3.34.61 \(\int \frac {-4096 x-64 x^3+(-256 x+60 x^3+x^5) \log (-4+x^2)+(4096 x+(256 x-64 x^3) \log (-4+x^2)) \log (\log (-4+x^2))}{(-4096+1024 x^2) \log (-4+x^2)} \, dx\)

Optimal. Leaf size=20 \[ \left (1+\frac {x^2}{64}-\log \left (\log \left (-4+x^2\right )\right )\right )^2 \]

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Rubi [A]  time = 0.24, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 74, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {6688, 12, 6686} \begin {gather*} \frac {\left (x^2-64 \log \left (\log \left (x^2-4\right )\right )+64\right )^2}{4096} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4096*x - 64*x^3 + (-256*x + 60*x^3 + x^5)*Log[-4 + x^2] + (4096*x + (256*x - 64*x^3)*Log[-4 + x^2])*Log[
Log[-4 + x^2]])/((-4096 + 1024*x^2)*Log[-4 + x^2]),x]

[Out]

(64 + x^2 - 64*Log[Log[-4 + x^2]])^2/4096

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 {x \left (64-\left (-4+x^2\right ) \log \left (-4+x^2\right )\right ) \left (64+x^2-64 \log \left (\log \left (-4+x^2\right )\right )\right )}{1024 \left (4-x^2\right ) \log \left (-4+x^2\right )} \, dx\\ &=\frac {\int \frac {x \left (64-\left (-4+x^2\right ) \log \left (-4+x^2\right )\right ) \left (64+x^2-64 \log \left (\log \left (-4+x^2\right )\right )\right )}{\left (4-x^2\right ) \log \left (-4+x^2\right )} \, dx}{1024}\\ &=\frac {\left (64+x^2-64 \log \left (\log \left (-4+x^2\right )\right )\right )^2}{4096}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 1.00 \begin {gather*} \frac {\left (64+x^2-64 \log \left (\log \left (-4+x^2\right )\right )\right )^2}{4096} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4096*x - 64*x^3 + (-256*x + 60*x^3 + x^5)*Log[-4 + x^2] + (4096*x + (256*x - 64*x^3)*Log[-4 + x^2]
)*Log[Log[-4 + x^2]])/((-4096 + 1024*x^2)*Log[-4 + x^2]),x]

[Out]

(64 + x^2 - 64*Log[Log[-4 + x^2]])^2/4096

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fricas [A]  time = 0.54, size = 34, normalized size = 1.70 \begin {gather*} \frac {1}{4096} \, x^{4} + \frac {1}{32} \, x^{2} - \frac {1}{32} \, {\left (x^{2} + 64\right )} \log \left (\log \left (x^{2} - 4\right )\right ) + \log \left (\log \left (x^{2} - 4\right )\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-64*x^3+256*x)*log(x^2-4)+4096*x)*log(log(x^2-4))+(x^5+60*x^3-256*x)*log(x^2-4)-64*x^3-4096*x)/(1
024*x^2-4096)/log(x^2-4),x, algorithm="fricas")

[Out]

1/4096*x^4 + 1/32*x^2 - 1/32*(x^2 + 64)*log(log(x^2 - 4)) + log(log(x^2 - 4))^2

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giac [B]  time = 0.22, size = 48, normalized size = 2.40 \begin {gather*} \frac {1}{4096} \, {\left (x^{2} - 4\right )}^{2} + \frac {17}{512} \, x^{2} - \frac {1}{32} \, {\left (x^{2} - 4\right )} \log \left (\log \left (x^{2} - 4\right )\right ) + \log \left (\log \left (x^{2} - 4\right )\right )^{2} - \frac {17}{8} \, \log \left (\log \left (x^{2} - 4\right )\right ) - \frac {17}{128} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-64*x^3+256*x)*log(x^2-4)+4096*x)*log(log(x^2-4))+(x^5+60*x^3-256*x)*log(x^2-4)-64*x^3-4096*x)/(1
024*x^2-4096)/log(x^2-4),x, algorithm="giac")

