3.49.33 \(\int \frac {-96+208 x^2+x^3-62 x^4-6 x^5+52 x^6+12 x^7+24 x^8-8 x^9-16 x^{10}+(-64+32 x^2+6 x^3-52 x^4-24 x^5-48 x^6+24 x^7+48 x^8) \log (x)+(12 x^3+24 x^4-24 x^5-48 x^6) \log ^2(x)+(8 x^3+16 x^4) \log ^3(x)}{x^3-6 x^5+12 x^7-8 x^9+(6 x^3-24 x^5+24 x^7) \log (x)+(12 x^3-24 x^5) \log ^2(x)+8 x^3 \log ^3(x)} \, dx\)

Optimal. Leaf size=27 \[ -2+x+\left (-x+\frac {2}{x \left (\frac {1}{2}-x^2+\log (x)\right )}\right )^2 \]

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Rubi [F]  time = 1.81, 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 {-96+208 x^2+x^3-62 x^4-6 x^5+52 x^6+12 x^7+24 x^8-8 x^9-16 x^{10}+\left (-64+32 x^2+6 x^3-52 x^4-24 x^5-48 x^6+24 x^7+48 x^8\right ) \log (x)+\left (12 x^3+24 x^4-24 x^5-48 x^6\right ) \log ^2(x)+\left (8 x^3+16 x^4\right ) \log ^3(x)}{x^3-6 x^5+12 x^7-8 x^9+\left (6 x^3-24 x^5+24 x^7\right ) \log (x)+\left (12 x^3-24 x^5\right ) \log ^2(x)+8 x^3 \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-96 + 208*x^2 + x^3 - 62*x^4 - 6*x^5 + 52*x^6 + 12*x^7 + 24*x^8 - 8*x^9 - 16*x^10 + (-64 + 32*x^2 + 6*x^3
- 52*x^4 - 24*x^5 - 48*x^6 + 24*x^7 + 48*x^8)*Log[x] + (12*x^3 + 24*x^4 - 24*x^5 - 48*x^6)*Log[x]^2 + (8*x^3
+ 16*x^4)*Log[x]^3)/(x^3 - 6*x^5 + 12*x^7 - 8*x^9 + (6*x^3 - 24*x^5 + 24*x^7)*Log[x] + (12*x^3 - 24*x^5)*Log[x
]^2 + 8*x^3*Log[x]^3),x]

[Out]

