3.54.99 \(\int \frac {(-7500 x-3750 x^3) \log (x)+(3750 x-1875 x^3+(-7500 x+3750 x^3) \log (x)) \log (\frac {16-16 x^2+4 x^4}{x^2})}{(-2+x^2) \log ^2(x) \log ^2(\frac {16-16 x^2+4 x^4}{x^2})} \, dx\)

Optimal. Leaf size=25 \[ \frac {1875 x^2}{\log (x) \log \left (\frac {\left (4-2 x^2\right )^2}{x^2}\right )} \]

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Rubi [F]  time = 1.42, 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 (-7500 x-3750 x^3\right ) \log (x)+\left (3750 x-1875 x^3+\left (-7500 x+3750 x^3\right ) \log (x)\right ) \log \left (\frac {16-16 x^2+4 x^4}{x^2}\right )}{\left (-2+x^2\right ) \log ^2(x) \log ^2\left (\frac {16-16 x^2+4 x^4}{x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-7500*x - 3750*x^3)*Log[x] + (3750*x - 1875*x^3 + (-7500*x + 3750*x^3)*Log[x])*Log[(16 - 16*x^2 + 4*x^4)
/x^2])/((-2 + x^2)*Log[x]^2*Log[(16 - 16*x^2 + 4*x^4)/x^2]^2),x]

[Out]

7500*Defer[Int][1/((Sqrt[2] - x)*Log[x]*Log[(4*(-2 + x^2)^2)/x^2]^2), x] - 3750*Defer[Int][x/(Log[x]*Log[(4*(-
2 + x^2)^2)/x^2]^2), x] - 7500*Defer[Int][1/((Sqrt[2] + x)*Log[x]*Log[(4*(-2 + x^2)^2)/x^2]^2), x] - 1875*Defe
r[Int][x/(Log[x]^2*Log[(4*(-2 + x^2)^2)/x^2]), x] + 3750*Defer[Int][x/(Log[x]*Log[(4*(-2 + x^2)^2)/x^2]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {3750 x \left (2+x^2\right )}{\left (-2+x^2\right ) \log (x) \log ^2\left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )}+\frac {1875 x (-1+2 \log (x))}{\log ^2(x) \log \left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )}\right ) \, dx\\ &=1875 \int \frac {x (-1+2 \log (x))}{\log ^2(x) \log \left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )} \, dx-3750 \int \frac {x \left (2+x^2\right )}{\left (-2+x^2\right ) \log (x) \log ^2\left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )} \, dx\\ &=1875 \int \left (-\frac {x}{\log ^2(x) \log \left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )}+\frac {2 x}{\log (x) \log \left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )}\right ) \, dx-3750 \int \left (\frac {x}{\log (x) \log ^2\left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )}+\frac {4 x}{\left (-2+x^2\right ) \log (x) \log ^2\left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )}\right ) \, dx\\ &=-\left (1875 \int \frac {x}{\log ^2(x) \log \left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )} \, dx\right )-3750 \int \frac {x}{\log (x) \log ^2\left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )} \, dx+3750 \int \frac {x}{\log (x) \log \left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )} \, dx-15000 \int \frac {x}{\left (-2+x^2\right ) \log (x) \log ^2\left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )} \, dx\\ &=-\left (1875 \int \frac {x}{\log ^2(x) \log \left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )} \, dx\right )-3750 \int \frac {x}{\log (x) \log ^2\left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )} \, dx+3750 \int \frac {x}{\log (x) \log \left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )} \, dx-15000 \int \left (-\frac {1}{2 \left (\sqrt {2}-x\right ) \log (x) \log ^2\left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )}+\frac {1}{2 \left (\sqrt {2}+x\right ) \log (x) \log ^2\left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )}\right ) \, dx\\ &=-\left (1875 \int \frac {x}{\log ^2(x) \log \left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )} \, dx\right )-3750 \int \frac {x}{\log (x) \log ^2\left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )} \, dx+3750 \int \frac {x}{\log (x) \log \left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )} \, dx+7500 \int \frac {1}{\left (\sqrt {2}-x\right ) \log (x) \log ^2\left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )} \, dx-7500 \int \frac {1}{\left (\sqrt {2}+x\right ) \log (x) \log ^2\left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.36, size = 24, normalized size = 0.96 \begin {gather*} \frac {1875 x^2}{\log (x) \log \left (\frac {4 \left (-2+x^2\right )^2}{x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-7500*x - 3750*x^3)*Log[x] + (3750*x - 1875*x^3 + (-7500*x + 3750*x^3)*Log[x])*Log[(16 - 16*x^2 +
4*x^4)/x^2])/((-2 + x^2)*Log[x]^2*Log[(16 - 16*x^2 + 4*x^4)/x^2]^2),x]

