3.54.19 \(\int \frac {-74+762 x-1320 x^2+754 x^3-132 x^4-12 x^5+4 x^6}{-8176-6428 x-120 x^2-1516 x^3+424 x^4+24 x^5-32 x^6+4 x^7+(-2044-1607 x-30 x^2-379 x^3+106 x^4+6 x^5-8 x^6+x^7) \log (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{64+32 x+4 x^2})} \, dx\)

Optimal. Leaf size=31 \[ \log \left (4+\log \left (8-\frac {\left ((-3+x)^2-\frac {1}{x}\right )^2 x^2}{4 (4+x)^2}\right )\right ) \]

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Rubi [F]  time = 2.05, 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 {-74+762 x-1320 x^2+754 x^3-132 x^4-12 x^5+4 x^6}{-8176-6428 x-120 x^2-1516 x^3+424 x^4+24 x^5-32 x^6+4 x^7+\left (-2044-1607 x-30 x^2-379 x^3+106 x^4+6 x^5-8 x^6+x^7\right ) \log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{64+32 x+4 x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-74 + 762*x - 1320*x^2 + 754*x^3 - 132*x^4 - 12*x^5 + 4*x^6)/(-8176 - 6428*x - 120*x^2 - 1516*x^3 + 424*x
^4 + 24*x^5 - 32*x^6 + 4*x^7 + (-2044 - 1607*x - 30*x^2 - 379*x^3 + 106*x^4 + 6*x^5 - 8*x^6 + x^7)*Log[(511 +
274*x - 61*x^2 + 110*x^3 - 54*x^4 + 12*x^5 - x^6)/(64 + 32*x + 4*x^2)]),x]

[Out]

-2*Defer[Int][1/((4 + x)*(4 + Log[(511 + 274*x - 61*x^2 + 110*x^3 - 54*x^4 + 12*x^5 - x^6)/(4*(4 + x)^2)])), x
] - 274*Defer[Int][1/((-511 - 274*x + 61*x^2 - 110*x^3 + 54*x^4 - 12*x^5 + x^6)*(4 + Log[(511 + 274*x - 61*x^2
+ 110*x^3 - 54*x^4 + 12*x^5 - x^6)/(4*(4 + x)^2)])), x] + 122*Defer[Int][x/((-511 - 274*x + 61*x^2 - 110*x^3
+ 54*x^4 - 12*x^5 + x^6)*(4 + Log[(511 + 274*x - 61*x^2 + 110*x^3 - 54*x^4 + 12*x^5 - x^6)/(4*(4 + x)^2)])), x
] - 330*Defer[Int][x^2/((-511 - 274*x + 61*x^2 - 110*x^3 + 54*x^4 - 12*x^5 + x^6)*(4 + Log[(511 + 274*x - 61*x
^2 + 110*x^3 - 54*x^4 + 12*x^5 - x^6)/(4*(4 + x)^2)])), x] + 216*Defer[Int][x^3/((-511 - 274*x + 61*x^2 - 110*
x^3 + 54*x^4 - 12*x^5 + x^6)*(4 + Log[(511 + 274*x - 61*x^2 + 110*x^3 - 54*x^4 + 12*x^5 - x^6)/(4*(4 + x)^2)])
), x] - 60*Defer[Int][x^4/((-511 - 274*x + 61*x^2 - 110*x^3 + 54*x^4 - 12*x^5 + x^6)*(4 + Log[(511 + 274*x - 6
1*x^2 + 110*x^3 - 54*x^4 + 12*x^5 - x^6)/(4*(4 + x)^2)])), x] + 6*Defer[Int][x^5/((-511 - 274*x + 61*x^2 - 110
*x^3 + 54*x^4 - 12*x^5 + x^6)*(4 + Log[(511 + 274*x - 61*x^2 + 110*x^3 - 54*x^4 + 12*x^5 - x^6)/(4*(4 + x)^2)]
)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (37-381 x+660 x^2-377 x^3+66 x^4+6 x^5-2 x^6\right )}{\left (2044+1607 x+30 x^2+379 x^3-106 x^4-6 x^5+8 x^6-x^7\right ) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )} \, dx\\ &=2 \int \frac {37-381 x+660 x^2-377 x^3+66 x^4+6 x^5-2 x^6}{\left (2044+1607 x+30 x^2+379 x^3-106 x^4-6 x^5+8 x^6-x^7\right ) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )} \, dx\\ &=2 \int \left (-\frac {1}{(4+x) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )}+\frac {-137+61 x-165 x^2+108 x^3-30 x^4+3 x^5}{\left (-511-274 x+61 x^2-110 x^3+54 x^4-12 x^5+x^6\right ) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{(4+x) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )} \, dx\right )+2 \int \frac {-137+61 x-165 x^2+108 x^3-30 x^4+3 x^5}{\left (-511-274 x+61 x^2-110 x^3+54 x^4-12 x^5+x^6\right ) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )} \, dx\\ &=-\left (2 \int \frac {1}{(4+x) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )} \, dx\right )+2 \int \left (-\frac {137}{\left (-511-274 x+61 x^2-110 x^3+54 x^4-12 x^5+x^6\right ) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )}+\frac {61 x}{\left (-511-274 x+61 x^2-110 x^3+54 x^4-12 x^5+x^6\right ) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )}-\frac {165 x^2}{\left (-511-274 x+61 x^2-110 x^3+54 x^4-12 x^5+x^6\right ) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )}+\frac {108 x^3}{\left (-511-274 x+61 x^2-110 x^3+54 x^4-12 x^5+x^6\right ) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )}-\frac {30 x^4}{\left (-511-274 x+61 x^2-110 x^3+54 x^4-12 x^5+x^6\right ) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )}+\frac {3 x^5}{\left (-511-274 x+61 x^2-110 x^3+54 x^4-12 x^5+x^6\right ) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{(4+x) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )} \, dx\right )+6 \int \frac {x^5}{\left (-511-274 x+61 x^2-110 x^3+54 x^4-12 x^5+x^6\right ) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )} \, dx-60 \int \frac {x^4}{\left (-511-274 x+61 x^2-110 x^3+54 x^4-12 x^5+x^6\right ) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )} \, dx+122 \int \frac {x}{\left (-511-274 x+61 x^2-110 x^3+54 x^4-12 x^5+x^6\right ) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )} \, dx+216 \int \frac {x^3}{\left (-511-274 x+61 x^2-110 x^3+54 x^4-12 x^5+x^6\right ) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )} \, dx-274 \int \frac {1}{\left (-511-274 x+61 x^2-110 x^3+54 x^4-12 x^5+x^6\right ) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )} \, dx-330 \int \frac {x^2}{\left (-511-274 x+61 x^2-110 x^3+54 x^4-12 x^5+x^6\right ) \left (4+\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 45, normalized size = 1.45 \begin {gather*} \log \left (-4-\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-74 + 762*x - 1320*x^2 + 754*x^3 - 132*x^4 - 12*x^5 + 4*x^6)/(-8176 - 6428*x - 120*x^2 - 1516*x^3 +
424*x^4 + 24*x^5 - 32*x^6 + 4*x^7 + (-2044 - 1607*x - 30*x^2 - 379*x^3 + 106*x^4 + 6*x^5 - 8*x^6 + x^7)*Log[(
511 + 274*x - 61*x^2 + 110*x^3 - 54*x^4 + 12*x^5 - x^6)/(64 + 32*x + 4*x^2)]),x]