x + x^2 + 64*Defer[Int][1/(x^3*(-1 + 2*x^2 - 2*Log[x])^3), x] - 128*Defer[Int][1/(x*(-1 + 2*x^2 - 2*Log[x])^3)
, x] - 32*Defer[Int][1/(x^3*(-1 + 2*x^2 - 2*Log[x])^2), x] + 16*Defer[Int][1/(x*(-1 + 2*x^2 - 2*Log[x])^2), x]
- 32*Defer[Int][x/(-1 + 2*x^2 - 2*Log[x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-96+208 x^2+x^3-62 x^4-6 x^5+52 x^6+12 x^7+24 x^8-8 x^9-16 x^{10}-\left (64-32 x^2-6 x^3+52 x^4+24 x^5+48 x^6-24 x^7-48 x^8\right ) \log (x)-12 x^3 \left (-1-2 x+2 x^2+4 x^3\right ) \log ^2(x)+8 x^3 (1+2 x) \log ^3(x)}{x^3 \left (1-2 x^2+2 \log (x)\right )^3} \, dx\\ &=\int \left (1+2 x-\frac {64 \left (-1+2 x^2\right )}{x^3 \left (-1+2 x^2-2 \log (x)\right )^3}-\frac {16 \left (2-x^2+2 x^4\right )}{x^3 \left (-1+2 x^2-2 \log (x)\right )^2}\right ) \, dx\\ &=x+x^2-16 \int \frac {2-x^2+2 x^4}{x^3 \left (-1+2 x^2-2 \log (x)\right )^2} \, dx-64 \int \frac {-1+2 x^2}{x^3 \left (-1+2 x^2-2 \log (x)\right )^3} \, dx\\ &=x+x^2-16 \int \left (\frac {2}{x^3 \left (-1+2 x^2-2 \log (x)\right )^2}-\frac {1}{x \left (-1+2 x^2-2 \log (x)\right )^2}+\frac {2 x}{\left (-1+2 x^2-2 \log (x)\right )^2}\right ) \, dx-64 \int \left (-\frac {1}{x^3 \left (-1+2 x^2-2 \log (x)\right )^3}+\frac {2}{x \left (-1+2 x^2-2 \log (x)\right )^3}\right ) \, dx\\ &=x+x^2+16 \int \frac {1}{x \left (-1+2 x^2-2 \log (x)\right )^2} \, dx-32 \int \frac {1}{x^3 \left (-1+2 x^2-2 \log (x)\right )^2} \, dx-32 \int \frac {x}{\left (-1+2 x^2-2 \log (x)\right )^2} \, dx+64 \int \frac {1}{x^3 \left (-1+2 x^2-2 \log (x)\right )^3} \, dx-128 \int \frac {1}{x \left (-1+2 x^2-2 \log (x)\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 38, normalized size = 1.41 \begin {gather*} x+x^2+\frac {16}{x^2 \left (1-2 x^2+2 \log (x)\right )^2}-\frac {8}{1-2 x^2+2 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-96 + 208*x^2 + x^3 - 62*x^4 - 6*x^5 + 52*x^6 + 12*x^7 + 24*x^8 - 8*x^9 - 16*x^10 + (-64 + 32*x^2 +
6*x^3 - 52*x^4 - 24*x^5 - 48*x^6 + 24*x^7 + 48*x^8)*Log[x] + (12*x^3 + 24*x^4 - 24*x^5 - 48*x^6)*Log[x]^2 + (
8*x^3 + 16*x^4)*Log[x]^3)/(x^3 - 6*x^5 + 12*x^7 - 8*x^9 + (6*x^3 - 24*x^5 + 24*x^7)*Log[x] + (12*x^3 - 24*x^5)
*Log[x]^2 + 8*x^3*Log[x]^3),x]

[Out]

x + x^2 + 16/(x^2*(1 - 2*x^2 + 2*Log[x])^2) - 8/(1 - 2*x^2 + 2*Log[x])

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fricas [B]  time = 0.53, size = 119, normalized size = 4.41 \begin {gather*} \frac {4 \, x^{8} + 4 \, x^{7} - 4 \, x^{6} - 4 \, x^{5} + 17 \, x^{4} + x^{3} + 4 \, {\left (x^{4} + x^{3}\right )} \log \relax (x)^{2} - 8 \, x^{2} - 4 \, {\left (2 \, x^{6} + 2 \, x^{5} - x^{4} - x^{3} + 4 \, x^{2}\right )} \log \relax (x) + 16}{4 \, x^{6} - 4 \, x^{4} + 4 \, x^{2} \log \relax (x)^{2} + x^{2} - 4 \, {\left (2 \, x^{4} - x^{2}\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^4+8*x^3)*log(x)^3+(-48*x^6-24*x^5+24*x^4+12*x^3)*log(x)^2+(48*x^8+24*x^7-48*x^6-24*x^5-52*x^4
+6*x^3+32*x^2-64)*log(x)-16*x^10-8*x^9+24*x^8+12*x^7+52*x^6-6*x^5-62*x^4+x^3+208*x^2-96)/(8*x^3*log(x)^3+(-24*
x^5+12*x^3)*log(x)^2+(24*x^7-24*x^5+6*x^3)*log(x)-8*x^9+12*x^7-6*x^5+x^3),x, algorithm="fricas")

[Out]

(4*x^8 + 4*x^7 - 4*x^6 - 4*x^5 + 17*x^4 + x^3 + 4*(x^4 + x^3)*log(x)^2 - 8*x^2 - 4*(2*x^6 + 2*x^5 - x^4 - x^3
+ 4*x^2)*log(x) + 16)/(4*x^6 - 4*x^4 + 4*x^2*log(x)^2 + x^2 - 4*(2*x^4 - x^2)*log(x))

