3.33.58 \(\int \frac {(5-2 x) \log (x)+(-5+x) \log (-5 x+x^2)+(5 x-x^2+(-5 x+x^2) \log (x)) \log ^2(-5 x+x^2)+(-5 x+x^2+(-5 x+x^2) \log (x)) \log (x^2) \log ^2(-5 x+x^2)}{(-5 x+x^2) \log ^2(-5 x+x^2)} \, dx\)

Optimal. Leaf size=21 \[ \log (x) \left (-x+\frac {1}{\log ((-5+x) x)}+x \log \left (x^2\right )\right ) \]

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Rubi [F]  time = 1.92, 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 {(5-2 x) \log (x)+(-5+x) \log \left (-5 x+x^2\right )+\left (5 x-x^2+\left (-5 x+x^2\right ) \log (x)\right ) \log ^2\left (-5 x+x^2\right )+\left (-5 x+x^2+\left (-5 x+x^2\right ) \log (x)\right ) \log \left (x^2\right ) \log ^2\left (-5 x+x^2\right )}{\left (-5 x+x^2\right ) \log ^2\left (-5 x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((5 - 2*x)*Log[x] + (-5 + x)*Log[-5*x + x^2] + (5*x - x^2 + (-5*x + x^2)*Log[x])*Log[-5*x + x^2]^2 + (-5*x
+ x^2 + (-5*x + x^2)*Log[x])*Log[x^2]*Log[-5*x + x^2]^2)/((-5*x + x^2)*Log[-5*x + x^2]^2),x]

[Out]

-(x*Log[x]) + x*Log[x]*Log[x^2] - Defer[Int][Log[x]/((-5 + x)*Log[(-5 + x)*x]^2), x] - Defer[Int][Log[x]/(x*Lo
g[(-5 + x)*x]^2), x] + Defer[Int][1/(x*Log[(-5 + x)*x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(5-2 x) \log (x)+(-5+x) \log \left (-5 x+x^2\right )+\left (5 x-x^2+\left (-5 x+x^2\right ) \log (x)\right ) \log ^2\left (-5 x+x^2\right )+\left (-5 x+x^2+\left (-5 x+x^2\right ) \log (x)\right ) \log \left (x^2\right ) \log ^2\left (-5 x+x^2\right )}{(-5+x) x \log ^2\left (-5 x+x^2\right )} \, dx\\ &=\int \frac {-\left ((-5+x) \log ((-5+x) x) \left (1+x \log ((-5+x) x) \left (-1+\log \left (x^2\right )\right )\right )\right )-\log (x) \left (5-2 x+(-5+x) x \log ^2((-5+x) x) \left (1+\log \left (x^2\right )\right )\right )}{(5-x) x \log ^2((-5+x) x)} \, dx\\ &=\int \left (\frac {5 \log (x)-2 x \log (x)-5 \log ((-5+x) x)+x \log ((-5+x) x)+5 x \log ^2((-5+x) x)-x^2 \log ^2((-5+x) x)-5 x \log (x) \log ^2((-5+x) x)+x^2 \log (x) \log ^2((-5+x) x)}{(-5+x) x \log ^2((-5+x) x)}+(1+\log (x)) \log \left (x^2\right )\right ) \, dx\\ &=\int \frac {5 \log (x)-2 x \log (x)-5 \log ((-5+x) x)+x \log ((-5+x) x)+5 x \log ^2((-5+x) x)-x^2 \log ^2((-5+x) x)-5 x \log (x) \log ^2((-5+x) x)+x^2 \log (x) \log ^2((-5+x) x)}{(-5+x) x \log ^2((-5+x) x)} \, dx+\int (1+\log (x)) \log \left (x^2\right ) \, dx\\ &=x \log (x) \log \left (x^2\right )-2 \int \log (x) \, dx+\int \frac {(-5+x) \log ((-5+x) x) (-1+x \log ((-5+x) x))-\log (x) \left (5-2 x+(-5+x) x \log ^2((-5+x) x)\right )}{(5-x) x \log ^2((-5+x) x)} \, dx\\ &=2 x-2 x \log (x)+x \log (x) \log \left (x^2\right )+\int \left (-1+\log (x)-\frac {(-5+2 x) \log (x)}{(-5+x) x \log ^2((-5+x) x)}+\frac {1}{x \log ((-5+x) x)}\right ) \, dx\\ &=x-2 x \log (x)+x \log (x) \log \left (x^2\right )+\int \log (x) \, dx-\int \frac {(-5+2 x) \log (x)}{(-5+x) x \log ^2((-5+x) x)} \, dx+\int \frac {1}{x \log ((-5+x) x)} \, dx\\ &=-x \log (x)+x \log (x) \log \left (x^2\right )-\int \left (\frac {\log (x)}{(-5+x) \log ^2((-5+x) x)}+\frac {\log (x)}{x \log ^2((-5+x) x)}\right ) \, dx+\int \frac {1}{x \log ((-5+x) x)} \, dx\\ &=-x \log (x)+x \log (x) \log \left (x^2\right )-\int \frac {\log (x)}{(-5+x) \log ^2((-5+x) x)} \, dx-\int \frac {\log (x)}{x \log ^2((-5+x) x)} \, dx+\int \frac {1}{x \log ((-5+x) x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 20, normalized size = 0.95 \begin {gather*} \log (x) \left (\frac {1}{\log ((-5+x) x)}+x \left (-1+\log \left (x^2\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - 2*x)*Log[x] + (-5 + x)*Log[-5*x + x^2] + (5*x - x^2 + (-5*x + x^2)*Log[x])*Log[-5*x + x^2]^2 +
(-5*x + x^2 + (-5*x + x^2)*Log[x])*Log[x^2]*Log[-5*x + x^2]^2)/((-5*x + x^2)*Log[-5*x + x^2]^2),x]

