3.14.16 \(\int \frac {20992-80852 x+116568 x^2-74529 x^3+17820 x^4+(-10512+35460 x-43308 x^2+22248 x^3-3888 x^4) \log (x)+(1296-3888 x+3888 x^2-1296 x^3) \log ^2(x)}{-8 x^3+24 x^4-24 x^5+8 x^6} \, dx\)

Optimal. Leaf size=33 \[ 5+\left (\frac {1}{4 (-1+x)}+\frac {5}{x}+9 \left (-3+\frac {3-\log (x)}{x}\right )\right )^2 \]

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Rubi [B]  time = 0.64, antiderivative size = 90, normalized size of antiderivative = 2.73, number of steps used = 19, number of rules used = 10, integrand size = 87, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6741, 12, 6742, 44, 37, 2357, 2314, 31, 2304, 2305} \begin {gather*} -\frac {4455 x^2}{4 (1-x)^2}+\frac {1024}{x^2}+\frac {81 \log ^2(x)}{x^2}-\frac {576 \log (x)}{x^2}-\frac {2230}{1-x}+\frac {17821}{16 (1-x)^2}-\frac {1744}{x}+\frac {9 x \log (x)}{2 (1-x)}+\frac {9 \log (x)}{2}+\frac {981 \log (x)}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(20992 - 80852*x + 116568*x^2 - 74529*x^3 + 17820*x^4 + (-10512 + 35460*x - 43308*x^2 + 22248*x^3 - 3888*x
^4)*Log[x] + (1296 - 3888*x + 3888*x^2 - 1296*x^3)*Log[x]^2)/(-8*x^3 + 24*x^4 - 24*x^5 + 8*x^6),x]

[Out]

17821/(16*(1 - x)^2) - 2230/(1 - x) + 1024/x^2 - 1744/x - (4455*x^2)/(4*(1 - x)^2) + (9*Log[x])/2 - (576*Log[x
])/x^2 + (981*Log[x])/(2*x) + (9*x*Log[x])/(2*(1 - x)) + (81*Log[x]^2)/x^2

Rule 12

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2314

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[(x*(d + e*x^r)^(q
+ 1)*(a + b*Log[c*x^n]))/d, x] - Dist[(b*n)/d, Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 6741

