Integrand size = 107, antiderivative size = 23 \[ \int \frac {-686 x-1064 x^2-672 x^3+128 x^4+\left (-686 x+56 x^2-32 x^3+128 x^4\right ) \log \left (\frac {2401 x^2+2352 x^3+2144 x^4+768 x^5+256 x^6}{2401-5488 x+4704 x^2-1792 x^3+256 x^4}\right )}{-343+28 x-16 x^2+64 x^3} \, dx=x^2 \log \left (\left (x+\frac {5 x^2}{\left (\frac {7}{4}-x\right )^2}\right )^2\right ) \] Output:
x^2*ln((x+5*x^2/(7/4-x)^2)^2)
Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {-686 x-1064 x^2-672 x^3+128 x^4+\left (-686 x+56 x^2-32 x^3+128 x^4\right ) \log \left (\frac {2401 x^2+2352 x^3+2144 x^4+768 x^5+256 x^6}{2401-5488 x+4704 x^2-1792 x^3+256 x^4}\right )}{-343+28 x-16 x^2+64 x^3} \, dx=x^2 \log \left (\frac {x^2 \left (49+24 x+16 x^2\right )^2}{(-7+4 x)^4}\right ) \] Input:
Integrate[(-686*x - 1064*x^2 - 672*x^3 + 128*x^4 + (-686*x + 56*x^2 - 32*x ^3 + 128*x^4)*Log[(2401*x^2 + 2352*x^3 + 2144*x^4 + 768*x^5 + 256*x^6)/(24 01 - 5488*x + 4704*x^2 - 1792*x^3 + 256*x^4)])/(-343 + 28*x - 16*x^2 + 64* x^3),x]
Output:
x^2*Log[(x^2*(49 + 24*x + 16*x^2)^2)/(-7 + 4*x)^4]
Time = 1.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {2463, 2009}
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.
\(\displaystyle \int \frac {128 x^4-672 x^3-1064 x^2+\left (128 x^4-32 x^3+56 x^2-686 x\right ) \log \left (\frac {256 x^6+768 x^5+2144 x^4+2352 x^3+2401 x^2}{256 x^4-1792 x^3+4704 x^2-5488 x+2401}\right )-686 x}{64 x^3-16 x^2+28 x-343} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {128 x^4-672 x^3-1064 x^2+\left (128 x^4-32 x^3+56 x^2-686 x\right ) \log \left (\frac {256 x^6+768 x^5+2144 x^4+2352 x^3+2401 x^2}{256 x^4-1792 x^3+4704 x^2-5488 x+2401}\right )-686 x}{140 (4 x-7)}+\frac {(-4 x-13) \left (128 x^4-672 x^3-1064 x^2+\left (128 x^4-32 x^3+56 x^2-686 x\right ) \log \left (\frac {256 x^6+768 x^5+2144 x^4+2352 x^3+2401 x^2}{256 x^4-1792 x^3+4704 x^2-5488 x+2401}\right )-686 x\right )}{140 \left (16 x^2+24 x+49\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^2 \log \left (\frac {x^2 \left (16 x^2+24 x+49\right )^2}{(7-4 x)^4}\right )\) |
Input:
Int[(-686*x - 1064*x^2 - 672*x^3 + 128*x^4 + (-686*x + 56*x^2 - 32*x^3 + 1 28*x^4)*Log[(2401*x^2 + 2352*x^3 + 2144*x^4 + 768*x^5 + 256*x^6)/(2401 - 5 488*x + 4704*x^2 - 1792*x^3 + 256*x^4)])/(-343 + 28*x - 16*x^2 + 64*x^3),x ]
Output:
x^2*Log[(x^2*(49 + 24*x + 16*x^2)^2)/(7 - 4*x)^4]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(21)=42\).
