3.21.78 \(\int \frac {1}{4} ((-400 x-800 x^2-400 x^3) \log (-\frac {12}{x})+(400 x+1200 x^2+800 x^3) \log ^2(-\frac {12}{x})+(-x^3-4 x^4-6 x^5-4 x^6-x^7) \log ^3(-\frac {12}{x})+(x^3+5 x^4+9 x^5+7 x^6+2 x^7) \log ^4(-\frac {12}{x})) \, dx\)

Optimal. Leaf size=23 \[ \left (100+\frac {1}{4} \left (x+x^2\right )^2 \log ^2\left (-\frac {12}{x}\right )\right )^2 \]

________________________________________________________________________________________

Rubi [B]  time = 0.72, antiderivative size = 176, normalized size of antiderivative = 7.65, number of steps used = 57, number of rules used = 9, integrand size = 117, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {12, 1594, 27, 43, 2334, 14, 2356, 2305, 2304} \begin {gather*} \frac {1}{16} x^8 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^7 \log ^4\left (-\frac {12}{x}\right )+\frac {3}{8} x^6 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^5 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{16} x^4 \log ^4\left (-\frac {12}{x}\right )+50 x^4 \log ^2\left (-\frac {12}{x}\right )+25 x^4 \log \left (-\frac {12}{x}\right )+100 x^3 \log ^2\left (-\frac {12}{x}\right )+\frac {200}{3} x^3 \log \left (-\frac {12}{x}\right )+50 x^2 \log ^2\left (-\frac {12}{x}\right )+50 x^2 \log \left (-\frac {12}{x}\right )-\frac {25}{3} \left (3 x^4+8 x^3+6 x^2\right ) \log \left (-\frac {12}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-400*x - 800*x^2 - 400*x^3)*Log[-12/x] + (400*x + 1200*x^2 + 800*x^3)*Log[-12/x]^2 + (-x^3 - 4*x^4 - 6*x
^5 - 4*x^6 - x^7)*Log[-12/x]^3 + (x^3 + 5*x^4 + 9*x^5 + 7*x^6 + 2*x^7)*Log[-12/x]^4)/4,x]

[Out]

50*x^2*Log[-12/x] + (200*x^3*Log[-12/x])/3 + 25*x^4*Log[-12/x] - (25*(6*x^2 + 8*x^3 + 3*x^4)*Log[-12/x])/3 + 5
0*x^2*Log[-12/x]^2 + 100*x^3*Log[-12/x]^2 + 50*x^4*Log[-12/x]^2 + (x^4*Log[-12/x]^4)/16 + (x^5*Log[-12/x]^4)/4
+ (3*x^6*Log[-12/x]^4)/8 + (x^7*Log[-12/x]^4)/4 + (x^8*Log[-12/x]^4)/16

Rule 12

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

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 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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 1594

