3.58.28 \(\int \frac {-32-112 x-152 x^2-100 x^3-96 x^4-164 x^5-128 x^6-32 x^7+(128 x^3+320 x^4+256 x^5+64 x^6) \log (3)+(-64 x^2-160 x^3-128 x^4-32 x^5) \log ^2(3)+(-64 x-160 x^2-144 x^3-312 x^4-712 x^5-576 x^6-128 x^7+(384 x^3+1088 x^4+896 x^5+192 x^6) \log (3)+(-128 x^2-384 x^3-320 x^4-64 x^5) \log ^2(3)) \log (x)}{8 x+12 x^2+6 x^3+x^4} \, dx\)

Optimal. Leaf size=31 \[ 3-2 (1+x)^2 \left (2+\frac {16 x^2 (-x+\log (3))^2}{(2+x)^2}\right ) \log (x) \]

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Rubi [B]  time = 1.33, antiderivative size = 246, normalized size of antiderivative = 7.94, number of steps used = 17, number of rules used = 11, integrand size = 192, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {6688, 12, 6742, 1612, 2357, 2304, 2295, 2314, 31, 2319, 44} \begin {gather*} -32 x^4 \log (x)+64 x^3 (1+\log (3)) \log (x)-4 x^2 \left (41+8 \log ^2(3)+32 \log (3)\right ) \log (x)-\frac {64 x \left (16-13 \log ^2(3)+2 \log (3) (7+4 \log (9))\right ) \log (x)}{x+2}+8 x \left (47+8 \log ^2(3)+40 \log (3)\right )-\frac {128 \left (4-3 \log ^2(3)+\log ^2(9)+\log (81)\right ) \log (x)}{(x+2)^2}+32 \left (4-3 \log ^2(3)+\log ^2(9)+\log (81)\right ) \log (x)-32 \left (4-3 \log ^2(3)+\log ^2(9)+\log (81)\right ) \log (x+2)+64 \left (16-13 \log ^2(3)+2 \log (3) (7+4 \log (9))\right ) \log (x+2)+\frac {64 \left (4-3 \log ^2(3)+\log ^2(9)+\log (81)\right )}{x+2}+8 x (47+8 \log (3) (5+\log (3))) \log (x)-8 x (47+8 \log (3) (5+\log (3)))-4 \log (x)-32 (2+\log (3)) (14+\log (243)) \log (x+2)-\frac {64 (2+\log (3))^2}{x+2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-32 - 112*x - 152*x^2 - 100*x^3 - 96*x^4 - 164*x^5 - 128*x^6 - 32*x^7 + (128*x^3 + 320*x^4 + 256*x^5 + 64
*x^6)*Log[3] + (-64*x^2 - 160*x^3 - 128*x^4 - 32*x^5)*Log[3]^2 + (-64*x - 160*x^2 - 144*x^3 - 312*x^4 - 712*x^
5 - 576*x^6 - 128*x^7 + (384*x^3 + 1088*x^4 + 896*x^5 + 192*x^6)*Log[3] + (-128*x^2 - 384*x^3 - 320*x^4 - 64*x
^5)*Log[3]^2)*Log[x])/(8*x + 12*x^2 + 6*x^3 + x^4),x]

[Out]

(-64*(2 + Log[3])^2)/(2 + x) + 8*x*(47 + 40*Log[3] + 8*Log[3]^2) - 8*x*(47 + 8*Log[3]*(5 + Log[3])) + (64*(4 -
3*Log[3]^2 + Log[9]^2 + Log[81]))/(2 + x) - 4*Log[x] - 32*x^4*Log[x] + 64*x^3*(1 + Log[3])*Log[x] - 4*x^2*(41
+ 32*Log[3] + 8*Log[3]^2)*Log[x] + 8*x*(47 + 8*Log[3]*(5 + Log[3]))*Log[x] - (64*x*(16 - 13*Log[3]^2 + 2*Log[
3]*(7 + 4*Log[9]))*Log[x])/(2 + x) + 32*(4 - 3*Log[3]^2 + Log[9]^2 + Log[81])*Log[x] - (128*(4 - 3*Log[3]^2 +
Log[9]^2 + Log[81])*Log[x])/(2 + x)^2 + 64*(16 - 13*Log[3]^2 + 2*Log[3]*(7 + 4*Log[9]))*Log[2 + x] - 32*(4 - 3
*Log[3]^2 + Log[9]^2 + Log[81])*Log[2 + x] - 32*(2 + Log[3])*(14 + Log[243])*Log[2 + x]