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giac [B]  time = 0.22, size = 65, normalized size = 2.41 \begin {gather*} x^{2} + x + \frac {8 \, {\left (2 \, x^{4} - 2 \, x^{2} \log \relax (x) - x^{2} + 2\right )}}{4 \, x^{6} - 8 \, x^{4} \log \relax (x) - 4 \, x^{4} + 4 \, x^{2} \log \relax (x)^{2} + 4 \, x^{2} \log \relax (x) + x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^4+8*x^3)*log(x)^3+(-48*x^6-24*x^5+24*x^4+12*x^3)*log(x)^2+(48*x^8+24*x^7-48*x^6-24*x^5-52*x^4
+6*x^3+32*x^2-64)*log(x)-16*x^10-8*x^9+24*x^8+12*x^7+52*x^6-6*x^5-62*x^4+x^3+208*x^2-96)/(8*x^3*log(x)^3+(-24*
x^5+12*x^3)*log(x)^2+(24*x^7-24*x^5+6*x^3)*log(x)-8*x^9+12*x^7-6*x^5+x^3),x, algorithm="giac")

[Out]

x^2 + x + 8*(2*x^4 - 2*x^2*log(x) - x^2 + 2)/(4*x^6 - 8*x^4*log(x) - 4*x^4 + 4*x^2*log(x)^2 + 4*x^2*log(x) + x
^2)

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maple [A]  time = 0.03, size = 43, normalized size = 1.59

method	result	size
risch	\(x^{2}+x +\frac {16 x^{4}-16 x^{2} \ln \relax (x )-8 x^{2}+16}{x^{2} \left (2 x^{2}-2 \ln \relax (x )-1\right )^{2}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*x^4+8*x^3)*ln(x)^3+(-48*x^6-24*x^5+24*x^4+12*x^3)*ln(x)^2+(48*x^8+24*x^7-48*x^6-24*x^5-52*x^4+6*x^3+3
2*x^2-64)*ln(x)-16*x^10-8*x^9+24*x^8+12*x^7+52*x^6-6*x^5-62*x^4+x^3+208*x^2-96)/(8*x^3*ln(x)^3+(-24*x^5+12*x^3
)*ln(x)^2+(24*x^7-24*x^5+6*x^3)*ln(x)-8*x^9+12*x^7-6*x^5+x^3),x,method=_RETURNVERBOSE)

[Out]

x^2+x+8*(2*x^4-2*x^2*ln(x)-x^2+2)/x^2/(2*x^2-2*ln(x)-1)^2

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maxima [B]  time = 0.40, size = 119, normalized size = 4.41 \begin {gather*} \frac {4 \, x^{8} + 4 \, x^{7} - 4 \, x^{6} - 4 \, x^{5} + 17 \, x^{4} + x^{3} + 4 \, {\left (x^{4} + x^{3}\right )} \log \relax (x)^{2} - 8 \, x^{2} - 4 \, {\left (2 \, x^{6} + 2 \, x^{5} - x^{4} - x^{3} + 4 \, x^{2}\right )} \log \relax (x) + 16}{4 \, x^{6} - 4 \, x^{4} + 4 \, x^{2} \log \relax (x)^{2} + x^{2} - 4 \, {\left (2 \, x^{4} - x^{2}\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^4+8*x^3)*log(x)^3+(-48*x^6-24*x^5+24*x^4+12*x^3)*log(x)^2+(48*x^8+24*x^7-48*x^6-24*x^5-52*x^4
+6*x^3+32*x^2-64)*log(x)-16*x^10-8*x^9+24*x^8+12*x^7+52*x^6-6*x^5-62*x^4+x^3+208*x^2-96)/(8*x^3*log(x)^3+(-24*
x^5+12*x^3)*log(x)^2+(24*x^7-24*x^5+6*x^3)*log(x)-8*x^9+12*x^7-6*x^5+x^3),x, algorithm="maxima")

[Out]