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

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 2334

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

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Polyx*(a + b*Log[c*
x^n])^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && PolynomialQ[Polyx, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \left (\left (-400 x-800 x^2-400 x^3\right ) \log \left (-\frac {12}{x}\right )+\left (400 x+1200 x^2+800 x^3\right ) \log ^2\left (-\frac {12}{x}\right )+\left (-x^3-4 x^4-6 x^5-4 x^6-x^7\right ) \log ^3\left (-\frac {12}{x}\right )+\left (x^3+5 x^4+9 x^5+7 x^6+2 x^7\right ) \log ^4\left (-\frac {12}{x}\right )\right ) \, dx\\ &=\frac {1}{4} \int \left (-400 x-800 x^2-400 x^3\right ) \log \left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int \left (400 x+1200 x^2+800 x^3\right ) \log ^2\left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int \left (-x^3-4 x^4-6 x^5-4 x^6-x^7\right ) \log ^3\left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int \left (x^3+5 x^4+9 x^5+7 x^6+2 x^7\right ) \log ^4\left (-\frac {12}{x}\right ) \, dx\\ &=\frac {1}{4} \int x \left (-400-800 x-400 x^2\right ) \log \left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int x \left (400+1200 x+800 x^2\right ) \log ^2\left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int \left (-x^3 \log ^3\left (-\frac {12}{x}\right )-4 x^4 \log ^3\left (-\frac {12}{x}\right )-6 x^5 \log ^3\left (-\frac {12}{x}\right )-4 x^6 \log ^3\left (-\frac {12}{x}\right )-x^7 \log ^3\left (-\frac {12}{x}\right )\right ) \, dx+\frac {1}{4} \int \left (x^3 \log ^4\left (-\frac {12}{x}\right )+5 x^4 \log ^4\left (-\frac {12}{x}\right )+9 x^5 \log ^4\left (-\frac {12}{x}\right )+7 x^6 \log ^4\left (-\frac {12}{x}\right )+2 x^7 \log ^4\left (-\frac {12}{x}\right )\right ) \, dx\\ &=\frac {1}{4} \int -400 x (1+x)^2 \log \left (-\frac {12}{x}\right ) \, dx-\frac {1}{4} \int x^3 \log ^3\left (-\frac {12}{x}\right ) \, dx-\frac {1}{4} \int x^7 \log ^3\left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int x^3 \log ^4\left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int \left (400 x \log ^2\left (-\frac {12}{x}\right )+1200 x^2 \log ^2\left (-\frac {12}{x}\right )+800 x^3 \log ^2\left (-\frac {12}{x}\right )\right ) \, dx+\frac {1}{2} \int x^7 \log ^4\left (-\frac {12}{x}\right ) \, dx+\frac {5}{4} \int x^4 \log ^4\left (-\frac {12}{x}\right ) \, dx-\frac {3}{2} \int x^5 \log ^3\left (-\frac {12}{x}\right ) \, dx+\frac {7}{4} \int x^6 \log ^4\left (-\frac {12}{x}\right ) \, dx+\frac {9}{4} \int x^5 \log ^4\left (-\frac {12}{x}\right ) \, dx-\int x^4 \log ^3\left (-\frac {12}{x}\right ) \, dx-\int x^6 \log ^3\left (-\frac {12}{x}\right ) \, dx\\ &=-\frac {1}{16} x^4 \log ^3\left (-\frac {12}{x}\right )-\frac {1}{5} x^5 \log ^3\left (-\frac {12}{x}\right )-\frac {1}{4} x^6 \log ^3\left (-\frac {12}{x}\right )-\frac {1}{7} x^7 \log ^3\left (-\frac {12}{x}\right )-\frac {1}{32} x^8 \log ^3\left (-\frac {12}{x}\right )+\frac {1}{16} x^4 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^5 \log ^4\left (-\frac {12}{x}\right )+\frac {3}{8} x^6 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^7 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{16} x^8 \log ^4\left (-\frac {12}{x}\right )-\frac {3}{32} \int x^7 \log ^2\left (-\frac {12}{x}\right ) \, dx-\frac {3}{16} \int x^3 \log ^2\left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int x^3 \log ^3\left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int x^7 \log ^3\left (-\frac {12}{x}\right ) \, dx-\frac {3}{7} \int x^6 \log ^2\left (-\frac {12}{x}\right ) \, dx-\frac {3}{5} \int x^4 \log ^2\left (-\frac {12}{x}\right ) \, dx-\frac {3}{4} \int x^5 \log ^2\left (-\frac {12}{x}\right ) \, dx+\frac {3}{2} \int x^5 \log ^3\left (-\frac {12}{x}\right ) \, dx-100 \int x (1+x)^2 \log \left (-\frac {12}{x}\right ) \, dx+100 \int x \log ^2\left (-\frac {12}{x}\right ) \, dx+200 \int x^3 \log ^2\left (-\frac {12}{x}\right ) \, dx+300 \int x^2 \log ^2\left (-\frac {12}{x}\right ) \, dx+\int x^4 \log ^3\left (-\frac {12}{x}\right ) \, dx+\int x^6 \log ^3\left (-\frac {12}{x}\right ) \, dx\\ &=-\frac {25}{3} \left (6 x^2+8 x^3+3 x^4\right ) \log \left (-\frac {12}{x}\right )+50 x^2 \log ^2\left (-\frac {12}{x}\right )+100 x^3 \log ^2\left (-\frac {12}{x}\right )+\frac {3197}{64} x^4 \log ^2\left (-\frac {12}{x}\right )-\frac {3}{25} x^5 \log ^2\left (-\frac {12}{x}\right )-\frac {1}{8} x^6 \log ^2\left (-\frac {12}{x}\right )-\frac {3}{49} x^7 \log ^2\left (-\frac {12}{x}\right )-\frac {3}{256} x^8 \log ^2\left (-\frac {12}{x}\right )+\frac {1}{16} x^4 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^5 \log ^4\left (-\frac {12}{x}\right )+\frac {3}{8} x^6 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^7 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{16} x^8 \log ^4\left (-\frac {12}{x}\right )-\frac {3}{128} \int x^7 \log \left (-\frac {12}{x}\right ) \, dx-\frac {3}{32} \int x^3 \log \left (-\frac {12}{x}\right ) \, dx+\frac {3}{32} \int x^7 \log ^2\left (-\frac {12}{x}\right ) \, dx-\frac {6}{49} \int x^6 \log \left (-\frac {12}{x}\right ) \, dx+\frac {3}{16} \int x^3 \log ^2\left (-\frac {12}{x}\right ) \, dx-\frac {6}{25} \int x^4 \log \left (-\frac {12}{x}\right ) \, dx-\frac {1}{4} \int x^5 \log \left (-\frac {12}{x}\right ) \, dx+\frac {3}{7} \int x^6 \log ^2\left (-\frac {12}{x}\right ) \, dx+\frac {3}{5} \int x^4 \log ^2\left (-\frac {12}{x}\right ) \, dx+\frac {3}{4} \int x^5 \log ^2\left (-\frac {12}{x}\right ) \, dx-100 \int \frac {1}{12} x \left (6+8 x+3 x^2\right ) \, dx+100 \int x \log \left (-\frac {12}{x}\right ) \, dx+100 \int x^3 \log \left (-\frac {12}{x}\right ) \, dx+200 \int x^2 \log \left (-\frac {12}{x}\right ) \, dx\\ &=25 x^2+\frac {200 x^3}{9}+\frac {3197 x^4}{512}-\frac {6 x^5}{625}-\frac {x^6}{144}-\frac {6 x^7}{2401}-\frac {3 x^8}{8192}+50 x^2 \log \left (-\frac {12}{x}\right )+\frac {200}{3} x^3 \log \left (-\frac {12}{x}\right )+\frac {3197}{128} x^4 \log \left (-\frac {12}{x}\right )-\frac {6}{125} x^5 \log \left (-\frac {12}{x}\right )-\frac {1}{24} x^6 \log \left (-\frac {12}{x}\right )-\frac {6}{343} x^7 \log \left (-\frac {12}{x}\right )-\frac {3 x^8 \log \left (-\frac {12}{x}\right )}{1024}-\frac {25}{3} \left (6 x^2+8 x^3+3 x^4\right ) \log \left (-\frac {12}{x}\right )+50 x^2 \log ^2\left (-\frac {12}{x}\right )+100 x^3 \log ^2\left (-\frac {12}{x}\right )+50 x^4 \log ^2\left (-\frac {12}{x}\right )+\frac {1}{16} x^4 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^5 \log ^4\left (-\frac {12}{x}\right )+\frac {3}{8} x^6 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^7 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{16} x^8 \log ^4\left (-\frac {12}{x}\right )+\frac {3}{128} \int x^7 \log \left (-\frac {12}{x}\right ) \, dx+\frac {3}{32} \int x^3 \log \left (-\frac {12}{x}\right ) \, dx+\frac {6}{49} \int x^6 \log \left (-\frac {12}{x}\right ) \, dx+\frac {6}{25} \int x^4 \log \left (-\frac {12}{x}\right ) \, dx+\frac {1}{4} \int x^5 \log \left (-\frac {12}{x}\right ) \, dx-\frac {25}{3} \int x \left (6+8 x+3 x^2\right ) \, dx\\ &=25 x^2+\frac {200 x^3}{9}+\frac {25 x^4}{4}+50 x^2 \log \left (-\frac {12}{x}\right )+\frac {200}{3} x^3 \log \left (-\frac {12}{x}\right )+25 x^4 \log \left (-\frac {12}{x}\right )-\frac {25}{3} \left (6 x^2+8 x^3+3 x^4\right ) \log \left (-\frac {12}{x}\right )+50 x^2 \log ^2\left (-\frac {12}{x}\right )+100 x^3 \log ^2\left (-\frac {12}{x}\right )+50 x^4 \log ^2\left (-\frac {12}{x}\right )+\frac {1}{16} x^4 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^5 \log ^4\left (-\frac {12}{x}\right )+\frac {3}{8} x^6 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^7 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{16} x^8 \log ^4\left (-\frac {12}{x}\right )-\frac {25}{3} \int \left (6 x+8 x^2+3 x^3\right ) \, dx\\ &=50 x^2 \log \left (-\frac {12}{x}\right )+\frac {200}{3} x^3 \log \left (-\frac {12}{x}\right )+25 x^4 \log \left (-\frac {12}{x}\right )-\frac {25}{3} \left (6 x^2+8 x^3+3 x^4\right ) \log \left (-\frac {12}{x}\right )+50 x^2 \log ^2\left (-\frac {12}{x}\right )+100 x^3 \log ^2\left (-\frac {12}{x}\right )+50 x^4 \log ^2\left (-\frac {12}{x}\right )+\frac {1}{16} x^4 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^5 \log ^4\left (-\frac {12}{x}\right )+\frac {3}{8} x^6 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^7 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{16} x^8 \log ^4\left (-\frac {12}{x}\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.25, size = 113, normalized size = 4.91 \begin {gather*} \frac {1}{4} \left (200 x^2 \log ^2\left (-\frac {12}{x}\right )+400 x^3 \log ^2\left (-\frac {12}{x}\right )+200 x^4 \log ^2\left (-\frac {12}{x}\right )+\frac {1}{4} x^4 \log ^4\left (-\frac {12}{x}\right )+x^5 \log ^4\left (-\frac {12}{x}\right )+\frac {3}{2} x^6 \log ^4\left (-\frac {12}{x}\right )+x^7 \log ^4\left (-\frac {12}{x}\right )+\frac {1}{4} x^8 \log ^4\left (-\frac {12}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-400*x - 800*x^2 - 400*x^3)*Log[-12/x] + (400*x + 1200*x^2 + 800*x^3)*Log[-12/x]^2 + (-x^3 - 4*x^4
- 6*x^5 - 4*x^6 - x^7)*Log[-12/x]^3 + (x^3 + 5*x^4 + 9*x^5 + 7*x^6 + 2*x^7)*Log[-12/x]^4)/4,x]

