3.21.64 \(\int \frac {78-48 x+390 x^2+(-48 x+5004 x^2-3072 x^3+384 x^4) \log (x)+(16224 x^2-19968 x^3+8640 x^4-1536 x^5+96 x^6) \log ^2(x)}{4 x+(52 x-32 x^2+4 x^3) \log (x)+(169 x-208 x^2+90 x^3-16 x^4+x^5) \log ^2(x)} \, dx\)

Optimal. Leaf size=27 \[ 3 \left (e^4+16 x^2-\frac {2}{2+\left (-3+(-4+x)^2\right ) \log (x)}\right ) \]

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Rubi [F]  time = 2.75, 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 {78-48 x+390 x^2+\left (-48 x+5004 x^2-3072 x^3+384 x^4\right ) \log (x)+\left (16224 x^2-19968 x^3+8640 x^4-1536 x^5+96 x^6\right ) \log ^2(x)}{4 x+\left (52 x-32 x^2+4 x^3\right ) \log (x)+\left (169 x-208 x^2+90 x^3-16 x^4+x^5\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(78 - 48*x + 390*x^2 + (-48*x + 5004*x^2 - 3072*x^3 + 384*x^4)*Log[x] + (16224*x^2 - 19968*x^3 + 8640*x^4
- 1536*x^5 + 96*x^6)*Log[x]^2)/(4*x + (52*x - 32*x^2 + 4*x^3)*Log[x] + (169*x - 208*x^2 + 90*x^3 - 16*x^4 + x^
5)*Log[x]^2),x]

[Out]

