3.16.81 \(\int \frac {-8 x^3-17 x^4+65 x^6+272 x^7+352 x^8+256 x^9+256 x^{10}+(2-x^2-32 x^3-83 x^4-32 x^5-128 x^6-256 x^7-256 x^8) \log (x)+(3 x^2+16 x^3+32 x^4) \log ^2(x)-\log ^3(x)}{x^2-2 x^4-16 x^5-31 x^6+16 x^7+96 x^8+256 x^9+256 x^{10}+(2 x^2-2 x^4-16 x^5-32 x^6) \log (x)+x^2 \log ^2(x)} \, dx\)

Optimal. Leaf size=29 \[ -16+x+\frac {\log (x)}{x-\frac {x}{\left (x+4 x^2\right )^2-\log (x)}} \]

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Rubi [F]  time = 2.82, 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 {-8 x^3-17 x^4+65 x^6+272 x^7+352 x^8+256 x^9+256 x^{10}+\left (2-x^2-32 x^3-83 x^4-32 x^5-128 x^6-256 x^7-256 x^8\right ) \log (x)+\left (3 x^2+16 x^3+32 x^4\right ) \log ^2(x)-\log ^3(x)}{x^2-2 x^4-16 x^5-31 x^6+16 x^7+96 x^8+256 x^9+256 x^{10}+\left (2 x^2-2 x^4-16 x^5-32 x^6\right ) \log (x)+x^2 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8*x^3 - 17*x^4 + 65*x^6 + 272*x^7 + 352*x^8 + 256*x^9 + 256*x^10 + (2 - x^2 - 32*x^3 - 83*x^4 - 32*x^5 -
128*x^6 - 256*x^7 - 256*x^8)*Log[x] + (3*x^2 + 16*x^3 + 32*x^4)*Log[x]^2 - Log[x]^3)/(x^2 - 2*x^4 - 16*x^5 -
31*x^6 + 16*x^7 + 96*x^8 + 256*x^9 + 256*x^10 + (2*x^2 - 2*x^4 - 16*x^5 - 32*x^6)*Log[x] + x^2*Log[x]^2),x]

[Out]