[Out]

Log[-4 - Log[(511 + 274*x - 61*x^2 + 110*x^3 - 54*x^4 + 12*x^5 - x^6)/(4*(4 + x)^2)]]

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fricas [A]  time = 0.97, size = 44, normalized size = 1.42 \begin {gather*} \log \left (\log \left (-\frac {x^{6} - 12 \, x^{5} + 54 \, x^{4} - 110 \, x^{3} + 61 \, x^{2} - 274 \, x - 511}{4 \, {\left (x^{2} + 8 \, x + 16\right )}}\right ) + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^6-12*x^5-132*x^4+754*x^3-1320*x^2+762*x-74)/((x^7-8*x^6+6*x^5+106*x^4-379*x^3-30*x^2-1607*x-204
4)*log((-x^6+12*x^5-54*x^4+110*x^3-61*x^2+274*x+511)/(4*x^2+32*x+64))+4*x^7-32*x^6+24*x^5+424*x^4-1516*x^3-120
*x^2-6428*x-8176),x, algorithm="fricas")

[Out]

log(log(-1/4*(x^6 - 12*x^5 + 54*x^4 - 110*x^3 + 61*x^2 - 274*x - 511)/(x^2 + 8*x + 16)) + 4)

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giac [A]  time = 0.23, size = 44, normalized size = 1.42 \begin {gather*} \log \left (\log \left (-\frac {x^{6} - 12 \, x^{5} + 54 \, x^{4} - 110 \, x^{3} + 61 \, x^{2} - 274 \, x - 511}{4 \, {\left (x^{2} + 8 \, x + 16\right )}}\right ) + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^6-12*x^5-132*x^4+754*x^3-1320*x^2+762*x-74)/((x^7-8*x^6+6*x^5+106*x^4-379*x^3-30*x^2-1607*x-204
4)*log((-x^6+12*x^5-54*x^4+110*x^3-61*x^2+274*x+511)/(4*x^2+32*x+64))+4*x^7-32*x^6+24*x^5+424*x^4-1516*x^3-120
*x^2-6428*x-8176),x, algorithm="giac")

