3.90.63 \(\int \frac {13122+34992 x+40824 x^2+27216 x^3+11340 x^4+3024 x^5+504 x^6+48 x^7+2 x^8+(162+216 x+108 x^2+24 x^3+2 x^4) \log (4 x)+(81+324 x+378 x^2+192 x^3+45 x^4+4 x^5) \log ^2(4 x)+2 x^2 \log ^4(4 x)}{6561+17496 x+20412 x^2+13608 x^3+5670 x^4+1512 x^5+252 x^6+24 x^7+x^8+(162 x+216 x^2+108 x^3+24 x^4+2 x^5) \log ^2(4 x)+x^2 \log ^4(4 x)} \, dx\)

Optimal. Leaf size=24 \[ 2 x+\frac {x}{x+\frac {(-3-x)^4}{\log ^2(4 x)}} \]

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Rubi [F]  time = 11.33, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {13122+34992 x+40824 x^2+27216 x^3+11340 x^4+3024 x^5+504 x^6+48 x^7+2 x^8+\left (162+216 x+108 x^2+24 x^3+2 x^4\right ) \log (4 x)+\left (81+324 x+378 x^2+192 x^3+45 x^4+4 x^5\right ) \log ^2(4 x)+2 x^2 \log ^4(4 x)}{6561+17496 x+20412 x^2+13608 x^3+5670 x^4+1512 x^5+252 x^6+24 x^7+x^8+\left (162 x+216 x^2+108 x^3+24 x^4+2 x^5\right ) \log ^2(4 x)+x^2 \log ^4(4 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(13122 + 34992*x + 40824*x^2 + 27216*x^3 + 11340*x^4 + 3024*x^5 + 504*x^6 + 48*x^7 + 2*x^8 + (162 + 216*x
+ 108*x^2 + 24*x^3 + 2*x^4)*Log[4*x] + (81 + 324*x + 378*x^2 + 192*x^3 + 45*x^4 + 4*x^5)*Log[4*x]^2 + 2*x^2*Lo
g[4*x]^4)/(6561 + 17496*x + 20412*x^2 + 13608*x^3 + 5670*x^4 + 1512*x^5 + 252*x^6 + 24*x^7 + x^8 + (162*x + 21
6*x^2 + 108*x^3 + 24*x^4 + 2*x^5)*Log[4*x]^2 + x^2*Log[4*x]^4),x]

[Out]

