)^3*log(81)^2 + log(3)^2*log(81)^4))/(150*log(81)^2 -
500*log(81) - 20*log(81)^3 + log(81)^4 + 625) + (((37500*log(3) - 71250*log(3)*log(81) + 32125*log(3)*log(81)^
2 - 14000*log(3)^2*log(81) - 6000*log(3)*log(81)^3 + 67200*log(3)^3*log(81) + 550*log(3)*log(81)^4 - 17920*log
(3)^4*log(81) - 30*log(3)*log(81)^5 + log(3)*log(81)^6 + 122500*log(3)^2 - 168000*log(3)^3 + 44800*log(3)^4 -
12600*log(3)^2*log(81)^2 + 2800*log(3)^2*log(81)^3 - 6720*log(3)^3*log(81)^2 - 140*log(3)^2*log(81)^4 + 1792*l
og(3)^4*log(81)^2)/(150*log(81)^2 - 500*log(81) - 20*log(81)^3 + log(81)^4 + 625) + (x*(19000*log(3)*log(81) -
15000*log(3) - 7100*log(3)*log(81)^2 + 7200*log(3)^2*log(81) + 1040*log(3)*log(81)^3 - 4480*log(3)^3*log(81)
- 52*log(3)*log(81)^4 - 23000*log(3)^2 + 11200*log(3)^3 + 80*log(3)^2*log(81)^2 - 160*log(3)^2*log(81)^3 + 448
*log(3)^3*log(81)^2 + 8*log(3)^2*log(81)^4))/(150*log(81)^2 - 500*log(81) - 20*log(81)^3 + log(81)^4 + 625) -
(((125000*log(3) + 187500*log(81) - 150000*log(3)*log(81) + 75000*log(3)*log(81)^2 - 20000*log(3)*log(81)^3 +
3000*log(3)*log(81)^4 - 240*log(3)*log(81)^5 + 8*log(3)*log(81)^6 - 93750*log(81)^2 + 25000*log(81)^3 - 3750*l
og(81)^4 + 300*log(81)^5 - 10*log(81)^6 - 156250)/(150*log(81)^2 - 500*log(81) - 20*log(81)^3 + log(81)^4 + 62
5) + (x*(12500*log(81) - 200000*log(3) + 160000*log(3)*log(81) - 48000*log(3)*log(81)^2 - 64000*log(3)^2*log(8
1) + 6400*log(3)*log(81)^3 - 320*log(3)*log(81)^4 + 80000*log(3)^2 - 26250*log(81)^2 + 11000*log(81)^3 - 2050*
log(81)^4 + 180*log(81)^5 - 6*log(81)^6 + 19200*log(3)^2*log(81)^2 - 2560*log(3)^2*log(81)^3 + 128*log(3)^2*lo
g(81)^4 + 31250))/(150*log(81)^2 - 500*log(81) - 20*log(81)^3 + log(81)^4 + 625))*(1500*log(3)*((4*log(3) - lo
g(81))*(4*log(3) + log(81) - 10))^(1/2) + 2100*log(3)^2*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2)
- 3360*log(3)^3*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + 896*log(3)^4*((4*log(3) - log(81))*(
4*log(3) + log(81) - 10))^(1/2) - 1400*log(3)*log(81)^2 + 11200*log(3)^2*log(81) + 280*log(3)*log(81)^3 - 4480
*log(3)^3*log(81) - 14*log(3)*log(81)^4 - 22400*log(3)^3 + 17920*log(3)^4 - 3584*log(3)^5 - 1120*log(3)^2*log(
81)^2 + 448*log(3)^3*log(81)^2 - 84*log(3)^2*log(81)^2*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2)
- 1550*log(3)*log(81)*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + 255*log(3)*log(81)^2*((4*log(3)
- log(81))*(4*log(3) + log(81) - 10))^(1/2) + 840*log(3)^2*log(81)*((4*log(3) - log(81))*(4*log(3) + log(81)
- 10))^(1/2) - 20*log(3)*log(81)^3*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + log(3)*log(81)^4*(
(4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2)))/((4*log(3) - log(81))*(4*log(3) + log(81) - 10)*(4*log
(3)*(4*log(3) - 10)*(8*log(3)*(4*log(3) - 10) + 100) - (4*log(3) - log(81))*(16*log(3)*(4*log(3) - 10) - 2*(4*
log(3) - log(81))*(4*log(3) + log(81) - 10) + 100)*(4*log(3) + log(81) - 10) + 1250)))*(1500*log(3)*((4*log(3)