[Out]

(200*x^2*Log[-12/x]^2 + 400*x^3*Log[-12/x]^2 + 200*x^4*Log[-12/x]^2 + (x^4*Log[-12/x]^4)/4 + x^5*Log[-12/x]^4
+ (3*x^6*Log[-12/x]^4)/2 + x^7*Log[-12/x]^4 + (x^8*Log[-12/x]^4)/4)/4

________________________________________________________________________________________

fricas [B]  time = 0.50, size = 55, normalized size = 2.39 \begin {gather*} \frac {1}{16} \, {\left (x^{8} + 4 \, x^{7} + 6 \, x^{6} + 4 \, x^{5} + x^{4}\right )} \log \left (-\frac {12}{x}\right )^{4} + 50 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (-\frac {12}{x}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(2*x^7+7*x^6+9*x^5+5*x^4+x^3)*log(-12/x)^4+1/4*(-x^7-4*x^6-6*x^5-4*x^4-x^3)*log(-12/x)^3+1/4*(80
0*x^3+1200*x^2+400*x)*log(-12/x)^2+1/4*(-400*x^3-800*x^2-400*x)*log(-12/x),x, algorithm="fricas")

[Out]

1/16*(x^8 + 4*x^7 + 6*x^6 + 4*x^5 + x^4)*log(-12/x)^4 + 50*(x^4 + 2*x^3 + x^2)*log(-12/x)^2

________________________________________________________________________________________

giac [B]  time = 0.99, size = 173, normalized size = 7.52 \begin {gather*} \frac {1}{1120} \, x^{8} {\left (\frac {160}{x} + \frac {280}{x^{2}} + \frac {224}{x^{3}} + \frac {70}{x^{4}} + 35\right )} \log \left (-\frac {12}{x}\right )^{3} + \frac {25}{3} \, x^{4} {\left (\frac {8}{x} + \frac {6}{x^{2}} + 3\right )} \log \left (-\frac {12}{x}\right ) + \frac {1}{16} \, {\left (x^{8} + 4 \, x^{7} + 6 \, x^{6} + 4 \, x^{5} + x^{4}\right )} \log \left (-\frac {12}{x}\right )^{4} - \frac {1}{1120} \, {\left (35 \, x^{8} + 160 \, x^{7} + 280 \, x^{6} + 224 \, x^{5} + 70 \, x^{4}\right )} \log \left (-\frac {12}{x}\right )^{3} + 50 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (-\frac {12}{x}\right )^{2} - \frac {25}{3} \, {\left (3 \, x^{4} + 8 \, x^{3} + 6 \, x^{2}\right )} \log \left (-\frac {12}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(2*x^7+7*x^6+9*x^5+5*x^4+x^3)*log(-12/x)^4+1/4*(-x^7-4*x^6-6*x^5-4*x^4-x^3)*log(-12/x)^3+1/4*(80
0*x^3+1200*x^2+400*x)*log(-12/x)^2+1/4*(-400*x^3-800*x^2-400*x)*log(-12/x),x, algorithm="giac")

[Out]

1/1120*x^8*(160/x + 280/x^2 + 224/x^3 + 70/x^4 + 35)*log(-12/x)^3 + 25/3*x^4*(8/x + 6/x^2 + 3)*log(-12/x) + 1/
16*(x^8 + 4*x^7 + 6*x^6 + 4*x^5 + x^4)*log(-12/x)^4 - 1/1120*(35*x^8 + 160*x^7 + 280*x^6 + 224*x^5 + 70*x^4)*l
og(-12/x)^3 + 50*(x^4 + 2*x^3 + x^2)*log(-12/x)^2 - 25/3*(3*x^4 + 8*x^3 + 6*x^2)*log(-12/x)

________________________________________________________________________________________

maple [B]  time = 0.09, size = 106, normalized size = 4.61

method	result	size
derivativedivides	\(\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{5}}{4}+100 \ln \left (-\frac {12}{x}\right )^{2} x^{3}+\frac {3 \ln \left (-\frac {12}{x}\right )^{4} x^{6}}{8}+50 \ln \left (-\frac {12}{x}\right )^{2} x^{4}+\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{7}}{4}+\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{8}}{16}+\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{4}}{16}+50 x^{2} \ln \left (-\frac {12}{x}\right )^{2}\)	\(106\)
default	\(\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{5}}{4}+100 \ln \left (-\frac {12}{x}\right )^{2} x^{3}+\frac {3 \ln \left (-\frac {12}{x}\right )^{4} x^{6}}{8}+50 \ln \left (-\frac {12}{x}\right )^{2} x^{4}+\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{7}}{4}+\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{8}}{16}+\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{4}}{16}+50 x^{2} \ln \left (-\frac {12}{x}\right )^{2}\)	\(106\)
risch	\(\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{5}}{4}+100 \ln \left (-\frac {12}{x}\right )^{2} x^{3}+\frac {3 \ln \left (-\frac {12}{x}\right )^{4} x^{6}}{8}+50 \ln \left (-\frac {12}{x}\right )^{2} x^{4}+\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{7}}{4}+\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{8}}{16}+\frac {\ln \left (-\frac {12}{x}\right )^{4} x^{4}}{16}+50 x^{2} \ln \left (-\frac {12}{x}\right )^{2}\)	\(106\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(2*x^7+7*x^6+9*x^5+5*x^4+x^3)*ln(-12/x)^4+1/4*(-x^7-4*x^6-6*x^5-4*x^4-x^3)*ln(-12/x)^3+1/4*(800*x^3+12
00*x^2+400*x)*ln(-12/x)^2+1/4*(-400*x^3-800*x^2-400*x)*ln(-12/x),x,method=_RETURNVERBOSE)