48*x^2 - 48*Defer[Int][(2 + 13*Log[x] - 8*x*Log[x] + x^2*Log[x])^(-2), x] - 32*Sqrt[3]*Defer[Int][1/((8 + 2*Sq
rt[3] - 2*x)*(2 + 13*Log[x] - 8*x*Log[x] + x^2*Log[x])^2), x] + 78*Defer[Int][1/(x*(2 + 13*Log[x] - 8*x*Log[x]
+ x^2*Log[x])^2), x] + 6*Defer[Int][x/(2 + 13*Log[x] - 8*x*Log[x] + x^2*Log[x])^2, x] - 8*(3 + 4*Sqrt[3])*Def
er[Int][1/((-8 - 2*Sqrt[3] + 2*x)*(2 + 13*Log[x] - 8*x*Log[x] + x^2*Log[x])^2), x] - 32*Sqrt[3]*Defer[Int][1/(
(-8 + 2*Sqrt[3] + 2*x)*(2 + 13*Log[x] - 8*x*Log[x] + x^2*Log[x])^2), x] - 8*(3 - 4*Sqrt[3])*Defer[Int][1/((-8
+ 2*Sqrt[3] + 2*x)*(2 + 13*Log[x] - 8*x*Log[x] + x^2*Log[x])^2), x] + 16*Sqrt[3]*Defer[Int][1/((8 + 2*Sqrt[3]
- 2*x)*(2 + 13*Log[x] - 8*x*Log[x] + x^2*Log[x])), x] + 4*(3 + 4*Sqrt[3])*Defer[Int][1/((-8 - 2*Sqrt[3] + 2*x)
*(2 + 13*Log[x] - 8*x*Log[x] + x^2*Log[x])), x] + 16*Sqrt[3]*Defer[Int][1/((-8 + 2*Sqrt[3] + 2*x)*(2 + 13*Log[
x] - 8*x*Log[x] + x^2*Log[x])), x] + 4*(3 - 4*Sqrt[3])*Defer[Int][1/((-8 + 2*Sqrt[3] + 2*x)*(2 + 13*Log[x] - 8
*x*Log[x] + x^2*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6 \left (13-8 x+65 x^2+2 x \left (-4+417 x-256 x^2+32 x^3\right ) \log (x)+16 x^2 \left (13-8 x+x^2\right )^2 \log ^2(x)\right )}{x \left (2+\left (13-8 x+x^2\right ) \log (x)\right )^2} \, dx\\ &=6 \int \frac {13-8 x+65 x^2+2 x \left (-4+417 x-256 x^2+32 x^3\right ) \log (x)+16 x^2 \left (13-8 x+x^2\right )^2 \log ^2(x)}{x \left (2+\left (13-8 x+x^2\right ) \log (x)\right )^2} \, dx\\ &=6 \int \left (16 x+\frac {169-192 x+86 x^2-16 x^3+x^4}{x \left (13-8 x+x^2\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2}+\frac {2 (-4+x)}{\left (13-8 x+x^2\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )}\right ) \, dx\\ &=48 x^2+6 \int \frac {169-192 x+86 x^2-16 x^3+x^4}{x \left (13-8 x+x^2\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx+12 \int \frac {-4+x}{\left (13-8 x+x^2\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )} \, dx\\ &=48 x^2+6 \int \left (-\frac {8}{\left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2}+\frac {13}{x \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2}+\frac {x}{\left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2}-\frac {4 (-4+x)}{\left (13-8 x+x^2\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2}\right ) \, dx+12 \int \left (-\frac {4}{\left (13-8 x+x^2\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )}+\frac {x}{\left (13-8 x+x^2\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )}\right ) \, dx\\ &=48 x^2+6 \int \frac {x}{\left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx+12 \int \frac {x}{\left (13-8 x+x^2\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )} \, dx-24 \int \frac {-4+x}{\left (13-8 x+x^2\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx-48 \int \frac {1}{\left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx-48 \int \frac {1}{\left (13-8 x+x^2\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )} \, dx+78 \int \frac {1}{x \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx\\ &=48 x^2+6 \int \frac {x}{\left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx+12 \int \left (\frac {1+\frac {4}{\sqrt {3}}}{\left (-8-2 \sqrt {3}+2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )}+\frac {1-\frac {4}{\sqrt {3}}}{\left (-8+2 \sqrt {3}+2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )}\right ) \, dx-24 \int \left (-\frac {4}{\left (13-8 x+x^2\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2}+\frac {x}{\left (13-8 x+x^2\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2}\right ) \, dx-48 \int \frac {1}{\left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx-48 \int \left (-\frac {1}{\sqrt {3} \left (8+2 \sqrt {3}-2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )}-\frac {1}{\sqrt {3} \left (-8+2 \sqrt {3}+2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )}\right ) \, dx+78 \int \frac {1}{x \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx\\ &=48 x^2+6 \int \frac {x}{\left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx-24 \int \frac {x}{\left (13-8 x+x^2\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx-48 \int \frac {1}{\left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx+78 \int \frac {1}{x \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx+96 \int \frac {1}{\left (13-8 x+x^2\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx+\left (16 \sqrt {3}\right ) \int \frac {1}{\left (8+2 \sqrt {3}-2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )} \, dx+\left (16 \sqrt {3}\right ) \int \frac {1}{\left (-8+2 \sqrt {3}+2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )} \, dx+\left (4 \left (3-4 \sqrt {3}\right )\right ) \int \frac {1}{\left (-8+2 \sqrt {3}+2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )} \, dx+\left (4 \left (3+4 \sqrt {3}\right )\right ) \int \frac {1}{\left (-8-2 \sqrt {3}+2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )} \, dx\\ &=48 x^2+6 \int \frac {x}{\left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx-24 \int \left (\frac {1+\frac {4}{\sqrt {3}}}{\left (-8-2 \sqrt {3}+2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2}+\frac {1-\frac {4}{\sqrt {3}}}{\left (-8+2 \sqrt {3}+2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2}\right ) \, dx-48 \int \frac {1}{\left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx+78 \int \frac {1}{x \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx+96 \int \left (-\frac {1}{\sqrt {3} \left (8+2 \sqrt {3}-2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2}-\frac {1}{\sqrt {3} \left (-8+2 \sqrt {3}+2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2}\right ) \, dx+\left (16 \sqrt {3}\right ) \int \frac {1}{\left (8+2 \sqrt {3}-2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )} \, dx+\left (16 \sqrt {3}\right ) \int \frac {1}{\left (-8+2 \sqrt {3}+2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )} \, dx+\left (4 \left (3-4 \sqrt {3}\right )\right ) \int \frac {1}{\left (-8+2 \sqrt {3}+2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )} \, dx+\left (4 \left (3+4 \sqrt {3}\right )\right ) \int \frac {1}{\left (-8-2 \sqrt {3}+2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )} \, dx\\ &=48 x^2+6 \int \frac {x}{\left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx-48 \int \frac {1}{\left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx+78 \int \frac {1}{x \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx+\left (16 \sqrt {3}\right ) \int \frac {1}{\left (8+2 \sqrt {3}-2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )} \, dx+\left (16 \sqrt {3}\right ) \int \frac {1}{\left (-8+2 \sqrt {3}+2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )} \, dx-\left (32 \sqrt {3}\right ) \int \frac {1}{\left (8+2 \sqrt {3}-2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx-\left (32 \sqrt {3}\right ) \int \frac {1}{\left (-8+2 \sqrt {3}+2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx+\left (4 \left (3-4 \sqrt {3}\right )\right ) \int \frac {1}{\left (-8+2 \sqrt {3}+2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )} \, dx-\left (8 \left (3-4 \sqrt {3}\right )\right ) \int \frac {1}{\left (-8+2 \sqrt {3}+2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx+\left (4 \left (3+4 \sqrt {3}\right )\right ) \int \frac {1}{\left (-8-2 \sqrt {3}+2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )} \, dx-\left (8 \left (3+4 \sqrt {3}\right )\right ) \int \frac {1}{\left (-8-2 \sqrt {3}+2 x\right ) \left (2+13 \log (x)-8 x \log (x)+x^2 \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.43, size = 25, normalized size = 0.93 \begin {gather*} 6 \left (8 x^2-\frac {1}{2+\left (13-8 x+x^2\right ) \log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(78 - 48*x + 390*x^2 + (-48*x + 5004*x^2 - 3072*x^3 + 384*x^4)*Log[x] + (16224*x^2 - 19968*x^3 + 864
0*x^4 - 1536*x^5 + 96*x^6)*Log[x]^2)/(4*x + (52*x - 32*x^2 + 4*x^3)*Log[x] + (169*x - 208*x^2 + 90*x^3 - 16*x^
4 + x^5)*Log[x]^2),x]

