\(\int \frac {36+372 x-1075 x^2+1250 x^3-777 x^4+262 x^5-42 x^6+2 x^7+e^x (-32 x^2+32 x^3-8 x^5+2 x^6)+(-2-5 x+11 x^2-6 x^3+x^4) \log (x)-x \log ^2(x)}{-16 x+24 x^2-12 x^3+2 x^4} \, dx\) [1826]

Optimal result

Integrand size = 110, antiderivative size = 39 \[ \int \frac {36+372 x-1075 x^2+1250 x^3-777 x^4+262 x^5-42 x^6+2 x^7+e^x \left (-32 x^2+32 x^3-8 x^5+2 x^6\right )+\left (-2-5 x+11 x^2-6 x^3+x^4\right ) \log (x)-x \log ^2(x)}{-16 x+24 x^2-12 x^3+2 x^4} \, dx=\left (1+e^x-2 x\right ) x^2+\frac {1}{4} \left (-(3-x)^2+\frac {\log (x)}{2-x}\right )^2 \]

[Out]

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

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.89 (sec) , antiderivative size = 164, normalized size of antiderivative = 4.21, number of steps used = 37, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.173, Rules used = {6873, 6874, 46, 37, 45, 2227, 2207, 2225, 2404, 2332, 2351, 31, 2353, 2352, 2338, 2356, 2389, 2379, 2438} \[ \int \frac {36+372 x-1075 x^2+1250 x^3-777 x^4+262 x^5-42 x^6+2 x^7+e^x \left (-32 x^2+32 x^3-8 x^5+2 x^6\right )+\left (-2-5 x+11 x^2-6 x^3+x^4\right ) \log (x)-x \log ^2(x)}{-16 x+24 x^2-12 x^3+2 x^4} \, dx=\frac {1}{8} \operatorname {PolyLog}\left (2,1-\frac {x}{2}\right )-\frac {\operatorname {PolyLog}\left (2,\frac {2}{x}\right )}{8}+\frac {x^4}{4}-5 x^3+e^x x^2+\frac {1075 x^2}{8 (2-x)^2}+\frac {29 x^2}{2}-27 x+\frac {1075}{2 (2-x)}-\frac {1075}{2 (2-x)^2}+\frac {\log ^2(x)}{4 (2-x)^2}+\frac {\log ^2(x)}{16}-\frac {x \log (x)}{4 (2-x)}+\frac {1}{2} x \log (x)-\frac {1}{8} \log (2) \log (x-2)+\frac {1}{8} \log \left (1-\frac {2}{x}\right ) \log (x)-\frac {9 \log (x)}{4} \]

[In]

Int[(36 + 372*x - 1075*x^2 + 1250*x^3 - 777*x^4 + 262*x^5 - 42*x^6 + 2*x^7 + E^x*(-32*x^2 + 32*x^3 - 8*x^5 + 2
*x^6) + (-2 - 5*x + 11*x^2 - 6*x^3 + x^4)*Log[x] - x*Log[x]^2)/(-16*x + 24*x^2 - 12*x^3 + 2*x^4),x]

[Out]

-1075/(2*(2 - x)^2) + 1075/(2*(2 - x)) - 27*x + (29*x^2)/2 + E^x*x^2 + (1075*x^2)/(8*(2 - x)^2) - 5*x^3 + x^4/
4 - (Log[2]*Log[-2 + x])/8 - (9*Log[x])/4 + (x*Log[x])/2 - (x*Log[x])/(4*(2 - x)) + (Log[1 - 2/x]*Log[x])/8 +
Log[x]^2/16 + Log[x]^2/(4*(2 - x)^2) + PolyLog[2, 1 - x/2]/8 - PolyLog[2, 2/x]/8

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 45

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 46

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] && Lt
Q[m + n + 2, 0])

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2227

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !TrueQ[$UseGamma]

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2351

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 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(a + b*Log[(-c)*(d/e)])*(Log[d + e*
x]/e), x] + Dist[b, Int[Log[(-e)*(x/d)]/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[(-c)*(d/e), 0]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2379

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

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2404

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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 6873

