3.80.77 \(\int \frac {-8+1260 x-2 x^2+(4-630 x+x^2) \log (4 x)+(630 x-2 x^2+(-630 x+2 x^2) \log (4 x)) \log (\frac {x}{-1+\log (4 x)})}{-2 x+2 x \log (4 x)} \, dx\)

Optimal. Leaf size=24 \[ \left (2-\frac {1}{2} (630-x) x\right ) \log \left (\frac {x}{-1+\log (4 x)}\right ) \]

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Rubi [F]  time = 0.84, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8 + 1260*x - 2*x^2 + (4 - 630*x + x^2)*Log[4*x] + (630*x - 2*x^2 + (-630*x + 2*x^2)*Log[4*x])*Log[x/(-1
+ Log[4*x])])/(-2*x + 2*x*Log[4*x]),x]

[Out]

(E^2*ExpIntegralEi[-2*(1 - Log[4*x])])/32 - (315*E*ExpIntegralEi[-1 + Log[4*x]])/4 + 2*Log[x] - 315*x*Log[-(x/
(1 - Log[4*x]))] + (x^2*Log[-(x/(1 - Log[4*x]))])/2 + Defer[Int][(-4 + 630*x - x^2)/(x*(-1 + Log[4*x])), x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{x (-2+2 \log (4 x))} \, dx\\ &=\int \frac {8-1260 x+2 x^2-\left (4-630 x+x^2\right ) \log (4 x)-\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{2 x (1-\log (4 x))} \, dx\\ &=\frac {1}{2} \int \frac {8-1260 x+2 x^2-\left (4-630 x+x^2\right ) \log (4 x)-\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{x (1-\log (4 x))} \, dx\\ &=\frac {1}{2} \int \left (\frac {\left (4-630 x+x^2\right ) (-2+\log (4 x))}{x (-1+\log (4 x))}+2 (-315+x) \log \left (\frac {x}{-1+\log (4 x)}\right )\right ) \, dx\\ &=\frac {1}{2} \int \frac {\left (4-630 x+x^2\right ) (-2+\log (4 x))}{x (-1+\log (4 x))} \, dx+\int (-315+x) \log \left (\frac {x}{-1+\log (4 x)}\right ) \, dx\\ &=-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} \int \left (\frac {4-630 x+x^2}{x}+\frac {-4+630 x-x^2}{x (-1+\log (4 x))}\right ) \, dx-\int \frac {(630-x) (-2+\log (4 x))}{2 (1-\log (4 x))} \, dx\\ &=-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} \int \frac {4-630 x+x^2}{x} \, dx-\frac {1}{2} \int \frac {(630-x) (-2+\log (4 x))}{1-\log (4 x)} \, dx+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx\\ &=-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} \int \left (-630+\frac {4}{x}+x\right ) \, dx-\frac {1}{2} \int \left (-630+x+\frac {630-x}{-1+\log (4 x)}\right ) \, dx+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx\\ &=2 \log (x)-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )-\frac {1}{2} \int \frac {630-x}{-1+\log (4 x)} \, dx+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx\\ &=2 \log (x)-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )-\frac {1}{2} \int \left (\frac {630}{-1+\log (4 x)}-\frac {x}{-1+\log (4 x)}\right ) \, dx+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx\\ &=2 \log (x)-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} \int \frac {x}{-1+\log (4 x)} \, dx+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx-315 \int \frac {1}{-1+\log (4 x)} \, dx\\ &=2 \log (x)-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{32} \operatorname {Subst}\left (\int \frac {e^{2 x}}{-1+x} \, dx,x,\log (4 x)\right )+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx-\frac {315}{4} \operatorname {Subst}\left (\int \frac {e^x}{-1+x} \, dx,x,\log (4 x)\right )\\ &=\frac {1}{32} e^2 \text {Ei}(-2 (1-\log (4 x)))-\frac {315}{4} e \text {Ei}(-1+\log (4 x))+2 \log (x)-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.31, size = 36, normalized size = 1.50 \begin {gather*} \frac {1}{2} \left (4 \log (x)-4 \log (1-\log (4 x))+(-630+x) x \log \left (\frac {x}{-1+\log (4 x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8 + 1260*x - 2*x^2 + (4 - 630*x + x^2)*Log[4*x] + (630*x - 2*x^2 + (-630*x + 2*x^2)*Log[4*x])*Log[
x/(-1 + Log[4*x])])/(-2*x + 2*x*Log[4*x]),x]

[Out]