[Out]

Log[x]*(Log[(-5 + x)*x]^(-1) + x*(-1 + Log[x^2]))

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fricas [A]  time = 0.56, size = 36, normalized size = 1.71 \begin {gather*} \frac {{\left (2 \, x \log \relax (x)^{2} - x \log \relax (x)\right )} \log \left (x^{2} - 5 \, x\right ) + \log \relax (x)}{\log \left (x^{2} - 5 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*x*log(x)^2 - x*log(x))*log(x^2 - 5*x) + log(x))/log(x^2 - 5*x)

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giac [A]  time = 0.35, size = 25, normalized size = 1.19 \begin {gather*} 2 \, x \log \relax (x)^{2} - x \log \relax (x) + \frac {\log \relax (x)}{\log \left (x - 5\right ) + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x*log(x)^2 - x*log(x) + log(x)/(log(x - 5) + log(x))

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maple [C]  time = 0.72, size = 167, normalized size = 7.95

method	result	size
risch	\(2 x \ln \relax (x )^{2}-\frac {i \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) \ln \relax (x )}{2}+i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} \ln \relax (x )-\frac {i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3} \ln \relax (x )}{2}-x \ln \relax (x )+\frac {2 \ln \relax (x )}{2 \ln \relax (x )+2 \ln \left (x -5\right )-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -5\right )\right ) \mathrm {csgn}\left (i x \left (x -5\right )\right )+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -5\right )\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (x -5\right )\right ) \mathrm {csgn}\left (i x \left (x -5\right )\right )^{2}-i \pi \mathrm {csgn}\left (i x \left (x -5\right )\right )^{3}}\)	\(167\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x*ln(x)^2-1/2*I*Pi*x*csgn(I*x)^2*csgn(I*x^2)*ln(x)+I*Pi*x*csgn(I*x)*csgn(I*x^2)^2*ln(x)-1/2*I*Pi*x*csgn(I*x^
2)^3*ln(x)-x*ln(x)+2*ln(x)/(2*ln(x)+2*ln(x-5)-I*Pi*csgn(I*x)*csgn(I*(x-5))*csgn(I*x*(x-5))+I*Pi*csgn(I*x)*csgn
(I*x*(x-5))^2+I*Pi*csgn(I*(x-5))*csgn(I*x*(x-5))^2-I*Pi*csgn(I*x*(x-5))^3)

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maxima [B]  time = 0.47, size = 45, normalized size = 2.14 \begin {gather*} \frac {2 \, x \log \relax (x)^{3} - x \log \relax (x)^{2} + {\left (2 \, x \log \relax (x)^{2} - x \log \relax (x)\right )} \log \left (x - 5\right ) + \log \relax (x)}{\log \left (x - 5\right ) + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x*log(x)^3 - x*log(x)^2 + (2*x*log(x)^2 - x*log(x))*log(x - 5) + log(x))/(log(x - 5) + log(x))

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mupad [B]  time = 2.18, size = 55, normalized size = 2.62 \begin {gather*} \frac {\ln \relax (x)}{\ln \left (x^2-5\,x\right )}-\frac {5}{4\,\left (x-\frac {5}{2}\right )}-\frac {x}{2\,x-5}+\frac {5}{2\,x-5}-x\,\ln \relax (x)+x\,\ln \left (x^2\right )\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(2*x - 5) - log(x^2 - 5*x)*(x - 5) + log(x^2 - 5*x)^2*(log(x)*(5*x - x^2) - 5*x + x^2) + log(x^2)*
log(x^2 - 5*x)^2*(5*x + log(x)*(5*x - x^2) - x^2))/(log(x^2 - 5*x)^2*(5*x - x^2)),x)

[Out]

log(x)/log(x^2 - 5*x) - 5/(4*(x - 5/2)) - x/(2*x - 5) + 5/(2*x - 5) - x*log(x) + x*log(x^2)*log(x)

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sympy [A]  time = 0.33, size = 24, normalized size = 1.14 \begin {gather*} 2 x \log {\relax (x )}^{2} - x \log {\relax (x )} + \frac {\log {\relax (x )}}{\log {\left (x^{2} - 5 x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**2-5*x)*ln(x)+x**2-5*x)*ln(x**2-5*x)**2*ln(x**2)+((x**2-5*x)*ln(x)-x**2+5*x)*ln(x**2-5*x)**2+(x
-5)*ln(x**2-5*x)+(-2*x+5)*ln(x))/(x**2-5*x)/ln(x**2-5*x)**2,x)

[Out]

2*x*log(x)**2 - x*log(x) + log(x)/log(x**2 - 5*x)

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