3.42.18 \(\int \frac {94 x+4 x^3+28 x \log (x)+2 x \log ^2(x)+(126 x-4 x^3+32 x \log (x)+2 x \log ^2(x)) \log (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x})}{(63-2 x^2+16 \log (x)+\log ^2(x)) \log ^3(\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x})} \, dx\)

Optimal. Leaf size=23 \[ \frac {x^2}{\log ^2\left (-2 x+\frac {-1+(8+\log (x))^2}{x}\right )} \]

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Rubi [F]  time = 2.40, 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 {94 x+4 x^3+28 x \log (x)+2 x \log ^2(x)+\left (126 x-4 x^3+32 x \log (x)+2 x \log ^2(x)\right ) \log \left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^3\left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(94*x + 4*x^3 + 28*x*Log[x] + 2*x*Log[x]^2 + (126*x - 4*x^3 + 32*x*Log[x] + 2*x*Log[x]^2)*Log[(63 - 2*x^2
+ 16*Log[x] + Log[x]^2)/x])/((63 - 2*x^2 + 16*Log[x] + Log[x]^2)*Log[(63 - 2*x^2 + 16*Log[x] + Log[x]^2)/x]^3)
,x]

[Out]

94*Defer[Int][x/((63 - 2*x^2 + 16*Log[x] + Log[x]^2)*Log[63/x - 2*x + (16*Log[x])/x + Log[x]^2/x]^3), x] + 4*D
efer[Int][x^3/((63 - 2*x^2 + 16*Log[x] + Log[x]^2)*Log[63/x - 2*x + (16*Log[x])/x + Log[x]^2/x]^3), x] + 28*De
fer[Int][(x*Log[x])/((63 - 2*x^2 + 16*Log[x] + Log[x]^2)*Log[63/x - 2*x + (16*Log[x])/x + Log[x]^2/x]^3), x] +
2*Defer[Int][(x*Log[x]^2)/((63 - 2*x^2 + 16*Log[x] + Log[x]^2)*Log[63/x - 2*x + (16*Log[x])/x + Log[x]^2/x]^3
), x] + 4*Defer[Int][x^3/((-63 + 2*x^2 - 16*Log[x] - Log[x]^2)*Log[63/x - 2*x + (16*Log[x])/x + Log[x]^2/x]^2)
, x] + 126*Defer[Int][x/((63 - 2*x^2 + 16*Log[x] + Log[x]^2)*Log[63/x - 2*x + (16*Log[x])/x + Log[x]^2/x]^2),
x] + 32*Defer[Int][(x*Log[x])/((63 - 2*x^2 + 16*Log[x] + Log[x]^2)*Log[63/x - 2*x + (16*Log[x])/x + Log[x]^2/x
]^2), x] + 2*Defer[Int][(x*Log[x]^2)/((63 - 2*x^2 + 16*Log[x] + Log[x]^2)*Log[63/x - 2*x + (16*Log[x])/x + Log
[x]^2/x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {94 x+4 x^3+28 x \log (x)+2 x \log ^2(x)+\left (126 x-4 x^3+32 x \log (x)+2 x \log ^2(x)\right ) \log \left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^3\left (\frac {63}{x}-2 x+\frac {16 \log (x)}{x}+\frac {\log ^2(x)}{x}\right )} \, dx\\ &=\int \left (\frac {94 x}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^3\left (\frac {63}{x}-2 x+\frac {16 \log (x)}{x}+\frac {\log ^2(x)}{x}\right )}+\frac {4 x^3}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^3\left (\frac {63}{x}-2 x+\frac {16 \log (x)}{x}+\frac {\log ^2(x)}{x}\right )}+\frac {28 x \log (x)}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^3\left (\frac {63}{x}-2 x+\frac {16 \log (x)}{x}+\frac {\log ^2(x)}{x}\right )}+\frac {2 x \log ^2(x)}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^3\left (\frac {63}{x}-2 x+\frac {16 \log (x)}{x}+\frac {\log ^2(x)}{x}\right )}+\frac {4 x^3}{\left (-63+2 x^2-16 \log (x)-\log ^2(x)\right ) \log ^2\left (\frac {63}{x}-2 x+\frac {16 \log (x)}{x}+\frac {\log ^2(x)}{x}\right )}+\frac {126 x}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {63}{x}-2 x+\frac {16 \log (x)}{x}+\frac {\log ^2(x)}{x}\right )}+\frac {32 x \log (x)}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {63}{x}-2 x+\frac {16 \log (x)}{x}+\frac {\log ^2(x)}{x}\right )}+\frac {2 x \log ^2(x)}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {63}{x}-2 x+\frac {16 \log (x)}{x}+\frac {\log ^2(x)}{x}\right )}\right ) \, dx\\ &=2 \int \frac {x \log ^2(x)}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^3\left (\frac {63}{x}-2 x+\frac {16 \log (x)}{x}+\frac {\log ^2(x)}{x}\right )} \, dx+2 \int \frac {x \log ^2(x)}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {63}{x}-2 x+\frac {16 \log (x)}{x}+\frac {\log ^2(x)}{x}\right )} \, dx+4 \int \frac {x^3}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^3\left (\frac {63}{x}-2 x+\frac {16 \log (x)}{x}+\frac {\log ^2(x)}{x}\right )} \, dx+4 \int \frac {x^3}{\left (-63+2 x^2-16 \log (x)-\log ^2(x)\right ) \log ^2\left (\frac {63}{x}-2 x+\frac {16 \log (x)}{x}+\frac {\log ^2(x)}{x}\right )} \, dx+28 \int \frac {x \log (x)}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^3\left (\frac {63}{x}-2 x+\frac {16 \log (x)}{x}+\frac {\log ^2(x)}{x}\right )} \, dx+32 \int \frac {x \log (x)}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {63}{x}-2 x+\frac {16 \log (x)}{x}+\frac {\log ^2(x)}{x}\right )} \, dx+94 \int \frac {x}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^3\left (\frac {63}{x}-2 x+\frac {16 \log (x)}{x}+\frac {\log ^2(x)}{x}\right )} \, dx+126 \int \frac {x}{\left (63-2 x^2+16 \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {63}{x}-2 x+\frac {16 \log (x)}{x}+\frac {\log ^2(x)}{x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 26, normalized size = 1.13 \begin {gather*} \frac {x^2}{\log ^2\left (\frac {63-2 x^2+16 \log (x)+\log ^2(x)}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(94*x + 4*x^3 + 28*x*Log[x] + 2*x*Log[x]^2 + (126*x - 4*x^3 + 32*x*Log[x] + 2*x*Log[x]^2)*Log[(63 -
2*x^2 + 16*Log[x] + Log[x]^2)/x])/((63 - 2*x^2 + 16*Log[x] + Log[x]^2)*Log[(63 - 2*x^2 + 16*Log[x] + Log[x]^2)
/x]^3),x]