[Out]

6*(8*x^2 - (2 + (13 - 8*x + x^2)*Log[x])^(-1))

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fricas [A]  time = 0.47, size = 42, normalized size = 1.56 \begin {gather*} \frac {6 \, {\left (16 \, x^{2} + 8 \, {\left (x^{4} - 8 \, x^{3} + 13 \, x^{2}\right )} \log \relax (x) - 1\right )}}{{\left (x^{2} - 8 \, x + 13\right )} \log \relax (x) + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((96*x^6-1536*x^5+8640*x^4-19968*x^3+16224*x^2)*log(x)^2+(384*x^4-3072*x^3+5004*x^2-48*x)*log(x)+390
*x^2-48*x+78)/((x^5-16*x^4+90*x^3-208*x^2+169*x)*log(x)^2+(4*x^3-32*x^2+52*x)*log(x)+4*x),x, algorithm="fricas
")

[Out]

6*(16*x^2 + 8*(x^4 - 8*x^3 + 13*x^2)*log(x) - 1)/((x^2 - 8*x + 13)*log(x) + 2)

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giac [A]  time = 0.31, size = 27, normalized size = 1.00 \begin {gather*} 48 \, x^{2} - \frac {6}{x^{2} \log \relax (x) - 8 \, x \log \relax (x) + 13 \, \log \relax (x) + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((96*x^6-1536*x^5+8640*x^4-19968*x^3+16224*x^2)*log(x)^2+(384*x^4-3072*x^3+5004*x^2-48*x)*log(x)+390
*x^2-48*x+78)/((x^5-16*x^4+90*x^3-208*x^2+169*x)*log(x)^2+(4*x^3-32*x^2+52*x)*log(x)+4*x),x, algorithm="giac")

[Out]

48*x^2 - 6/(x^2*log(x) - 8*x*log(x) + 13*log(x) + 2)