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 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 1612

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

Rule 2295

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

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 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 2319

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 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 6688

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

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 {4 (1+x) \left (-\left (\left (2+3 x+x^2\right ) \left (4+4 x+8 x^4-16 x^3 \log (3)+x^2 \left (1+8 \log ^2(3)\right )\right )\right )-2 x \left (8+16 x^5+4 x \left (3+4 \log ^2(3)\right )+x^3 \left (33-88 \log (3)+8 \log ^2(3)\right )+x^2 \left (6-48 \log (3)+32 \log ^2(3)\right )-8 x^4 (-7+\log (27))\right ) \log (x)\right )}{x (2+x)^3} \, dx\\ &=4 \int \frac {(1+x) \left (-\left (\left (2+3 x+x^2\right ) \left (4+4 x+8 x^4-16 x^3 \log (3)+x^2 \left (1+8 \log ^2(3)\right )\right )\right )-2 x \left (8+16 x^5+4 x \left (3+4 \log ^2(3)\right )+x^3 \left (33-88 \log (3)+8 \log ^2(3)\right )+x^2 \left (6-48 \log (3)+32 \log ^2(3)\right )-8 x^4 (-7+\log (27))\right ) \log (x)\right )}{x (2+x)^3} \, dx\\ &=4 \int \left (\frac {(1+x)^2 \left (-4-4 x-8 x^4+16 x^3 \log (3)-x^2 \left (1+8 \log ^2(3)\right )\right )}{x (2+x)^2}+\frac {2 (1+x) \left (-8-16 x^5-56 x^4 \left (1-\frac {3 \log (3)}{7}\right )-33 x^3 \left (1+\frac {8}{33} (-11+\log (3)) \log (3)\right )-12 x \left (1+\frac {4 \log ^2(3)}{3}\right )-6 x^2 \left (1+\frac {8}{3} \log (3) (-3+\log (9))\right )\right ) \log (x)}{(2+x)^3}\right ) \, dx\\ &=4 \int \frac {(1+x)^2 \left (-4-4 x-8 x^4+16 x^3 \log (3)-x^2 \left (1+8 \log ^2(3)\right )\right )}{x (2+x)^2} \, dx+8 \int \frac {(1+x) \left (-8-16 x^5-56 x^4 \left (1-\frac {3 \log (3)}{7}\right )-33 x^3 \left (1+\frac {8}{33} (-11+\log (3)) \log (3)\right )-12 x \left (1+\frac {4 \log ^2(3)}{3}\right )-6 x^2 \left (1+\frac {8}{3} \log (3) (-3+\log (9))\right )\right ) \log (x)}{(2+x)^3} \, dx\\ &=4 \int \left (-\frac {1}{x}-8 x^3+16 x^2 (1+\log (3))+\frac {8 (-14-5 \log (3)) (2+\log (3))}{2+x}+\frac {16 (2+\log (3))^2}{(2+x)^2}-x \left (41+32 \log (3)+8 \log ^2(3)\right )+2 \left (47+40 \log (3)+8 \log ^2(3)\right )\right ) \, dx+8 \int \left (-16 x^3 \log (x)+24 x^2 (1+\log (3)) \log (x)-x \left (41+32 \log (3)+8 \log ^2(3)\right ) \log (x)+47 \left (1+\frac {8}{47} \log (3) (5+\log (3))\right ) \log (x)+\frac {16 \left (-16+13 \log ^2(3)-2 \log (3) (7+4 \log (9))\right ) \log (x)}{(2+x)^2}-\frac {32 \left (-4+3 \log ^2(3)-\log ^2(9)-\log (81)\right ) \log (x)}{(2+x)^3}\right ) \, dx\\ &=-8 x^4+\frac {64}{3} x^3 (1+\log (3))-\frac {64 (2+\log (3))^2}{2+x}-2 x^2 \left (41+32 \log (3)+8 \log ^2(3)\right )+8 x \left (47+40 \log (3)+8 \log ^2(3)\right )-4 \log (x)-32 (2+\log (3)) (14+\log (243)) \log (2+x)-128 \int x^3 \log (x) \, dx+(192 (1+\log (3))) \int x^2 \log (x) \, dx-\left (8 \left (41+32 \log (3)+8 \log ^2(3)\right )\right ) \int x \log (x) \, dx+(8 (47+8 \log (3) (5+\log (3)))) \int \log (x) \, dx-\left (128 \left (16-13 \log ^2(3)+2 \log (3) (7+4 \log (9))\right )\right ) \int \frac {\log (x)}{(2+x)^2} \, dx+\left (256 \left (4-3 \log ^2(3)+\log ^2(9)+\log (81)\right )\right ) \int \frac {\log (x)}{(2+x)^3} \, dx\\ &=-\frac {64 (2+\log (3))^2}{2+x}+8 x \left (47+40 \log (3)+8 \log ^2(3)\right )-8 x (47+8 \log (3) (5+\log (3)))-4 \log (x)-32 x^4 \log (x)+64 x^3 (1+\log (3)) \log (x)-4 x^2 \left (41+32 \log (3)+8 \log ^2(3)\right ) \log (x)+8 x (47+8 \log (3) (5+\log (3))) \log (x)-\frac {64 x \left (16-13 \log ^2(3)+2 \log (3) (7+4 \log (9))\right ) \log (x)}{2+x}-\frac {128 \left (4-3 \log ^2(3)+\log ^2(9)+\log (81)\right ) \log (x)}{(2+x)^2}-32 (2+\log (3)) (14+\log (243)) \log (2+x)+\left (64 \left (16-13 \log ^2(3)+2 \log (3) (7+4 \log (9))\right )\right ) \int \frac {1}{2+x} \, dx+\left (128 \left (4-3 \log ^2(3)+\log ^2(9)+\log (81)\right )\right ) \int \frac {1}{x (2+x)^2} \, dx\\ &=-\frac {64 (2+\log (3))^2}{2+x}+8 x \left (47+40 \log (3)+8 \log ^2(3)\right )-8 x (47+8 \log (3) (5+\log (3)))-4 \log (x)-32 x^4 \log (x)+64 x^3 (1+\log (3)) \log (x)-4 x^2 \left (41+32 \log (3)+8 \log ^2(3)\right ) \log (x)+8 x (47+8 \log (3) (5+\log (3))) \log (x)-\frac {64 x \left (16-13 \log ^2(3)+2 \log (3) (7+4 \log (9))\right ) \log (x)}{2+x}-\frac {128 \left (4-3 \log ^2(3)+\log ^2(9)+\log (81)\right ) \log (x)}{(2+x)^2}+64 \left (16-13 \log ^2(3)+2 \log (3) (7+4 \log (9))\right ) \log (2+x)-32 (2+\log (3)) (14+\log (243)) \log (2+x)+\left (128 \left (4-3 \log ^2(3)+\log ^2(9)+\log (81)\right )\right ) \int \left (\frac {1}{4 x}-\frac {1}{2 (2+x)^2}-\frac {1}{4 (2+x)}\right ) \, dx\\ &=-\frac {64 (2+\log (3))^2}{2+x}+8 x \left (47+40 \log (3)+8 \log ^2(3)\right )-8 x (47+8 \log (3) (5+\log (3)))+\frac {64 \left (4-3 \log ^2(3)+\log ^2(9)+\log (81)\right )}{2+x}-4 \log (x)-32 x^4 \log (x)+64 x^3 (1+\log (3)) \log (x)-4 x^2 \left (41+32 \log (3)+8 \log ^2(3)\right ) \log (x)+8 x (47+8 \log (3) (5+\log (3))) \log (x)-\frac {64 x \left (16-13 \log ^2(3)+2 \log (3) (7+4 \log (9))\right ) \log (x)}{2+x}+32 \left (4-3 \log ^2(3)+\log ^2(9)+\log (81)\right ) \log (x)-\frac {128 \left (4-3 \log ^2(3)+\log ^2(9)+\log (81)\right ) \log (x)}{(2+x)^2}+64 \left (16-13 \log ^2(3)+2 \log (3) (7+4 \log (9))\right ) \log (2+x)-32 \left (4-3 \log ^2(3)+\log ^2(9)+\log (81)\right ) \log (2+x)-32 (2+\log (3)) (14+\log (243)) \log (2+x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.13, size = 84, normalized size = 2.71 \begin {gather*} -\frac {4 \left (12+36 x+24 x^6+3 x^2 \left (13+8 \log ^2(3)\right )+2 x^3 \left (9+648 \log (3)+24 \log ^2(3)-224 \log (27)\right )-16 x^5 (-3+\log (27))+x^4 \left (27-264 \log (3)+24 \log ^2(3)+56 \log (27)\right )\right ) \log (x)}{3 (2+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-32 - 112*x - 152*x^2 - 100*x^3 - 96*x^4 - 164*x^5 - 128*x^6 - 32*x^7 + (128*x^3 + 320*x^4 + 256*x^
5 + 64*x^6)*Log[3] + (-64*x^2 - 160*x^3 - 128*x^4 - 32*x^5)*Log[3]^2 + (-64*x - 160*x^2 - 144*x^3 - 312*x^4 -
712*x^5 - 576*x^6 - 128*x^7 + (384*x^3 + 1088*x^4 + 896*x^5 + 192*x^6)*Log[3] + (-128*x^2 - 384*x^3 - 320*x^4
- 64*x^5)*Log[3]^2)*Log[x])/(8*x + 12*x^2 + 6*x^3 + x^4),x]

