Integrand size = 125, antiderivative size = 28 \[ \int \frac {288-96 x-568 x^2+208 x^3+304 x^4-96 x^5-24 x^6+\left (-72 x+24 x^2+142 x^3-52 x^4-76 x^5+24 x^6+6 x^7\right ) \log \left (\frac {9-3 x-8 x^2+3 x^3}{-3+3 x^2}\right )}{-9 x^3+3 x^4+17 x^5-6 x^6-8 x^7+3 x^8} \, dx=\left (\frac {4}{x}-\log \left (-3+x-\frac {x}{3 \left (\frac {1}{x}-x\right )}\right )\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(71\) vs. \(2(28)=56\).
Time = 0.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.54 \[ \int \frac {288-96 x-568 x^2+208 x^3+304 x^4-96 x^5-24 x^6+\left (-72 x+24 x^2+142 x^3-52 x^4-76 x^5+24 x^6+6 x^7\right ) \log \left (\frac {9-3 x-8 x^2+3 x^3}{-3+3 x^2}\right )}{-9 x^3+3 x^4+17 x^5-6 x^6-8 x^7+3 x^8} \, dx=2 \left (\frac {8}{x^2}-\frac {4 \log \left (\frac {9-3 x-8 x^2+3 x^3}{-3+3 x^2}\right )}{x}+\frac {1}{2} \log ^2\left (\frac {9-3 x-8 x^2+3 x^3}{-3+3 x^2}\right )\right ) \]
Integrate[(288 - 96*x - 568*x^2 + 208*x^3 + 304*x^4 - 96*x^5 - 24*x^6 + (- 72*x + 24*x^2 + 142*x^3 - 52*x^4 - 76*x^5 + 24*x^6 + 6*x^7)*Log[(9 - 3*x - 8*x^2 + 3*x^3)/(-3 + 3*x^2)])/(-9*x^3 + 3*x^4 + 17*x^5 - 6*x^6 - 8*x^7 + 3*x^8),x]
2*(8/x^2 - (4*Log[(9 - 3*x - 8*x^2 + 3*x^3)/(-3 + 3*x^2)])/x + Log[(9 - 3* x - 8*x^2 + 3*x^3)/(-3 + 3*x^2)]^2/2)
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {-24 x^6-96 x^5+304 x^4+208 x^3-568 x^2+\left (6 x^7+24 x^6-76 x^5-52 x^4+142 x^3+24 x^2-72 x\right ) \log \left (\frac {3 x^3-8 x^2-3 x+9}{3 x^2-3}\right )-96 x+288}{3 x^8-8 x^7-6 x^6+17 x^5+3 x^4-9 x^3} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-24 x^6-96 x^5+304 x^4+208 x^3-568 x^2+\left (6 x^7+24 x^6-76 x^5-52 x^4+142 x^3+24 x^2-72 x\right ) \log \left (\frac {3 x^3-8 x^2-3 x+9}{3 x^2-3}\right )-96 x+288}{x^3 \left (3 x^5-8 x^4-6 x^3+17 x^2+3 x-9\right )}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {-24 x^6-96 x^5+304 x^4+208 x^3-568 x^2+\left (6 x^7+24 x^6-76 x^5-52 x^4+142 x^3+24 x^2-72 x\right ) \log \left (\frac {3 x^3-8 x^2-3 x+9}{3 x^2-3}\right )-96 x+288}{x^3 \left (x^2-1\right )}+\frac {(8-3 x) \left (-24 x^6-96 x^5+304 x^4+208 x^3-568 x^2+\left (6 x^7+24 x^6-76 x^5-52 x^4+142 x^3+24 x^2-72 x\right ) \log \left (\frac {3 x^3-8 x^2-3 x+9}{3 x^2-3}\right )-96 x+288\right )}{x^3 \left (3 x^3-8 x^2-3 x+9\right )}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 \left (-3 x^6-12 x^5+38 x^4+26 x^3-71 x^2-12 x+36\right ) \left (x \log \left (\frac {3 x^3-8 x^2-3 x+9}{3 x^2-3}\right )-4\right )}{x^3 \left (1-x^2\right ) \left (3 x^3-8 x^2-3 x+9\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int -\frac {\left (-3 x^6-12 x^5+38 x^4+26 x^3-71 x^2-12 x+36\right ) \left (4-x \log \left (-\frac {3 x^3-8 x^2-3 x+9}{3 \left (1-x^2\right )}\right )\right )}{x^3 \left (1-x^2\right ) \left (3 x^3-8 x^2-3 x+9\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {\left (-3 x^6-12 x^5+38 x^4+26 x^3-71 x^2-12 x+36\right ) \left (4-x \log \left (-\frac {3 x^3-8 x^2-3 x+9}{3 \left (1-x^2\right )}\right )\right )}{x^3 \left (1-x^2\right ) \left (3 x^3-8 x^2-3 x+9\right )}dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -2 \int \left (\frac {4 \left (3 x^6+12 x^5-38 x^4-26 x^3+71 x^2+12 x-36\right )}{(x-1) x^3 (x+1) \left (3 x^3-8 x^2-3 x+9\right )}-\frac {\left (3 x^6+12 x^5-38 x^4-26 x^3+71 x^2+12 x-36\right ) \log \left (\frac {3 x^3-8 x^2-3 x+9}{3 \left (x^2-1\right )}\right )}{(x-1) x^2 (x+1) \left (3 x^3-8 x^2-3 x+9\right )}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle -2 \int \left (\frac {4 \left (3 x^6+12 x^5-38 x^4-26 x^3+71 x^2+12 x-36\right )}{(x-1) x^3 (x+1) \left (3 x^3-8 x^2-3 x+9\right )}-\frac {\left (3 x^6+12 x^5-38 x^4-26 x^3+71 x^2+12 x-36\right ) \log \left (\frac {3 x^3-8 x^2-3 x+9}{3 \left (x^2-1\right )}\right )}{(x-1) x^2 (x+1) \left (3 x^3-8 x^2-3 x+9\right )}\right )dx\) |
Int[(288 - 96*x - 568*x^2 + 208*x^3 + 304*x^4 - 96*x^5 - 24*x^6 + (-72*x + 24*x^2 + 142*x^3 - 52*x^4 - 76*x^5 + 24*x^6 + 6*x^7)*Log[(9 - 3*x - 8*x^2 + 3*x^3)/(-3 + 3*x^2)])/(-9*x^3 + 3*x^4 + 17*x^5 - 6*x^6 - 8*x^7 + 3*x^8) ,x]
3.1.94.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs. \(2(28)=56\).
Time = 0.95 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.43
method | result | size |
norman | \(\frac {16+x^{2} \ln \left (\frac {3 x^{3}-8 x^{2}-3 x +9}{3 x^{2}-3}\right )^{2}-8 x \ln \left (\frac {3 x^{3}-8 x^{2}-3 x +9}{3 x^{2}-3}\right )}{x^{2}}\) | \(68\) |
parallelrisch | \(-\frac {-144-9 x^{2} \ln \left (\frac {3 x^{3}-8 x^{2}-3 x +9}{3 x^{2}-3}\right )^{2}+72 x \ln \left (\frac {3 x^{3}-8 x^{2}-3 x +9}{3 x^{2}-3}\right )}{9 x^{2}}\) | \(68\) |