- log(81))*(4*log(3) + log(81) - 10))^(1/2) + 2100*log(3)^2*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^
(1/2) - 3360*log(3)^3*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + 896*log(3)^4*((4*log(3) - log(8
1))*(4*log(3) + log(81) - 10))^(1/2) - 1400*log(3)*log(81)^2 + 11200*log(3)^2*log(81) + 280*log(3)*log(81)^3 -
4480*log(3)^3*log(81) - 14*log(3)*log(81)^4 - 22400*log(3)^3 + 17920*log(3)^4 - 3584*log(3)^5 - 1120*log(3)^2
*log(81)^2 + 448*log(3)^3*log(81)^2 - 84*log(3)^2*log(81)^2*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(
1/2) - 1550*log(3)*log(81)*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + 255*log(3)*log(81)^2*((4*l
og(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + 840*log(3)^2*log(81)*((4*log(3) - log(81))*(4*log(3) + log
(81) - 10))^(1/2) - 20*log(3)*log(81)^3*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + log(3)*log(81
)^4*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2)))/((4*log(3) - log(81))*(4*log(3) + log(81) - 10)*(
4*log(3)*(4*log(3) - 10)*(8*log(3)*(4*log(3) - 10) + 100) - (4*log(3) - log(81))*(16*log(3)*(4*log(3) - 10) -
2*(4*log(3) - log(81))*(4*log(3) + log(81) - 10) + 100)*(4*log(3) + log(81) - 10) + 1250)))*(1500*log(3)*((4*l
og(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + 2100*log(3)^2*((4*log(3) - log(81))*(4*log(3) + log(81) -
10))^(1/2) - 3360*log(3)^3*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + 896*log(3)^4*((4*log(3) -
log(81))*(4*log(3) + log(81) - 10))^(1/2) - 1400*log(3)*log(81)^2 + 11200*log(3)^2*log(81) + 280*log(3)*log(81
)^3 - 4480*log(3)^3*log(81) - 14*log(3)*log(81)^4 - 22400*log(3)^3 + 17920*log(3)^4 - 3584*log(3)^5 - 1120*log
(3)^2*log(81)^2 + 448*log(3)^3*log(81)^2 - 84*log(3)^2*log(81)^2*((4*log(3) - log(81))*(4*log(3) + log(81) - 1
0))^(1/2) - 1550*log(3)*log(81)*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + 255*log(3)*log(81)^2*
((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + 840*log(3)^2*log(81)*((4*log(3) - log(81))*(4*log(3)
+ log(81) - 10))^(1/2) - 20*log(3)*log(81)^3*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + log(3)*l
og(81)^4*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2)))/((4*log(3) - log(81))*(4*log(3) + log(81) -
10)*(4*log(3)*(4*log(3) - 10)*(8*log(3)*(4*log(3) - 10) + 100) - (4*log(3) - log(81))*(16*log(3)*(4*log(3) - 1
0) - 2*(4*log(3) - log(81))*(4*log(3) + log(81) - 10) + 100)*(4*log(3) + log(81) - 10) + 1250)) + (log((2240*l
og(3)^4*log(81) - 1680*log(3)^3*log(81) - 8400*log(3)^2*log(81) + 33600*log(3)^3 - 29120*log(3)^4 + 6272*log(3
)^5 + 2240*log(3)^2*log(81)^2 - 280*log(3)^2*log(81)^3 + 168*log(3)^3*log(81)^2 + 14*log(3)^2*log(81)^4 - 224*
log(3)^4*log(81)^2)/(150*log(81)^2 - 500*log(81) - 20*log(81)^3 + log(81)^4 + 625) + (x*(560*log(3)^3*log(81)
- 1200*log(3)^2*log(81) + 3600*log(3)^2 - 3360*log(3)^3 + 784*log(3)^4 + 220*log(3)^2*log(81)^2 - 20*log(3)^2*