[Out]

(-4*(12 + 36*x + 24*x^6 + 3*x^2*(13 + 8*Log[3]^2) + 2*x^3*(9 + 648*Log[3] + 24*Log[3]^2 - 224*Log[27]) - 16*x^
5*(-3 + Log[27]) + x^4*(27 - 264*Log[3] + 24*Log[3]^2 + 56*Log[27]))*Log[x])/(3*(2 + x)^2)

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fricas [B]  time = 0.59, size = 78, normalized size = 2.52 \begin {gather*} -\frac {4 \, {\left (8 \, x^{6} + 16 \, x^{5} + 9 \, x^{4} + 6 \, x^{3} + 8 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \relax (3)^{2} + 13 \, x^{2} - 16 \, {\left (x^{5} + 2 \, x^{4} + x^{3}\right )} \log \relax (3) + 12 \, x + 4\right )} \log \relax (x)}{x^{2} + 4 \, x + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-64*x^5-320*x^4-384*x^3-128*x^2)*log(3)^2+(192*x^6+896*x^5+1088*x^4+384*x^3)*log(3)-128*x^7-576*x
^6-712*x^5-312*x^4-144*x^3-160*x^2-64*x)*log(x)+(-32*x^5-128*x^4-160*x^3-64*x^2)*log(3)^2+(64*x^6+256*x^5+320*
x^4+128*x^3)*log(3)-32*x^7-128*x^6-164*x^5-96*x^4-100*x^3-152*x^2-112*x-32)/(x^4+6*x^3+12*x^2+8*x),x, algorith
m="fricas")

[Out]

-4*(8*x^6 + 16*x^5 + 9*x^4 + 6*x^3 + 8*(x^4 + 2*x^3 + x^2)*log(3)^2 + 13*x^2 - 16*(x^5 + 2*x^4 + x^3)*log(3) +
12*x + 4)*log(x)/(x^2 + 4*x + 4)

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giac [B]  time = 0.14, size = 106, normalized size = 3.42 \begin {gather*} -4 \, {\left (8 \, x^{4} - 16 \, x^{3} {\left (\log \relax (3) + 1\right )} + {\left (8 \, \log \relax (3)^{2} + 32 \, \log \relax (3) + 41\right )} x^{2} - 2 \, {\left (8 \, \log \relax (3)^{2} + 40 \, \log \relax (3) + 47\right )} x - \frac {32 \, {\left (3 \, x \log \relax (3)^{2} + 14 \, x \log \relax (3) + 5 \, \log \relax (3)^{2} + 16 \, x + 24 \, \log \relax (3) + 28\right )}}{x^{2} + 4 \, x + 4}\right )} \log \relax (x) - 4 \, {\left (40 \, \log \relax (3)^{2} + 192 \, \log \relax (3) + 225\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-64*x^5-320*x^4-384*x^3-128*x^2)*log(3)^2+(192*x^6+896*x^5+1088*x^4+384*x^3)*log(3)-128*x^7-576*x
^6-712*x^5-312*x^4-144*x^3-160*x^2-64*x)*log(x)+(-32*x^5-128*x^4-160*x^3-64*x^2)*log(3)^2+(64*x^6+256*x^5+320*
x^4+128*x^3)*log(3)-32*x^7-128*x^6-164*x^5-96*x^4-100*x^3-152*x^2-112*x-32)/(x^4+6*x^3+12*x^2+8*x),x, algorith
m="giac")