[Out]

1/4*ln(-12/x)^4*x^5+100*ln(-12/x)^2*x^3+3/8*ln(-12/x)^4*x^6+50*ln(-12/x)^2*x^4+1/4*ln(-12/x)^4*x^7+1/16*ln(-12
/x)^4*x^8+1/16*ln(-12/x)^4*x^4+50*x^2*ln(-12/x)^2

________________________________________________________________________________________

maxima [B]  time = 0.35, size = 55, normalized size = 2.39 \begin {gather*} \frac {1}{16} \, {\left (x^{8} + 4 \, x^{7} + 6 \, x^{6} + 4 \, x^{5} + x^{4}\right )} \log \left (-\frac {12}{x}\right )^{4} + 50 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (-\frac {12}{x}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(2*x^7+7*x^6+9*x^5+5*x^4+x^3)*log(-12/x)^4+1/4*(-x^7-4*x^6-6*x^5-4*x^4-x^3)*log(-12/x)^3+1/4*(80
0*x^3+1200*x^2+400*x)*log(-12/x)^2+1/4*(-400*x^3-800*x^2-400*x)*log(-12/x),x, algorithm="maxima")

[Out]

1/16*(x^8 + 4*x^7 + 6*x^6 + 4*x^5 + x^4)*log(-12/x)^4 + 50*(x^4 + 2*x^3 + x^2)*log(-12/x)^2

________________________________________________________________________________________

mupad [B]  time = 1.26, size = 57, normalized size = 2.48 \begin {gather*} \frac {x^2\,{\ln \left (-\frac {12}{x}\right )}^2\,{\left (x+1\right )}^2\,\left (x^4\,{\ln \left (-\frac {12}{x}\right )}^2+2\,x^3\,{\ln \left (-\frac {12}{x}\right )}^2+x^2\,{\ln \left (-\frac {12}{x}\right )}^2+800\right )}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(-12/x)^2*(400*x + 1200*x^2 + 800*x^3))/4 - (log(-12/x)^3*(x^3 + 4*x^4 + 6*x^5 + 4*x^6 + x^7))/4 + (lo
g(-12/x)^4*(x^3 + 5*x^4 + 9*x^5 + 7*x^6 + 2*x^7))/4 - (log(-12/x)*(400*x + 800*x^2 + 400*x^3))/4,x)

[Out]

(x^2*log(-12/x)^2*(x + 1)^2*(x^2*log(-12/x)^2 + 2*x^3*log(-12/x)^2 + x^4*log(-12/x)^2 + 800))/16

________________________________________________________________________________________

sympy [B]  time = 0.22, size = 58, normalized size = 2.52 \begin {gather*} \left (50 x^{4} + 100 x^{3} + 50 x^{2}\right ) \log {\left (- \frac {12}{x} \right )}^{2} + \left (\frac {x^{8}}{16} + \frac {x^{7}}{4} + \frac {3 x^{6}}{8} + \frac {x^{5}}{4} + \frac {x^{4}}{16}\right ) \log {\left (- \frac {12}{x} \right )}^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(2*x**7+7*x**6+9*x**5+5*x**4+x**3)*ln(-12/x)**4+1/4*(-x**7-4*x**6-6*x**5-4*x**4-x**3)*ln(-12/x)*
*3+1/4*(800*x**3+1200*x**2+400*x)*ln(-12/x)**2+1/4*(-400*x**3-800*x**2-400*x)*ln(-12/x),x)

[Out]

(50*x**4 + 100*x**3 + 50*x**2)*log(-12/x)**2 + (x**8/16 + x**7/4 + 3*x**6/8 + x**5/4 + x**4/16)*log(-12/x)**4

________________________________________________________________________________________