3.32.15 \(\int \frac {(2 x^2+5 x^3-2 x^5+2 x^6) \log (4-\log (6))+(4-x^2+x^3) \log (4-x^2+x^3) \log (4-\log (6))}{(-4 x^4+x^6-x^7+(4 x-x^3+x^4) \log (4-x^2+x^3)) \log ^2(\frac {-x^3+\log (4-x^2+x^3)}{x})} \, dx\)

Optimal. Leaf size=33 \[ \frac {\log (4-\log (6))}{\log \left (-x^2+\frac {\log \left (4+x \left (-x+x^2\right )\right )}{x}\right )} \]

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Rubi [A]  time = 0.16, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 124, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {6686} \begin {gather*} \frac {\log (4-\log (6))}{\log \left (-\frac {x^3-\log \left (x^3-x^2+4\right )}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2*x^2 + 5*x^3 - 2*x^5 + 2*x^6)*Log[4 - Log[6]] + (4 - x^2 + x^3)*Log[4 - x^2 + x^3]*Log[4 - Log[6]])/((-
4*x^4 + x^6 - x^7 + (4*x - x^3 + x^4)*Log[4 - x^2 + x^3])*Log[(-x^3 + Log[4 - x^2 + x^3])/x]^2),x]

[Out]

Log[4 - Log[6]]/Log[-((x^3 - Log[4 - x^2 + x^3])/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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\log (4-\log (6))}{\log \left (-\frac {x^3-\log \left (4-x^2+x^3\right )}{x}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 32, normalized size = 0.97 \begin {gather*} \frac {\log (4-\log (6))}{\log \left (\frac {-x^3+\log \left (4-x^2+x^3\right )}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2*x^2 + 5*x^3 - 2*x^5 + 2*x^6)*Log[4 - Log[6]] + (4 - x^2 + x^3)*Log[4 - x^2 + x^3]*Log[4 - Log[6]
])/((-4*x^4 + x^6 - x^7 + (4*x - x^3 + x^4)*Log[4 - x^2 + x^3])*Log[(-x^3 + Log[4 - x^2 + x^3])/x]^2),x]

[Out]

Log[4 - Log[6]]/Log[(-x^3 + Log[4 - x^2 + x^3])/x]

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fricas [A]  time = 0.62, size = 33, normalized size = 1.00 \begin {gather*} \frac {\log \left (-\log \relax (6) + 4\right )}{\log \left (-\frac {x^{3} - \log \left (x^{3} - x^{2} + 4\right )}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-x^2+4)*log(-log(6)+4)*log(x^3-x^2+4)+(2*x^6-2*x^5+5*x^3+2*x^2)*log(-log(6)+4))/((x^4-x^3+4*x)*
log(x^3-x^2+4)-x^7+x^6-4*x^4)/log((log(x^3-x^2+4)-x^3)/x)^2,x, algorithm="fricas")

[Out]

log(-log(6) + 4)/log(-(x^3 - log(x^3 - x^2 + 4))/x)

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giac [A]  time = 0.73, size = 33, normalized size = 1.00 \begin {gather*} \frac {\log \left (-\log \relax (6) + 4\right )}{\log \left (-x^{3} + \log \left (x^{3} - x^{2} + 4\right )\right ) - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-x^2+4)*log(-log(6)+4)*log(x^3-x^2+4)+(2*x^6-2*x^5+5*x^3+2*x^2)*log(-log(6)+4))/((x^4-x^3+4*x)*
log(x^3-x^2+4)-x^7+x^6-4*x^4)/log((log(x^3-x^2+4)-x^3)/x)^2,x, algorithm="giac")

[Out]

log(-log(6) + 4)/(log(-x^3 + log(x^3 - x^2 + 4)) - log(x))