-x^(-1) + x + Log[x]/x + 3*Defer[Int][(-1 + x^2 + 8*x^3 + 16*x^4 - Log[x])^(-2), x] - Defer[Int][1/(x^2*(-1 +
x^2 + 8*x^3 + 16*x^4 - Log[x])^2), x] + 32*Defer[Int][x/(-1 + x^2 + 8*x^3 + 16*x^4 - Log[x])^2, x] + 78*Defer[
Int][x^2/(-1 + x^2 + 8*x^3 + 16*x^4 - Log[x])^2, x] - 40*Defer[Int][x^3/(-1 + x^2 + 8*x^3 + 16*x^4 - Log[x])^2
, x] - 288*Defer[Int][x^4/(-1 + x^2 + 8*x^3 + 16*x^4 - Log[x])^2, x] - 896*Defer[Int][x^5/(-1 + x^2 + 8*x^3 +
16*x^4 - Log[x])^2, x] - 1024*Defer[Int][x^6/(-1 + x^2 + 8*x^3 + 16*x^4 - Log[x])^2, x] + Defer[Int][(-1 + x^2
+ 8*x^3 + 16*x^4 - Log[x])^(-1), x] + Defer[Int][1/(x^2*(-1 + x^2 + 8*x^3 + 16*x^4 - Log[x])), x] + 16*Defer[
Int][x/(-1 + x^2 + 8*x^3 + 16*x^4 - Log[x]), x] + 48*Defer[Int][x^2/(-1 + x^2 + 8*x^3 + 16*x^4 - Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^3 \left (-8-17 x+65 x^3+272 x^4+352 x^5+256 x^6+256 x^7\right )-\left (-2+x^2+32 x^3+83 x^4+32 x^5+128 x^6+256 x^7+256 x^8\right ) \log (x)+x^2 \left (3+16 x+32 x^2\right ) \log ^2(x)-\log ^3(x)}{x^2 \left (1-x^2-8 x^3-16 x^4+\log (x)\right )^2} \, dx\\ &=\int \left (\frac {2+x^2}{x^2}+\frac {-1+3 x^2+32 x^3+78 x^4-40 x^5-288 x^6-896 x^7-1024 x^8}{x^2 \left (-1+x^2+8 x^3+16 x^4-\log (x)\right )^2}+\frac {1+x^2+16 x^3+48 x^4}{x^2 \left (-1+x^2+8 x^3+16 x^4-\log (x)\right )}-\frac {\log (x)}{x^2}\right ) \, dx\\ &=\int \frac {2+x^2}{x^2} \, dx+\int \frac {-1+3 x^2+32 x^3+78 x^4-40 x^5-288 x^6-896 x^7-1024 x^8}{x^2 \left (-1+x^2+8 x^3+16 x^4-\log (x)\right )^2} \, dx+\int \frac {1+x^2+16 x^3+48 x^4}{x^2 \left (-1+x^2+8 x^3+16 x^4-\log (x)\right )} \, dx-\int \frac {\log (x)}{x^2} \, dx\\ &=\frac {1}{x}+\frac {\log (x)}{x}+\int \left (1+\frac {2}{x^2}\right ) \, dx+\int \left (\frac {3}{\left (-1+x^2+8 x^3+16 x^4-\log (x)\right )^2}-\frac {1}{x^2 \left (-1+x^2+8 x^3+16 x^4-\log (x)\right )^2}+\frac {32 x}{\left (-1+x^2+8 x^3+16 x^4-\log (x)\right )^2}+\frac {78 x^2}{\left (-1+x^2+8 x^3+16 x^4-\log (x)\right )^2}-\frac {40 x^3}{\left (-1+x^2+8 x^3+16 x^4-\log (x)\right )^2}-\frac {288 x^4}{\left (-1+x^2+8 x^3+16 x^4-\log (x)\right )^2}-\frac {896 x^5}{\left (-1+x^2+8 x^3+16 x^4-\log (x)\right )^2}-\frac {1024 x^6}{\left (-1+x^2+8 x^3+16 x^4-\log (x)\right )^2}\right ) \, dx+\int \left (\frac {1}{-1+x^2+8 x^3+16 x^4-\log (x)}+\frac {1}{x^2 \left (-1+x^2+8 x^3+16 x^4-\log (x)\right )}+\frac {16 x}{-1+x^2+8 x^3+16 x^4-\log (x)}+\frac {48 x^2}{-1+x^2+8 x^3+16 x^4-\log (x)}\right ) \, dx\\ &=-\frac {1}{x}+x+\frac {\log (x)}{x}+3 \int \frac {1}{\left (-1+x^2+8 x^3+16 x^4-\log (x)\right )^2} \, dx+16 \int \frac {x}{-1+x^2+8 x^3+16 x^4-\log (x)} \, dx+32 \int \frac {x}{\left (-1+x^2+8 x^3+16 x^4-\log (x)\right )^2} \, dx-40 \int \frac {x^3}{\left (-1+x^2+8 x^3+16 x^4-\log (x)\right )^2} \, dx+48 \int \frac {x^2}{-1+x^2+8 x^3+16 x^4-\log (x)} \, dx+78 \int \frac {x^2}{\left (-1+x^2+8 x^3+16 x^4-\log (x)\right )^2} \, dx-288 \int \frac {x^4}{\left (-1+x^2+8 x^3+16 x^4-\log (x)\right )^2} \, dx-896 \int \frac {x^5}{\left (-1+x^2+8 x^3+16 x^4-\log (x)\right )^2} \, dx-1024 \int \frac {x^6}{\left (-1+x^2+8 x^3+16 x^4-\log (x)\right )^2} \, dx-\int \frac {1}{x^2 \left (-1+x^2+8 x^3+16 x^4-\log (x)\right )^2} \, dx+\int \frac {1}{-1+x^2+8 x^3+16 x^4-\log (x)} \, dx+\int \frac {1}{x^2 \left (-1+x^2+8 x^3+16 x^4-\log (x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 48, normalized size = 1.66 \begin {gather*} \frac {-1+x^2+\frac {-1+x^2+8 x^3+16 x^4}{-1+x^2+8 x^3+16 x^4-\log (x)}+\log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*x^3 - 17*x^4 + 65*x^6 + 272*x^7 + 352*x^8 + 256*x^9 + 256*x^10 + (2 - x^2 - 32*x^3 - 83*x^4 - 32
*x^5 - 128*x^6 - 256*x^7 - 256*x^8)*Log[x] + (3*x^2 + 16*x^3 + 32*x^4)*Log[x]^2 - Log[x]^3)/(x^2 - 2*x^4 - 16*
x^5 - 31*x^6 + 16*x^7 + 96*x^8 + 256*x^9 + 256*x^10 + (2*x^2 - 2*x^4 - 16*x^5 - 32*x^6)*Log[x] + x^2*Log[x]^2)
,x]