[Out]

1/4096*(x^2 - 4)^2 + 17/512*x^2 - 1/32*(x^2 - 4)*log(log(x^2 - 4)) + log(log(x^2 - 4))^2 - 17/8*log(log(x^2 -
4)) - 17/128

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maple [B]  time = 0.48, size = 42, normalized size = 2.10

method	result	size
risch	\(\ln \left (\ln \left (x^{2}-4\right )\right )^{2}-\frac {\ln \left (\ln \left (x^{2}-4\right )\right ) x^{2}}{32}+\frac {x^{4}}{4096}+\frac {x^{2}}{32}-2 \ln \left (\ln \left (x^{2}-4\right )\right )\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-64*x^3+256*x)*ln(x^2-4)+4096*x)*ln(ln(x^2-4))+(x^5+60*x^3-256*x)*ln(x^2-4)-64*x^3-4096*x)/(1024*x^2-40
96)/ln(x^2-4),x,method=_RETURNVERBOSE)

[Out]

ln(ln(x^2-4))^2-1/32*ln(ln(x^2-4))*x^2+1/4096*x^4+1/32*x^2-2*ln(ln(x^2-4))

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maxima [B]  time = 0.49, size = 50, normalized size = 2.50 \begin {gather*} \frac {1}{4096} \, x^{4} - \frac {1}{32} \, x^{2} \log \left (\log \left (x + 2\right ) + \log \left (x - 2\right )\right ) + \frac {1}{32} \, x^{2} + \log \left (\log \left (x + 2\right ) + \log \left (x - 2\right )\right )^{2} - 2 \, \log \left (\log \left (x + 2\right ) + \log \left (x - 2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-64*x^3+256*x)*log(x^2-4)+4096*x)*log(log(x^2-4))+(x^5+60*x^3-256*x)*log(x^2-4)-64*x^3-4096*x)/(1
024*x^2-4096)/log(x^2-4),x, algorithm="maxima")

[Out]

1/4096*x^4 - 1/32*x^2*log(log(x + 2) + log(x - 2)) + 1/32*x^2 + log(log(x + 2) + log(x - 2))^2 - 2*log(log(x +
2) + log(x - 2))

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mupad [B]  time = 2.16, size = 41, normalized size = 2.05 \begin {gather*} \frac {x^4}{4096}-\frac {x^2\,\ln \left (\ln \left (x^2-4\right )\right )}{32}+\frac {x^2}{32}+{\ln \left (\ln \left (x^2-4\right )\right )}^2-2\,\ln \left (\ln \left (x^2-4\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4096*x - log(log(x^2 - 4))*(4096*x + log(x^2 - 4)*(256*x - 64*x^3)) - log(x^2 - 4)*(60*x^3 - 256*x + x^5
) + 64*x^3)/(log(x^2 - 4)*(1024*x^2 - 4096)),x)

[Out]

log(log(x^2 - 4))^2 - (x^2*log(log(x^2 - 4)))/32 - 2*log(log(x^2 - 4)) + x^2/32 + x^4/4096

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sympy [B]  time = 0.42, size = 42, normalized size = 2.10 \begin {gather*} \frac {x^{4}}{4096} - \frac {x^{2} \log {\left (\log {\left (x^{2} - 4 \right )} \right )}}{32} + \frac {x^{2}}{32} + \log {\left (\log {\left (x^{2} - 4 \right )} \right )}^{2} - 2 \log {\left (\log {\left (x^{2} - 4 \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-64*x**3+256*x)*ln(x**2-4)+4096*x)*ln(ln(x**2-4))+(x**5+60*x**3-256*x)*ln(x**2-4)-64*x**3-4096*x)
/(1024*x**2-4096)/ln(x**2-4),x)

[Out]

x**4/4096 - x**2*log(log(x**2 - 4))/32 + x**2/32 + log(log(x**2 - 4))**2 - 2*log(log(x**2 - 4))

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