3.37.72 \(\int \frac {(-32+366 x-90 x^2+(-16+4 x) \log (-4+x)) \log (-\frac {9 x^2}{2-45 x+\log (-4+x)})}{-8 x+182 x^2-45 x^3+(-4 x+x^2) \log (-4+x)} \, dx\)

Optimal. Leaf size=22 \[ \log ^2\left (\frac {x}{5-\frac {2+\log (-4+x)}{9 x}}\right ) \]

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Rubi [A]  time = 0.54, antiderivative size = 19, normalized size of antiderivative = 0.86, number of steps used = 1, number of rules used = 4, integrand size = 66, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {6741, 6742, 6684, 6686} \begin {gather*} \log ^2\left (-\frac {9 x^2}{-45 x+\log (x-4)+2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-32 + 366*x - 90*x^2 + (-16 + 4*x)*Log[-4 + x])*Log[(-9*x^2)/(2 - 45*x + Log[-4 + x])])/(-8*x + 182*x^2
- 45*x^3 + (-4*x + x^2)*Log[-4 + x]),x]

[Out]

Log[(-9*x^2)/(2 - 45*x + Log[-4 + x])]^2

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

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 6741

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log ^2\left (-\frac {9 x^2}{2-45 x+\log (-4+x)}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 19, normalized size = 0.86 \begin {gather*} \log ^2\left (-\frac {9 x^2}{2-45 x+\log (-4+x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-32 + 366*x - 90*x^2 + (-16 + 4*x)*Log[-4 + x])*Log[(-9*x^2)/(2 - 45*x + Log[-4 + x])])/(-8*x + 18
2*x^2 - 45*x^3 + (-4*x + x^2)*Log[-4 + x]),x]

[Out]

Log[(-9*x^2)/(2 - 45*x + Log[-4 + x])]^2

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fricas [A]  time = 0.61, size = 21, normalized size = 0.95 \begin {gather*} \log \left (\frac {9 \, x^{2}}{45 \, x - \log \left (x - 4\right ) - 2}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x-16)*log(x-4)-90*x^2+366*x-32)*log(-9*x^2/(log(x-4)-45*x+2))/((x^2-4*x)*log(x-4)-45*x^3+182*x^2
-8*x),x, algorithm="fricas")

[Out]

log(9*x^2/(45*x - log(x - 4) - 2))^2

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giac [B]  time = 0.21, size = 72, normalized size = 3.27 \begin {gather*} -2 \, {\left (2 \, \log \relax (x) - \log \left (-45 \, x + \log \left (x - 4\right ) + 2\right )\right )} \log \left (45 \, x - \log \left (x - 4\right ) - 2\right ) + 8 \, \log \relax (3) \log \relax (x) + 4 \, \log \relax (x)^{2} - 4 \, \log \relax (3) \log \left (-45 \, x + \log \left (x - 4\right ) + 2\right ) - \log \left (-45 \, x + \log \left (x - 4\right ) + 2\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x-16)*log(x-4)-90*x^2+366*x-32)*log(-9*x^2/(log(x-4)-45*x+2))/((x^2-4*x)*log(x-4)-45*x^3+182*x^2
-8*x),x, algorithm="giac")

[Out]

-2*(2*log(x) - log(-45*x + log(x - 4) + 2))*log(45*x - log(x - 4) - 2) + 8*log(3)*log(x) + 4*log(x)^2 - 4*log(
3)*log(-45*x + log(x - 4) + 2) - log(-45*x + log(x - 4) + 2)^2

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maple [C]  time = 0.28, size = 536, normalized size = 24.36

method	result	size
risch	\(-i \ln \left (\ln \left (x -4\right )-45 x +2\right ) \pi \mathrm {csgn}\left (\frac {i x^{2}}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right )^{3}-i \ln \left (\ln \left (x -4\right )-45 x +2\right ) \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right )^{2}+4 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+i \ln \left (\ln \left (x -4\right )-45 x +2\right ) \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-2 i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right )^{2}+i \ln \left (\ln \left (x -4\right )-45 x +2\right ) \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right )+\ln \left (-\frac {\ln \left (x -4\right )}{45}+x -\frac {2}{45}\right )^{2}-2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right )^{3}-2 i \ln \left (\ln \left (x -4\right )-45 x +2\right ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+i \ln \left (\ln \left (x -4\right )-45 x +2\right ) \pi \,\mathrm {csgn}\left (\frac {i}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right )^{2}+4 \ln \relax (x )^{2}+2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right )^{2}-2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right )-4 \ln \relax (x ) \ln \left (-\frac {\ln \left (x -4\right )}{45}+x -\frac {2}{45}\right )+2 \ln \left (\ln \left (x -4\right )-45 x +2\right ) \ln \relax (5)+2 i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i x^{2}}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right )^{3}+i \ln \left (\ln \left (x -4\right )-45 x +2\right ) \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-4 \ln \relax (5) \ln \relax (x )\)	\(536\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x-16)*ln(x-4)-90*x^2+366*x-32)*ln(-9*x^2/(ln(x-4)-45*x+2))/((x^2-4*x)*ln(x-4)-45*x^3+182*x^2-8*x),x,me
thod=_RETURNVERBOSE)

