3.43.62 \(\int \frac {-3168+4032 x-228 x^2+342 x^3-2 x^4+8 x^5+(1728-1728 x-12 x^4+12 x^5) \log (x^2-x^3)}{-x^3+x^4} \, dx\)

Optimal. Leaf size=26 \[ \left (-5+\left (\frac {12}{x}+x\right )^2\right ) \left (-5+6 \log \left (x \left (x-x^2\right )\right )\right ) \]

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Rubi [B]  time = 0.50, antiderivative size = 115, normalized size of antiderivative = 4.42, number of steps used = 22, number of rules used = 13, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {1593, 6742, 1620, 2513, 2418, 2395, 44, 43, 2357, 2304, 446, 73, 14} \begin {gather*} -5 x^2-\frac {720}{x^2}+6 x^2 \log (1-x)+12 x^2 \log (x)-6 x^2 \left (-\log \left ((1-x) x^2\right )+\log (1-x)+2 \log (x)\right )+\frac {864 \log (1-x)}{x^2}+\frac {1728 \log (x)}{x^2}-\frac {864 \left (-\log \left ((1-x) x^2\right )+\log (1-x)+2 \log (x)\right )}{x^2}+114 \log (1-x)+228 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3168 + 4032*x - 228*x^2 + 342*x^3 - 2*x^4 + 8*x^5 + (1728 - 1728*x - 12*x^4 + 12*x^5)*Log[x^2 - x^3])/(-
x^3 + x^4),x]

[Out]

-720/x^2 - 5*x^2 + 114*Log[1 - x] + (864*Log[1 - x])/x^2 + 6*x^2*Log[1 - x] + 228*Log[x] + (1728*Log[x])/x^2 +
12*x^2*Log[x] - (864*(Log[1 - x] + 2*Log[x] - Log[(1 - x)*x^2]))/x^2 - 6*x^2*(Log[1 - x] + 2*Log[x] - Log[(1
- x)*x^2])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 73

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[(a*c + b*
d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && Integer
Q[m]

Rule 446

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

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

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 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 2395

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2513

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*(RFx_.), x_Symbol] :> Dist[
p*r, Int[RFx*Log[a + b*x], x], x] + (Dist[q*r, Int[RFx*Log[c + d*x], x], x] - Dist[p*r*Log[a + b*x] + q*r*Log[
c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r], Int[RFx, x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] &&
RationalFunctionQ[RFx, x] && NeQ[b*c - a*d, 0] &&  !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; Integ
ersQ[m, n]]

