3.82.18 \(\int \frac {(2560-512 x) \log (x)+(-1280+256 x+(-1280-256 x) \log (x)) \log (x^2)+(-250 x+150 x^2-30 x^3+2 x^4) \log ^2(x^2)}{(-125+75 x-15 x^2+x^3) \log ^2(x^2)} \, dx\)

Optimal. Leaf size=31 \[ 4+x^2+\frac {x \left (2+\frac {2 (3+x)}{5-x}\right )^2 \log (x)}{\log \left (x^2\right )} \]

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Rubi [F]  time = 0.81, 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 {(2560-512 x) \log (x)+(-1280+256 x+(-1280-256 x) \log (x)) \log \left (x^2\right )+\left (-250 x+150 x^2-30 x^3+2 x^4\right ) \log ^2\left (x^2\right )}{\left (-125+75 x-15 x^2+x^3\right ) \log ^2\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((2560 - 512*x)*Log[x] + (-1280 + 256*x + (-1280 - 256*x)*Log[x])*Log[x^2] + (-250*x + 150*x^2 - 30*x^3 +
2*x^4)*Log[x^2]^2)/((-125 + 75*x - 15*x^2 + x^3)*Log[x^2]^2),x]

[Out]

x^2 - 512*Defer[Int][Log[x]/((-5 + x)^2*Log[x^2]^2), x] - 1280*Defer[Int][1/((-5 + x)^3*Log[x^2]), x] + 256*De
fer[Int][x/((-5 + x)^3*Log[x^2]), x] - 2560*Defer[Int][Log[x]/((-5 + x)^3*Log[x^2]), x] - 256*Defer[Int][Log[x
]/((-5 + x)^2*Log[x^2]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 (-5+x) \log \left (x^2\right ) \left (128+(-5+x)^2 x \log \left (x^2\right )\right )+256 \log (x) \left (2 (-5+x)+(5+x) \log \left (x^2\right )\right )}{(5-x)^3 \log ^2\left (x^2\right )} \, dx\\ &=\int \left (2 x-\frac {512 \log (x)}{(-5+x)^2 \log ^2\left (x^2\right )}-\frac {256 (5-x+5 \log (x)+x \log (x))}{(-5+x)^3 \log \left (x^2\right )}\right ) \, dx\\ &=x^2-256 \int \frac {5-x+5 \log (x)+x \log (x)}{(-5+x)^3 \log \left (x^2\right )} \, dx-512 \int \frac {\log (x)}{(-5+x)^2 \log ^2\left (x^2\right )} \, dx\\ &=x^2-256 \int \left (\frac {5}{(-5+x)^3 \log \left (x^2\right )}-\frac {x}{(-5+x)^3 \log \left (x^2\right )}+\frac {5 \log (x)}{(-5+x)^3 \log \left (x^2\right )}+\frac {x \log (x)}{(-5+x)^3 \log \left (x^2\right )}\right ) \, dx-512 \int \frac {\log (x)}{(-5+x)^2 \log ^2\left (x^2\right )} \, dx\\ &=x^2+256 \int \frac {x}{(-5+x)^3 \log \left (x^2\right )} \, dx-256 \int \frac {x \log (x)}{(-5+x)^3 \log \left (x^2\right )} \, dx-512 \int \frac {\log (x)}{(-5+x)^2 \log ^2\left (x^2\right )} \, dx-1280 \int \frac {1}{(-5+x)^3 \log \left (x^2\right )} \, dx-1280 \int \frac {\log (x)}{(-5+x)^3 \log \left (x^2\right )} \, dx\\ &=x^2-256 \int \left (\frac {5 \log (x)}{(-5+x)^3 \log \left (x^2\right )}+\frac {\log (x)}{(-5+x)^2 \log \left (x^2\right )}\right ) \, dx+256 \int \frac {x}{(-5+x)^3 \log \left (x^2\right )} \, dx-512 \int \frac {\log (x)}{(-5+x)^2 \log ^2\left (x^2\right )} \, dx-1280 \int \frac {1}{(-5+x)^3 \log \left (x^2\right )} \, dx-1280 \int \frac {\log (x)}{(-5+x)^3 \log \left (x^2\right )} \, dx\\ &=x^2+256 \int \frac {x}{(-5+x)^3 \log \left (x^2\right )} \, dx-256 \int \frac {\log (x)}{(-5+x)^2 \log \left (x^2\right )} \, dx-512 \int \frac {\log (x)}{(-5+x)^2 \log ^2\left (x^2\right )} \, dx-1280 \int \frac {1}{(-5+x)^3 \log \left (x^2\right )} \, dx-2 \left (1280 \int \frac {\log (x)}{(-5+x)^3 \log \left (x^2\right )} \, dx\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 29, normalized size = 0.94 \begin {gather*} \frac {x \left (256 \log (x)+(-5+x)^2 x \log \left (x^2\right )\right )}{(-5+x)^2 \log \left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2560 - 512*x)*Log[x] + (-1280 + 256*x + (-1280 - 256*x)*Log[x])*Log[x^2] + (-250*x + 150*x^2 - 30*
x^3 + 2*x^4)*Log[x^2]^2)/((-125 + 75*x - 15*x^2 + x^3)*Log[x^2]^2),x]