log(81)^3 - 56*log(3)^3*log(81)^2 + log(3)^2*log(81)^4))/(150*log(81)^2 - 500*log(81) - 20*log(81)^3 + log(81)
^4 + 625) - (((37500*log(3) - 71250*log(3)*log(81) + 32125*log(3)*log(81)^2 - 14000*log(3)^2*log(81) - 6000*lo
g(3)*log(81)^3 + 67200*log(3)^3*log(81) + 550*log(3)*log(81)^4 - 17920*log(3)^4*log(81) - 30*log(3)*log(81)^5
+ log(3)*log(81)^6 + 122500*log(3)^2 - 168000*log(3)^3 + 44800*log(3)^4 - 12600*log(3)^2*log(81)^2 + 2800*log(
3)^2*log(81)^3 - 6720*log(3)^3*log(81)^2 - 140*log(3)^2*log(81)^4 + 1792*log(3)^4*log(81)^2)/(150*log(81)^2 -
500*log(81) - 20*log(81)^3 + log(81)^4 + 625) + (x*(19000*log(3)*log(81) - 15000*log(3) - 7100*log(3)*log(81)^
2 + 7200*log(3)^2*log(81) + 1040*log(3)*log(81)^3 - 4480*log(3)^3*log(81) - 52*log(3)*log(81)^4 - 23000*log(3)
^2 + 11200*log(3)^3 + 80*log(3)^2*log(81)^2 - 160*log(3)^2*log(81)^3 + 448*log(3)^3*log(81)^2 + 8*log(3)^2*log
(81)^4))/(150*log(81)^2 - 500*log(81) - 20*log(81)^3 + log(81)^4 + 625) + (((125000*log(3) + 187500*log(81) -
150000*log(3)*log(81) + 75000*log(3)*log(81)^2 - 20000*log(3)*log(81)^3 + 3000*log(3)*log(81)^4 - 240*log(3)*l
og(81)^5 + 8*log(3)*log(81)^6 - 93750*log(81)^2 + 25000*log(81)^3 - 3750*log(81)^4 + 300*log(81)^5 - 10*log(81
)^6 - 156250)/(150*log(81)^2 - 500*log(81) - 20*log(81)^3 + log(81)^4 + 625) + (x*(12500*log(81) - 200000*log(
3) + 160000*log(3)*log(81) - 48000*log(3)*log(81)^2 - 64000*log(3)^2*log(81) + 6400*log(3)*log(81)^3 - 320*log
(3)*log(81)^4 + 80000*log(3)^2 - 26250*log(81)^2 + 11000*log(81)^3 - 2050*log(81)^4 + 180*log(81)^5 - 6*log(81
)^6 + 19200*log(3)^2*log(81)^2 - 2560*log(3)^2*log(81)^3 + 128*log(3)^2*log(81)^4 + 31250))/(150*log(81)^2 - 5
00*log(81) - 20*log(81)^3 + log(81)^4 + 625))*(1500*log(3)*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1
/2) + 2100*log(3)^2*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) - 3360*log(3)^3*((4*log(3) - log(81
))*(4*log(3) + log(81) - 10))^(1/2) + 896*log(3)^4*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + 14
00*log(3)*log(81)^2 - 11200*log(3)^2*log(81) - 280*log(3)*log(81)^3 + 4480*log(3)^3*log(81) + 14*log(3)*log(81
)^4 + 22400*log(3)^3 - 17920*log(3)^4 + 3584*log(3)^5 + 1120*log(3)^2*log(81)^2 - 448*log(3)^3*log(81)^2 - 84*
log(3)^2*log(81)^2*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) - 1550*log(3)*log(81)*((4*log(3) - l
og(81))*(4*log(3) + log(81) - 10))^(1/2) + 255*log(3)*log(81)^2*((4*log(3) - log(81))*(4*log(3) + log(81) - 10
))^(1/2) + 840*log(3)^2*log(81)*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) - 20*log(3)*log(81)^3*(
(4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + log(3)*log(81)^4*((4*log(3) - log(81))*(4*log(3) + log
(81) - 10))^(1/2)))/((4*log(3) - log(81))*(4*log(3) + log(81) - 10)*(4*log(3)*(4*log(3) - 10)*(8*log(3)*(4*log
(3) - 10) + 100) - (4*log(3) - log(81))*(16*log(3)*(4*log(3) - 10) - 2*(4*log(3) - log(81))*(4*log(3) + log(81
) - 10) + 100)*(4*log(3) + log(81) - 10) + 1250)))*(1500*log(3)*((4*log(3) - log(81))*(4*log(3) + log(81) - 10