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maple [A]  time = 0.06, size = 28, normalized size = 1.04

method	result	size
risch	\(48 x^{2}-\frac {6}{x^{2} \ln \relax (x )-8 x \ln \relax (x )+13 \ln \relax (x )+2}\)	\(28\)
norman	\(\frac {39 \ln \relax (x )-24 x \ln \relax (x )+627 x^{2} \ln \relax (x )+96 x^{2}-384 x^{3} \ln \relax (x )+48 x^{4} \ln \relax (x )}{x^{2} \ln \relax (x )-8 x \ln \relax (x )+13 \ln \relax (x )+2}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((96*x^6-1536*x^5+8640*x^4-19968*x^3+16224*x^2)*ln(x)^2+(384*x^4-3072*x^3+5004*x^2-48*x)*ln(x)+390*x^2-48*
x+78)/((x^5-16*x^4+90*x^3-208*x^2+169*x)*ln(x)^2+(4*x^3-32*x^2+52*x)*ln(x)+4*x),x,method=_RETURNVERBOSE)

[Out]

48*x^2-6/(x^2*ln(x)-8*x*ln(x)+13*ln(x)+2)

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maxima [A]  time = 0.57, size = 42, normalized size = 1.56 \begin {gather*} \frac {6 \, {\left (16 \, x^{2} + 8 \, {\left (x^{4} - 8 \, x^{3} + 13 \, x^{2}\right )} \log \relax (x) - 1\right )}}{{\left (x^{2} - 8 \, x + 13\right )} \log \relax (x) + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((96*x^6-1536*x^5+8640*x^4-19968*x^3+16224*x^2)*log(x)^2+(384*x^4-3072*x^3+5004*x^2-48*x)*log(x)+390
*x^2-48*x+78)/((x^5-16*x^4+90*x^3-208*x^2+169*x)*log(x)^2+(4*x^3-32*x^2+52*x)*log(x)+4*x),x, algorithm="maxima
")

[Out]

6*(16*x^2 + 8*(x^4 - 8*x^3 + 13*x^2)*log(x) - 1)/((x^2 - 8*x + 13)*log(x) + 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\ln \relax (x)}^2\,\left (96\,x^6-1536\,x^5+8640\,x^4-19968\,x^3+16224\,x^2\right )-48\,x-\ln \relax (x)\,\left (-384\,x^4+3072\,x^3-5004\,x^2+48\,x\right )+390\,x^2+78}{\left (x^5-16\,x^4+90\,x^3-208\,x^2+169\,x\right )\,{\ln \relax (x)}^2+\left (4\,x^3-32\,x^2+52\,x\right )\,\ln \relax (x)+4\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^2*(16224*x^2 - 19968*x^3 + 8640*x^4 - 1536*x^5 + 96*x^6) - 48*x - log(x)*(48*x - 5004*x^2 + 3072*x
^3 - 384*x^4) + 390*x^2 + 78)/(4*x + log(x)^2*(169*x - 208*x^2 + 90*x^3 - 16*x^4 + x^5) + log(x)*(52*x - 32*x^
2 + 4*x^3)),x)

[Out]

int((log(x)^2*(16224*x^2 - 19968*x^3 + 8640*x^4 - 1536*x^5 + 96*x^6) - 48*x - log(x)*(48*x - 5004*x^2 + 3072*x
^3 - 384*x^4) + 390*x^2 + 78)/(4*x + log(x)^2*(169*x - 208*x^2 + 90*x^3 - 16*x^4 + x^5) + log(x)*(52*x - 32*x^
2 + 4*x^3)), x)

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sympy [A]  time = 0.20, size = 19, normalized size = 0.70 \begin {gather*} 48 x^{2} - \frac {6}{\left (x^{2} - 8 x + 13\right ) \log {\relax (x )} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((96*x**6-1536*x**5+8640*x**4-19968*x**3+16224*x**2)*ln(x)**2+(384*x**4-3072*x**3+5004*x**2-48*x)*ln
(x)+390*x**2-48*x+78)/((x**5-16*x**4+90*x**3-208*x**2+169*x)*ln(x)**2+(4*x**3-32*x**2+52*x)*ln(x)+4*x),x)

[Out]

48*x**2 - 6/((x**2 - 8*x + 13)*log(x) + 2)

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