(4*Log[x] - 4*Log[1 - Log[4*x]] + (-630 + x)*x*Log[x/(-1 + Log[4*x])])/2

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fricas [A]  time = 0.56, size = 21, normalized size = 0.88 \begin {gather*} \frac {1}{2} \, {\left (x^{2} - 630 \, x + 4\right )} \log \left (\frac {x}{\log \left (4 \, x\right ) - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-630*x)*log(4*x)-2*x^2+630*x)*log(x/(log(4*x)-1))+(x^2-630*x+4)*log(4*x)-2*x^2+1260*x-8)/(2*
x*log(4*x)-2*x),x, algorithm="fricas")

[Out]

1/2*(x^2 - 630*x + 4)*log(x/(log(4*x) - 1))

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giac [B]  time = 0.24, size = 43, normalized size = 1.79 \begin {gather*} \frac {1}{2} \, {\left (x^{2} - 630 \, x\right )} \log \relax (x) - \frac {1}{2} \, {\left (x^{2} - 630 \, x\right )} \log \left (\log \left (4 \, x\right ) - 1\right ) + 2 \, \log \relax (x) - 2 \, \log \left (2 \, \log \relax (2) + \log \relax (x) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-630*x)*log(4*x)-2*x^2+630*x)*log(x/(log(4*x)-1))+(x^2-630*x+4)*log(4*x)-2*x^2+1260*x-8)/(2*
x*log(4*x)-2*x),x, algorithm="giac")

[Out]

1/2*(x^2 - 630*x)*log(x) - 1/2*(x^2 - 630*x)*log(log(4*x) - 1) + 2*log(x) - 2*log(2*log(2) + log(x) - 1)

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maple [A]  time = 0.07, size = 45, normalized size = 1.88

method	result	size
norman	\(-315 x \ln \left (\frac {x}{\ln \left (4 x \right )-1}\right )+\frac {\ln \left (\frac {x}{\ln \left (4 x \right )-1}\right ) x^{2}}{2}-2 \ln \left (\ln \left (4 x \right )-1\right )+2 \ln \relax (x )\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^2-630*x)*ln(4*x)-2*x^2+630*x)*ln(x/(ln(4*x)-1))+(x^2-630*x+4)*ln(4*x)-2*x^2+1260*x-8)/(2*x*ln(4*x)-
2*x),x,method=_RETURNVERBOSE)

[Out]

-315*x*ln(x/(ln(4*x)-1))+1/2*ln(x/(ln(4*x)-1))*x^2-2*ln(ln(4*x)-1)+2*ln(x)

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maxima [B]  time = 0.48, size = 43, normalized size = 1.79 \begin {gather*} \frac {1}{2} \, {\left (x^{2} - 630 \, x + 4\right )} \log \relax (x) - \frac {1}{2} \, {\left (x^{2} - 630 \, x - 4\right )} \log \left (2 \, \log \relax (2) + \log \relax (x) - 1\right ) - 4 \, \log \left (2 \, \log \relax (2) + \log \relax (x) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-630*x)*log(4*x)-2*x^2+630*x)*log(x/(log(4*x)-1))+(x^2-630*x+4)*log(4*x)-2*x^2+1260*x-8)/(2*
x*log(4*x)-2*x),x, algorithm="maxima")

[Out]

1/2*(x^2 - 630*x + 4)*log(x) - 1/2*(x^2 - 630*x - 4)*log(2*log(2) + log(x) - 1) - 4*log(2*log(2) + log(x) - 1)

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mupad [B]  time = 8.40, size = 36, normalized size = 1.50 \begin {gather*} 2\,\ln \relax (x)-2\,\ln \left (\ln \left (4\,x\right )-1\right )-\ln \left (\frac {x}{\ln \left (4\,x\right )-1}\right )\,\left (315\,x-\frac {x^2}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x/(log(4*x) - 1))*(log(4*x)*(630*x - 2*x^2) - 630*x + 2*x^2) - 1260*x - log(4*x)*(x^2 - 630*x + 4) +
2*x^2 + 8)/(2*x - 2*x*log(4*x)),x)

[Out]

2*log(x) - 2*log(log(4*x) - 1) - log(x/(log(4*x) - 1))*(315*x - x^2/2)

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sympy [A]  time = 0.43, size = 32, normalized size = 1.33 \begin {gather*} \left (\frac {x^{2}}{2} - 315 x\right ) \log {\left (\frac {x}{\log {\left (4 x \right )} - 1} \right )} + 2 \log {\relax (x )} - 2 \log {\left (\log {\left (4 x \right )} - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**2-630*x)*ln(4*x)-2*x**2+630*x)*ln(x/(ln(4*x)-1))+(x**2-630*x+4)*ln(4*x)-2*x**2+1260*x-8)/(2*
x*ln(4*x)-2*x),x)

[Out]

(x**2/2 - 315*x)*log(x/(log(4*x) - 1)) + 2*log(x) - 2*log(log(4*x) - 1)

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