[Out]

-4*(8*x^4 - 16*x^3*(log(3) + 1) + (8*log(3)^2 + 32*log(3) + 41)*x^2 - 2*(8*log(3)^2 + 40*log(3) + 47)*x - 32*(
3*x*log(3)^2 + 14*x*log(3) + 5*log(3)^2 + 16*x + 24*log(3) + 28)/(x^2 + 4*x + 4))*log(x) - 4*(40*log(3)^2 + 19
2*log(3) + 225)*log(x)

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maple [B]  time = 0.15, size = 86, normalized size = 2.77

method	result	size
norman	\(\frac {-16 \ln \relax (x )+\left (-64+64 \ln \relax (3)\right ) x^{5} \ln \relax (x )-48 x \ln \relax (x )+\left (-52-32 \ln \relax (3)^{2}\right ) x^{2} \ln \relax (x )+\left (-36+128 \ln \relax (3)-32 \ln \relax (3)^{2}\right ) x^{4} \ln \relax (x )+\left (-24+64 \ln \relax (3)-64 \ln \relax (3)^{2}\right ) x^{3} \ln \relax (x )-32 x^{6} \ln \relax (x )}{\left (2+x \right )^{2}}\)	\(86\)
risch	\(-\frac {4 \left (8 x^{4} \ln \relax (3)^{2}-16 x^{5} \ln \relax (3)+8 x^{6}+16 x^{3} \ln \relax (3)^{2}-32 x^{4} \ln \relax (3)+16 x^{5}-32 x^{2} \ln \relax (3)^{2}-16 x^{3} \ln \relax (3)+9 x^{4}-160 x \ln \relax (3)^{2}-192 x^{2} \ln \relax (3)+6 x^{3}-160 \ln \relax (3)^{2}-768 x \ln \relax (3)-212 x^{2}-768 \ln \relax (3)-888 x -896\right ) \ln \relax (x )}{x^{2}+4 x +4}-160 \ln \relax (3)^{2} \ln \relax (x )-768 \ln \relax (3) \ln \relax (x )-900 \ln \relax (x )\)	\(141\)
default	\(320 x \ln \relax (3) \ln \relax (x )-128 x^{2} \ln \relax (3) \ln \relax (x )-164 x^{2} \ln \relax (x )-32 x^{2} \ln \relax (3)^{2} \ln \relax (x )-4 \ln \relax (x )+64 x^{3} \ln \relax (x )-32 x^{4} \ln \relax (x )+376 x \ln \relax (x )+\frac {128 \ln \relax (x ) x \left (4+x \right )}{\left (2+x \right )^{2}}+\frac {128 \ln \relax (3) \ln \relax (x ) x^{2}}{\left (2+x \right )^{2}}+\frac {512 \ln \relax (3) \ln \relax (x ) x}{\left (2+x \right )^{2}}-\frac {896 \ln \relax (3) \ln \relax (x ) x}{2+x}+\frac {32 \ln \relax (3)^{2} \ln \relax (x ) x^{2}}{\left (2+x \right )^{2}}+\frac {128 \ln \relax (3)^{2} \ln \relax (x ) x}{\left (2+x \right )^{2}}-\frac {192 \ln \relax (3)^{2} \ln \relax (x ) x}{2+x}+64 \ln \relax (x ) \ln \relax (3) x^{3}-\frac {1024 \ln \relax (x ) x}{2+x}+64 \ln \relax (x ) \ln \relax (3)^{2} x\)	\(182\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-64*x^5-320*x^4-384*x^3-128*x^2)*ln(3)^2+(192*x^6+896*x^5+1088*x^4+384*x^3)*ln(3)-128*x^7-576*x^6-712*x
^5-312*x^4-144*x^3-160*x^2-64*x)*ln(x)+(-32*x^5-128*x^4-160*x^3-64*x^2)*ln(3)^2+(64*x^6+256*x^5+320*x^4+128*x^
3)*ln(3)-32*x^7-128*x^6-164*x^5-96*x^4-100*x^3-152*x^2-112*x-32)/(x^4+6*x^3+12*x^2+8*x),x,method=_RETURNVERBOS
E)