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

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 \frac {-20992+80852 x-116568 x^2+74529 x^3-17820 x^4-\left (-10512+35460 x-43308 x^2+22248 x^3-3888 x^4\right ) \log (x)-\left (1296-3888 x+3888 x^2-1296 x^3\right ) \log ^2(x)}{8 (1-x)^3 x^3} \, dx\\ &=\frac {1}{8} \int \frac {-20992+80852 x-116568 x^2+74529 x^3-17820 x^4-\left (-10512+35460 x-43308 x^2+22248 x^3-3888 x^4\right ) \log (x)-\left (1296-3888 x+3888 x^2-1296 x^3\right ) \log ^2(x)}{(1-x)^3 x^3} \, dx\\ &=\frac {1}{8} \int \left (-\frac {74529}{(-1+x)^3}+\frac {20992}{(-1+x)^3 x^3}-\frac {80852}{(-1+x)^3 x^2}+\frac {116568}{(-1+x)^3 x}+\frac {17820 x}{(-1+x)^3}-\frac {36 \left (-292+693 x-510 x^2+108 x^3\right ) \log (x)}{(-1+x)^2 x^3}-\frac {1296 \log ^2(x)}{x^3}\right ) \, dx\\ &=\frac {74529}{16 (1-x)^2}-\frac {9}{2} \int \frac {\left (-292+693 x-510 x^2+108 x^3\right ) \log (x)}{(-1+x)^2 x^3} \, dx-162 \int \frac {\log ^2(x)}{x^3} \, dx+\frac {4455}{2} \int \frac {x}{(-1+x)^3} \, dx+2624 \int \frac {1}{(-1+x)^3 x^3} \, dx-\frac {20213}{2} \int \frac {1}{(-1+x)^3 x^2} \, dx+14571 \int \frac {1}{(-1+x)^3 x} \, dx\\ &=\frac {74529}{16 (1-x)^2}-\frac {4455 x^2}{4 (1-x)^2}+\frac {81 \log ^2(x)}{x^2}-\frac {9}{2} \int \left (-\frac {\log (x)}{(-1+x)^2}-\frac {292 \log (x)}{x^3}+\frac {109 \log (x)}{x^2}\right ) \, dx-162 \int \frac {\log (x)}{x^3} \, dx+2624 \int \left (\frac {1}{(-1+x)^3}-\frac {3}{(-1+x)^2}+\frac {6}{-1+x}-\frac {1}{x^3}-\frac {3}{x^2}-\frac {6}{x}\right ) \, dx-\frac {20213}{2} \int \left (\frac {1}{(-1+x)^3}-\frac {2}{(-1+x)^2}+\frac {3}{-1+x}-\frac {1}{x^2}-\frac {3}{x}\right ) \, dx+14571 \int \left (\frac {1}{(-1+x)^3}-\frac {1}{(-1+x)^2}+\frac {1}{-1+x}-\frac {1}{x}\right ) \, dx\\ &=\frac {17821}{16 (1-x)^2}-\frac {2230}{1-x}+\frac {2705}{2 x^2}-\frac {4469}{2 x}-\frac {4455 x^2}{4 (1-x)^2}-\frac {9}{2} \log (1-x)+\frac {9 \log (x)}{2}+\frac {81 \log (x)}{x^2}+\frac {81 \log ^2(x)}{x^2}+\frac {9}{2} \int \frac {\log (x)}{(-1+x)^2} \, dx-\frac {981}{2} \int \frac {\log (x)}{x^2} \, dx+1314 \int \frac {\log (x)}{x^3} \, dx\\ &=\frac {17821}{16 (1-x)^2}-\frac {2230}{1-x}+\frac {1024}{x^2}-\frac {1744}{x}-\frac {4455 x^2}{4 (1-x)^2}-\frac {9}{2} \log (1-x)+\frac {9 \log (x)}{2}-\frac {576 \log (x)}{x^2}+\frac {981 \log (x)}{2 x}+\frac {9 x \log (x)}{2 (1-x)}+\frac {81 \log ^2(x)}{x^2}+\frac {9}{2} \int \frac {1}{-1+x} \, dx\\ &=\frac {17821}{16 (1-x)^2}-\frac {2230}{1-x}+\frac {1024}{x^2}-\frac {1744}{x}-\frac {4455 x^2}{4 (1-x)^2}+\frac {9 \log (x)}{2}-\frac {576 \log (x)}{x^2}+\frac {981 \log (x)}{2 x}+\frac {9 x \log (x)}{2 (1-x)}+\frac {81 \log ^2(x)}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 57, normalized size = 1.73 \begin {gather*} \frac {16384-60672 x+72153 x^2-27864 x^3+72 \left (-128+365 x-345 x^2+108 x^3\right ) \log (x)+1296 (-1+x)^2 \log ^2(x)}{16 (-1+x)^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20992 - 80852*x + 116568*x^2 - 74529*x^3 + 17820*x^4 + (-10512 + 35460*x - 43308*x^2 + 22248*x^3 -
3888*x^4)*Log[x] + (1296 - 3888*x + 3888*x^2 - 1296*x^3)*Log[x]^2)/(-8*x^3 + 24*x^4 - 24*x^5 + 8*x^6),x]

[Out]

(16384 - 60672*x + 72153*x^2 - 27864*x^3 + 72*(-128 + 365*x - 345*x^2 + 108*x^3)*Log[x] + 1296*(-1 + x)^2*Log[
x]^2)/(16*(-1 + x)^2*x^2)

