3.32.98 \(\int \frac {256 x-192 x^2+32 x^3-8 x^5+22 x^6-10 x^7+(4 x^6-2 x^7) \log (-2+x)+(-4 x^6+2 x^7) \log (x^2)+(256-192 x+32 x^2-8 x^4+22 x^5-10 x^6+(4 x^5-2 x^6) \log (-2+x)+(-4 x^5+2 x^6) \log (x^2)) \log (\frac {16-5 x^4-x^4 \log (-2+x)+x^4 \log (x^2)}{x^4})}{-32 x+16 x^2+10 x^5-5 x^6+(2 x^5-x^6) \log (-2+x)+(-2 x^5+x^6) \log (x^2)} \, dx\)

Optimal. Leaf size=22 \[ \left (x+\log \left (-5+\frac {16}{x^4}-\log (-2+x)+\log \left (x^2\right )\right )\right )^2 \]

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Rubi [A]  time = 0.64, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 200, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {6688, 12, 6686} \begin {gather*} \left (\log \left (\frac {16}{x^4}+\log \left (x^2\right )-\log (x-2)-5\right )+x\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(256*x - 192*x^2 + 32*x^3 - 8*x^5 + 22*x^6 - 10*x^7 + (4*x^6 - 2*x^7)*Log[-2 + x] + (-4*x^6 + 2*x^7)*Log[x
^2] + (256 - 192*x + 32*x^2 - 8*x^4 + 22*x^5 - 10*x^6 + (4*x^5 - 2*x^6)*Log[-2 + x] + (-4*x^5 + 2*x^6)*Log[x^2
])*Log[(16 - 5*x^4 - x^4*Log[-2 + x] + x^4*Log[x^2])/x^4])/(-32*x + 16*x^2 + 10*x^5 - 5*x^6 + (2*x^5 - x^6)*Lo
g[-2 + x] + (-2*x^5 + x^6)*Log[x^2]),x]

[Out]

(x + Log[-5 + 16/x^4 - Log[-2 + x] + Log[x^2]])^2

Rule 12

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

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (128-96 x+16 x^2-4 x^4+11 x^5-5 x^6-(-2+x) x^5 \log (-2+x)+(-2+x) x^5 \log \left (x^2\right )\right ) \left (-x-\log \left (-5+\frac {16}{x^4}-\log (-2+x)+\log \left (x^2\right )\right )\right )}{(2-x) x \left (16-5 x^4-x^4 \log (-2+x)+x^4 \log \left (x^2\right )\right )} \, dx\\ &=2 \int \frac {\left (128-96 x+16 x^2-4 x^4+11 x^5-5 x^6-(-2+x) x^5 \log (-2+x)+(-2+x) x^5 \log \left (x^2\right )\right ) \left (-x-\log \left (-5+\frac {16}{x^4}-\log (-2+x)+\log \left (x^2\right )\right )\right )}{(2-x) x \left (16-5 x^4-x^4 \log (-2+x)+x^4 \log \left (x^2\right )\right )} \, dx\\ &=\left (x+\log \left (-5+\frac {16}{x^4}-\log (-2+x)+\log \left (x^2\right )\right )\right )^2\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.12, size = 54, normalized size = 2.45 \begin {gather*} 2 \left (\frac {x^2}{2}+x \log \left (-5+\frac {16}{x^4}-\log (-2+x)+\log \left (x^2\right )\right )+\frac {1}{2} \log ^2\left (-5+\frac {16}{x^4}-\log (-2+x)+\log \left (x^2\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(256*x - 192*x^2 + 32*x^3 - 8*x^5 + 22*x^6 - 10*x^7 + (4*x^6 - 2*x^7)*Log[-2 + x] + (-4*x^6 + 2*x^7)
*Log[x^2] + (256 - 192*x + 32*x^2 - 8*x^4 + 22*x^5 - 10*x^6 + (4*x^5 - 2*x^6)*Log[-2 + x] + (-4*x^5 + 2*x^6)*L
og[x^2])*Log[(16 - 5*x^4 - x^4*Log[-2 + x] + x^4*Log[x^2])/x^4])/(-32*x + 16*x^2 + 10*x^5 - 5*x^6 + (2*x^5 - x
^6)*Log[-2 + x] + (-2*x^5 + x^6)*Log[x^2]),x]