[Out]

(-16*ln(x)+(-64+64*ln(3))*x^5*ln(x)-48*x*ln(x)+(-52-32*ln(3)^2)*x^2*ln(x)+(-36+128*ln(3)-32*ln(3)^2)*x^4*ln(x)
+(-24+64*ln(3)-64*ln(3)^2)*x^3*ln(x)-32*x^6*ln(x))/(2+x)^2

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maxima [B]  time = 0.51, size = 638, normalized size = 20.58 \begin {gather*} -8 \, x^{4} + \frac {64}{3} \, x^{3} - 16 \, {\left (x^{2} - 12 \, x + \frac {16 \, {\left (4 \, x + 7\right )}}{x^{2} + 4 \, x + 4} + 48 \, \log \left (x + 2\right )\right )} \log \relax (3)^{2} - 128 \, {\left (x - \frac {4 \, {\left (3 \, x + 5\right )}}{x^{2} + 4 \, x + 4} - 6 \, \log \left (x + 2\right )\right )} \log \relax (3)^{2} + 32 \, {\left (\frac {4 \, {\left (x + 1\right )} \log \relax (x)}{x^{2} + 4 \, x + 4} + \frac {2}{x + 2} + \log \left (x + 2\right ) - \log \relax (x)\right )} \log \relax (3)^{2} - 160 \, {\left (\frac {2 \, {\left (2 \, x + 3\right )}}{x^{2} + 4 \, x + 4} + \log \left (x + 2\right )\right )} \log \relax (3)^{2} - 82 \, x^{2} + \frac {64}{3} \, {\left (x^{3} - 9 \, x^{2} + 72 \, x - \frac {48 \, {\left (5 \, x + 9\right )}}{x^{2} + 4 \, x + 4} - 240 \, \log \left (x + 2\right )\right )} \log \relax (3) + 128 \, {\left (x^{2} - 12 \, x + \frac {16 \, {\left (4 \, x + 7\right )}}{x^{2} + 4 \, x + 4} + 48 \, \log \left (x + 2\right )\right )} \log \relax (3) + 320 \, {\left (x - \frac {4 \, {\left (3 \, x + 5\right )}}{x^{2} + 4 \, x + 4} - 6 \, \log \left (x + 2\right )\right )} \log \relax (3) + 128 \, {\left (\frac {2 \, {\left (2 \, x + 3\right )}}{x^{2} + 4 \, x + 4} + \log \left (x + 2\right )\right )} \log \relax (3) + \frac {64 \, {\left (x + 1\right )} \log \relax (3)^{2}}{x^{2} + 4 \, x + 4} + 16 \, {\left (8 \, \log \relax (3)^{2} + 48 \, \log \relax (3) + 53\right )} \log \left (x + 2\right ) + 376 \, x + \frac {160 \, {\left (x + 1\right )} \log \relax (x)}{x^{2} + 4 \, x + 4} + \frac {2 \, {\left (12 \, x^{6} - 16 \, x^{5} {\left (2 \, \log \relax (3) - 1\right )} + {\left (24 \, \log \relax (3)^{2} - 32 \, \log \relax (3) + 43\right )} x^{4} - 8 \, x^{3} {\left (28 \, \log \relax (3) + 25\right )} - 12 \, {\left (24 \, \log \relax (3)^{2} + 128 \, \log \relax (3) + 147\right )} x^{2} - 48 \, {\left (8 \, \log \relax (3)^{2} + 32 \, \log \relax (3) + 41\right )} x - 6 \, {\left (8 \, x^{6} - 16 \, x^{5} {\left (\log \relax (3) - 1\right )} + {\left (8 \, \log \relax (3)^{2} - 32 \, \log \relax (3) + 9\right )} x^{4} + 2 \, {\left (8 \, \log \relax (3)^{2} - 8 \, \log \relax (3) + 3\right )} x^{3}\right )} \log \relax (x) + 768 \, \log \relax (3) + 576\right )}}{3 \, {\left (x^{2} + 4 \, x + 4\right )}} - \frac {1024 \, {\left (6 \, x + 11\right )}}{x^{2} + 4 \, x + 4} + \frac {2048 \, {\left (5 \, x + 9\right )}}{x^{2} + 4 \, x + 4} - \frac {1312 \, {\left (4 \, x + 7\right )}}{x^{2} + 4 \, x + 4} + \frac {384 \, {\left (3 \, x + 5\right )}}{x^{2} + 4 \, x + 4} - \frac {200 \, {\left (2 \, x + 3\right )}}{x^{2} + 4 \, x + 4} - \frac {8 \, {\left (x + 3\right )}}{x^{2} + 4 \, x + 4} + \frac {152 \, {\left (x + 1\right )}}{x^{2} + 4 \, x + 4} + \frac {32 \, \log \relax (x)}{x^{2} + 4 \, x + 4} + \frac {56}{x^{2} + 4 \, x + 4} + \frac {64}{x + 2} - 848 \, \log \left (x + 2\right ) - 52 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-64*x^5-320*x^4-384*x^3-128*x^2)*log(3)^2+(192*x^6+896*x^5+1088*x^4+384*x^3)*log(3)-128*x^7-576*x
^6-712*x^5-312*x^4-144*x^3-160*x^2-64*x)*log(x)+(-32*x^5-128*x^4-160*x^3-64*x^2)*log(3)^2+(64*x^6+256*x^5+320*
x^4+128*x^3)*log(3)-32*x^7-128*x^6-164*x^5-96*x^4-100*x^3-152*x^2-112*x-32)/(x^4+6*x^3+12*x^2+8*x),x, algorith
m="maxima")