2*x - 8748*Defer[Int][((3 + x)^4 + x*Log[4*x]^2)^(-2), x] - 6561*Defer[Int][1/(x*((3 + x)^4 + x*Log[4*x]^2)^2)
, x] + 6804*Defer[Int][x^2/((3 + x)^4 + x*Log[4*x]^2)^2, x] + 5670*Defer[Int][x^3/((3 + x)^4 + x*Log[4*x]^2)^2
, x] + 2268*Defer[Int][x^4/((3 + x)^4 + x*Log[4*x]^2)^2, x] + 504*Defer[Int][x^5/((3 + x)^4 + x*Log[4*x]^2)^2,
x] + 60*Defer[Int][x^6/((3 + x)^4 + x*Log[4*x]^2)^2, x] + 3*Defer[Int][x^7/((3 + x)^4 + x*Log[4*x]^2)^2, x] +
162*Defer[Int][Log[4*x]/((3 + x)^4 + x*Log[4*x]^2)^2, x] + 216*Defer[Int][(x*Log[4*x])/((3 + x)^4 + x*Log[4*x
]^2)^2, x] + 108*Defer[Int][(x^2*Log[4*x])/((3 + x)^4 + x*Log[4*x]^2)^2, x] + 24*Defer[Int][(x^3*Log[4*x])/((3
+ x)^4 + x*Log[4*x]^2)^2, x] + 2*Defer[Int][(x^4*Log[4*x])/((3 + x)^4 + x*Log[4*x]^2)^2, x] + 81*Defer[Int][1
/(x*((3 + x)^4 + x*Log[4*x]^2)), x] - 54*Defer[Int][x/((3 + x)^4 + x*Log[4*x]^2), x] - 24*Defer[Int][x^2/((3 +
x)^4 + x*Log[4*x]^2), x] - 3*Defer[Int][x^3/((3 + x)^4 + x*Log[4*x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 (3+x)^8+2 (3+x)^4 \log (4 x)+(3+x)^3 \left (3+9 x+4 x^2\right ) \log ^2(4 x)+2 x^2 \log ^4(4 x)}{\left ((3+x)^4+x \log ^2(4 x)\right )^2} \, dx\\ &=\int \left (2+\frac {-6561-8748 x+6804 x^3+5670 x^4+2268 x^5+504 x^6+60 x^7+3 x^8+162 x \log (4 x)+216 x^2 \log (4 x)+108 x^3 \log (4 x)+24 x^4 \log (4 x)+2 x^5 \log (4 x)}{x \left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2}-\frac {3 (-1+x) (3+x)^3}{x \left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )}\right ) \, dx\\ &=2 x-3 \int \frac {(-1+x) (3+x)^3}{x \left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )} \, dx+\int \frac {-6561-8748 x+6804 x^3+5670 x^4+2268 x^5+504 x^6+60 x^7+3 x^8+162 x \log (4 x)+216 x^2 \log (4 x)+108 x^3 \log (4 x)+24 x^4 \log (4 x)+2 x^5 \log (4 x)}{x \left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2} \, dx\\ &=2 x-3 \int \frac {(-1+x) (3+x)^3}{x \left ((3+x)^4+x \log ^2(4 x)\right )} \, dx+\int \frac {3 (-1+x) (3+x)^7+2 x (3+x)^4 \log (4 x)}{x \left ((3+x)^4+x \log ^2(4 x)\right )^2} \, dx\\ &=2 x-3 \int \left (-\frac {27}{x \left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )}+\frac {18 x}{81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)}+\frac {8 x^2}{81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)}+\frac {x^3}{81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)}\right ) \, dx+\int \left (-\frac {8748}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2}-\frac {6561}{x \left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2}+\frac {6804 x^2}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2}+\frac {5670 x^3}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2}+\frac {2268 x^4}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2}+\frac {504 x^5}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2}+\frac {60 x^6}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2}+\frac {3 x^7}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2}+\frac {162 \log (4 x)}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2}+\frac {216 x \log (4 x)}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2}+\frac {108 x^2 \log (4 x)}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2}+\frac {24 x^3 \log (4 x)}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2}+\frac {2 x^4 \log (4 