Time = 0.49 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.26
method | result | size |
default | \(\ln \left (\frac {x^{2} \left (256 x^{4}+768 x^{3}+2144 x^{2}+2352 x +2401\right )}{256 x^{4}-1792 x^{3}+4704 x^{2}-5488 x +2401}\right ) x^{2}\) | \(52\) |
parallelrisch | \(\ln \left (\frac {x^{2} \left (256 x^{4}+768 x^{3}+2144 x^{2}+2352 x +2401\right )}{256 x^{4}-1792 x^{3}+4704 x^{2}-5488 x +2401}\right ) x^{2}\) | \(52\) |
parts | \(\ln \left (\frac {x^{2} \left (256 x^{4}+768 x^{3}+2144 x^{2}+2352 x +2401\right )}{256 x^{4}-1792 x^{3}+4704 x^{2}-5488 x +2401}\right ) x^{2}\) | \(52\) |
norman | \(x^{2} \ln \left (\frac {256 x^{6}+768 x^{5}+2144 x^{4}+2352 x^{3}+2401 x^{2}}{256 x^{4}-1792 x^{3}+4704 x^{2}-5488 x +2401}\right )\) | \(55\) |
risch | \(x^{2} \ln \left (\frac {256 x^{6}+768 x^{5}+2144 x^{4}+2352 x^{3}+2401 x^{2}}{256 x^{4}-1792 x^{3}+4704 x^{2}-5488 x +2401}\right )\) | \(55\) |
orering | \(\frac {\left (196608 x^{8}-753664 x^{7}+401408 x^{6}-2709504 x^{5}+9103360 x^{4}+5412736 x^{3}+7211232 x^{2}-2554664 x +15647317\right ) \left (\left (128 x^{4}-32 x^{3}+56 x^{2}-686 x \right ) \ln \left (\frac {256 x^{6}+768 x^{5}+2144 x^{4}+2352 x^{3}+2401 x^{2}}{256 x^{4}-1792 x^{3}+4704 x^{2}-5488 x +2401}\right )+128 x^{4}-672 x^{3}-1064 x^{2}-686 x \right )}{64 x \left (4096 x^{6}-12288 x^{5}+8960 x^{4}-53760 x^{3}+215600 x^{2}+268912 x +117649\right ) \left (64 x^{3}-16 x^{2}+28 x -343\right )}-\frac {\left (16 x^{2}+24 x +49\right ) \left (-7+4 x \right ) \left (1024 x^{5}-5376 x^{4}-7616 x^{2}+3724 x -45619\right ) \left (\frac {\left (512 x^{3}-96 x^{2}+112 x -686\right ) \ln \left (\frac {256 x^{6}+768 x^{5}+2144 x^{4}+2352 x^{3}+2401 x^{2}}{256 x^{4}-1792 x^{3}+4704 x^{2}-5488 x +2401}\right )+\frac {\left (128 x^{4}-32 x^{3}+56 x^{2}-686 x \right ) \left (\frac {1536 x^{5}+3840 x^{4}+8576 x^{3}+7056 x^{2}+4802 x}{256 x^{4}-1792 x^{3}+4704 x^{2}-5488 x +2401}-\frac {\left (256 x^{6}+768 x^{5}+2144 x^{4}+2352 x^{3}+2401 x^{2}\right ) \left (1024 x^{3}-5376 x^{2}+9408 x -5488\right )}{\left (256 x^{4}-1792 x^{3}+4704 x^{2}-5488 x +2401\right )^{2}}\right ) \left (256 x^{4}-1792 x^{3}+4704 x^{2}-5488 x +2401\right )}{256 x^{6}+768 x^{5}+2144 x^{4}+2352 x^{3}+2401 x^{2}}+512 x^{3}-2016 x^{2}-2128 x -686}{64 x^{3}-16 x^{2}+28 x -343}-\frac {\left (\left (128 x^{4}-32 x^{3}+56 x^{2}-686 x \right ) \ln \left (\frac {256 x^{6}+768 x^{5}+2144 x^{4}+2352 x^{3}+2401 x^{2}}{256 x^{4}-1792 x^{3}+4704 x^{2}-5488 x +2401}\right )+128 x^{4}-672 x^{3}-1064 x^{2}-686 x \right ) \left (192 x^{2}-32 x +28\right )}{\left (64 x^{3}-16 x^{2}+28 x -343\right )^{2}}\right )}{64 \left (4096 x^{6}-12288 x^{5}+8960 x^{4}-53760 x^{3}+215600 x^{2}+268912 x +117649\right )}\) | \(653\) |
Input:
int(((128*x^4-32*x^3+56*x^2-686*x)*ln((256*x^6+768*x^5+2144*x^4+2352*x^3+2 401*x^2)/(256*x^4-1792*x^3+4704*x^2-5488*x+2401))+128*x^4-672*x^3-1064*x^2 -686*x)/(64*x^3-16*x^2+28*x-343),x,method=_RETURNVERBOSE)
Output:
ln(x^2*(256*x^4+768*x^3+2144*x^2+2352*x+2401)/(256*x^4-1792*x^3+4704*x^2-5 488*x+2401))*x^2
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (21) = 42\).
Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.35 \[ \int \frac {-686 x-1064 x^2-672 x^3+128 x^4+\left (-686 x+56 x^2-32 x^3+128 x^4\right ) \log \left (\frac {2401 x^2+2352 x^3+2144 x^4+768 x^5+256 x^6}{2401-5488 x+4704 x^2-1792 x^3+256 x^4}\right )}{-343+28 x-16 x^2+64 x^3} \, dx=x^{2} \log \left (\frac {256 \, x^{6} + 768 \, x^{5} + 2144 \, x^{4} + 2352 \, x^{3} + 2401 \, x^{2}}{256 \, x^{4} - 1792 \, x^{3} + 4704 \, x^{2} - 5488 \, x + 2401}\right ) \] Input:
integrate(((128*x^4-32*x^3+56*x^2-686*x)*log((256*x^6+768*x^5+2144*x^4+235 2*x^3+2401*x^2)/(256*x^4-1792*x^3+4704*x^2-5488*x+2401))+128*x^4-672*x^3-1 064*x^2-686*x)/(64*x^3-16*x^2+28*x-343),x, algorithm="fricas")
Output:
x^2*log((256*x^6 + 768*x^5 + 2144*x^4 + 2352*x^3 + 2401*x^2)/(256*x^4 - 17 92*x^3 + 4704*x^2 - 5488*x + 2401))
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (19) = 38\).
Time = 0.13 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13 \[ \int \frac {-686 x-1064 x^2-672 x^3+128 x^4+\left (-686 x+56 x^2-32 x^3+128 x^4\right ) \log \left (\frac {2401 x^2+2352 x^3+2144 x^4+768 x^5+256 x^6}{2401-5488 x+4704 x^2-1792 x^3+256 x^4}\right )}{-343+28 x-16 x^2+64 x^3} \, dx=x^{2} \log {\left (\frac {256 x^{6} + 768 x^{5} + 2144 x^{4} + 2352 x^{3} + 2401 x^{2}}{256 x^{4} - 1792 x^{3} + 4704 x^{2} - 5488 x + 2401} \right )} \] Input:
integrate(((128*x**4-32*x**3+56*x**2-686*x)*ln((256*x**6+768*x**5+2144*x** 4+2352*x**3+2401*x**2)/(256*x**4-1792*x**3+4704*x**2-5488*x+2401))+128*x** 4-672*x**3-1064*x**2-686*x)/(64*x**3-16*x**2+28*x-343),x)
Output:
x**2*log((256*x**6 + 768*x**5 + 2144*x**4 + 2352*x**3 + 2401*x**2)/(256*x* *4 - 1792*x**3 + 4704*x**2 - 5488*x + 2401))
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (21) = 42\).
Time = 0.75 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.78 \[ \int \frac {-686 x-1064 x^2-672 x^3+128 x^4+\left (-686 x+56 x^2-32 x^3+128 x^4\right ) \log \left (\frac {2401 x^2+2352 x^3+2144 x^4+768 x^5+256 x^6}{2401-5488 x+4704 x^2-1792 x^3+256 x^4}\right )}{-343+28 x-16 x^2+64 x^3} \, dx=2 \, x^{2} \log \left (x\right ) + \frac {1}{8} \, {\left (16 \, x^{2} + 31\right )} \log \left (16 \, x^{2} + 24 \, x + 49\right ) - \frac {1}{4} \, {\left (16 \, x^{2} - 49\right )} \log \left (4 \, x - 7\right ) - \frac {31}{8} \, \log \left (16 \, x^{2} + 24 \, x + 49\right ) - \frac {49}{4} \, \log \left (4 \, x - 7\right ) \] Input:
integrate(((128*x^4-32*x^3+56*x^2-686*x)*log((256*x^6+768*x^5+2144*x^4+235 2*x^3+2401*x^2)/(256*x^4-1792*x^3+4704*x^2-5488*x+2401))+128*x^4-672*x^3-1 064*x^2-686*x)/(64*x^3-16*x^2+28*x-343),x, algorithm="maxima")
Output:
2*x^2*log(x) + 1/8*(16*x^2 + 31)*log(16*x^2 + 24*x + 49) - 1/4*(16*x^2 - 4 9)*log(4*x - 7) - 31/8*log(16*x^2 + 24*x + 49) - 49/4*log(4*x - 7)
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (21) = 42\).