[Out]

-8*x^4 + 64/3*x^3 - 16*(x^2 - 12*x + 16*(4*x + 7)/(x^2 + 4*x + 4) + 48*log(x + 2))*log(3)^2 - 128*(x - 4*(3*x
+ 5)/(x^2 + 4*x + 4) - 6*log(x + 2))*log(3)^2 + 32*(4*(x + 1)*log(x)/(x^2 + 4*x + 4) + 2/(x + 2) + log(x + 2)
- log(x))*log(3)^2 - 160*(2*(2*x + 3)/(x^2 + 4*x + 4) + log(x + 2))*log(3)^2 - 82*x^2 + 64/3*(x^3 - 9*x^2 + 72
*x - 48*(5*x + 9)/(x^2 + 4*x + 4) - 240*log(x + 2))*log(3) + 128*(x^2 - 12*x + 16*(4*x + 7)/(x^2 + 4*x + 4) +
48*log(x + 2))*log(3) + 320*(x - 4*(3*x + 5)/(x^2 + 4*x + 4) - 6*log(x + 2))*log(3) + 128*(2*(2*x + 3)/(x^2 +
4*x + 4) + log(x + 2))*log(3) + 64*(x + 1)*log(3)^2/(x^2 + 4*x + 4) + 16*(8*log(3)^2 + 48*log(3) + 53)*log(x +
2) + 376*x + 160*(x + 1)*log(x)/(x^2 + 4*x + 4) + 2/3*(12*x^6 - 16*x^5*(2*log(3) - 1) + (24*log(3)^2 - 32*log
(3) + 43)*x^4 - 8*x^3*(28*log(3) + 25) - 12*(24*log(3)^2 + 128*log(3) + 147)*x^2 - 48*(8*log(3)^2 + 32*log(3)
+ 41)*x - 6*(8*x^6 - 16*x^5*(log(3) - 1) + (8*log(3)^2 - 32*log(3) + 9)*x^4 + 2*(8*log(3)^2 - 8*log(3) + 3)*x^
3)*log(x) + 768*log(3) + 576)/(x^2 + 4*x + 4) - 1024*(6*x + 11)/(x^2 + 4*x + 4) + 2048*(5*x + 9)/(x^2 + 4*x +
4) - 1312*(4*x + 7)/(x^2 + 4*x + 4) + 384*(3*x + 5)/(x^2 + 4*x + 4) - 200*(2*x + 3)/(x^2 + 4*x + 4) - 8*(x + 3
)/(x^2 + 4*x + 4) + 152*(x + 1)/(x^2 + 4*x + 4) + 32*log(x)/(x^2 + 4*x + 4) + 56/(x^2 + 4*x + 4) + 64/(x + 2)
- 848*log(x + 2) - 52*log(x)