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fricas [B]  time = 0.81, size = 64, normalized size = 1.94 \begin {gather*} -\frac {27864 \, x^{3} - 1296 \, {\left (x^{2} - 2 \, x + 1\right )} \log \relax (x)^{2} - 72153 \, x^{2} - 72 \, {\left (108 \, x^{3} - 345 \, x^{2} + 365 \, x - 128\right )} \log \relax (x) + 60672 \, x - 16384}{16 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1296*x^3+3888*x^2-3888*x+1296)*log(x)^2+(-3888*x^4+22248*x^3-43308*x^2+35460*x-10512)*log(x)+1782
0*x^4-74529*x^3+116568*x^2-80852*x+20992)/(8*x^6-24*x^5+24*x^4-8*x^3),x, algorithm="fricas")

[Out]

-1/16*(27864*x^3 - 1296*(x^2 - 2*x + 1)*log(x)^2 - 72153*x^2 - 72*(108*x^3 - 345*x^2 + 365*x - 128)*log(x) + 6
0672*x - 16384)/(x^4 - 2*x^3 + x^2)

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giac [A]  time = 0.32, size = 57, normalized size = 1.73 \begin {gather*} -\frac {9}{2} \, {\left (\frac {1}{x - 1} - \frac {109 \, x - 128}{x^{2}}\right )} \log \relax (x) + \frac {40 \, x - 39}{16 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {81 \, \log \relax (x)^{2}}{x^{2}} - \frac {16 \, {\left (109 \, x - 64\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1296*x^3+3888*x^2-3888*x+1296)*log(x)^2+(-3888*x^4+22248*x^3-43308*x^2+35460*x-10512)*log(x)+1782
0*x^4-74529*x^3+116568*x^2-80852*x+20992)/(8*x^6-24*x^5+24*x^4-8*x^3),x, algorithm="giac")

[Out]

-9/2*(1/(x - 1) - (109*x - 128)/x^2)*log(x) + 1/16*(40*x - 39)/(x^2 - 2*x + 1) + 81*log(x)^2/x^2 - 16*(109*x -
64)/x^2

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maple [A]  time = 0.05, size = 58, normalized size = 1.76

method	result	size
risch	\(\frac {81 \ln \relax (x )^{2}}{x^{2}}+\frac {9 \left (108 x^{2}-237 x +128\right ) \ln \relax (x )}{2 x^{2} \left (x -1\right )}-\frac {27864 x^{3}-72153 x^{2}+60672 x -16384}{16 x^{2} \left (x -1\right )^{2}}\)	\(58\)
default	\(\frac {1}{16 \left (x -1\right )^{2}}+\frac {5}{2 \left (x -1\right )}+\frac {1024}{x^{2}}-\frac {1744}{x}+\frac {9 \ln \relax (x )}{2}+\frac {81 \ln \relax (x )^{2}}{x^{2}}-\frac {576 \ln \relax (x )}{x^{2}}+\frac {981 \ln \relax (x )}{2 x}-\frac {9 \ln \relax (x ) x}{2 \left (x -1\right )}\)	\(63\)
norman	\(\frac {1024-\frac {3483 x^{4}}{4}+\frac {58221 x^{2}}{16}+243 x^{4} \ln \relax (x )-\frac {2619 x^{2} \ln \relax (x )}{2}-3792 x +81 \ln \relax (x )^{2}+\frac {3285 x \ln \relax (x )}{2}-162 x \ln \relax (x )^{2}+81 x^{2} \ln \relax (x )^{2}-576 \ln \relax (x )}{x^{2} \left (x -1\right )^{2}}-243 \ln \relax (x )\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-1296*x^3+3888*x^2-3888*x+1296)*ln(x)^2+(-3888*x^4+22248*x^3-43308*x^2+35460*x-10512)*ln(x)+17820*x^4-74
529*x^3+116568*x^2-80852*x+20992)/(8*x^6-24*x^5+24*x^4-8*x^3),x,method=_RETURNVERBOSE)

[Out]

81*ln(x)^2/x^2+9/2*(108*x^2-237*x+128)/x^2/(x-1)*ln(x)-1/16*(27864*x^3-72153*x^2+60672*x-16384)/x^2/(x-1)^2