[Out]

2*(x^2/2 + x*Log[-5 + 16/x^4 - Log[-2 + x] + Log[x^2]] + Log[-5 + 16/x^4 - Log[-2 + x] + Log[x^2]]^2/2)

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fricas [B]  time = 0.71, size = 67, normalized size = 3.05 \begin {gather*} x^{2} + 2 \, x \log \left (\frac {x^{4} \log \left (x^{2}\right ) - x^{4} \log \left (x - 2\right ) - 5 \, x^{4} + 16}{x^{4}}\right ) + \log \left (\frac {x^{4} \log \left (x^{2}\right ) - x^{4} \log \left (x - 2\right ) - 5 \, x^{4} + 16}{x^{4}}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^6-4*x^5)*log(x^2)+(-2*x^6+4*x^5)*log(x-2)-10*x^6+22*x^5-8*x^4+32*x^2-192*x+256)*log((x^4*log(
x^2)-x^4*log(x-2)-5*x^4+16)/x^4)+(2*x^7-4*x^6)*log(x^2)+(-2*x^7+4*x^6)*log(x-2)-10*x^7+22*x^6-8*x^5+32*x^3-192
*x^2+256*x)/((x^6-2*x^5)*log(x^2)+(-x^6+2*x^5)*log(x-2)-5*x^6+10*x^5+16*x^2-32*x),x, algorithm="fricas")

[Out]

x^2 + 2*x*log((x^4*log(x^2) - x^4*log(x - 2) - 5*x^4 + 16)/x^4) + log((x^4*log(x^2) - x^4*log(x - 2) - 5*x^4 +
16)/x^4)^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (5 \, x^{7} - 11 \, x^{6} + 4 \, x^{5} - 16 \, x^{3} + 96 \, x^{2} - {\left (x^{7} - 2 \, x^{6}\right )} \log \left (x^{2}\right ) + {\left (x^{7} - 2 \, x^{6}\right )} \log \left (x - 2\right ) + {\left (5 \, x^{6} - 11 \, x^{5} + 4 \, x^{4} - 16 \, x^{2} - {\left (x^{6} - 2 \, x^{5}\right )} \log \left (x^{2}\right ) + {\left (x^{6} - 2 \, x^{5}\right )} \log \left (x - 2\right ) + 96 \, x - 128\right )} \log \left (\frac {x^{4} \log \left (x^{2}\right ) - x^{4} \log \left (x - 2\right ) - 5 \, x^{4} + 16}{x^{4}}\right ) - 128 \, x\right )}}{5 \, x^{6} - 10 \, x^{5} - 16 \, x^{2} - {\left (x^{6} - 2 \, x^{5}\right )} \log \left (x^{2}\right ) + {\left (x^{6} - 2 \, x^{5}\right )} \log \left (x - 2\right ) + 32 \, x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^6-4*x^5)*log(x^2)+(-2*x^6+4*x^5)*log(x-2)-10*x^6+22*x^5-8*x^4+32*x^2-192*x+256)*log((x^4*log(
x^2)-x^4*log(x-2)-5*x^4+16)/x^4)+(2*x^7-4*x^6)*log(x^2)+(-2*x^7+4*x^6)*log(x-2)-10*x^7+22*x^6-8*x^5+32*x^3-192
*x^2+256*x)/((x^6-2*x^5)*log(x^2)+(-x^6+2*x^5)*log(x-2)-5*x^6+10*x^5+16*x^2-32*x),x, algorithm="giac")