[Out]

-I*ln(ln(x-4)-45*x+2)*Pi*csgn(I*x^2/(1/45*ln(x-4)-x+2/45))^3-I*ln(ln(x-4)-45*x+2)*Pi*csgn(I*x^2)*csgn(I*x^2/(1
/45*ln(x-4)-x+2/45))^2+4*I*Pi*ln(x)*csgn(I*x)*csgn(I*x^2)^2+I*ln(ln(x-4)-45*x+2)*Pi*csgn(I*x^2)^3-2*I*Pi*ln(x)
*csgn(I/(1/45*ln(x-4)-x+2/45))*csgn(I*x^2/(1/45*ln(x-4)-x+2/45))^2+I*ln(ln(x-4)-45*x+2)*Pi*csgn(I*x^2)*csgn(I/
(1/45*ln(x-4)-x+2/45))*csgn(I*x^2/(1/45*ln(x-4)-x+2/45))+ln(-1/45*ln(x-4)+x-2/45)^2-2*I*Pi*ln(x)*csgn(I*x)^2*c
sgn(I*x^2)-2*I*Pi*ln(x)*csgn(I*x^2)^3-2*I*ln(ln(x-4)-45*x+2)*Pi*csgn(I*x)*csgn(I*x^2)^2+I*ln(ln(x-4)-45*x+2)*P
i*csgn(I/(1/45*ln(x-4)-x+2/45))*csgn(I*x^2/(1/45*ln(x-4)-x+2/45))^2+4*ln(x)^2+2*I*Pi*ln(x)*csgn(I*x^2)*csgn(I*
x^2/(1/45*ln(x-4)-x+2/45))^2-2*I*Pi*ln(x)*csgn(I*x^2)*csgn(I/(1/45*ln(x-4)-x+2/45))*csgn(I*x^2/(1/45*ln(x-4)-x
+2/45))-4*ln(x)*ln(-1/45*ln(x-4)+x-2/45)+2*ln(ln(x-4)-45*x+2)*ln(5)+2*I*Pi*ln(x)*csgn(I*x^2/(1/45*ln(x-4)-x+2/
45))^3+I*ln(ln(x-4)-45*x+2)*Pi*csgn(I*x)^2*csgn(I*x^2)-4*ln(5)*ln(x)

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maxima [B]  time = 0.42, size = 73, normalized size = 3.32 \begin {gather*} -4 \, \log \relax (x)^{2} + 4 \, \log \relax (x) \log \left (-45 \, x + \log \left (x - 4\right ) + 2\right ) - \log \left (-45 \, x + \log \left (x - 4\right ) + 2\right )^{2} + 2 \, {\left (2 \, \log \relax (x) - \log \left (-45 \, x + \log \left (x - 4\right ) + 2\right )\right )} \log \left (\frac {9 \, x^{2}}{45 \, x - \log \left (x - 4\right ) - 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x-16)*log(x-4)-90*x^2+366*x-32)*log(-9*x^2/(log(x-4)-45*x+2))/((x^2-4*x)*log(x-4)-45*x^3+182*x^2
-8*x),x, algorithm="maxima")

[Out]

-4*log(x)^2 + 4*log(x)*log(-45*x + log(x - 4) + 2) - log(-45*x + log(x - 4) + 2)^2 + 2*(2*log(x) - log(-45*x +
log(x - 4) + 2))*log(9*x^2/(45*x - log(x - 4) - 2))

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mupad [B]  time = 3.21, size = 19, normalized size = 0.86 \begin {gather*} {\ln \left (-\frac {9\,x^2}{\ln \left (x-4\right )-45\,x+2}\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(-(9*x^2)/(log(x - 4) - 45*x + 2))*(366*x - 90*x^2 + log(x - 4)*(4*x - 16) - 32))/(8*x + log(x - 4)*(
4*x - x^2) - 182*x^2 + 45*x^3),x)

[Out]

log(-(9*x^2)/(log(x - 4) - 45*x + 2))^2

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sympy [A]  time = 0.51, size = 19, normalized size = 0.86 \begin {gather*} \log {\left (- \frac {9 x^{2}}{- 45 x + \log {\left (x - 4 \right )} + 2} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x-16)*ln(x-4)-90*x**2+366*x-32)*ln(-9*x**2/(ln(x-4)-45*x+2))/((x**2-4*x)*ln(x-4)-45*x**3+182*x**
2-8*x),x)

[Out]

log(-9*x**2/(-45*x + log(x - 4) + 2))**2

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