[Out]

(-1 + x^2 + (-1 + x^2 + 8*x^3 + 16*x^4)/(-1 + x^2 + 8*x^3 + 16*x^4 - Log[x]) + Log[x])/x

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fricas [B]  time = 0.81, size = 63, normalized size = 2.17 \begin {gather*} \frac {16 \, x^{6} + 8 \, x^{5} + x^{4} - x^{2} + 8 \, {\left (2 \, x^{4} + x^{3}\right )} \log \relax (x) - \log \relax (x)^{2}}{16 \, x^{5} + 8 \, x^{4} + x^{3} - x \log \relax (x) - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)^3+(32*x^4+16*x^3+3*x^2)*log(x)^2+(-256*x^8-256*x^7-128*x^6-32*x^5-83*x^4-32*x^3-x^2+2)*log(
x)+256*x^10+256*x^9+352*x^8+272*x^7+65*x^6-17*x^4-8*x^3)/(x^2*log(x)^2+(-32*x^6-16*x^5-2*x^4+2*x^2)*log(x)+256
*x^10+256*x^9+96*x^8+16*x^7-31*x^6-16*x^5-2*x^4+x^2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.36, size = 53, normalized size = 1.83 \begin {gather*} x + \frac {16 \, x^{4} + 8 \, x^{3} + x^{2} - 1}{16 \, x^{5} + 8 \, x^{4} + x^{3} - x \log \relax (x) - x} + \frac {\log \relax (x)}{x} - \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)^3+(32*x^4+16*x^3+3*x^2)*log(x)^2+(-256*x^8-256*x^7-128*x^6-32*x^5-83*x^4-32*x^3-x^2+2)*log(
x)+256*x^10+256*x^9+352*x^8+272*x^7+65*x^6-17*x^4-8*x^3)/(x^2*log(x)^2+(-32*x^6-16*x^5-2*x^4+2*x^2)*log(x)+256
*x^10+256*x^9+96*x^8+16*x^7-31*x^6-16*x^5-2*x^4+x^2),x, algorithm="giac")

[Out]

x + (16*x^4 + 8*x^3 + x^2 - 1)/(16*x^5 + 8*x^4 + x^3 - x*log(x) - x) + log(x)/x - 1/x

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maple [A]  time = 0.03, size = 57, normalized size = 1.97

method	result	size
risch	\(\frac {\ln \relax (x )}{x}+\frac {x^{2}-1}{x}+\frac {16 x^{4}+8 x^{3}+x^{2}-1}{x \left (16 x^{4}+8 x^{3}+x^{2}-\ln \relax (x )-1\right )}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-ln(x)^3+(32*x^4+16*x^3+3*x^2)*ln(x)^2+(-256*x^8-256*x^7-128*x^6-32*x^5-83*x^4-32*x^3-x^2+2)*ln(x)+256*x^
10+256*x^9+352*x^8+272*x^7+65*x^6-17*x^4-8*x^3)/(x^2*ln(x)^2+(-32*x^6-16*x^5-2*x^4+2*x^2)*ln(x)+256*x^10+256*x
^9+96*x^8+16*x^7-31*x^6-16*x^5-2*x^4+x^2),x,method=_RETURNVERBOSE)

[Out]

ln(x)/x+(x^2-1)/x+(16*x^4+8*x^3+x^2-1)/x/(16*x^4+8*x^3+x^2-ln(x)-1)