x)}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2}\right ) \, dx\\ &=2 x+2 \int \frac {x^4 \log (4 x)}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2} \, dx+3 \int \frac {x^7}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2} \, dx-3 \int \frac {x^3}{81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)} \, dx+24 \int \frac {x^3 \log (4 x)}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2} \, dx-24 \int \frac {x^2}{81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)} \, dx-54 \int \frac {x}{81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)} \, dx+60 \int \frac {x^6}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2} \, dx+81 \int \frac {1}{x \left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )} \, dx+108 \int \frac {x^2 \log (4 x)}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2} \, dx+162 \int \frac {\log (4 x)}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2} \, dx+216 \int \frac {x \log (4 x)}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2} \, dx+504 \int \frac {x^5}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2} \, dx+2268 \int \frac {x^4}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2} \, dx+5670 \int \frac {x^3}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2} \, dx-6561 \int \frac {1}{x \left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2} \, dx+6804 \int \frac {x^2}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2} \, dx-8748 \int \frac {1}{\left (81+108 x+54 x^2+12 x^3+x^4+x \log ^2(4 x)\right )^2} \, dx\\ &=2 x+2 \int \frac {x^4 \log (4 x)}{\left ((3+x)^4+x \log ^2(4 x)\right )^2} \, dx+3 \int \frac {x^7}{\left ((3+x)^4+x \log ^2(4 x)\right )^2} \, dx-3 \int \frac {x^3}{(3+x)^4+x \log ^2(4 x)} \, dx+24 \int \frac {x^3 \log (4 x)}{\left ((3+x)^4+x \log ^2(4 x)\right )^2} \, dx-24 \int \frac {x^2}{(3+x)^4+x \log ^2(4 x)} \, dx-54 \int \frac {x}{(3+x)^4+x \log ^2(4 x)} \, dx+60 \int \frac {x^6}{\left ((3+x)^4+x \log ^2(4 x)\right )^2} \, dx+81 \int \frac {1}{x \left ((3+x)^4+x \log ^2(4 x)\right )} \, dx+108 \int \frac {x^2 \log (4 x)}{\left ((3+x)^4+x \log ^2(4 x)\right )^2} \, dx+162 \int \frac {\log (4 x)}{\left ((3+x)^4+x \log ^2(4 x)\right )^2} \, dx+216 \int \frac {x \log (4 x)}{\left ((3+x)^4+x \log ^2(4 x)\right )^2} \, dx+504 \int \frac {x^5}{\left ((3+x)^4+x \log ^2(4 x)\right )^2} \, dx+2268 \int \frac {x^4}{\left ((3+x)^4+x \log ^2(4 x)\right )^2} \, dx+5670 \int \frac {x^3}{\left ((3+x)^4+x \log ^2(4 x)\right )^2} \, dx-6561 \int \frac {1}{x \left ((3+x)^4+x \log ^2(4 x)\right )^2} \, dx+6804 \int \frac {x^2}{\left ((3+x)^4+x \log ^2(4 x)\right )^2} \, dx-8748 \int \frac {1}{\left ((3+x)^4+x \log ^2(4 x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.69, size = 40, normalized size = 1.67 \begin {gather*} \frac {(3+x)^4 (-1+2 x)+2 x^2 \log ^2(4 x)}{(3+x)^4+x \log ^2(4 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(13122 + 34992*x + 40824*x^2 + 27216*x^3 + 11340*x^4 + 3024*x^5 + 504*x^6 + 48*x^7 + 2*x^8 + (162 +
216*x + 108*x^2 + 24*x^3 + 2*x^4)*Log[4*x] + (81 + 324*x + 378*x^2 + 192*x^3 + 45*x^4 + 4*x^5)*Log[4*x]^2 + 2*
x^2*Log[4*x]^4)/(6561 + 17496*x + 20412*x^2 + 13608*x^3 + 5670*x^4 + 1512*x^5 + 252*x^6 + 24*x^7 + x^8 + (162*
x + 216*x^2 + 108*x^3 + 24*x^4 + 2*x^5)*Log[4*x]^2 + x^2*Log[4*x]^4),x]