[Out]

(1875*x^2)/(Log[x]*Log[(4*(-2 + x^2)^2)/x^2])

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fricas [A]  time = 0.63, size = 27, normalized size = 1.08 \begin {gather*} \frac {1875 \, x^{2}}{\log \relax (x) \log \left (\frac {4 \, {\left (x^{4} - 4 \, x^{2} + 4\right )}}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3750*x^3-7500*x)*log(x)-1875*x^3+3750*x)*log((4*x^4-16*x^2+16)/x^2)+(-3750*x^3-7500*x)*log(x))/(x
^2-2)/log(x)^2/log((4*x^4-16*x^2+16)/x^2)^2,x, algorithm="fricas")

[Out]

1875*x^2/(log(x)*log(4*(x^4 - 4*x^2 + 4)/x^2))

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giac [A]  time = 0.33, size = 30, normalized size = 1.20 \begin {gather*} \frac {1875 \, x^{2}}{\log \left (4 \, x^{4} - 16 \, x^{2} + 16\right ) \log \relax (x) - 2 \, \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3750*x^3-7500*x)*log(x)-1875*x^3+3750*x)*log((4*x^4-16*x^2+16)/x^2)+(-3750*x^3-7500*x)*log(x))/(x
^2-2)/log(x)^2/log((4*x^4-16*x^2+16)/x^2)^2,x, algorithm="giac")

[Out]

1875*x^2/(log(4*x^4 - 16*x^2 + 16)*log(x) - 2*log(x)^2)

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maple [C]  time = 0.35, size = 251, normalized size = 10.04

method	result	size
risch	\(-\frac {3750 i x^{2}}{\ln \relax (x ) \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}-\pi \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i \left (x^{2}-2\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}-2\right )^{2}}{x^{2}}\right )+\pi \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}-2\right )^{2}}{x^{2}}\right )^{2}-\pi \mathrm {csgn}\left (i \left (x^{2}-2\right )\right )^{2} \mathrm {csgn}\left (i \left (x^{2}-2\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (x^{2}-2\right )\right ) \mathrm {csgn}\left (i \left (x^{2}-2\right )^{2}\right )^{2}-\pi \mathrm {csgn}\left (i \left (x^{2}-2\right )^{2}\right )^{3}+\pi \,\mathrm {csgn}\left (i \left (x^{2}-2\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}-2\right )^{2}}{x^{2}}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i \left (x^{2}-2\right )^{2}}{x^{2}}\right )^{3}+4 i \ln \relax (x )-4 i \ln \left (x^{2}-2\right )-4 i \ln \relax (2)\right )}\)	\(251\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((3750*x^3-7500*x)*ln(x)-1875*x^3+3750*x)*ln((4*x^4-16*x^2+16)/x^2)+(-3750*x^3-7500*x)*ln(x))/(x^2-2)/ln(
x)^2/ln((4*x^4-16*x^2+16)/x^2)^2,x,method=_RETURNVERBOSE)

[Out]

-3750*I*x^2/ln(x)/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3-Pi*csgn(I/x^2)*csg
n(I*(x^2-2)^2)*csgn(I/x^2*(x^2-2)^2)+Pi*csgn(I/x^2)*csgn(I/x^2*(x^2-2)^2)^2-Pi*csgn(I*(x^2-2))^2*csgn(I*(x^2-2
)^2)+2*Pi*csgn(I*(x^2-2))*csgn(I*(x^2-2)^2)^2-Pi*csgn(I*(x^2-2)^2)^3+Pi*csgn(I*(x^2-2)^2)*csgn(I/x^2*(x^2-2)^2
)^2-Pi*csgn(I/x^2*(x^2-2)^2)^3+4*I*ln(x)-4*I*ln(x^2-2)-4*I*ln(2))