[Out]

integrate(2*(5*x^7 - 11*x^6 + 4*x^5 - 16*x^3 + 96*x^2 - (x^7 - 2*x^6)*log(x^2) + (x^7 - 2*x^6)*log(x - 2) + (5
*x^6 - 11*x^5 + 4*x^4 - 16*x^2 - (x^6 - 2*x^5)*log(x^2) + (x^6 - 2*x^5)*log(x - 2) + 96*x - 128)*log((x^4*log(
x^2) - x^4*log(x - 2) - 5*x^4 + 16)/x^4) - 128*x)/(5*x^6 - 10*x^5 - 16*x^2 - (x^6 - 2*x^5)*log(x^2) + (x^6 - 2
*x^5)*log(x - 2) + 32*x), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (2 x^{6}-4 x^{5}\right ) \ln \left (x^{2}\right )+\left (-2 x^{6}+4 x^{5}\right ) \ln \left (x -2\right )-10 x^{6}+22 x^{5}-8 x^{4}+32 x^{2}-192 x +256\right ) \ln \left (\frac {x^{4} \ln \left (x^{2}\right )-x^{4} \ln \left (x -2\right )-5 x^{4}+16}{x^{4}}\right )+\left (2 x^{7}-4 x^{6}\right ) \ln \left (x^{2}\right )+\left (-2 x^{7}+4 x^{6}\right ) \ln \left (x -2\right )-10 x^{7}+22 x^{6}-8 x^{5}+32 x^{3}-192 x^{2}+256 x}{\left (x^{6}-2 x^{5}\right ) \ln \left (x^{2}\right )+\left (-x^{6}+2 x^{5}\right ) \ln \left (x -2\right )-5 x^{6}+10 x^{5}+16 x^{2}-32 x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^6-4*x^5)*ln(x^2)+(-2*x^6+4*x^5)*ln(x-2)-10*x^6+22*x^5-8*x^4+32*x^2-192*x+256)*ln((x^4*ln(x^2)-x^4*l
n(x-2)-5*x^4+16)/x^4)+(2*x^7-4*x^6)*ln(x^2)+(-2*x^7+4*x^6)*ln(x-2)-10*x^7+22*x^6-8*x^5+32*x^3-192*x^2+256*x)/(
(x^6-2*x^5)*ln(x^2)+(-x^6+2*x^5)*ln(x-2)-5*x^6+10*x^5+16*x^2-32*x),x)

[Out]

int((((2*x^6-4*x^5)*ln(x^2)+(-2*x^6+4*x^5)*ln(x-2)-10*x^6+22*x^5-8*x^4+32*x^2-192*x+256)*ln((x^4*ln(x^2)-x^4*l
n(x-2)-5*x^4+16)/x^4)+(2*x^7-4*x^6)*ln(x^2)+(-2*x^7+4*x^6)*ln(x-2)-10*x^7+22*x^6-8*x^5+32*x^3-192*x^2+256*x)/(
(x^6-2*x^5)*ln(x^2)+(-x^6+2*x^5)*ln(x-2)-5*x^6+10*x^5+16*x^2-32*x),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 2 \, \int \frac {5 \, x^{7} - 11 \, x^{6} + 4 \, x^{5} - 16 \, x^{3} + 96 \, x^{2} - {\left (x^{7} - 2 \, x^{6}\right )} \log \left (x^{2}\right ) + {\left (x^{7} - 2 \, x^{6}\right )} \log \left (x - 2\right ) + {\left (5 \, x^{6} - 11 \, x^{5} + 4 \, x^{4} - 16 \, x^{2} - {\left (x^{6} - 2 \, x^{5}\right )} \log \left (x^{2}\right ) + {\left (x^{6} - 2 \, x^{5}\right )} \log \left (x - 2\right ) + 96 \, x - 128\right )} \log \left (\frac {x^{4} \log \left (x^{2}\right ) - x^{4} \log \left (x - 2\right ) - 5 \, x^{4} + 16}{x^{4}}\right ) - 128 \, x}{5 \, x^{6} - 10 \, x^{5} - 16 \, x^{2} - {\left (x^{6} - 2 \, x^{5}\right )} \log \left (x^{2}\right ) + {\left (x^{6} - 2 \, x^{5}\right )} \log \left (x - 2\right ) + 32 \, x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^6-4*x^5)*log(x^2)+(-2*x^6+4*x^5)*log(x-2)-10*x^6+22*x^5-8*x^4+32*x^2-192*x+256)*log((x^4*log(
x^2)-x^4*log(x-2)-5*x^4+16)/x^4)+(2*x^7-4*x^6)*log(x^2)+(-2*x^7+4*x^6)*log(x-2)-10*x^7+22*x^6-8*x^5+32*x^3-192
*x^2+256*x)/((x^6-2*x^5)*log(x^2)+(-x^6+2*x^5)*log(x-2)-5*x^6+10*x^5+16*x^2-32*x),x, algorithm="maxima")

