3.102.31 \(\int \frac {(7 x-7 x^3) \log (x)+(28 x-28 x^2) \log ^2(x)+(2-2 x^2+4 x^2 \log (x)+8 x \log ^2(x)) \log (\frac {1-x^2+(4-4 x) \log (x)}{4 \log (x)})}{(-x+x^3) \log (x)+(-4 x+4 x^2) \log ^2(x)} \, dx\) [10131]

Optimal. Leaf size=25 \[ -7 x+\log ^2\left (1-x-\frac {-1+x^2}{4 \log (x)}\right ) \]

[Out]

ln(1-x-1/4*(x^2-1)/ln(x))^2-7*x

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Rubi [F]
time = 10.83, 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 {\left (7 x-7 x^3\right ) \log (x)+\left (28 x-28 x^2\right ) \log ^2(x)+\left (2-2 x^2+4 x^2 \log (x)+8 x \log ^2(x)\right ) \log \left (\frac {1-x^2+(4-4 x) \log (x)}{4 \log (x)}\right )}{\left (-x+x^3\right ) \log (x)+\left (-4 x+4 x^2\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((7*x - 7*x^3)*Log[x] + (28*x - 28*x^2)*Log[x]^2 + (2 - 2*x^2 + 4*x^2*Log[x] + 8*x*Log[x]^2)*Log[(1 - x^2
+ (4 - 4*x)*Log[x])/(4*Log[x])])/((-x + x^3)*Log[x] + (-4*x + 4*x^2)*Log[x]^2),x]

[Out]

-7*x + 4*Defer[Int][Log[-1/4*((-1 + x)*(1 + x + 4*Log[x]))/Log[x]]/(1 + x + 4*Log[x]), x] + 4*Defer[Int][Log[-
1/4*((-1 + x)*(1 + x + 4*Log[x]))/Log[x]]/((-1 + x)*(1 + x + 4*Log[x])), x] - 2*Defer[Int][Log[-1/4*((-1 + x)*
(1 + x + 4*Log[x]))/Log[x]]/(Log[x]*(1 + x + 4*Log[x])), x] - 2*Defer[Int][Log[-1/4*((-1 + x)*(1 + x + 4*Log[x
]))/Log[x]]/(x*Log[x]*(1 + x + 4*Log[x])), x] + 8*Defer[Int][(Log[x]*Log[-1/4*((-1 + x)*(1 + x + 4*Log[x]))/Lo
g[x]])/((-1 + x)*(1 + x + 4*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-\left (\left (7 x-7 x^3\right ) \log (x)\right )-\left (28 x-28 x^2\right ) \log ^2(x)-\left (2-2 x^2+4 x^2 \log (x)+8 x \log ^2(x)\right ) \log \left (\frac {1-x^2+(4-4 x) \log (x)}{4 \log (x)}\right )}{(1-x) x \log (x) (1+x+4 \log (x))} \, dx\\ &=\int \left (-7+\frac {2 \left (1-x^2+2 x^2 \log (x)+4 x \log ^2(x)\right ) \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{(-1+x) x \log (x) (1+x+4 \log (x))}\right ) \, dx\\ &=-7 x+2 \int \frac {\left (1-x^2+2 x^2 \log (x)+4 x \log ^2(x)\right ) \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{(-1+x) x \log (x) (1+x+4 \log (x))} \, dx\\ &=-7 x+2 \int \left (\frac {\left (1-x^2+2 x^2 \log (x)+4 x \log ^2(x)\right ) \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{(-1+x) \log (x) (1+x+4 \log (x))}-\frac {\left (1-x^2+2 x^2 \log (x)+4 x \log ^2(x)\right ) \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{x \log (x) (1+x+4 \log (x))}\right ) \, dx\\ &=-7 x+2 \int \frac {\left (1-x^2+2 x^2 \log (x)+4 x \log ^2(x)\right ) \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{(-1+x) \log (x) (1+x+4 \log (x))} \, dx-2 \int \frac {\left (1-x^2+2 x^2 \log (x)+4 x \log ^2(x)\right ) \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{x \log (x) (1+x+4 \log (x))} \, dx\\ &=-7 x-2 \int \left (\frac {2 x \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{1+x+4 \log (x)}+\frac {\log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{x \log (x) (1+x+4 \log (x))}-\frac {x \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{\log (x) (1+x+4 \log (x))}+\frac {4 \log (x) \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{1+x+4 \log (x)}\right ) \, dx+2 \int \left (\frac {2 x^2 \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{(-1+x) (1+x+4 \log (x))}+\frac {\log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{(-1+x) \log (x) (1+x+4 \log (x))}-\frac {x^2 \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{(-1+x) \log (x) (1+x+4 \log (x))}+\frac {4 x \log (x) \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{(-1+x) (1+x+4 \log (x))}\right ) \, dx\\ &=-7 x+2 \int \frac {\log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{(-1+x) \log (x) (1+x+4 \log (x))} \, dx-2 \int \frac {\log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{x \log (x) (1+x+4 \log (x))} \, dx+2 \int \frac {x \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{\log (x) (1+x+4 \log (x))} \, dx-2 \int \frac {x^2 \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{(-1+x) \log (x) (1+x+4 \log (x))} \, dx-4 \int \frac {x \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{1+x+4 \log (x)} \, dx+4 \int \frac {x^2 \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{(-1+x) (1+x+4 \log (x))} \, dx-8 \int \frac {\log (x) \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{1+x+4 \log (x)} \, dx+8 \int \frac {x \log (x) \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{(-1+x) (1+x+4 \log (x))} \, dx\\ &=-7 x+2 \int \frac {\log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{(-1+x) \log (x) (1+x+4 \log (x))} \, dx-2 \int \frac {\log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{x \log (x) (1+x+4 \log (x))} \, dx+2 \int \frac {x \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{\log (x) (1+x+4 \log (x))} \, dx-2 \int \left (\frac {\log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{\log (x) (1+x+4 \log (x))}+\frac {\log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{(-1+x) \log (x) (1+x+4 \log (x))}+\frac {x \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{\log (x) (1+x+4 \log (x))}\right ) \, dx-4 \int \frac {x \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{1+x+4 \log (x)} \, dx+4 \int \left (\frac {\log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{1+x+4 \log (x)}+\frac {\log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{(-1+x) (1+x+4 \log (x))}+\frac {x \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{1+x+4 \log (x)}\right ) \, dx-8 \int \frac {\log (x) \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{1+x+4 \log (x)} \, dx+8 \int \left (\frac {\log (x) \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{1+x+4 \log (x)}+\frac {\log (x) \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{(-1+x) (1+x+4 \log (x))}\right ) \, dx\\ &=-7 x-2 \int \frac {\log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{\log (x) (1+x+4 \log (x))} \, dx-2 \int \frac {\log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{x \log (x) (1+x+4 \log (x))} \, dx+4 \int \frac {\log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{1+x+4 \log (x)} \, dx+4 \int \frac {\log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{(-1+x) (1+x+4 \log (x))} \, dx+8 \int \frac {\log (x) \log \left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right )}{(-1+x) (1+x+4 \log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.05, size = 25, normalized size = 1.00 \begin {gather*} -7 x+\log ^2\left (-\frac {(-1+x) (1+x+4 \log (x))}{4 \log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((7*x - 7*x^3)*Log[x] + (28*x - 28*x^2)*Log[x]^2 + (2 - 2*x^2 + 4*x^2*Log[x] + 8*x*Log[x]^2)*Log[(1
- x^2 + (4 - 4*x)*Log[x])/(4*Log[x])])/((-x + x^3)*Log[x] + (-4*x + 4*x^2)*Log[x]^2),x]