[Out]

x^2/Log[(63 - 2*x^2 + 16*Log[x] + Log[x]^2)/x]^2

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fricas [A]  time = 0.77, size = 29, normalized size = 1.26 \begin {gather*} \frac {x^{2}}{\log \left (-\frac {2 \, x^{2} - \log \relax (x)^{2} - 16 \, \log \relax (x) - 63}{x}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)^2+32*x*log(x)-4*x^3+126*x)*log((log(x)^2+16*log(x)-2*x^2+63)/x)+2*x*log(x)^2+28*x*log(x
)+4*x^3+94*x)/(log(x)^2+16*log(x)-2*x^2+63)/log((log(x)^2+16*log(x)-2*x^2+63)/x)^3,x, algorithm="fricas")

[Out]

x^2/log(-(2*x^2 - log(x)^2 - 16*log(x) - 63)/x)^2

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giac [B]  time = 0.32, size = 230, normalized size = 10.00 \begin {gather*} \frac {2 \, x^{4} + x^{2} \log \relax (x)^{2} + 14 \, x^{2} \log \relax (x) + 47 \, x^{2}}{2 \, x^{2} \log \left (-2 \, x^{2} + \log \relax (x)^{2} + 16 \, \log \relax (x) + 63\right )^{2} - 4 \, x^{2} \log \left (-2 \, x^{2} + \log \relax (x)^{2} + 16 \, \log \relax (x) + 63\right ) \log \relax (x) + 2 \, x^{2} \log \relax (x)^{2} + \log \left (-2 \, x^{2} + \log \relax (x)^{2} + 16 \, \log \relax (x) + 63\right )^{2} \log \relax (x)^{2} - 2 \, \log \left (-2 \, x^{2} + \log \relax (x)^{2} + 16 \, \log \relax (x) + 63\right ) \log \relax (x)^{3} + \log \relax (x)^{4} + 14 \, \log \left (-2 \, x^{2} + \log \relax (x)^{2} + 16 \, \log \relax (x) + 63\right )^{2} \log \relax (x) - 28 \, \log \left (-2 \, x^{2} + \log \relax (x)^{2} + 16 \, \log \relax (x) + 63\right ) \log \relax (x)^{2} + 14 \, \log \relax (x)^{3} + 47 \, \log \left (-2 \, x^{2} + \log \relax (x)^{2} + 16 \, \log \relax (x) + 63\right )^{2} - 94 \, \log \left (-2 \, x^{2} + \log \relax (x)^{2} + 16 \, \log \relax (x) + 63\right ) \log \relax (x) + 47 \, \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)^2+32*x*log(x)-4*x^3+126*x)*log((log(x)^2+16*log(x)-2*x^2+63)/x)+2*x*log(x)^2+28*x*log(x
)+4*x^3+94*x)/(log(x)^2+16*log(x)-2*x^2+63)/log((log(x)^2+16*log(x)-2*x^2+63)/x)^3,x, algorithm="giac")

[Out]

(2*x^4 + x^2*log(x)^2 + 14*x^2*log(x) + 47*x^2)/(2*x^2*log(-2*x^2 + log(x)^2 + 16*log(x) + 63)^2 - 4*x^2*log(-
2*x^2 + log(x)^2 + 16*log(x) + 63)*log(x) + 2*x^2*log(x)^2 + log(-2*x^2 + log(x)^2 + 16*log(x) + 63)^2*log(x)^
2 - 2*log(-2*x^2 + log(x)^2 + 16*log(x) + 63)*log(x)^3 + log(x)^4 + 14*log(-2*x^2 + log(x)^2 + 16*log(x) + 63)
^2*log(x) - 28*log(-2*x^2 + log(x)^2 + 16*log(x) + 63)*log(x)^2 + 14*log(x)^3 + 47*log(-2*x^2 + log(x)^2 + 16*
log(x) + 63)^2 - 94*log(-2*x^2 + log(x)^2 + 16*log(x) + 63)*log(x) + 47*log(x)^2)

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maple [C]  time = 0.18, size = 237, normalized size = 10.30

method	result	size
risch	\(-\frac {4 x^{2}}{\left (\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (-x^{2}+\frac {\ln \relax (x )^{2}}{2}+8 \ln \relax (x )+\frac {63}{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+\frac {\ln \relax (x )^{2}}{2}+8 \ln \relax (x )+\frac {63}{2}\right )}{x}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+\frac {\ln \relax (x )^{2}}{2}+8 \ln \relax (x )+\frac {63}{2}\right )}{x}\right )^{2}+2 \pi \mathrm {csgn}\left (\frac {i \left (-x^{2}+\frac {\ln \relax (x )^{2}}{2}+8 \ln \relax (x )+\frac {63}{2}\right )}{x}\right )^{2}+\pi \,\mathrm {csgn}\left (i \left (-x^{2}+\frac {\ln \relax (x )^{2}}{2}+8 \ln \relax (x )+\frac {63}{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+\frac {\ln \relax (x )^{2}}{2}+8 \ln \relax (x )+\frac {63}{2}\right )}{x}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (-x^{2}+\frac {\ln \relax (x )^{2}}{2}+8 \ln \relax (x )+\frac {63}{2}\right )}{x}\right )^{3}-2 \pi +2 i \ln \relax (2)-2 i \ln \relax (x )+2 i \ln \left (x^{2}-\frac {\ln \relax (x )^{2}}{2}-8 \ln \relax (x )-\frac {63}{2}\right )\right )^{2}}\)	\(237\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x*ln(x)^2+32*x*ln(x)-4*x^3+126*x)*ln((ln(x)^2+16*ln(x)-2*x^2+63)/x)+2*x*ln(x)^2+28*x*ln(x)+4*x^3+94*x)
/(ln(x)^2+16*ln(x)-2*x^2+63)/ln((ln(x)^2+16*ln(x)-2*x^2+63)/x)^3,x,method=_RETURNVERBOSE)

[Out]