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maple [C]  time = 0.12, size = 245, normalized size = 7.42

method	result	size
risch	\(\frac {2 i \ln \left (-\ln \relax (2)-\ln \relax (3)+4\right )}{\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (-\ln \left (x^{3}-x^{2}+4\right )+x^{3}\right )\right ) \mathrm {csgn}\left (\frac {i \left (-\ln \left (x^{3}-x^{2}+4\right )+x^{3}\right )}{x}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (-\ln \left (x^{3}-x^{2}+4\right )+x^{3}\right )}{x}\right )^{2}+2 \pi \mathrm {csgn}\left (\frac {i \left (-\ln \left (x^{3}-x^{2}+4\right )+x^{3}\right )}{x}\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (-\ln \left (x^{3}-x^{2}+4\right )+x^{3}\right )\right ) \mathrm {csgn}\left (\frac {i \left (-\ln \left (x^{3}-x^{2}+4\right )+x^{3}\right )}{x}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i \left (-\ln \left (x^{3}-x^{2}+4\right )+x^{3}\right )}{x}\right )^{3}-2 \pi -2 i \ln \relax (x )+2 i \ln \left (-\ln \left (x^{3}-x^{2}+4\right )+x^{3}\right )}\)	\(245\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3-x^2+4)*ln(-ln(6)+4)*ln(x^3-x^2+4)+(2*x^6-2*x^5+5*x^3+2*x^2)*ln(-ln(6)+4))/((x^4-x^3+4*x)*ln(x^3-x^2+
4)-x^7+x^6-4*x^4)/ln((ln(x^3-x^2+4)-x^3)/x)^2,x,method=_RETURNVERBOSE)

[Out]

2*I*ln(-ln(2)-ln(3)+4)/(Pi*csgn(I/x)*csgn(I*(-ln(x^3-x^2+4)+x^3))*csgn(I/x*(-ln(x^3-x^2+4)+x^3))-Pi*csgn(I/x)*
csgn(I/x*(-ln(x^3-x^2+4)+x^3))^2+2*Pi*csgn(I/x*(-ln(x^3-x^2+4)+x^3))^2-Pi*csgn(I*(-ln(x^3-x^2+4)+x^3))*csgn(I/
x*(-ln(x^3-x^2+4)+x^3))^2-Pi*csgn(I/x*(-ln(x^3-x^2+4)+x^3))^3-2*Pi-2*I*ln(x)+2*I*ln(-ln(x^3-x^2+4)+x^3))

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maxima [A]  time = 0.89, size = 37, normalized size = 1.12 \begin {gather*} \frac {\log \left (-\log \relax (3) - \log \relax (2) + 4\right )}{\log \left (-x^{3} + \log \left (x^{3} - x^{2} + 4\right )\right ) - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-x^2+4)*log(-log(6)+4)*log(x^3-x^2+4)+(2*x^6-2*x^5+5*x^3+2*x^2)*log(-log(6)+4))/((x^4-x^3+4*x)*
log(x^3-x^2+4)-x^7+x^6-4*x^4)/log((log(x^3-x^2+4)-x^3)/x)^2,x, algorithm="maxima")

[Out]

log(-log(3) - log(2) + 4)/(log(-x^3 + log(x^3 - x^2 + 4)) - log(x))

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mupad [B]  time = 2.51, size = 32, normalized size = 0.97 \begin {gather*} \frac {\ln \left (4-\ln \relax (6)\right )}{\ln \left (\frac {\ln \left (x^3-x^2+4\right )-x^3}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(4 - log(6))*(2*x^2 + 5*x^3 - 2*x^5 + 2*x^6) + log(4 - log(6))*log(x^3 - x^2 + 4)*(x^3 - x^2 + 4))/(lo
g((log(x^3 - x^2 + 4) - x^3)/x)^2*(log(x^3 - x^2 + 4)*(4*x - x^3 + x^4) - 4*x^4 + x^6 - x^7)),x)

[Out]

log(4 - log(6))/log((log(x^3 - x^2 + 4) - x^3)/x)

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sympy [A]  time = 0.61, size = 22, normalized size = 0.67 \begin {gather*} \frac {\log {\left (4 - \log {\relax (6 )} \right )}}{\log {\left (\frac {- x^{3} + \log {\left (x^{3} - x^{2} + 4 \right )}}{x} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**3-x**2+4)*ln(-ln(6)+4)*ln(x**3-x**2+4)+(2*x**6-2*x**5+5*x**3+2*x**2)*ln(-ln(6)+4))/((x**4-x**3+
4*x)*ln(x**3-x**2+4)-x**7+x**6-4*x**4)/ln((ln(x**3-x**2+4)-x**3)/x)**2,x)

[Out]

log(4 - log(6))/log((-x**3 + log(x**3 - x**2 + 4))/x)

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