[Out]

-7*x + Log[-1/4*((-1 + x)*(1 + x + 4*Log[x]))/Log[x]]^2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs. \(2(23)=46\).
time = 1.18, size = 55, normalized size = 2.20

method	result	size
default	\(-7 x -4 \ln \left (2\right ) \ln \left (x -1\right )-4 \ln \left (2\right ) \ln \left (x +4 \ln \left (x \right )+1\right )+4 \ln \left (2\right ) \ln \left (\ln \left (x \right )\right )+\ln \left (-\frac {4 x \ln \left (x \right )+x^{2}-4 \ln \left (x \right )-1}{\ln \left (x \right )}\right )^{2}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x*ln(x)^2+4*x^2*ln(x)-2*x^2+2)*ln(1/4*((-4*x+4)*ln(x)-x^2+1)/ln(x))+(-28*x^2+28*x)*ln(x)^2+(-7*x^3+7*x
)*ln(x))/((4*x^2-4*x)*ln(x)^2+(x^3-x)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

-7*x-4*ln(2)*ln(x-1)-4*ln(2)*ln(x+4*ln(x)+1)+4*ln(2)*ln(ln(x))+ln(-(4*x*ln(x)+x^2-4*ln(x)-1)/ln(x))^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (23) = 46\).
time = 0.51, size = 76, normalized size = 3.04 \begin {gather*} -2 \, {\left (2 \, \log \left (2\right ) - \log \left (-x + 1\right ) + \log \left (\log \left (x\right )\right )\right )} \log \left (x + 4 \, \log \left (x\right ) + 1\right ) + \log \left (x + 4 \, \log \left (x\right ) + 1\right )^{2} - 2 \, {\left (2 \, \log \left (2\right ) + \log \left (\log \left (x\right )\right )\right )} \log \left (-x + 1\right ) + \log \left (-x + 1\right )^{2} + 4 \, \log \left (2\right ) \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2} - 7 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x*log(x)^2+4*x^2*log(x)-2*x^2+2)*log(1/4*((-4*x+4)*log(x)-x^2+1)/log(x))+(-28*x^2+28*x)*log(x)^2
+(-7*x^3+7*x)*log(x))/((4*x^2-4*x)*log(x)^2+(x^3-x)*log(x)),x, algorithm="maxima")

[Out]