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mupad [B]  time = 3.94, size = 43, normalized size = 1.39 \begin {gather*} -\frac {4\,\ln \relax (x)\,{\left (x+1\right )}^2\,\left (4\,x+8\,x^2\,{\ln \relax (3)}^2-16\,x^3\,\ln \relax (3)+x^2+8\,x^4+4\right )}{{\left (x+2\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(112*x - log(3)*(128*x^3 + 320*x^4 + 256*x^5 + 64*x^6) + log(x)*(64*x - log(3)*(384*x^3 + 1088*x^4 + 896*
x^5 + 192*x^6) + log(3)^2*(128*x^2 + 384*x^3 + 320*x^4 + 64*x^5) + 160*x^2 + 144*x^3 + 312*x^4 + 712*x^5 + 576
*x^6 + 128*x^7) + log(3)^2*(64*x^2 + 160*x^3 + 128*x^4 + 32*x^5) + 152*x^2 + 100*x^3 + 96*x^4 + 164*x^5 + 128*
x^6 + 32*x^7 + 32)/(8*x + 12*x^2 + 6*x^3 + x^4),x)

[Out]

-(4*log(x)*(x + 1)^2*(4*x + 8*x^2*log(3)^2 - 16*x^3*log(3) + x^2 + 8*x^4 + 4))/(x + 2)^2

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sympy [B]  time = 0.48, size = 151, normalized size = 4.87 \begin {gather*} \left (-900 - 768 \log {\relax (3 )} - 160 \log {\relax (3 )}^{2}\right ) \log {\relax (x )} + \frac {\left (- 32 x^{6} - 64 x^{5} + 64 x^{5} \log {\relax (3 )} - 32 x^{4} \log {\relax (3 )}^{2} - 36 x^{4} + 128 x^{4} \log {\relax (3 )} - 64 x^{3} \log {\relax (3 )}^{2} - 24 x^{3} + 64 x^{3} \log {\relax (3 )} + 128 x^{2} \log {\relax (3 )}^{2} + 768 x^{2} \log {\relax (3 )} + 848 x^{2} + 640 x \log {\relax (3 )}^{2} + 3072 x \log {\relax (3 )} + 3552 x + 640 \log {\relax (3 )}^{2} + 3072 \log {\relax (3 )} + 3584\right ) \log {\relax (x )}}{x^{2} + 4 x + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-64*x**5-320*x**4-384*x**3-128*x**2)*ln(3)**2+(192*x**6+896*x**5+1088*x**4+384*x**3)*ln(3)-128*x*
*7-576*x**6-712*x**5-312*x**4-144*x**3-160*x**2-64*x)*ln(x)+(-32*x**5-128*x**4-160*x**3-64*x**2)*ln(3)**2+(64*
x**6+256*x**5+320*x**4+128*x**3)*ln(3)-32*x**7-128*x**6-164*x**5-96*x**4-100*x**3-152*x**2-112*x-32)/(x**4+6*x
**3+12*x**2+8*x),x)

[Out]

(-900 - 768*log(3) - 160*log(3)**2)*log(x) + (-32*x**6 - 64*x**5 + 64*x**5*log(3) - 32*x**4*log(3)**2 - 36*x**
4 + 128*x**4*log(3) - 64*x**3*log(3)**2 - 24*x**3 + 64*x**3*log(3) + 128*x**2*log(3)**2 + 768*x**2*log(3) + 84
8*x**2 + 640*x*log(3)**2 + 3072*x*log(3) + 3552*x + 640*log(3)**2 + 3072*log(3) + 3584)*log(x)/(x**2 + 4*x + 4
)

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