(4*x^8 + 4*x^7 - 4*x^6 - 4*x^5 + 17*x^4 + x^3 + 4*(x^4 + x^3)*log(x)^2 - 8*x^2 - 4*(2*x^6 + 2*x^5 - x^4 - x^3
+ 4*x^2)*log(x) + 16)/(4*x^6 - 4*x^4 + 4*x^2*log(x)^2 + x^2 - 4*(2*x^4 - x^2)*log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\ln \relax (x)\,\left (48\,x^8+24\,x^7-48\,x^6-24\,x^5-52\,x^4+6\,x^3+32\,x^2-64\right )+{\ln \relax (x)}^3\,\left (16\,x^4+8\,x^3\right )+208\,x^2+x^3-62\,x^4-6\,x^5+52\,x^6+12\,x^7+24\,x^8-8\,x^9-16\,x^{10}+{\ln \relax (x)}^2\,\left (-48\,x^6-24\,x^5+24\,x^4+12\,x^3\right )-96}{\ln \relax (x)\,\left (24\,x^7-24\,x^5+6\,x^3\right )+{\ln \relax (x)}^2\,\left (12\,x^3-24\,x^5\right )+8\,x^3\,{\ln \relax (x)}^3+x^3-6\,x^5+12\,x^7-8\,x^9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(32*x^2 + 6*x^3 - 52*x^4 - 24*x^5 - 48*x^6 + 24*x^7 + 48*x^8 - 64) + log(x)^3*(8*x^3 + 16*x^4) + 2
08*x^2 + x^3 - 62*x^4 - 6*x^5 + 52*x^6 + 12*x^7 + 24*x^8 - 8*x^9 - 16*x^10 + log(x)^2*(12*x^3 + 24*x^4 - 24*x^
5 - 48*x^6) - 96)/(log(x)*(6*x^3 - 24*x^5 + 24*x^7) + log(x)^2*(12*x^3 - 24*x^5) + 8*x^3*log(x)^3 + x^3 - 6*x^
5 + 12*x^7 - 8*x^9),x)

[Out]

int((log(x)*(32*x^2 + 6*x^3 - 52*x^4 - 24*x^5 - 48*x^6 + 24*x^7 + 48*x^8 - 64) + log(x)^3*(8*x^3 + 16*x^4) + 2
08*x^2 + x^3 - 62*x^4 - 6*x^5 + 52*x^6 + 12*x^7 + 24*x^8 - 8*x^9 - 16*x^10 + log(x)^2*(12*x^3 + 24*x^4 - 24*x^
5 - 48*x^6) - 96)/(log(x)*(6*x^3 - 24*x^5 + 24*x^7) + log(x)^2*(12*x^3 - 24*x^5) + 8*x^3*log(x)^3 + x^3 - 6*x^
5 + 12*x^7 - 8*x^9), x)

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sympy [B]  time = 0.21, size = 61, normalized size = 2.26 \begin {gather*} x^{2} + x + \frac {16 x^{4} - 16 x^{2} \log {\relax (x )} - 8 x^{2} + 16}{4 x^{6} - 4 x^{4} + 4 x^{2} \log {\relax (x )}^{2} + x^{2} + \left (- 8 x^{4} + 4 x^{2}\right ) \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x**4+8*x**3)*ln(x)**3+(-48*x**6-24*x**5+24*x**4+12*x**3)*ln(x)**2+(48*x**8+24*x**7-48*x**6-24*x
**5-52*x**4+6*x**3+32*x**2-64)*ln(x)-16*x**10-8*x**9+24*x**8+12*x**7+52*x**6-6*x**5-62*x**4+x**3+208*x**2-96)/
(8*x**3*ln(x)**3+(-24*x**5+12*x**3)*ln(x)**2+(24*x**7-24*x**5+6*x**3)*ln(x)-8*x**9+12*x**7-6*x**5+x**3),x)

[Out]

x**2 + x + (16*x**4 - 16*x**2*log(x) - 8*x**2 + 16)/(4*x**6 - 4*x**4 + 4*x**2*log(x)**2 + x**2 + (-8*x**4 + 4*
x**2)*log(x))

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