3.61.80 \(\int \frac {-256 x-128 x^2+(256+128 x-352 x^2-128 x^3-8 x^4) \log (x)+(-256+640 x+672 x^2+176 x^3+16 x^4) \log ^2(x)+(-384-64 x+624 x^2+560 x^3+184 x^4+24 x^5+x^6) \log ^3(x)}{(32+48 x+24 x^2+4 x^3) \log ^3(x)} \, dx\)

Optimal. Leaf size=27 \[ \left (2-3 x-x \left (\frac {x}{4}+\frac {4}{(2+x) \log (x)}\right )\right )^2 \]

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Rubi [F]  time = 0.35, 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 {-256 x-128 x^2+\left (256+128 x-352 x^2-128 x^3-8 x^4\right ) \log (x)+\left (-256+640 x+672 x^2+176 x^3+16 x^4\right ) \log ^2(x)+\left (-384-64 x+624 x^2+560 x^3+184 x^4+24 x^5+x^6\right ) \log ^3(x)}{\left (32+48 x+24 x^2+4 x^3\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-256*x - 128*x^2 + (256 + 128*x - 352*x^2 - 128*x^3 - 8*x^4)*Log[x] + (-256 + 640*x + 672*x^2 + 176*x^3 +
16*x^4)*Log[x]^2 + (-384 - 64*x + 624*x^2 + 560*x^3 + 184*x^4 + 24*x^5 + x^6)*Log[x]^3)/((32 + 48*x + 24*x^2
+ 4*x^3)*Log[x]^3),x]

[Out]

(8 - 12*x - x^2)^2/16 - 32*Defer[Int][x/((2 + x)^2*Log[x]^3), x] - 2*Defer[Int][(-32 - 16*x + 44*x^2 + 16*x^3
+ x^4)/((2 + x)^3*Log[x]^2), x] + 4*Defer[Int][(-8 + 24*x + 9*x^2 + x^3)/((2 + x)^2*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{4} (6+x) \left (-8+12 x+x^2\right )-\frac {32 x}{(2+x)^2 \log ^3(x)}-\frac {2 \left (-32-16 x+44 x^2+16 x^3+x^4\right )}{(2+x)^3 \log ^2(x)}+\frac {4 \left (-8+24 x+9 x^2+x^3\right )}{(2+x)^2 \log (x)}\right ) \, dx\\ &=\frac {1}{4} \int (6+x) \left (-8+12 x+x^2\right ) \, dx-2 \int \frac {-32-16 x+44 x^2+16 x^3+x^4}{(2+x)^3 \log ^2(x)} \, dx+4 \int \frac {-8+24 x+9 x^2+x^3}{(2+x)^2 \log (x)} \, dx-32 \int \frac {x}{(2+x)^2 \log ^3(x)} \, dx\\ &=\frac {1}{16} \left (8-12 x-x^2\right )^2-2 \int \frac {-32-16 x+44 x^2+16 x^3+x^4}{(2+x)^3 \log ^2(x)} \, dx+4 \int \frac {-8+24 x+9 x^2+x^3}{(2+x)^2 \log (x)} \, dx-32 \int \frac {x}{(2+x)^2 \log ^3(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 49, normalized size = 1.81 \begin {gather*} \frac {1}{16} x \left (-192+128 x+24 x^2+x^3+\frac {256 x}{(2+x)^2 \log ^2(x)}+\frac {32 \left (-8+12 x+x^2\right )}{(2+x) \log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-256*x - 128*x^2 + (256 + 128*x - 352*x^2 - 128*x^3 - 8*x^4)*Log[x] + (-256 + 640*x + 672*x^2 + 176
*x^3 + 16*x^4)*Log[x]^2 + (-384 - 64*x + 624*x^2 + 560*x^3 + 184*x^4 + 24*x^5 + x^6)*Log[x]^3)/((32 + 48*x + 2
4*x^2 + 4*x^3)*Log[x]^3),x]

[Out]

(x*(-192 + 128*x + 24*x^2 + x^3 + (256*x)/((2 + x)^2*Log[x]^2) + (32*(-8 + 12*x + x^2))/((2 + x)*Log[x])))/16

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fricas [B]  time = 0.49, size = 75, normalized size = 2.78 \begin {gather*} \frac {{\left (x^{6} + 28 \, x^{5} + 228 \, x^{4} + 416 \, x^{3} - 256 \, x^{2} - 768 \, x\right )} \log \relax (x)^{2} + 256 \, x^{2} + 32 \, {\left (x^{4} + 14 \, x^{3} + 16 \, x^{2} - 16 \, x\right )} \log \relax (x)}{16 \, {\left (x^{2} + 4 \, x + 4\right )} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^6+24*x^5+184*x^4+560*x^3+624*x^2-64*x-384)*log(x)^3+(16*x^4+176*x^3+672*x^2+640*x-256)*log(x)^2+
(-8*x^4-128*x^3-352*x^2+128*x+256)*log(x)-128*x^2-256*x)/(4*x^3+24*x^2+48*x+32)/log(x)^3,x, algorithm="fricas"
)

[Out]

1/16*((x^6 + 28*x^5 + 228*x^4 + 416*x^3 - 256*x^2 - 768*x)*log(x)^2 + 256*x^2 + 32*(x^4 + 14*x^3 + 16*x^2 - 16
*x)*log(x))/((x^2 + 4*x + 4)*log(x)^2)