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-36-372 x+1075 x^2-1250 x^3+777 x^4-262 x^5+42 x^6-2 x^7-e^x \left (-32 x^2+32 x^3-8 x^5+2 x^6\right )-\left (-2-5 x+11 x^2-6 x^3+x^4\right ) \log (x)+x \log ^2(x)}{x \left (16-24 x+12 x^2-2 x^3\right )} \, dx \\ & = \int \left (\frac {186}{(-2+x)^3}+\frac {18}{(-2+x)^3 x}-\frac {1075 x}{2 (-2+x)^3}+\frac {625 x^2}{(-2+x)^3}-\frac {777 x^3}{2 (-2+x)^3}+\frac {131 x^4}{(-2+x)^3}-\frac {21 x^5}{(-2+x)^3}+\frac {x^6}{(-2+x)^3}+e^x x (2+x)+\frac {\left (1+3 x-4 x^2+x^3\right ) \log (x)}{2 (-2+x)^2 x}-\frac {\log ^2(x)}{2 (-2+x)^3}\right ) \, dx \\ & = -\frac {93}{(2-x)^2}+\frac {1}{2} \int \frac {\left (1+3 x-4 x^2+x^3\right ) \log (x)}{(-2+x)^2 x} \, dx-\frac {1}{2} \int \frac {\log ^2(x)}{(-2+x)^3} \, dx+18 \int \frac {1}{(-2+x)^3 x} \, dx-21 \int \frac {x^5}{(-2+x)^3} \, dx+131 \int \frac {x^4}{(-2+x)^3} \, dx-\frac {777}{2} \int \frac {x^3}{(-2+x)^3} \, dx-\frac {1075}{2} \int \frac {x}{(-2+x)^3} \, dx+625 \int \frac {x^2}{(-2+x)^3} \, dx+\int \frac {x^6}{(-2+x)^3} \, dx+\int e^x x (2+x) \, dx \\ & = -\frac {93}{(2-x)^2}+\frac {1075 x^2}{8 (2-x)^2}+\frac {\log ^2(x)}{4 (2-x)^2}-\frac {1}{2} \int \frac {\log (x)}{(-2+x)^2 x} \, dx+\frac {1}{2} \int \left (\log (x)-\frac {\log (x)}{2 (-2+x)^2}-\frac {\log (x)}{4 (-2+x)}+\frac {\log (x)}{4 x}\right ) \, dx+18 \int \left (\frac {1}{2 (-2+x)^3}-\frac {1}{4 (-2+x)^2}+\frac {1}{8 (-2+x)}-\frac {1}{8 x}\right ) \, dx-21 \int \left (24+\frac {32}{(-2+x)^3}+\frac {80}{(-2+x)^2}+\frac {80}{-2+x}+6 x+x^2\right ) \, dx+131 \int \left (6+\frac {16}{(-2+x)^3}+\frac {32}{(-2+x)^2}+\frac {24}{-2+x}+x\right ) \, dx-\frac {777}{2} \int \left (1+\frac {8}{(-2+x)^3}+\frac {12}{(-2+x)^2}+\frac {6}{-2+x}\right ) \, dx+625 \int \left (\frac {4}{(-2+x)^3}+\frac {4}{(-2+x)^2}+\frac {1}{-2+x}\right ) \, dx+\int \left (2 e^x x+e^x x^2\right ) \, dx+\int \left (80+\frac {64}{(-2+x)^3}+\frac {192}{(-2+x)^2}+\frac {240}{-2+x}+24 x+6 x^2+x^3\right ) \, dx \\ & = -\frac {1075}{2 (2-x)^2}+\frac {1075}{2 (2-x)}-\frac {53 x}{2}+\frac {29 x^2}{2}+\frac {1075 x^2}{8 (2-x)^2}-5 x^3+\frac {x^4}{4}+\frac {1}{4} \log (2-x)-\frac {9 \log (x)}{4}+\frac {\log ^2(x)}{4 (2-x)^2}-\frac {1}{8} \int \frac {\log (x)}{-2+x} \, dx+\frac {1}{8} \int \frac {\log (x)}{x} \, dx-2 \left (\frac {1}{4} \int \frac {\log (x)}{(-2+x)^2} \, dx\right )+\frac {1}{4} \int \frac {\log (x)}{(-2+x) x} \, dx+\frac {1}{2} \int \log (x) \, dx+2 \int e^x x \, dx+\int e^x x^2 \, dx \\ & = -\frac {1075}{2 (2-x)^2}+\frac {1075}{2 (2-x)}-27 x+2 e^x x+\frac {29 x^2}{2}+e^x x^2+\frac {1075 x^2}{8 (2-x)^2}-5 x^3+\frac {x^4}{4}+\frac {1}{4} \log (2-x)-\frac {1}{8} \log (2) \log (-2+x)-\frac {9 \log (x)}{4}+\frac {1}{2} x \log (x)+\frac {1}{8} \log \left (1-\frac {2}{x}\right ) \log (x)+\frac {\log ^2(x)}{16}+\frac {\log ^2(x)}{4 (2-x)^2}-2 \left (\frac {x \log (x)}{8 (2-x)}+\frac {1}{8} \int \frac {1}{-2+x} \, dx\right )-\frac {1}{8} \int \frac {\log \left (1-\frac {2}{x}\right )}{x} \, dx-\frac {1}{8} \int \frac {\log \left (\frac {x}{2}\right )}{-2+x} \, dx-2 \int e^x \, dx-2 \int e^x x \, dx \\ & = -2 e^x-\frac {1075}{2 (2-x)^2}+\frac {1075}{2 (2-x)}-27 x+\frac {29 x^2}{2}+e^x x^2+\frac {1075 x^2}{8 (2-x)^2}-5 x^3+\frac {x^4}{4}+\frac {1}{4} \log (2-x)-\frac {1}{8} \log (2) \log (-2+x)-\frac {9 \log (x)}{4}+\frac {1}{2} x \log (x)+\frac {1}{8} \log \left (1-\frac {2}{x}\right ) \log (x)+\frac {\log ^2(x)}{16}+\frac {\log ^2(x)}{4 (2-x)^2}-2 \left (\frac {1}{8} \log (2-x)+\frac {x \log (x)}{8 (2-x)}\right )+\frac {1}{8} \operatorname {PolyLog}\left (2,1-\frac {x}{2}\right )-\frac {\operatorname {PolyLog}\left (2,\frac {2}{x}\right )}{8}+2 \int e^x \, dx \\ & = -\frac {1075}{2 (2-x)^2}+\frac {1075}{2 (2-x)}-27 x+\frac {29 x^2}{2}+e^x x^2+\frac {1075 x^2}{8 (2-x)^2}-5 x^3+\frac {x^4}{4}+\frac {1}{4} \log (2-x)-\frac {1}{8} \log (2) \log (-2+x)-\frac {9 \log (x)}{4}+\frac {1}{2} x \log (x)+\frac {1}{8} \log \left (1-\frac {2}{x}\right ) \log (x)+\frac {\log ^2(x)}{16}+\frac {\log ^2(x)}{4 (2-x)^2}-2 \left (\frac {1}{8} \log (2-x)+\frac {x \log (x)}{8 (2-x)}\right )+\frac {1}{8} \operatorname {PolyLog}\left (2,1-\frac {x}{2}\right )-\frac {\operatorname {PolyLog}\left (2,\frac {2}{x}\right )}{8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.28 \[ \int \frac {36+372 x-1075 x^2+1250 x^3-777 x^4+262 x^5-42 x^6+2 x^7+e^x \left (-32 x^2+32 x^3-8 x^5+2 x^6\right )+\left (-2-5 x+11 x^2-6 x^3+x^4\right ) \log (x)-x \log ^2(x)}{-16 x+24 x^2-12 x^3+2 x^4} \, dx=\frac {1}{4} \left (x \left (-108+\left (58+4 e^x\right ) x-20 x^2+x^3\right )+\frac {2 (-3+x)^2 \log (x)}{-2+x}+\frac {\log ^2(x)}{(-2+x)^2}\right ) \]