-2*(2*log(2) - log(-x + 1) + log(log(x)))*log(x + 4*log(x) + 1) + log(x + 4*log(x) + 1)^2 - 2*(2*log(2) + log(
log(x)))*log(-x + 1) + log(-x + 1)^2 + 4*log(2)*log(log(x)) + log(log(x))^2 - 7*x

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Fricas [A]
time = 0.36, size = 25, normalized size = 1.00 \begin {gather*} \log \left (-\frac {x^{2} + 4 \, {\left (x - 1\right )} \log \left (x\right ) - 1}{4 \, \log \left (x\right )}\right )^{2} - 7 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x*log(x)^2+4*x^2*log(x)-2*x^2+2)*log(1/4*((-4*x+4)*log(x)-x^2+1)/log(x))+(-28*x^2+28*x)*log(x)^2
+(-7*x^3+7*x)*log(x))/((4*x^2-4*x)*log(x)^2+(x^3-x)*log(x)),x, algorithm="fricas")

[Out]

log(-1/4*(x^2 + 4*(x - 1)*log(x) - 1)/log(x))^2 - 7*x

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Sympy [A]
time = 0.19, size = 27, normalized size = 1.08 \begin {gather*} - 7 x + \log {\left (\frac {- \frac {x^{2}}{4} + \frac {\left (4 - 4 x\right ) \log {\left (x \right )}}{4} + \frac {1}{4}}{\log {\left (x \right )}} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x*ln(x)**2+4*x**2*ln(x)-2*x**2+2)*ln(1/4*((-4*x+4)*ln(x)-x**2+1)/ln(x))+(-28*x**2+28*x)*ln(x)**2
+(-7*x**3+7*x)*ln(x))/((4*x**2-4*x)*ln(x)**2+(x**3-x)*ln(x)),x)

[Out]

-7*x + log((-x**2/4 + (4 - 4*x)*log(x)/4 + 1/4)/log(x))**2

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (23) = 46\).
time = 0.47, size = 148, normalized size = 5.92 \begin {gather*} 2 \, {\left (\log \left (x + 4 \, \log \left (x\right ) + 1\right ) + \log \left (x - 1\right ) - \log \left (\log \left (x\right )\right )\right )} \log \left (-x^{2} - 4 \, x \log \left (x\right ) + 4 \, \log \left (x\right ) + 1\right ) - 2 \, {\left (\log \left (x + 4 \, \log \left (x\right ) + 1\right ) + \log \left (x - 1\right )\right )} \log \left (x + 4 \, \log \left (x\right ) + 1\right ) - 4 \, \log \left (2\right ) \log \left (x + 4 \, \log \left (x\right ) + 1\right ) + 2 \, \log \left (x + 4 \, \log \left (x\right ) + 1\right )^{2} - 4 \, \log \left (2\right ) \log \left (x - 1\right ) - \log \left (x - 1\right )^{2} - 2 \, \log \left (x + 4 \, \log \left (x\right ) + 1\right ) \log \left (-x - 4 \, \log \left (x\right ) - 1\right ) + \log \left (-x - 4 \, \log \left (x\right ) - 1\right )^{2} + 4 \, \log \left (2\right ) \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2} - 7 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x*log(x)^2+4*x^2*log(x)-2*x^2+2)*log(1/4*((-4*x+4)*log(x)-x^2+1)/log(x))+(-28*x^2+28*x)*log(x)^2
+(-7*x^3+7*x)*log(x))/((4*x^2-4*x)*log(x)^2+(x^3-x)*log(x)),x, algorithm="giac")

[Out]

2*(log(x + 4*log(x) + 1) + log(x - 1) - log(log(x)))*log(-x^2 - 4*x*log(x) + 4*log(x) + 1) - 2*(log(x + 4*log(
x) + 1) + log(x - 1))*log(x + 4*log(x) + 1) - 4*log(2)*log(x + 4*log(x) + 1) + 2*log(x + 4*log(x) + 1)^2 - 4*l
og(2)*log(x - 1) - log(x - 1)^2 - 2*log(x + 4*log(x) + 1)*log(-x - 4*log(x) - 1) + log(-x - 4*log(x) - 1)^2 +
4*log(2)*log(log(x)) + log(log(x))^2 - 7*x

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Mupad [B]
time = 8.63, size = 29, normalized size = 1.16 \begin {gather*} {\ln \left (-\frac {\frac {\ln \left (x\right )\,\left (4\,x-4\right )}{4}+\frac {x^2}{4}-\frac {1}{4}}{\ln \left (x\right )}\right )}^2-7\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)^2*(28*x - 28*x^2) + log(-((log(x)*(4*x - 4))/4 + x^2/4 - 1/4)/log(x))*(8*x*log(x)^2 + 4*x^2*log(x
) - 2*x^2 + 2) + log(x)*(7*x - 7*x^3))/(log(x)^2*(4*x - 4*x^2) + log(x)*(x - x^3)),x)

[Out]

log(-((log(x)*(4*x - 4))/4 + x^2/4 - 1/4)/log(x))^2 - 7*x

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