Time = 0.18 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.35 \[ \int \frac {-686 x-1064 x^2-672 x^3+128 x^4+\left (-686 x+56 x^2-32 x^3+128 x^4\right ) \log \left (\frac {2401 x^2+2352 x^3+2144 x^4+768 x^5+256 x^6}{2401-5488 x+4704 x^2-1792 x^3+256 x^4}\right )}{-343+28 x-16 x^2+64 x^3} \, dx=x^{2} \log \left (\frac {256 \, x^{6} + 768 \, x^{5} + 2144 \, x^{4} + 2352 \, x^{3} + 2401 \, x^{2}}{256 \, x^{4} - 1792 \, x^{3} + 4704 \, x^{2} - 5488 \, x + 2401}\right ) \] Input:
integrate(((128*x^4-32*x^3+56*x^2-686*x)*log((256*x^6+768*x^5+2144*x^4+235 2*x^3+2401*x^2)/(256*x^4-1792*x^3+4704*x^2-5488*x+2401))+128*x^4-672*x^3-1 064*x^2-686*x)/(64*x^3-16*x^2+28*x-343),x, algorithm="giac")
Output:
x^2*log((256*x^6 + 768*x^5 + 2144*x^4 + 2352*x^3 + 2401*x^2)/(256*x^4 - 17 92*x^3 + 4704*x^2 - 5488*x + 2401))
Time = 3.32 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39 \[ \int \frac {-686 x-1064 x^2-672 x^3+128 x^4+\left (-686 x+56 x^2-32 x^3+128 x^4\right ) \log \left (\frac {2401 x^2+2352 x^3+2144 x^4+768 x^5+256 x^6}{2401-5488 x+4704 x^2-1792 x^3+256 x^4}\right )}{-343+28 x-16 x^2+64 x^3} \, dx=x^2\,\left (\ln \left (\frac {1}{256\,x^4-1792\,x^3+4704\,x^2-5488\,x+2401}\right )+\ln \left (256\,x^6+768\,x^5+2144\,x^4+2352\,x^3+2401\,x^2\right )\right ) \] Input:
int(-(686*x + log((2401*x^2 + 2352*x^3 + 2144*x^4 + 768*x^5 + 256*x^6)/(47 04*x^2 - 5488*x - 1792*x^3 + 256*x^4 + 2401))*(686*x - 56*x^2 + 32*x^3 - 1 28*x^4) + 1064*x^2 + 672*x^3 - 128*x^4)/(28*x - 16*x^2 + 64*x^3 - 343),x)
Output:
x^2*(log(1/(4704*x^2 - 5488*x - 1792*x^3 + 256*x^4 + 2401)) + log(2401*x^2 + 2352*x^3 + 2144*x^4 + 768*x^5 + 256*x^6))
Time = 0.15 (sec) , antiderivative size = 132, normalized size of antiderivative = 5.74 \[ \int \frac {-686 x-1064 x^2-672 x^3+128 x^4+\left (-686 x+56 x^2-32 x^3+128 x^4\right ) \log \left (\frac {2401 x^2+2352 x^3+2144 x^4+768 x^5+256 x^6}{2401-5488 x+4704 x^2-1792 x^3+256 x^4}\right )}{-343+28 x-16 x^2+64 x^3} \, dx=-\frac {31 \,\mathrm {log}\left (16 x^{2}+24 x +49\right )}{8}+\frac {31 \,\mathrm {log}\left (4 x -7\right )}{4}+\mathrm {log}\left (\frac {256 x^{6}+768 x^{5}+2144 x^{4}+2352 x^{3}+2401 x^{2}}{256 x^{4}-1792 x^{3}+4704 x^{2}-5488 x +2401}\right ) x^{2}+\frac {31 \,\mathrm {log}\left (\frac {256 x^{6}+768 x^{5}+2144 x^{4}+2352 x^{3}+2401 x^{2}}{256 x^{4}-1792 x^{3}+4704 x^{2}-5488 x +2401}\right )}{16}-\frac {31 \,\mathrm {log}\left (x \right )}{8} \] Input:
int(((128*x^4-32*x^3+56*x^2-686*x)*log((256*x^6+768*x^5+2144*x^4+2352*x^3+ 2401*x^2)/(256*x^4-1792*x^3+4704*x^2-5488*x+2401))+128*x^4-672*x^3-1064*x^ 2-686*x)/(64*x^3-16*x^2+28*x-343),x)
Output:
( - 62*log(16*x**2 + 24*x + 49) + 124*log(4*x - 7) + 16*log((256*x**6 + 76 8*x**5 + 2144*x**4 + 2352*x**3 + 2401*x**2)/(256*x**4 - 1792*x**3 + 4704*x **2 - 5488*x + 2401))*x**2 + 31*log((256*x**6 + 768*x**5 + 2144*x**4 + 235 2*x**3 + 2401*x**2)/(256*x**4 - 1792*x**3 + 4704*x**2 - 5488*x + 2401)) - 62*log(x))/16