3.39.1 \(\int \frac {20480+(-40960-14080 x) \log (x)}{(15129 x-5412 x^2+484 x^3) \log (x)+(-7872 x+1408 x^2) \log (x) \log (\frac {x^2}{\log (x)})+1024 x \log (x) \log ^2(\frac {x^2}{\log (x)})} \, dx\)

Optimal. Leaf size=27 \[ \frac {20}{-4+\frac {5}{16} \left (\frac {1}{2}-x\right )+x+\log \left (\frac {x^2}{\log (x)}\right )} \]

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Rubi [A]  time = 0.16, antiderivative size = 20, normalized size of antiderivative = 0.74, number of steps used = 3, number of rules used = 3, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {6688, 12, 6686} \begin {gather*} -\frac {640}{-32 \log \left (\frac {x^2}{\log (x)}\right )-22 x+123} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(20480 + (-40960 - 14080*x)*Log[x])/((15129*x - 5412*x^2 + 484*x^3)*Log[x] + (-7872*x + 1408*x^2)*Log[x]*L
og[x^2/Log[x]] + 1024*x*Log[x]*Log[x^2/Log[x]]^2),x]

[Out]

-640/(123 - 22*x - 32*Log[x^2/Log[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 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 {1280 (16-(32+11 x) \log (x))}{x \log (x) \left (123-22 x-32 \log \left (\frac {x^2}{\log (x)}\right )\right )^2} \, dx\\ &=1280 \int \frac {16-(32+11 x) \log (x)}{x \log (x) \left (123-22 x-32 \log \left (\frac {x^2}{\log (x)}\right )\right )^2} \, dx\\ &=-\frac {640}{123-22 x-32 \log \left (\frac {x^2}{\log (x)}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 0.74 \begin {gather*} -\frac {1280}{246-44 x-64 \log \left (\frac {x^2}{\log (x)}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20480 + (-40960 - 14080*x)*Log[x])/((15129*x - 5412*x^2 + 484*x^3)*Log[x] + (-7872*x + 1408*x^2)*Lo
g[x]*Log[x^2/Log[x]] + 1024*x*Log[x]*Log[x^2/Log[x]]^2),x]

[Out]

-1280/(246 - 44*x - 64*Log[x^2/Log[x]])

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fricas [A]  time = 0.77, size = 20, normalized size = 0.74 \begin {gather*} \frac {640}{22 \, x + 32 \, \log \left (\frac {x^{2}}{\log \relax (x)}\right ) - 123} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-14080*x-40960)*log(x)+20480)/(1024*x*log(x)*log(x^2/log(x))^2+(1408*x^2-7872*x)*log(x)*log(x^2/lo
g(x))+(484*x^3-5412*x^2+15129*x)*log(x)),x, algorithm="fricas")

[Out]

640/(22*x + 32*log(x^2/log(x)) - 123)

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giac [A]  time = 0.35, size = 18, normalized size = 0.67 \begin {gather*} \frac {640}{22 \, x + 64 \, \log \relax (x) - 32 \, \log \left (\log \relax (x)\right ) - 123} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-14080*x-40960)*log(x)+20480)/(1024*x*log(x)*log(x^2/log(x))^2+(1408*x^2-7872*x)*log(x)*log(x^2/lo
g(x))+(484*x^3-5412*x^2+15129*x)*log(x)),x, algorithm="giac")

[Out]

640/(22*x + 64*log(x) - 32*log(log(x)) - 123)

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maple [C]  time = 0.08, size = 162, normalized size = 6.00

method	result	size
risch	\(\frac {640 i}{16 \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-32 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+16 \pi \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \relax (x )}\right )-16 \pi \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \relax (x )}\right )^{2}+16 \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-16 \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \relax (x )}\right )^{2}+16 \pi \mathrm {csgn}\left (\frac {i x^{2}}{\ln \relax (x )}\right )^{3}+22 i x +64 i \ln \relax (x )-32 i \ln \left (\ln \relax (x )\right )-123 i}\)	\(162\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-14080*x-40960)*ln(x)+20480)/(1024*x*ln(x)*ln(x^2/ln(x))^2+(1408*x^2-7872*x)*ln(x)*ln(x^2/ln(x))+(484*x^
3-5412*x^2+15129*x)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

640*I/(16*Pi*csgn(I*x)^2*csgn(I*x^2)-32*Pi*csgn(I*x)*csgn(I*x^2)^2+16*Pi*csgn(I/ln(x))*csgn(I*x^2)*csgn(I*x^2/
ln(x))-16*Pi*csgn(I/ln(x))*csgn(I*x^2/ln(x))^2+16*Pi*csgn(I*x^2)^3-16*Pi*csgn(I*x^2)*csgn(I*x^2/ln(x))^2+16*Pi
*csgn(I*x^2/ln(x))^3+22*I*x+64*I*ln(x)-32*I*ln(ln(x))-123*I)

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maxima [A]  time = 0.38, size = 18, normalized size = 0.67 \begin {gather*} \frac {640}{22 \, x + 64 \, \log \relax (x) - 32 \, \log \left (\log \relax (x)\right ) - 123} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-14080*x-40960)*log(x)+20480)/(1024*x*log(x)*log(x^2/log(x))^2+(1408*x^2-7872*x)*log(x)*log(x^2/lo
g(x))+(484*x^3-5412*x^2+15129*x)*log(x)),x, algorithm="maxima")

[Out]

640/(22*x + 64*log(x) - 32*log(log(x)) - 123)

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mupad [B]  time = 2.93, size = 20, normalized size = 0.74 \begin {gather*} \frac {640}{22\,x+32\,\ln \left (\frac {x^2}{\ln \relax (x)}\right )-123} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(14080*x + 40960) - 20480)/(log(x)*(15129*x - 5412*x^2 + 484*x^3) + 1024*x*log(x)*log(x^2/log(x))
^2 - log(x)*log(x^2/log(x))*(7872*x - 1408*x^2)),x)

[Out]

640/(22*x + 32*log(x^2/log(x)) - 123)

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sympy [A]  time = 0.29, size = 17, normalized size = 0.63 \begin {gather*} \frac {20}{\frac {11 x}{16} + \log {\left (\frac {x^{2}}{\log {\relax (x )}} \right )} - \frac {123}{32}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-14080*x-40960)*ln(x)+20480)/(1024*x*ln(x)*ln(x**2/ln(x))**2+(1408*x**2-7872*x)*ln(x)*ln(x**2/ln(x
))+(484*x**3-5412*x**2+15129*x)*ln(x)),x)

[Out]

20/(11*x/16 + log(x**2/log(x)) - 123/32)

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