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maxima [B]  time = 0.47, size = 147, normalized size = 4.45 \begin {gather*} \frac {1312 \, {\left (12 \, x^{3} - 18 \, x^{2} + 4 \, x + 1\right )}}{x^{4} - 2 \, x^{3} + x^{2}} + \frac {9 \, {\left (18 \, {\left (x - 1\right )} \log \relax (x)^{2} + 109 \, x^{2} + {\left (108 \, x^{2} - 237 \, x + 128\right )} \log \relax (x) - 173 \, x + 64\right )}}{2 \, {\left (x^{3} - x^{2}\right )}} - \frac {20213 \, {\left (6 \, x^{2} - 9 \, x + 2\right )}}{4 \, {\left (x^{3} - 2 \, x^{2} + x\right )}} - \frac {4455 \, {\left (2 \, x - 1\right )}}{4 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {14571 \, {\left (2 \, x - 3\right )}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {74529}{16 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1296*x^3+3888*x^2-3888*x+1296)*log(x)^2+(-3888*x^4+22248*x^3-43308*x^2+35460*x-10512)*log(x)+1782
0*x^4-74529*x^3+116568*x^2-80852*x+20992)/(8*x^6-24*x^5+24*x^4-8*x^3),x, algorithm="maxima")

[Out]

1312*(12*x^3 - 18*x^2 + 4*x + 1)/(x^4 - 2*x^3 + x^2) + 9/2*(18*(x - 1)*log(x)^2 + 109*x^2 + (108*x^2 - 237*x +
128)*log(x) - 173*x + 64)/(x^3 - x^2) - 20213/4*(6*x^2 - 9*x + 2)/(x^3 - 2*x^2 + x) - 4455/4*(2*x - 1)/(x^2 -
2*x + 1) + 14571/2*(2*x - 3)/(x^2 - 2*x + 1) + 74529/16/(x^2 - 2*x + 1)

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mupad [B]  time = 1.03, size = 62, normalized size = 1.88 \begin {gather*} \frac {81\,{\ln \relax (x)}^2-x\,\left (162\,{\ln \relax (x)}^2-\frac {3285\,\ln \relax (x)}{2}+3792\right )-576\,\ln \relax (x)+x^2\,\left (81\,{\ln \relax (x)}^2-\frac {3105\,\ln \relax (x)}{2}+\frac {72153}{16}\right )+x^3\,\left (486\,\ln \relax (x)-\frac {3483}{2}\right )+1024}{x^2\,{\left (x-1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((80852*x + log(x)*(43308*x^2 - 35460*x - 22248*x^3 + 3888*x^4 + 10512) + log(x)^2*(3888*x - 3888*x^2 + 129
6*x^3 - 1296) - 116568*x^2 + 74529*x^3 - 17820*x^4 - 20992)/(8*x^3 - 24*x^4 + 24*x^5 - 8*x^6),x)

[Out]

(81*log(x)^2 - x*(162*log(x)^2 - (3285*log(x))/2 + 3792) - 576*log(x) + x^2*(81*log(x)^2 - (3105*log(x))/2 + 7
2153/16) + x^3*(486*log(x) - 3483/2) + 1024)/(x^2*(x - 1)^2)

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sympy [B]  time = 0.30, size = 63, normalized size = 1.91 \begin {gather*} \frac {- 27864 x^{3} + 72153 x^{2} - 60672 x + 16384}{16 x^{4} - 32 x^{3} + 16 x^{2}} + \frac {\left (972 x^{2} - 2133 x + 1152\right ) \log {\relax (x )}}{2 x^{3} - 2 x^{2}} + \frac {81 \log {\relax (x )}^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1296*x**3+3888*x**2-3888*x+1296)*ln(x)**2+(-3888*x**4+22248*x**3-43308*x**2+35460*x-10512)*ln(x)+
17820*x**4-74529*x**3+116568*x**2-80852*x+20992)/(8*x**6-24*x**5+24*x**4-8*x**3),x)

[Out]

(-27864*x**3 + 72153*x**2 - 60672*x + 16384)/(16*x**4 - 32*x**3 + 16*x**2) + (972*x**2 - 2133*x + 1152)*log(x)
/(2*x**3 - 2*x**2) + 81*log(x)**2/x**2

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