[Out]

2*integrate((5*x^7 - 11*x^6 + 4*x^5 - 16*x^3 + 96*x^2 - (x^7 - 2*x^6)*log(x^2) + (x^7 - 2*x^6)*log(x - 2) + (5
*x^6 - 11*x^5 + 4*x^4 - 16*x^2 - (x^6 - 2*x^5)*log(x^2) + (x^6 - 2*x^5)*log(x - 2) + 96*x - 128)*log((x^4*log(
x^2) - x^4*log(x - 2) - 5*x^4 + 16)/x^4) - 128*x)/(5*x^6 - 10*x^5 - 16*x^2 - (x^6 - 2*x^5)*log(x^2) + (x^6 - 2
*x^5)*log(x - 2) + 32*x), x)

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mupad [B]  time = 2.45, size = 34, normalized size = 1.55 \begin {gather*} {\left (x+\ln \left (-\frac {x^4\,\ln \left (x-2\right )-x^4\,\ln \left (x^2\right )+5\,x^4-16}{x^4}\right )\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x^2)*(4*x^6 - 2*x^7) - log(x - 2)*(4*x^6 - 2*x^7) - 256*x + log(-(x^4*log(x - 2) - x^4*log(x^2) + 5*x
^4 - 16)/x^4)*(192*x - log(x - 2)*(4*x^5 - 2*x^6) + log(x^2)*(4*x^5 - 2*x^6) - 32*x^2 + 8*x^4 - 22*x^5 + 10*x^
6 - 256) + 192*x^2 - 32*x^3 + 8*x^5 - 22*x^6 + 10*x^7)/(32*x - log(x - 2)*(2*x^5 - x^6) + log(x^2)*(2*x^5 - x^
6) - 16*x^2 - 10*x^5 + 5*x^6),x)

[Out]

(x + log(-(x^4*log(x - 2) - x^4*log(x^2) + 5*x^4 - 16)/x^4))^2

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sympy [B]  time = 2.73, size = 65, normalized size = 2.95 \begin {gather*} x^{2} + 2 x \log {\left (\frac {x^{4} \log {\left (x^{2} \right )} - x^{4} \log {\left (x - 2 \right )} - 5 x^{4} + 16}{x^{4}} \right )} + \log {\left (\frac {x^{4} \log {\left (x^{2} \right )} - x^{4} \log {\left (x - 2 \right )} - 5 x^{4} + 16}{x^{4}} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**6-4*x**5)*ln(x**2)+(-2*x**6+4*x**5)*ln(x-2)-10*x**6+22*x**5-8*x**4+32*x**2-192*x+256)*ln((x*
*4*ln(x**2)-x**4*ln(x-2)-5*x**4+16)/x**4)+(2*x**7-4*x**6)*ln(x**2)+(-2*x**7+4*x**6)*ln(x-2)-10*x**7+22*x**6-8*
x**5+32*x**3-192*x**2+256*x)/((x**6-2*x**5)*ln(x**2)+(-x**6+2*x**5)*ln(x-2)-5*x**6+10*x**5+16*x**2-32*x),x)

[Out]

x**2 + 2*x*log((x**4*log(x**2) - x**4*log(x - 2) - 5*x**4 + 16)/x**4) + log((x**4*log(x**2) - x**4*log(x - 2)
- 5*x**4 + 16)/x**4)**2

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