[In]

Integrate[(36 + 372*x - 1075*x^2 + 1250*x^3 - 777*x^4 + 262*x^5 - 42*x^6 + 2*x^7 + E^x*(-32*x^2 + 32*x^3 - 8*x
^5 + 2*x^6) + (-2 - 5*x + 11*x^2 - 6*x^3 + x^4)*Log[x] - x*Log[x]^2)/(-16*x + 24*x^2 - 12*x^3 + 2*x^4),x]

[Out]

(x*(-108 + (58 + 4*E^x)*x - 20*x^2 + x^3) + (2*(-3 + x)^2*Log[x])/(-2 + x) + Log[x]^2/(-2 + x)^2)/4

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.62

method	result	size
risch	\(\frac {\ln \left (x \right )^{2}}{4 x^{2}-16 x +16}+\frac {\left (x^{2}-2 x +1\right ) \ln \left (x \right )}{2 x -4}+\frac {x^{4}}{4}-5 x^{3}+\frac {29 x^{2}}{2}-27 x -2 \ln \left (x \right )+{\mathrm e}^{x} x^{2}\)	\(63\)
parallelrisch	\(\frac {-864+8 \,{\mathrm e}^{x} x^{4}+32 \,{\mathrm e}^{x} x^{2}-32 \,{\mathrm e}^{x} x^{3}+84 x \ln \left (x \right )+2 \ln \left (x \right )^{2}+284 x^{4}-840 x^{3}+1112 x^{2}+2 x^{6}-48 x^{5}-72 \ln \left (x \right )+4 x^{3} \ln \left (x \right )-32 x^{2} \ln \left (x \right )}{8 x^{2}-32 x +32}\)	\(90\)

[In]

int((-x*ln(x)^2+(x^4-6*x^3+11*x^2-5*x-2)*ln(x)+(2*x^6-8*x^5+32*x^3-32*x^2)*exp(x)+2*x^7-42*x^6+262*x^5-777*x^4
+1250*x^3-1075*x^2+372*x+36)/(2*x^4-12*x^3+24*x^2-16*x),x,method=_RETURNVERBOSE)

[Out]

1/4/(x^2-4*x+4)*ln(x)^2+1/2*(x^2-2*x+1)/(-2+x)*ln(x)+1/4*x^4-5*x^3+29/2*x^2-27*x-2*ln(x)+exp(x)*x^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (33) = 66\).

Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.00 \[ \int \frac {36+372 x-1075 x^2+1250 x^3-777 x^4+262 x^5-42 x^6+2 x^7+e^x \left (-32 x^2+32 x^3-8 x^5+2 x^6\right )+\left (-2-5 x+11 x^2-6 x^3+x^4\right ) \log (x)-x \log ^2(x)}{-16 x+24 x^2-12 x^3+2 x^4} \, dx=\frac {x^{6} - 24 \, x^{5} + 142 \, x^{4} - 420 \, x^{3} + 664 \, x^{2} + 4 \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )} e^{x} + 2 \, {\left (x^{3} - 8 \, x^{2} + 21 \, x - 18\right )} \log \left (x\right ) + \log \left (x\right )^{2} - 432 \, x}{4 \, {\left (x^{2} - 4 \, x + 4\right )}} \]

[In]

integrate((-x*log(x)^2+(x^4-6*x^3+11*x^2-5*x-2)*log(x)+(2*x^6-8*x^5+32*x^3-32*x^2)*exp(x)+2*x^7-42*x^6+262*x^5
-777*x^4+1250*x^3-1075*x^2+372*x+36)/(2*x^4-12*x^3+24*x^2-16*x),x, algorithm="fricas")

[Out]

1/4*(x^6 - 24*x^5 + 142*x^4 - 420*x^3 + 664*x^2 + 4*(x^4 - 4*x^3 + 4*x^2)*e^x + 2*(x^3 - 8*x^2 + 21*x - 18)*lo
g(x) + log(x)^2 - 432*x)/(x^2 - 4*x + 4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (26) = 52\).