[Out]

((3 + x)^4*(-1 + 2*x) + 2*x^2*Log[4*x]^2)/((3 + x)^4 + x*Log[4*x]^2)

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fricas [B]  time = 0.74, size = 65, normalized size = 2.71 \begin {gather*} \frac {2 \, x^{5} + 23 \, x^{4} + 2 \, x^{2} \log \left (4 \, x\right )^{2} + 96 \, x^{3} + 162 \, x^{2} + 54 \, x - 81}{x^{4} + 12 \, x^{3} + x \log \left (4 \, x\right )^{2} + 54 \, x^{2} + 108 \, x + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*log(4*x)^4+(4*x^5+45*x^4+192*x^3+378*x^2+324*x+81)*log(4*x)^2+(2*x^4+24*x^3+108*x^2+216*x+162
)*log(4*x)+2*x^8+48*x^7+504*x^6+3024*x^5+11340*x^4+27216*x^3+40824*x^2+34992*x+13122)/(x^2*log(4*x)^4+(2*x^5+2
4*x^4+108*x^3+216*x^2+162*x)*log(4*x)^2+x^8+24*x^7+252*x^6+1512*x^5+5670*x^4+13608*x^3+20412*x^2+17496*x+6561)
,x, algorithm="fricas")

[Out]

(2*x^5 + 23*x^4 + 2*x^2*log(4*x)^2 + 96*x^3 + 162*x^2 + 54*x - 81)/(x^4 + 12*x^3 + x*log(4*x)^2 + 54*x^2 + 108
*x + 81)

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giac [B]  time = 5.53, size = 52, normalized size = 2.17 \begin {gather*} 2 \, x - \frac {x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81}{x^{4} + 12 \, x^{3} + x \log \left (4 \, x\right )^{2} + 54 \, x^{2} + 108 \, x + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*log(4*x)^4+(4*x^5+45*x^4+192*x^3+378*x^2+324*x+81)*log(4*x)^2+(2*x^4+24*x^3+108*x^2+216*x+162
)*log(4*x)+2*x^8+48*x^7+504*x^6+3024*x^5+11340*x^4+27216*x^3+40824*x^2+34992*x+13122)/(x^2*log(4*x)^4+(2*x^5+2
4*x^4+108*x^3+216*x^2+162*x)*log(4*x)^2+x^8+24*x^7+252*x^6+1512*x^5+5670*x^4+13608*x^3+20412*x^2+17496*x+6561)
,x, algorithm="giac")

[Out]

2*x - (x^4 + 12*x^3 + 54*x^2 + 108*x + 81)/(x^4 + 12*x^3 + x*log(4*x)^2 + 54*x^2 + 108*x + 81)

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maple [B]  time = 0.06, size = 53, normalized size = 2.21

method	result	size
risch	\(2 x -\frac {x^{4}+12 x^{3}+54 x^{2}+108 x +81}{x^{4}+x \ln \left (4 x \right )^{2}+12 x^{3}+54 x^{2}+108 x +81}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2*ln(4*x)^4+(4*x^5+45*x^4+192*x^3+378*x^2+324*x+81)*ln(4*x)^2+(2*x^4+24*x^3+108*x^2+216*x+162)*ln(4*x
)+2*x^8+48*x^7+504*x^6+3024*x^5+11340*x^4+27216*x^3+40824*x^2+34992*x+13122)/(x^2*ln(4*x)^4+(2*x^5+24*x^4+108*
x^3+216*x^2+162*x)*ln(4*x)^2+x^8+24*x^7+252*x^6+1512*x^5+5670*x^4+13608*x^3+20412*x^2+17496*x+6561),x,method=_
RETURNVERBOSE)

[Out]

2*x-(x^4+12*x^3+54*x^2+108*x+81)/(x^4+x*ln(4*x)^2+12*x^3+54*x^2+108*x+81)

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maxima [B]  time = 0.49, size = 91, normalized size = 3.79 \begin {gather*} \frac {2 \, x^{5} + 23 \, x^{4} + 8 \, x^{2} \log \relax (2) \log \relax (x) + 2 \, x^{2} \log \relax (x)^{2} + 2 \, {\left (4 \, \log \relax (2)^{2} + 81\right )} x^{2} + 96 \, x^{3} + 54 \, x - 81}{x^{4} + 12 \, x^{3} + 4 \, x \log \relax (2) \log \relax (x) + x \log \relax (x)^{2} + 4 \, {\left (\log \relax (2)^{2} + 27\right )} x + 54 \, x^{2} + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*log(4*x)^4+(4*x^5+45*x^4+192*x^3+378*x^2+324*x+81)*log(4*x)^2+(2*x^4+24*x^3+108*x^2+216*x+162
)*log(4*x)+2*x^8+48*x^7+504*x^6+3024*x^5+11340*x^4+27216*x^3+40824*x^2+34992*x+13122)/(x^2*log(4*x)^4+(2*x^5+2
4*x^4+108*x^3+216*x^2+162*x)*log(4*x)^2+x^8+24*x^7+252*x^6+1512*x^5+5670*x^4+13608*x^3+20412*x^2+17496*x+6561)
,x, algorithm="maxima")

[Out]