default | \(\text {Expression too large to display}\) | \(1072\) |
parts | \(\text {Expression too large to display}\) | \(1072\) |
risch | \(\text {Expression too large to display}\) | \(6674\) |
int(((6*x^7+24*x^6-76*x^5-52*x^4+142*x^3+24*x^2-72*x)*ln((3*x^3-8*x^2-3*x+ 9)/(3*x^2-3))-24*x^6-96*x^5+304*x^4+208*x^3-568*x^2-96*x+288)/(3*x^8-8*x^7 -6*x^6+17*x^5+3*x^4-9*x^3),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.32 \[ \int \frac {288-96 x-568 x^2+208 x^3+304 x^4-96 x^5-24 x^6+\left (-72 x+24 x^2+142 x^3-52 x^4-76 x^5+24 x^6+6 x^7\right ) \log \left (\frac {9-3 x-8 x^2+3 x^3}{-3+3 x^2}\right )}{-9 x^3+3 x^4+17 x^5-6 x^6-8 x^7+3 x^8} \, dx=\frac {x^{2} \log \left (\frac {3 \, x^{3} - 8 \, x^{2} - 3 \, x + 9}{3 \, {\left (x^{2} - 1\right )}}\right )^{2} - 8 \, x \log \left (\frac {3 \, x^{3} - 8 \, x^{2} - 3 \, x + 9}{3 \, {\left (x^{2} - 1\right )}}\right ) + 16}{x^{2}} \]
integrate(((6*x^7+24*x^6-76*x^5-52*x^4+142*x^3+24*x^2-72*x)*log((3*x^3-8*x ^2-3*x+9)/(3*x^2-3))-24*x^6-96*x^5+304*x^4+208*x^3-568*x^2-96*x+288)/(3*x^ 8-8*x^7-6*x^6+17*x^5+3*x^4-9*x^3),x, algorithm=\
(x^2*log(1/3*(3*x^3 - 8*x^2 - 3*x + 9)/(x^2 - 1))^2 - 8*x*log(1/3*(3*x^3 - 8*x^2 - 3*x + 9)/(x^2 - 1)) + 16)/x^2
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (17) = 34\).
Time = 0.15 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00 \[ \int \frac {288-96 x-568 x^2+208 x^3+304 x^4-96 x^5-24 x^6+\left (-72 x+24 x^2+142 x^3-52 x^4-76 x^5+24 x^6+6 x^7\right ) \log \left (\frac {9-3 x-8 x^2+3 x^3}{-3+3 x^2}\right )}{-9 x^3+3 x^4+17 x^5-6 x^6-8 x^7+3 x^8} \, dx=\log {\left (\frac {3 x^{3} - 8 x^{2} - 3 x + 9}{3 x^{2} - 3} \right )}^{2} - \frac {8 \log {\left (\frac {3 x^{3} - 8 x^{2} - 3 x + 9}{3 x^{2} - 3} \right )}}{x} + \frac {16}{x^{2}} \]
integrate(((6*x**7+24*x**6-76*x**5-52*x**4+142*x**3+24*x**2-72*x)*ln((3*x* *3-8*x**2-3*x+9)/(3*x**2-3))-24*x**6-96*x**5+304*x**4+208*x**3-568*x**2-96 *x+288)/(3*x**8-8*x**7-6*x**6+17*x**5+3*x**4-9*x**3),x)
log((3*x**3 - 8*x**2 - 3*x + 9)/(3*x**2 - 3))**2 - 8*log((3*x**3 - 8*x**2 - 3*x + 9)/(3*x**2 - 3))/x + 16/x**2
Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (26) = 52\).
Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 4.89 \[ \int \frac {288-96 x-568 x^2+208 x^3+304 x^4-96 x^5-24 x^6+\left (-72 x+24 x^2+142 x^3-52 x^4-76 x^5+24 x^6+6 x^7\right ) \log \left (\frac {9-3 x-8 x^2+3 x^3}{-3+3 x^2}\right )}{-9 x^3+3 x^4+17 x^5-6 x^6-8 x^7+3 x^8} \, dx=\frac {x^{2} \log \left (3 \, x^{3} - 8 \, x^{2} - 3 \, x + 9\right )^{2} + x^{2} \log \left (x + 1\right )^{2} + x^{2} \log \left (x - 1\right )^{2} + 8 \, x \log \left (3\right ) - 2 \, {\left (x^{2} \log \left (3\right ) + x^{2} \log \left (x + 1\right ) + x^{2} \log \left (x - 1\right ) + 4 \, x\right )} \log \left (3 \, x^{3} - 8 \, x^{2} - 3 \, x + 9\right ) + 2 \, {\left (x^{2} \log \left (3\right ) + x^{2} \log \left (x - 1\right ) + 4 \, x\right )} \log \left (x + 1\right ) + 2 \, {\left (x^{2} \log \left (3\right ) + 4 \, x\right )} \log \left (x - 1\right ) + 16}{x^{2}} \]
integrate(((6*x^7+24*x^6-76*x^5-52*x^4+142*x^3+24*x^2-72*x)*log((3*x^3-8*x ^2-3*x+9)/(3*x^2-3))-24*x^6-96*x^5+304*x^4+208*x^3-568*x^2-96*x+288)/(3*x^ 8-8*x^7-6*x^6+17*x^5+3*x^4-9*x^3),x, algorithm=\
(x^2*log(3*x^3 - 8*x^2 - 3*x + 9)^2 + x^2*log(x + 1)^2 + x^2*log(x - 1)^2 + 8*x*log(3) - 2*(x^2*log(3) + x^2*log(x + 1) + x^2*log(x - 1) + 4*x)*log( 3*x^3 - 8*x^2 - 3*x + 9) + 2*(x^2*log(3) + x^2*log(x - 1) + 4*x)*log(x + 1 ) + 2*(x^2*log(3) + 4*x)*log(x - 1) + 16)/x^2
\[ \int \frac {288-96 x-568 x^2+208 x^3+304 x^4-96 x^5-24 x^6+\left (-72 x+24 x^2+142 x^3-52 x^4-76 x^5+24 x^6+6 x^7\right ) \log \left (\frac {9-3 x-8 x^2+3 x^3}{-3+3 x^2}\right )}{-9 x^3+3 x^4+17 x^5-6 x^6-8 x^7+3 x^8} \, dx=\int { -\frac {2 \, {\left (12 \, x^{6} + 48 \, x^{5} - 152 \, x^{4} - 104 \, x^{3} + 284 \, x^{2} - {\left (3 \, x^{7} + 12 \, x^{6} - 38 \, x^{5} - 26 \, x^{4} + 71 \, x^{3} + 12 \, x^{2} - 36 \, x\right )} \log \left (\frac {3 \, x^{3} - 8 \, x^{2} - 3 \, x + 9}{3 \, {\left (x^{2} - 1\right )}}\right ) + 48 \, x - 144\right )}}{3 \, x^{8} - 8 \, x^{7} - 6 \, x^{6} + 17 \, x^{5} + 3 \, x^{4} - 9 \, x^{3}} \,d x } \]
integrate(((6*x^7+24*x^6-76*x^5-52*x^4+142*x^3+24*x^2-72*x)*log((3*x^3-8*x ^2-3*x+9)/(3*x^2-3))-24*x^6-96*x^5+304*x^4+208*x^3-568*x^2-96*x+288)/(3*x^ 8-8*x^7-6*x^6+17*x^5+3*x^4-9*x^3),x, algorithm=\
integrate(-2*(12*x^6 + 48*x^5 - 152*x^4 - 104*x^3 + 284*x^2 - (3*x^7 + 12* x^6 - 38*x^5 - 26*x^4 + 71*x^3 + 12*x^2 - 36*x)*log(1/3*(3*x^3 - 8*x^2 - 3 *x + 9)/(x^2 - 1)) + 48*x - 144)/(3*x^8 - 8*x^7 - 6*x^6 + 17*x^5 + 3*x^4 - 9*x^3), x)
Time = 9.88 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {288-96 x-568 x^2+208 x^3+304 x^4-96 x^5-24 x^6+\left (-72 x+24 x^2+142 x^3-52 x^4-76 x^5+24 x^6+6 x^7\right ) \log \left (\frac {9-3 x-8 x^2+3 x^3}{-3+3 x^2}\right )}{-9 x^3+3 x^4+17 x^5-6 x^6-8 x^7+3 x^8} \, dx=\frac {{\left (x\,\ln \left (-\frac {-3\,x^3+8\,x^2+3\,x-9}{3\,x^2-3}\right )-4\right )}^2}{x^2} \]