[Out]

log(log(-1/4*(x^6 - 12*x^5 + 54*x^4 - 110*x^3 + 61*x^2 - 274*x - 511)/(x^2 + 8*x + 16)) + 4)

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maple [A]  time = 0.12, size = 48, normalized size = 1.55

method	result	size
norman	\(\ln \left (\ln \left (\frac {-x^{6}+12 x^{5}-54 x^{4}+110 x^{3}-61 x^{2}+274 x +511}{4 x^{2}+32 x +64}\right )+4\right )\)	\(48\)
risch	\(\ln \left (\ln \left (\frac {-x^{6}+12 x^{5}-54 x^{4}+110 x^{3}-61 x^{2}+274 x +511}{4 x^{2}+32 x +64}\right )+4\right )\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^6-12*x^5-132*x^4+754*x^3-1320*x^2+762*x-74)/((x^7-8*x^6+6*x^5+106*x^4-379*x^3-30*x^2-1607*x-2044)*ln(
(-x^6+12*x^5-54*x^4+110*x^3-61*x^2+274*x+511)/(4*x^2+32*x+64))+4*x^7-32*x^6+24*x^5+424*x^4-1516*x^3-120*x^2-64
28*x-8176),x,method=_RETURNVERBOSE)

[Out]

ln(ln((-x^6+12*x^5-54*x^4+110*x^3-61*x^2+274*x+511)/(4*x^2+32*x+64))+4)

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maxima [A]  time = 0.53, size = 44, normalized size = 1.42 \begin {gather*} \log \left (-2 \, \log \relax (2) + \log \left (-x^{6} + 12 \, x^{5} - 54 \, x^{4} + 110 \, x^{3} - 61 \, x^{2} + 274 \, x + 511\right ) - 2 \, \log \left (x + 4\right ) + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^6-12*x^5-132*x^4+754*x^3-1320*x^2+762*x-74)/((x^7-8*x^6+6*x^5+106*x^4-379*x^3-30*x^2-1607*x-204
4)*log((-x^6+12*x^5-54*x^4+110*x^3-61*x^2+274*x+511)/(4*x^2+32*x+64))+4*x^7-32*x^6+24*x^5+424*x^4-1516*x^3-120
*x^2-6428*x-8176),x, algorithm="maxima")

[Out]

log(-2*log(2) + log(-x^6 + 12*x^5 - 54*x^4 + 110*x^3 - 61*x^2 + 274*x + 511) - 2*log(x + 4) + 4)

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mupad [B]  time = 3.92, size = 47, normalized size = 1.52 \begin {gather*} \ln \left (\ln \left (\frac {-x^6+12\,x^5-54\,x^4+110\,x^3-61\,x^2+274\,x+511}{4\,x^2+32\,x+64}\right )+4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1320*x^2 - 762*x - 754*x^3 + 132*x^4 + 12*x^5 - 4*x^6 + 74)/(6428*x + log((274*x - 61*x^2 + 110*x^3 - 54*
x^4 + 12*x^5 - x^6 + 511)/(32*x + 4*x^2 + 64))*(1607*x + 30*x^2 + 379*x^3 - 106*x^4 - 6*x^5 + 8*x^6 - x^7 + 20
44) + 120*x^2 + 1516*x^3 - 424*x^4 - 24*x^5 + 32*x^6 - 4*x^7 + 8176),x)

[Out]

log(log((274*x - 61*x^2 + 110*x^3 - 54*x^4 + 12*x^5 - x^6 + 511)/(32*x + 4*x^2 + 64)) + 4)

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sympy [A]  time = 0.69, size = 42, normalized size = 1.35 \begin {gather*} \log {\left (\log {\left (\frac {- x^{6} + 12 x^{5} - 54 x^{4} + 110 x^{3} - 61 x^{2} + 274 x + 511}{4 x^{2} + 32 x + 64} \right )} + 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**6-12*x**5-132*x**4+754*x**3-1320*x**2+762*x-74)/((x**7-8*x**6+6*x**5+106*x**4-379*x**3-30*x**2
-1607*x-2044)*ln((-x**6+12*x**5-54*x**4+110*x**3-61*x**2+274*x+511)/(4*x**2+32*x+64))+4*x**7-32*x**6+24*x**5+4
24*x**4-1516*x**3-120*x**2-6428*x-8176),x)

[Out]

log(log((-x**6 + 12*x**5 - 54*x**4 + 110*x**3 - 61*x**2 + 274*x + 511)/(4*x**2 + 32*x + 64)) + 4)

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