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giac [B]  time = 4.46, size = 76, normalized size = 2.81 \begin {gather*} \frac {1}{16} \, x^{4} + \frac {3}{2} \, x^{3} + 8 \, x^{2} - 12 \, x + \frac {2 \, {\left (x^{4} \log \relax (x) + 14 \, x^{3} \log \relax (x) + 16 \, x^{2} \log \relax (x) + 8 \, x^{2} - 16 \, x \log \relax (x)\right )}}{x^{2} \log \relax (x)^{2} + 4 \, x \log \relax (x)^{2} + 4 \, \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^6+24*x^5+184*x^4+560*x^3+624*x^2-64*x-384)*log(x)^3+(16*x^4+176*x^3+672*x^2+640*x-256)*log(x)^2+
(-8*x^4-128*x^3-352*x^2+128*x+256)*log(x)-128*x^2-256*x)/(4*x^3+24*x^2+48*x+32)/log(x)^3,x, algorithm="giac")

[Out]

1/16*x^4 + 3/2*x^3 + 8*x^2 - 12*x + 2*(x^4*log(x) + 14*x^3*log(x) + 16*x^2*log(x) + 8*x^2 - 16*x*log(x))/(x^2*
log(x)^2 + 4*x*log(x)^2 + 4*log(x)^2)

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maple [B]  time = 0.04, size = 58, normalized size = 2.15

method	result	size
risch	\(-12 x +\frac {x^{4}}{16}+8 x^{2}+\frac {3 x^{3}}{2}+\frac {2 x \left (x^{3} \ln \relax (x )+14 x^{2} \ln \relax (x )+16 x \ln \relax (x )+8 x -16 \ln \relax (x )\right )}{\left (2+x \right )^{2} \ln \relax (x )^{2}}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^6+24*x^5+184*x^4+560*x^3+624*x^2-64*x-384)*ln(x)^3+(16*x^4+176*x^3+672*x^2+640*x-256)*ln(x)^2+(-8*x^4-
128*x^3-352*x^2+128*x+256)*ln(x)-128*x^2-256*x)/(4*x^3+24*x^2+48*x+32)/ln(x)^3,x,method=_RETURNVERBOSE)

[Out]

-12*x+1/16*x^4+8*x^2+3/2*x^3+2*x*(x^3*ln(x)+14*x^2*ln(x)+16*x*ln(x)+8*x-16*ln(x))/(2+x)^2/ln(x)^2

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maxima [B]  time = 0.38, size = 75, normalized size = 2.78 \begin {gather*} \frac {{\left (x^{6} + 28 \, x^{5} + 228 \, x^{4} + 416 \, x^{3} - 256 \, x^{2} - 768 \, x\right )} \log \relax (x)^{2} + 256 \, x^{2} + 32 \, {\left (x^{4} + 14 \, x^{3} + 16 \, x^{2} - 16 \, x\right )} \log \relax (x)}{16 \, {\left (x^{2} + 4 \, x + 4\right )} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^6+24*x^5+184*x^4+560*x^3+624*x^2-64*x-384)*log(x)^3+(16*x^4+176*x^3+672*x^2+640*x-256)*log(x)^2+
(-8*x^4-128*x^3-352*x^2+128*x+256)*log(x)-128*x^2-256*x)/(4*x^3+24*x^2+48*x+32)/log(x)^3,x, algorithm="maxima"
)

[Out]

1/16*((x^6 + 28*x^5 + 228*x^4 + 416*x^3 - 256*x^2 - 768*x)*log(x)^2 + 256*x^2 + 32*(x^4 + 14*x^3 + 16*x^2 - 16
*x)*log(x))/((x^2 + 4*x + 4)*log(x)^2)

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mupad [B]  time = 4.53, size = 68, normalized size = 2.52 \begin {gather*} \frac {x\,\left (x^5+28\,x^4+228\,x^3+416\,x^2-256\,x-768\right )}{16\,{\left (x+2\right )}^2}+\frac {16\,x^2+\frac {x\,\ln \relax (x)\,\left (32\,x^3+448\,x^2+512\,x-512\right )}{16}}{{\ln \relax (x)}^2\,{\left (x+2\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(256*x + log(x)*(352*x^2 - 128*x + 128*x^3 + 8*x^4 - 256) - log(x)^2*(640*x + 672*x^2 + 176*x^3 + 16*x^4
- 256) - log(x)^3*(624*x^2 - 64*x + 560*x^3 + 184*x^4 + 24*x^5 + x^6 - 384) + 128*x^2)/(log(x)^3*(48*x + 24*x^
2 + 4*x^3 + 32)),x)

[Out]

(x*(416*x^2 - 256*x + 228*x^3 + 28*x^4 + x^5 - 768))/(16*(x + 2)^2) + (16*x^2 + (x*log(x)*(512*x + 448*x^2 + 3
2*x^3 - 512))/16)/(log(x)^2*(x + 2)^2)

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sympy [B]  time = 0.20, size = 60, normalized size = 2.22 \begin {gather*} \frac {x^{4}}{16} + \frac {3 x^{3}}{2} + 8 x^{2} - 12 x + \frac {16 x^{2} + \left (2 x^{4} + 28 x^{3} + 32 x^{2} - 32 x\right ) \log {\relax (x )}}{\left (x^{2} + 4 x + 4\right ) \log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**6+24*x**5+184*x**4+560*x**3+624*x**2-64*x-384)*ln(x)**3+(16*x**4+176*x**3+672*x**2+640*x-256)*l
n(x)**2+(-8*x**4-128*x**3-352*x**2+128*x+256)*ln(x)-128*x**2-256*x)/(4*x**3+24*x**2+48*x+32)/ln(x)**3,x)

[Out]

x**4/16 + 3*x**3/2 + 8*x**2 - 12*x + (16*x**2 + (2*x**4 + 28*x**3 + 32*x**2 - 32*x)*log(x))/((x**2 + 4*x + 4)*
log(x)**2)

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