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 {-3168+4032 x-228 x^2+342 x^3-2 x^4+8 x^5+\left (1728-1728 x-12 x^4+12 x^5\right ) \log \left (x^2-x^3\right )}{(-1+x) x^3} \, dx\\ &=\int \left (\frac {2 \left (-1584+2016 x-114 x^2+171 x^3-x^4+4 x^5\right )}{(-1+x) x^3}+\frac {12 \left (-12-x^2\right ) \left (12-x^2\right ) \log \left ((1-x) x^2\right )}{x^3}\right ) \, dx\\ &=2 \int \frac {-1584+2016 x-114 x^2+171 x^3-x^4+4 x^5}{(-1+x) x^3} \, dx+12 \int \frac {\left (-12-x^2\right ) \left (12-x^2\right ) \log \left ((1-x) x^2\right )}{x^3} \, dx\\ &=2 \int \left (3+\frac {492}{-1+x}+\frac {1584}{x^3}-\frac {432}{x^2}-\frac {318}{x}+4 x\right ) \, dx+12 \int \frac {\left (-12-x^2\right ) \left (12-x^2\right ) \log (1-x)}{x^3} \, dx+24 \int \frac {\left (-12-x^2\right ) \left (12-x^2\right ) \log (x)}{x^3} \, dx-\left (12 \left (\log (1-x)+2 \log (x)-\log \left ((1-x) x^2\right )\right )\right ) \int \frac {\left (-12-x^2\right ) \left (12-x^2\right )}{x^3} \, dx\\ &=-\frac {1584}{x^2}+\frac {864}{x}+6 x+4 x^2+984 \log (1-x)-636 \log (x)+12 \int \left (-\frac {144 \log (1-x)}{x^3}+x \log (1-x)\right ) \, dx+24 \int \left (-\frac {144 \log (x)}{x^3}+x \log (x)\right ) \, dx-\left (6 \left (\log (1-x)+2 \log (x)-\log \left ((1-x) x^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {(-12-x) (12-x)}{x^2} \, dx,x,x^2\right )\\ &=-\frac {1584}{x^2}+\frac {864}{x}+6 x+4 x^2+984 \log (1-x)-636 \log (x)+12 \int x \log (1-x) \, dx+24 \int x \log (x) \, dx-1728 \int \frac {\log (1-x)}{x^3} \, dx-3456 \int \frac {\log (x)}{x^3} \, dx-\left (6 \left (\log (1-x)+2 \log (x)-\log \left ((1-x) x^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {-144+x^2}{x^2} \, dx,x,x^2\right )\\ &=-\frac {720}{x^2}+\frac {864}{x}+6 x-2 x^2+984 \log (1-x)+\frac {864 \log (1-x)}{x^2}+6 x^2 \log (1-x)-636 \log (x)+\frac {1728 \log (x)}{x^2}+12 x^2 \log (x)+6 \int \frac {x^2}{1-x} \, dx+864 \int \frac {1}{(1-x) x^2} \, dx-\left (6 \left (\log (1-x)+2 \log (x)-\log \left ((1-x) x^2\right )\right )\right ) \operatorname {Subst}\left (\int \left (1-\frac {144}{x^2}\right ) \, dx,x,x^2\right )\\ &=-\frac {720}{x^2}+\frac {864}{x}+6 x-2 x^2+984 \log (1-x)+\frac {864 \log (1-x)}{x^2}+6 x^2 \log (1-x)-636 \log (x)+\frac {1728 \log (x)}{x^2}+12 x^2 \log (x)-\frac {864 \left (\log (1-x)+2 \log (x)-\log \left ((1-x) x^2\right )\right )}{x^2}-6 x^2 \left (\log (1-x)+2 \log (x)-\log \left ((1-x) x^2\right )\right )+6 \int \left (-1+\frac {1}{1-x}-x\right ) \, dx+864 \int \left (\frac {1}{1-x}+\frac {1}{x^2}+\frac {1}{x}\right ) \, dx\\ &=-\frac {720}{x^2}-5 x^2+114 \log (1-x)+\frac {864 \log (1-x)}{x^2}+6 x^2 \log (1-x)+228 \log (x)+\frac {1728 \log (x)}{x^2}+12 x^2 \log (x)-\frac {864 \left (\log (1-x)+2 \log (x)-\log \left ((1-x) x^2\right )\right )}{x^2}-6 x^2 \left (\log (1-x)+2 \log (x)-\log \left ((1-x) x^2\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.09, size = 53, normalized size = 2.04 \begin {gather*} -\frac {720}{x^2}-5 x^2+114 \log (1-x)+228 \log (x)+\frac {864 \log \left ((1-x) x^2\right )}{x^2}+6 x^2 \log \left ((1-x) x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3168 + 4032*x - 228*x^2 + 342*x^3 - 2*x^4 + 8*x^5 + (1728 - 1728*x - 12*x^4 + 12*x^5)*Log[x^2 - x^
3])/(-x^3 + x^4),x]

[Out]

-720/x^2 - 5*x^2 + 114*Log[1 - x] + 228*Log[x] + (864*Log[(1 - x)*x^2])/x^2 + 6*x^2*Log[(1 - x)*x^2]

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fricas [A]  time = 0.95, size = 34, normalized size = 1.31 \begin {gather*} -\frac {5 \, x^{4} - 6 \, {\left (x^{4} + 19 \, x^{2} + 144\right )} \log \left (-x^{3} + x^{2}\right ) + 720}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^5-12*x^4-1728*x+1728)*log(-x^3+x^2)+8*x^5-2*x^4+342*x^3-228*x^2+4032*x-3168)/(x^4-x^3),x, alg
orithm="fricas")

[Out]

-(5*x^4 - 6*(x^4 + 19*x^2 + 144)*log(-x^3 + x^2) + 720)/x^2

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giac [A]  time = 0.14, size = 42, normalized size = 1.62 \begin {gather*} -5 \, x^{2} + 6 \, {\left (x^{2} + \frac {144}{x^{2}}\right )} \log \left (-x^{3} + x^{2}\right ) - \frac {720}{x^{2}} + 114 \, \log \left (x - 1\right ) + 228 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^5-12*x^4-1728*x+1728)*log(-x^3+x^2)+8*x^5-2*x^4+342*x^3-228*x^2+4032*x-3168)/(x^4-x^3),x, alg
orithm="giac")