(2*x^5 + 23*x^4 + 8*x^2*log(2)*log(x) + 2*x^2*log(x)^2 + 2*(4*log(2)^2 + 81)*x^2 + 96*x^3 + 54*x - 81)/(x^4 +
12*x^3 + 4*x*log(2)*log(x) + x*log(x)^2 + 4*(log(2)^2 + 27)*x + 54*x^2 + 81)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {34992\,x+{\ln \left (4\,x\right )}^2\,\left (4\,x^5+45\,x^4+192\,x^3+378\,x^2+324\,x+81\right )+\ln \left (4\,x\right )\,\left (2\,x^4+24\,x^3+108\,x^2+216\,x+162\right )+40824\,x^2+27216\,x^3+11340\,x^4+3024\,x^5+504\,x^6+48\,x^7+2\,x^8+2\,x^2\,{\ln \left (4\,x\right )}^4+13122}{17496\,x+{\ln \left (4\,x\right )}^2\,\left (2\,x^5+24\,x^4+108\,x^3+216\,x^2+162\,x\right )+20412\,x^2+13608\,x^3+5670\,x^4+1512\,x^5+252\,x^6+24\,x^7+x^8+x^2\,{\ln \left (4\,x\right )}^4+6561} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((34992*x + log(4*x)^2*(324*x + 378*x^2 + 192*x^3 + 45*x^4 + 4*x^5 + 81) + log(4*x)*(216*x + 108*x^2 + 24*x
^3 + 2*x^4 + 162) + 40824*x^2 + 27216*x^3 + 11340*x^4 + 3024*x^5 + 504*x^6 + 48*x^7 + 2*x^8 + 2*x^2*log(4*x)^4
+ 13122)/(17496*x + log(4*x)^2*(162*x + 216*x^2 + 108*x^3 + 24*x^4 + 2*x^5) + 20412*x^2 + 13608*x^3 + 5670*x^
4 + 1512*x^5 + 252*x^6 + 24*x^7 + x^8 + x^2*log(4*x)^4 + 6561),x)

[Out]

int((34992*x + log(4*x)^2*(324*x + 378*x^2 + 192*x^3 + 45*x^4 + 4*x^5 + 81) + log(4*x)*(216*x + 108*x^2 + 24*x
^3 + 2*x^4 + 162) + 40824*x^2 + 27216*x^3 + 11340*x^4 + 3024*x^5 + 504*x^6 + 48*x^7 + 2*x^8 + 2*x^2*log(4*x)^4
+ 13122)/(17496*x + log(4*x)^2*(162*x + 216*x^2 + 108*x^3 + 24*x^4 + 2*x^5) + 20412*x^2 + 13608*x^3 + 5670*x^
4 + 1512*x^5 + 252*x^6 + 24*x^7 + x^8 + x^2*log(4*x)^4 + 6561), x)

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sympy [B]  time = 0.36, size = 49, normalized size = 2.04 \begin {gather*} 2 x + \frac {- x^{4} - 12 x^{3} - 54 x^{2} - 108 x - 81}{x^{4} + 12 x^{3} + 54 x^{2} + x \log {\left (4 x \right )}^{2} + 108 x + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2*ln(4*x)**4+(4*x**5+45*x**4+192*x**3+378*x**2+324*x+81)*ln(4*x)**2+(2*x**4+24*x**3+108*x**2+2
16*x+162)*ln(4*x)+2*x**8+48*x**7+504*x**6+3024*x**5+11340*x**4+27216*x**3+40824*x**2+34992*x+13122)/(x**2*ln(4
*x)**4+(2*x**5+24*x**4+108*x**3+216*x**2+162*x)*ln(4*x)**2+x**8+24*x**7+252*x**6+1512*x**5+5670*x**4+13608*x**
3+20412*x**2+17496*x+6561),x)

[Out]

2*x + (-x**4 - 12*x**3 - 54*x**2 - 108*x - 81)/(x**4 + 12*x**3 + 54*x**2 + x*log(4*x)**2 + 108*x + 81)

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