-4*x^2/(Pi*csgn(I/x)*csgn(I*(-x^2+1/2*ln(x)^2+8*ln(x)+63/2))*csgn(I*(-x^2+1/2*ln(x)^2+8*ln(x)+63/2)/x)-Pi*csgn
(I/x)*csgn(I*(-x^2+1/2*ln(x)^2+8*ln(x)+63/2)/x)^2+2*Pi*csgn(I*(-x^2+1/2*ln(x)^2+8*ln(x)+63/2)/x)^2+Pi*csgn(I*(
-x^2+1/2*ln(x)^2+8*ln(x)+63/2))*csgn(I*(-x^2+1/2*ln(x)^2+8*ln(x)+63/2)/x)^2+Pi*csgn(I*(-x^2+1/2*ln(x)^2+8*ln(x
)+63/2)/x)^3-2*Pi+2*I*ln(2)-2*I*ln(x)+2*I*ln(x^2-1/2*ln(x)^2-8*ln(x)-63/2))^2

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maxima [B]  time = 0.41, size = 49, normalized size = 2.13 \begin {gather*} \frac {x^{2}}{\log \left (-2 \, x^{2} + \log \relax (x)^{2} + 16 \, \log \relax (x) + 63\right )^{2} - 2 \, \log \left (-2 \, x^{2} + \log \relax (x)^{2} + 16 \, \log \relax (x) + 63\right ) \log \relax (x) + \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)^2+32*x*log(x)-4*x^3+126*x)*log((log(x)^2+16*log(x)-2*x^2+63)/x)+2*x*log(x)^2+28*x*log(x
)+4*x^3+94*x)/(log(x)^2+16*log(x)-2*x^2+63)/log((log(x)^2+16*log(x)-2*x^2+63)/x)^3,x, algorithm="maxima")

[Out]

x^2/(log(-2*x^2 + log(x)^2 + 16*log(x) + 63)^2 - 2*log(-2*x^2 + log(x)^2 + 16*log(x) + 63)*log(x) + log(x)^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {94\,x+2\,x\,{\ln \relax (x)}^2+\ln \left (\frac {-2\,x^2+{\ln \relax (x)}^2+16\,\ln \relax (x)+63}{x}\right )\,\left (-4\,x^3+2\,x\,{\ln \relax (x)}^2+32\,x\,\ln \relax (x)+126\,x\right )+28\,x\,\ln \relax (x)+4\,x^3}{{\ln \left (\frac {-2\,x^2+{\ln \relax (x)}^2+16\,\ln \relax (x)+63}{x}\right )}^3\,\left (-2\,x^2+{\ln \relax (x)}^2+16\,\ln \relax (x)+63\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((94*x + 2*x*log(x)^2 + log((16*log(x) + log(x)^2 - 2*x^2 + 63)/x)*(126*x + 2*x*log(x)^2 + 32*x*log(x) - 4*
x^3) + 28*x*log(x) + 4*x^3)/(log((16*log(x) + log(x)^2 - 2*x^2 + 63)/x)^3*(16*log(x) + log(x)^2 - 2*x^2 + 63))
,x)

[Out]

int((94*x + 2*x*log(x)^2 + log((16*log(x) + log(x)^2 - 2*x^2 + 63)/x)*(126*x + 2*x*log(x)^2 + 32*x*log(x) - 4*
x^3) + 28*x*log(x) + 4*x^3)/(log((16*log(x) + log(x)^2 - 2*x^2 + 63)/x)^3*(16*log(x) + log(x)^2 - 2*x^2 + 63))
, x)

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sympy [A]  time = 0.37, size = 24, normalized size = 1.04 \begin {gather*} \frac {x^{2}}{\log {\left (\frac {- 2 x^{2} + \log {\relax (x )}^{2} + 16 \log {\relax (x )} + 63}{x} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*ln(x)**2+32*x*ln(x)-4*x**3+126*x)*ln((ln(x)**2+16*ln(x)-2*x**2+63)/x)+2*x*ln(x)**2+28*x*ln(x)+
4*x**3+94*x)/(ln(x)**2+16*ln(x)-2*x**2+63)/ln((ln(x)**2+16*ln(x)-2*x**2+63)/x)**3,x)

[Out]

x**2/log((-2*x**2 + log(x)**2 + 16*log(x) + 63)/x)**2

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