))^(1/2) + 2100*log(3)^2*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) - 3360*log(3)^3*((4*log(3) - l
og(81))*(4*log(3) + log(81) - 10))^(1/2) + 896*log(3)^4*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2)
+ 1400*log(3)*log(81)^2 - 11200*log(3)^2*log(81) - 280*log(3)*log(81)^3 + 4480*log(3)^3*log(81) + 14*log(3)*l
og(81)^4 + 22400*log(3)^3 - 17920*log(3)^4 + 3584*log(3)^5 + 1120*log(3)^2*log(81)^2 - 448*log(3)^3*log(81)^2
- 84*log(3)^2*log(81)^2*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) - 1550*log(3)*log(81)*((4*log(3
) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + 255*log(3)*log(81)^2*((4*log(3) - log(81))*(4*log(3) + log(81)
- 10))^(1/2) + 840*log(3)^2*log(81)*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) - 20*log(3)*log(81
)^3*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + log(3)*log(81)^4*((4*log(3) - log(81))*(4*log(3)
+ log(81) - 10))^(1/2)))/((4*log(3) - log(81))*(4*log(3) + log(81) - 10)*(4*log(3)*(4*log(3) - 10)*(8*log(3)*(
4*log(3) - 10) + 100) - (4*log(3) - log(81))*(16*log(3)*(4*log(3) - 10) - 2*(4*log(3) - log(81))*(4*log(3) + l
og(81) - 10) + 100)*(4*log(3) + log(81) - 10) + 1250)))*(1500*log(3)*((4*log(3) - log(81))*(4*log(3) + log(81)
- 10))^(1/2) + 2100*log(3)^2*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) - 3360*log(3)^3*((4*log(3
) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + 896*log(3)^4*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^
(1/2) + 1400*log(3)*log(81)^2 - 11200*log(3)^2*log(81) - 280*log(3)*log(81)^3 + 4480*log(3)^3*log(81) + 14*log
(3)*log(81)^4 + 22400*log(3)^3 - 17920*log(3)^4 + 3584*log(3)^5 + 1120*log(3)^2*log(81)^2 - 448*log(3)^3*log(8
1)^2 - 84*log(3)^2*log(81)^2*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) - 1550*log(3)*log(81)*((4*
log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + 255*log(3)*log(81)^2*((4*log(3) - log(81))*(4*log(3) + lo
g(81) - 10))^(1/2) + 840*log(3)^2*log(81)*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) - 20*log(3)*l
og(81)^3*((4*log(3) - log(81))*(4*log(3) + log(81) - 10))^(1/2) + log(3)*log(81)^4*((4*log(3) - log(81))*(4*lo
g(3) + log(81) - 10))^(1/2)))/((4*log(3) - log(81))*(4*log(3) + log(81) - 10)*(4*log(3)*(4*log(3) - 10)*(8*log
(3)*(4*log(3) - 10) + 100) - (4*log(3) - log(81))*(16*log(3)*(4*log(3) - 10) - 2*(4*log(3) - log(81))*(4*log(3
) + log(81) - 10) + 100)*(4*log(3) + log(81) - 10) + 1250))

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sympy [A]  time = 0.54, size = 22, normalized size = 0.81 \begin {gather*} \frac {- x \log {\relax (3 )} - 7 \log {\relax (3 )}}{x^{2} + x \left (-5 + 4 \log {\relax (3 )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((28*ln(3)**2+(x**2+14*x-35)*ln(3))/(16*x**2*ln(3)**2+(8*x**3-40*x**2)*ln(3)+x**4-10*x**3+25*x**2),x)

[Out]

(-x*log(3) - 7*log(3))/(x**2 + x*(-5 + 4*log(3)))

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