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maxima [A]  time = 0.53, size = 28, normalized size = 1.12 \begin {gather*} \frac {1875 \, x^{2}}{2 \, {\left (\log \relax (2) \log \relax (x) + \log \left (x^{2} - 2\right ) \log \relax (x) - \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3750*x^3-7500*x)*log(x)-1875*x^3+3750*x)*log((4*x^4-16*x^2+16)/x^2)+(-3750*x^3-7500*x)*log(x))/(x
^2-2)/log(x)^2/log((4*x^4-16*x^2+16)/x^2)^2,x, algorithm="maxima")

[Out]

1875/2*x^2/(log(2)*log(x) + log(x^2 - 2)*log(x) - log(x)^2)

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mupad [B]  time = 4.00, size = 292, normalized size = 11.68 \begin {gather*} 22500\,\ln \relax (x)-\frac {45000\,x^4+30000\,x^2}{x^6+6\,x^4+12\,x^2+8}-\frac {\frac {3750\,x^4}{{\left (x^2+2\right )}^2}-\frac {30000\,x^4\,\ln \relax (x)}{{\left (x^2+2\right )}^3}+\frac {3750\,x^2\,{\ln \relax (x)}^2\,\left (x^6+6\,x^4+20\,x^2-8\right )}{{\left (x^2+2\right )}^3}}{\ln \relax (x)}+\frac {\frac {1875\,x^2}{\ln \relax (x)}-\frac {1875\,x^2\,\ln \left (\frac {4\,x^4-16\,x^2+16}{x^2}\right )\,\left (x^2-2\right )\,\left (2\,\ln \relax (x)-1\right )}{2\,{\ln \relax (x)}^2\,\left (x^2+2\right )}}{\ln \left (\frac {4\,x^4-16\,x^2+16}{x^2}\right )}-\frac {\frac {1875\,x^2\,\left (x^2-2\right )}{2\,\left (x^2+2\right )}-\frac {1875\,x^2\,\ln \relax (x)\,\left (x^4+2\,x^2-4\right )}{{\left (x^2+2\right )}^2}+\frac {1875\,x^2\,{\ln \relax (x)}^2\,\left (x^4+4\,x^2-4\right )}{{\left (x^2+2\right )}^2}}{{\ln \relax (x)}^2}+1875\,x^2-\frac {\ln \relax (x)\,\left (-3750\,x^8+60000\,x^4+300000\,x^2+180000\right )}{x^6+6\,x^4+12\,x^2+8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(7500*x + 3750*x^3) + log((4*x^4 - 16*x^2 + 16)/x^2)*(log(x)*(7500*x - 3750*x^3) - 3750*x + 1875*
x^3))/(log((4*x^4 - 16*x^2 + 16)/x^2)^2*log(x)^2*(x^2 - 2)),x)

[Out]

22500*log(x) - (30000*x^2 + 45000*x^4)/(12*x^2 + 6*x^4 + x^6 + 8) - ((3750*x^4)/(x^2 + 2)^2 - (30000*x^4*log(x
))/(x^2 + 2)^3 + (3750*x^2*log(x)^2*(20*x^2 + 6*x^4 + x^6 - 8))/(x^2 + 2)^3)/log(x) + ((1875*x^2)/log(x) - (18
75*x^2*log((4*x^4 - 16*x^2 + 16)/x^2)*(x^2 - 2)*(2*log(x) - 1))/(2*log(x)^2*(x^2 + 2)))/log((4*x^4 - 16*x^2 +
16)/x^2) - ((1875*x^2*(x^2 - 2))/(2*(x^2 + 2)) - (1875*x^2*log(x)*(2*x^2 + x^4 - 4))/(x^2 + 2)^2 + (1875*x^2*l
og(x)^2*(4*x^2 + x^4 - 4))/(x^2 + 2)^2)/log(x)^2 + 1875*x^2 - (log(x)*(300000*x^2 + 60000*x^4 - 3750*x^8 + 180
000))/(12*x^2 + 6*x^4 + x^6 + 8)

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sympy [A]  time = 0.28, size = 24, normalized size = 0.96 \begin {gather*} \frac {1875 x^{2}}{\log {\relax (x )} \log {\left (\frac {4 x^{4} - 16 x^{2} + 16}{x^{2}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3750*x**3-7500*x)*ln(x)-1875*x**3+3750*x)*ln((4*x**4-16*x**2+16)/x**2)+(-3750*x**3-7500*x)*ln(x))
/(x**2-2)/ln(x)**2/ln((4*x**4-16*x**2+16)/x**2)**2,x)

[Out]

1875*x**2/(log(x)*log((4*x**4 - 16*x**2 + 16)/x**2))

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