Time = 0.22 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.62 \[ \int \frac {36+372 x-1075 x^2+1250 x^3-777 x^4+262 x^5-42 x^6+2 x^7+e^x \left (-32 x^2+32 x^3-8 x^5+2 x^6\right )+\left (-2-5 x+11 x^2-6 x^3+x^4\right ) \log (x)-x \log ^2(x)}{-16 x+24 x^2-12 x^3+2 x^4} \, dx=\frac {x^{4}}{4} - 5 x^{3} + x^{2} e^{x} + \frac {29 x^{2}}{2} - 27 x - 2 \log {\left (x \right )} + \frac {\log {\left (x \right )}^{2}}{4 x^{2} - 16 x + 16} + \frac {\left (x^{2} - 2 x + 1\right ) \log {\left (x \right )}}{2 x - 4} \]

[In]

integrate((-x*ln(x)**2+(x**4-6*x**3+11*x**2-5*x-2)*ln(x)+(2*x**6-8*x**5+32*x**3-32*x**2)*exp(x)+2*x**7-42*x**6
+262*x**5-777*x**4+1250*x**3-1075*x**2+372*x+36)/(2*x**4-12*x**3+24*x**2-16*x),x)

[Out]

x**4/4 - 5*x**3 + x**2*exp(x) + 29*x**2/2 - 27*x - 2*log(x) + log(x)**2/(4*x**2 - 16*x + 16) + (x**2 - 2*x + 1
)*log(x)/(2*x - 4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (33) = 66\).

Time = 0.26 (sec) , antiderivative size = 217, normalized size of antiderivative = 5.56 \[ \int \frac {36+372 x-1075 x^2+1250 x^3-777 x^4+262 x^5-42 x^6+2 x^7+e^x \left (-32 x^2+32 x^3-8 x^5+2 x^6\right )+\left (-2-5 x+11 x^2-6 x^3+x^4\right ) \log (x)-x \log ^2(x)}{-16 x+24 x^2-12 x^3+2 x^4} \, dx=\frac {1}{4} \, x^{4} - 5 \, x^{3} + \frac {29}{2} \, x^{2} - \frac {53}{2} \, x - \frac {2 \, x^{3} - 8 \, x^{2} - 4 \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )} e^{x} - 2 \, {\left (x^{3} - 4 \, x^{2} + 5 \, x - 2\right )} \log \left (x\right ) - \log \left (x\right )^{2} + 8 \, x}{4 \, {\left (x^{2} - 4 \, x + 4\right )}} - \frac {32 \, {\left (6 \, x - 11\right )}}{x^{2} - 4 \, x + 4} + \frac {336 \, {\left (5 \, x - 9\right )}}{x^{2} - 4 \, x + 4} - \frac {1048 \, {\left (4 \, x - 7\right )}}{x^{2} - 4 \, x + 4} + \frac {1554 \, {\left (3 \, x - 5\right )}}{x^{2} - 4 \, x + 4} - \frac {1250 \, {\left (2 \, x - 3\right )}}{x^{2} - 4 \, x + 4} + \frac {1075 \, {\left (x - 1\right )}}{2 \, {\left (x^{2} - 4 \, x + 4\right )}} + \frac {9 \, {\left (x - 3\right )}}{2 \, {\left (x^{2} - 4 \, x + 4\right )}} - \frac {93}{x^{2} - 4 \, x + 4} - 2 \, \log \left (x\right ) \]

[In]

integrate((-x*log(x)^2+(x^4-6*x^3+11*x^2-5*x-2)*log(x)+(2*x^6-8*x^5+32*x^3-32*x^2)*exp(x)+2*x^7-42*x^6+262*x^5
-777*x^4+1250*x^3-1075*x^2+372*x+36)/(2*x^4-12*x^3+24*x^2-16*x),x, algorithm="maxima")

[Out]