[Out]

-5*x^2 + 6*(x^2 + 144/x^2)*log(-x^3 + x^2) - 720/x^2 + 114*log(x - 1) + 228*log(x)

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maple [A]  time = 0.12, size = 49, normalized size = 1.88

method	result	size
risch	\(\frac {6 \left (x^{4}+144\right ) \ln \left (-x^{3}+x^{2}\right )}{x^{2}}+\frac {-5 x^{4}+228 x^{2} \ln \relax (x )+114 \ln \left (x -1\right ) x^{2}-720}{x^{2}}\)	\(49\)
default	\(6 \ln \left (-x^{3}+x^{2}\right ) x^{2}-5 x^{2}+114 \ln \left (x -1\right )+\frac {864 \ln \left (-x^{3}+x^{2}\right )}{x^{2}}-\frac {720}{x^{2}}+228 \ln \relax (x )\)	\(52\)
norman	\(\frac {-720+114 \ln \left (-x^{3}+x^{2}\right ) x^{2}-5 x^{4}+6 \ln \left (-x^{3}+x^{2}\right ) x^{4}+864 \ln \left (-x^{3}+x^{2}\right )}{x^{2}}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*x^5-12*x^4-1728*x+1728)*ln(-x^3+x^2)+8*x^5-2*x^4+342*x^3-228*x^2+4032*x-3168)/(x^4-x^3),x,method=_RET
URNVERBOSE)

[Out]

6*(x^4+144)/x^2*ln(-x^3+x^2)+(-5*x^4+228*x^2*ln(x)+114*ln(x-1)*x^2-720)/x^2

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maxima [B]  time = 0.40, size = 86, normalized size = 3.31 \begin {gather*} 4 \, x^{2} + 6 \, x - \frac {3 \, {\left (3 \, x^{4} + 2 \, x^{3} - 4 \, {\left (x^{4} + 72 \, x^{2} + 144\right )} \log \relax (x) - 2 \, {\left (x^{4} - 145 \, x^{2} + 144\right )} \log \left (-x + 1\right ) + 288 \, x - 288\right )}}{x^{2}} - \frac {1584 \, {\left (2 \, x + 1\right )}}{x^{2}} + \frac {4032}{x} + 984 \, \log \left (x - 1\right ) - 636 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^5-12*x^4-1728*x+1728)*log(-x^3+x^2)+8*x^5-2*x^4+342*x^3-228*x^2+4032*x-3168)/(x^4-x^3),x, alg
orithm="maxima")

[Out]

4*x^2 + 6*x - 3*(3*x^4 + 2*x^3 - 4*(x^4 + 72*x^2 + 144)*log(x) - 2*(x^4 - 145*x^2 + 144)*log(-x + 1) + 288*x -
288)/x^2 - 1584*(2*x + 1)/x^2 + 4032/x + 984*log(x - 1) - 636*log(x)

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mupad [B]  time = 3.05, size = 47, normalized size = 1.81 \begin {gather*} 114\,\ln \left (x^2\,\left (x-1\right )\right )+x^2\,\left (6\,\ln \left (x^2-x^3\right )-5\right )+\frac {864\,\ln \left (x^2-x^3\right )-720}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((228*x^2 - 4032*x - 342*x^3 + 2*x^4 - 8*x^5 + log(x^2 - x^3)*(1728*x + 12*x^4 - 12*x^5 - 1728) + 3168)/(x^
3 - x^4),x)

[Out]

114*log(x^2*(x - 1)) + x^2*(6*log(x^2 - x^3) - 5) + (864*log(x^2 - x^3) - 720)/x^2

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sympy [B]  time = 0.22, size = 39, normalized size = 1.50 \begin {gather*} - 5 x^{2} + 228 \log {\relax (x )} + 114 \log {\left (x - 1 \right )} + \frac {\left (6 x^{4} + 864\right ) \log {\left (- x^{3} + x^{2} \right )}}{x^{2}} - \frac {720}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x**5-12*x**4-1728*x+1728)*ln(-x**3+x**2)+8*x**5-2*x**4+342*x**3-228*x**2+4032*x-3168)/(x**4-x**
3),x)

[Out]

-5*x**2 + 228*log(x) + 114*log(x - 1) + (6*x**4 + 864)*log(-x**3 + x**2)/x**2 - 720/x**2

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