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maxima [B]  time = 0.56, size = 63, normalized size = 2.17 \begin {gather*} \frac {16 \, x^{6} + 8 \, x^{5} + x^{4} - x^{2} + 8 \, {\left (2 \, x^{4} + x^{3}\right )} \log \relax (x) - \log \relax (x)^{2}}{16 \, x^{5} + 8 \, x^{4} + x^{3} - x \log \relax (x) - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)^3+(32*x^4+16*x^3+3*x^2)*log(x)^2+(-256*x^8-256*x^7-128*x^6-32*x^5-83*x^4-32*x^3-x^2+2)*log(
x)+256*x^10+256*x^9+352*x^8+272*x^7+65*x^6-17*x^4-8*x^3)/(x^2*log(x)^2+(-32*x^6-16*x^5-2*x^4+2*x^2)*log(x)+256
*x^10+256*x^9+96*x^8+16*x^7-31*x^6-16*x^5-2*x^4+x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.29, size = 169, normalized size = 5.83 \begin {gather*} x+\frac {\ln \relax (x)}{x}-\frac {\frac {x^4}{4}+\frac {x^3}{8}+\frac {x^2}{64}-\frac {1}{32}}{x^5+\frac {3\,x^4}{8}+\frac {x^3}{32}-\frac {x}{64}}+\frac {\frac {256\,x^8+256\,x^7+96\,x^6+16\,x^5-47\,x^4-24\,x^3-3\,x^2+2}{x\,\left (64\,x^4+24\,x^3+2\,x^2-1\right )}+\frac {\ln \relax (x)\,\left (48\,x^4+16\,x^3+x^2+1\right )}{x\,\left (64\,x^4+24\,x^3+2\,x^2-1\right )}}{x^2-\ln \relax (x)+8\,x^3+16\,x^4-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^2*(3*x^2 + 16*x^3 + 32*x^4) - log(x)^3 - log(x)*(x^2 + 32*x^3 + 83*x^4 + 32*x^5 + 128*x^6 + 256*x^
7 + 256*x^8 - 2) - 8*x^3 - 17*x^4 + 65*x^6 + 272*x^7 + 352*x^8 + 256*x^9 + 256*x^10)/(x^2*log(x)^2 - log(x)*(2
*x^4 - 2*x^2 + 16*x^5 + 32*x^6) + x^2 - 2*x^4 - 16*x^5 - 31*x^6 + 16*x^7 + 96*x^8 + 256*x^9 + 256*x^10),x)

[Out]

x + log(x)/x - (x^2/64 + x^3/8 + x^4/4 - 1/32)/(x^3/32 - x/64 + (3*x^4)/8 + x^5) + ((16*x^5 - 24*x^3 - 47*x^4
- 3*x^2 + 96*x^6 + 256*x^7 + 256*x^8 + 2)/(x*(2*x^2 + 24*x^3 + 64*x^4 - 1)) + (log(x)*(x^2 + 16*x^3 + 48*x^4 +
1))/(x*(2*x^2 + 24*x^3 + 64*x^4 - 1)))/(x^2 - log(x) + 8*x^3 + 16*x^4 - 1)

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sympy [B]  time = 0.22, size = 44, normalized size = 1.52 \begin {gather*} x + \frac {- 16 x^{4} - 8 x^{3} - x^{2} + 1}{- 16 x^{5} - 8 x^{4} - x^{3} + x \log {\relax (x )} + x} + \frac {\log {\relax (x )}}{x} - \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-ln(x)**3+(32*x**4+16*x**3+3*x**2)*ln(x)**2+(-256*x**8-256*x**7-128*x**6-32*x**5-83*x**4-32*x**3-x*
*2+2)*ln(x)+256*x**10+256*x**9+352*x**8+272*x**7+65*x**6-17*x**4-8*x**3)/(x**2*ln(x)**2+(-32*x**6-16*x**5-2*x*
*4+2*x**2)*ln(x)+256*x**10+256*x**9+96*x**8+16*x**7-31*x**6-16*x**5-2*x**4+x**2),x)

[Out]

x + (-16*x**4 - 8*x**3 - x**2 + 1)/(-16*x**5 - 8*x**4 - x**3 + x*log(x) + x) + log(x)/x - 1/x

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