1/4*x^4 - 5*x^3 + 29/2*x^2 - 53/2*x - 1/4*(2*x^3 - 8*x^2 - 4*(x^4 - 4*x^3 + 4*x^2)*e^x - 2*(x^3 - 4*x^2 + 5*x
- 2)*log(x) - log(x)^2 + 8*x)/(x^2 - 4*x + 4) - 32*(6*x - 11)/(x^2 - 4*x + 4) + 336*(5*x - 9)/(x^2 - 4*x + 4)
- 1048*(4*x - 7)/(x^2 - 4*x + 4) + 1554*(3*x - 5)/(x^2 - 4*x + 4) - 1250*(2*x - 3)/(x^2 - 4*x + 4) + 1075/2*(x
- 1)/(x^2 - 4*x + 4) + 9/2*(x - 3)/(x^2 - 4*x + 4) - 93/(x^2 - 4*x + 4) - 2*log(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (33) = 66\).

Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.23 \[ \int \frac {36+372 x-1075 x^2+1250 x^3-777 x^4+262 x^5-42 x^6+2 x^7+e^x \left (-32 x^2+32 x^3-8 x^5+2 x^6\right )+\left (-2-5 x+11 x^2-6 x^3+x^4\right ) \log (x)-x \log ^2(x)}{-16 x+24 x^2-12 x^3+2 x^4} \, dx=\frac {x^{6} - 24 \, x^{5} + 4 \, x^{4} e^{x} + 142 \, x^{4} - 16 \, x^{3} e^{x} + 2 \, x^{3} \log \left (x\right ) - 420 \, x^{3} + 16 \, x^{2} e^{x} - 16 \, x^{2} \log \left (x\right ) + 664 \, x^{2} + 42 \, x \log \left (x\right ) + \log \left (x\right )^{2} - 432 \, x - 36 \, \log \left (x\right )}{4 \, {\left (x^{2} - 4 \, x + 4\right )}} \]

[In]

integrate((-x*log(x)^2+(x^4-6*x^3+11*x^2-5*x-2)*log(x)+(2*x^6-8*x^5+32*x^3-32*x^2)*exp(x)+2*x^7-42*x^6+262*x^5
-777*x^4+1250*x^3-1075*x^2+372*x+36)/(2*x^4-12*x^3+24*x^2-16*x),x, algorithm="giac")

[Out]

1/4*(x^6 - 24*x^5 + 4*x^4*e^x + 142*x^4 - 16*x^3*e^x + 2*x^3*log(x) - 420*x^3 + 16*x^2*e^x - 16*x^2*log(x) + 6
64*x^2 + 42*x*log(x) + log(x)^2 - 432*x - 36*log(x))/(x^2 - 4*x + 4)

Mupad [B] (verification not implemented)

Time = 9.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.79 \[ \int \frac {36+372 x-1075 x^2+1250 x^3-777 x^4+262 x^5-42 x^6+2 x^7+e^x \left (-32 x^2+32 x^3-8 x^5+2 x^6\right )+\left (-2-5 x+11 x^2-6 x^3+x^4\right ) \log (x)-x \log ^2(x)}{-16 x+24 x^2-12 x^3+2 x^4} \, dx=\frac {{\ln \left (x\right )}^2}{4\,\left (x^2-4\,x+4\right )}-3\,\ln \left (x\right )-27\,x+x^2\,{\mathrm {e}}^x-\frac {3\,\ln \left (x\right )}{2\,\left (x-2\right )}+\frac {29\,x^2}{2}-5\,x^3+\frac {x^4}{4}+\frac {x^2\,\ln \left (x\right )}{2\,\left (x-2\right )} \]

[In]

int((x*log(x)^2 - 372*x + log(x)*(5*x - 11*x^2 + 6*x^3 - x^4 + 2) + exp(x)*(32*x^2 - 32*x^3 + 8*x^5 - 2*x^6) +
1075*x^2 - 1250*x^3 + 777*x^4 - 262*x^5 + 42*x^6 - 2*x^7 - 36)/(16*x - 24*x^2 + 12*x^3 - 2*x^4),x)

[Out]

log(x)^2/(4*(x^2 - 4*x + 4)) - 3*log(x) - 27*x + x^2*exp(x) - (3*log(x))/(2*(x - 2)) + (29*x^2)/2 - 5*x^3 + x^
4/4 + (x^2*log(x))/(2*(x - 2))