[Out]

(x*(256*Log[x] + (-5 + x)^2*x*Log[x^2]))/((-5 + x)^2*Log[x^2])

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fricas [A]  time = 0.59, size = 28, normalized size = 0.90 \begin {gather*} \frac {x^{4} - 10 \, x^{3} + 25 \, x^{2} + 128 \, x}{x^{2} - 10 \, x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4-30*x^3+150*x^2-250*x)*log(x^2)^2+((-256*x-1280)*log(x)+256*x-1280)*log(x^2)+(-512*x+2560)*lo
g(x))/(x^3-15*x^2+75*x-125)/log(x^2)^2,x, algorithm="fricas")

[Out]

(x^4 - 10*x^3 + 25*x^2 + 128*x)/(x^2 - 10*x + 25)

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giac [A]  time = 0.20, size = 17, normalized size = 0.55 \begin {gather*} x^{2} + \frac {128 \, x}{x^{2} - 10 \, x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4-30*x^3+150*x^2-250*x)*log(x^2)^2+((-256*x-1280)*log(x)+256*x-1280)*log(x^2)+(-512*x+2560)*lo
g(x))/(x^3-15*x^2+75*x-125)/log(x^2)^2,x, algorithm="giac")

[Out]

x^2 + 128*x/(x^2 - 10*x + 25)

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maple [C]  time = 0.11, size = 131, normalized size = 4.23

method	result	size
risch	\(\frac {x \left (x^{3}-10 x^{2}+25 x +128\right )}{x^{2}-10 x +25}-\frac {128 x \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (\mathrm {csgn}\left (i x \right )^{2}-2 \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )+\mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{\left (x^{2}-10 x +25\right ) \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}\)	\(131\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^4-30*x^3+150*x^2-250*x)*ln(x^2)^2+((-256*x-1280)*ln(x)+256*x-1280)*ln(x^2)+(-512*x+2560)*ln(x))/(x^3
-15*x^2+75*x-125)/ln(x^2)^2,x,method=_RETURNVERBOSE)

[Out]

x*(x^3-10*x^2+25*x+128)/(x^2-10*x+25)-128*x*Pi*csgn(I*x^2)*(csgn(I*x)^2-2*csgn(I*x^2)*csgn(I*x)+csgn(I*x^2)^2)
/(x^2-10*x+25)/(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)

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maxima [A]  time = 0.41, size = 28, normalized size = 0.90 \begin {gather*} \frac {x^{4} - 10 \, x^{3} + 25 \, x^{2} + 128 \, x}{x^{2} - 10 \, x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4-30*x^3+150*x^2-250*x)*log(x^2)^2+((-256*x-1280)*log(x)+256*x-1280)*log(x^2)+(-512*x+2560)*lo
g(x))/(x^3-15*x^2+75*x-125)/log(x^2)^2,x, algorithm="maxima")

[Out]

(x^4 - 10*x^3 + 25*x^2 + 128*x)/(x^2 - 10*x + 25)

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mupad [B]  time = 5.90, size = 42, normalized size = 1.35 \begin {gather*} \frac {x\,\left (256\,\ln \relax (x)+25\,x\,\ln \left (x^2\right )-10\,x^2\,\ln \left (x^2\right )+x^3\,\ln \left (x^2\right )\right )}{\ln \left (x^2\right )\,{\left (x-5\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x^2)*(log(x)*(256*x + 1280) - 256*x + 1280) + log(x)*(512*x - 2560) + log(x^2)^2*(250*x - 150*x^2 +
30*x^3 - 2*x^4))/(log(x^2)^2*(75*x - 15*x^2 + x^3 - 125)),x)

[Out]

(x*(256*log(x) + 25*x*log(x^2) - 10*x^2*log(x^2) + x^3*log(x^2)))/(log(x^2)*(x - 5)^2)

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sympy [A]  time = 0.27, size = 14, normalized size = 0.45 \begin {gather*} x^{2} + \frac {128 x}{x^{2} - 10 x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**4-30*x**3+150*x**2-250*x)*ln(x**2)**2+((-256*x-1280)*ln(x)+256*x-1280)*ln(x**2)+(-512*x+2560)
*ln(x))/(x**3-15*x**2+75*x-125)/ln(x**2)**2,x)

[Out]

x**2 + 128*x/(x**2 - 10*x + 25)

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