3.3.26 \(\int \frac {1088-8 x-48 x^2-194 x^3+48 x^4-3 x^5}{-544 x+4 x^2-1064 x^3+6 x^4+48 x^5-4 x^6+(816 x^3-6 x^4-36 x^5+3 x^6) \log (\frac {-32+4 x}{-34-4 x+x^2})} \, dx\)

Optimal. Leaf size=27 \[ \log \left (-4-\frac {2}{x^2}+3 \log \left (\frac {4}{4+\frac {2}{8-x}+x}\right )\right ) \]

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Rubi [F]  time = 1.94, 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 {1088-8 x-48 x^2-194 x^3+48 x^4-3 x^5}{-544 x+4 x^2-1064 x^3+6 x^4+48 x^5-4 x^6+\left (816 x^3-6 x^4-36 x^5+3 x^6\right ) \log \left (\frac {-32+4 x}{-34-4 x+x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1088 - 8*x - 48*x^2 - 194*x^3 + 48*x^4 - 3*x^5)/(-544*x + 4*x^2 - 1064*x^3 + 6*x^4 + 48*x^5 - 4*x^6 + (81
6*x^3 - 6*x^4 - 36*x^5 + 3*x^6)*Log[(-32 + 4*x)/(-34 - 4*x + x^2)]),x]

[Out]

12*Defer[Int][(-2 - 4*x^2 + 3*x^2*Log[(4*(-8 + x))/(-34 - 4*x + x^2)])^(-1), x] + 204*Sqrt[2/19]*Defer[Int][1/
((4 + 2*Sqrt[38] - 2*x)*(-2 - 4*x^2 + 3*x^2*Log[(4*(-8 + x))/(-34 - 4*x + x^2)])), x] + 192*Defer[Int][1/((-8
+ x)*(-2 - 4*x^2 + 3*x^2*Log[(4*(-8 + x))/(-34 - 4*x + x^2)])), x] + 4*Defer[Int][1/(x*(-2 - 4*x^2 + 3*x^2*Log
[(4*(-8 + x))/(-34 - 4*x + x^2)])), x] - 3*Defer[Int][x/(-2 - 4*x^2 + 3*x^2*Log[(4*(-8 + x))/(-34 - 4*x + x^2)
]), x] - (252*(19 + Sqrt[38])*Defer[Int][1/((-4 - 2*Sqrt[38] + 2*x)*(-2 - 4*x^2 + 3*x^2*Log[(4*(-8 + x))/(-34
- 4*x + x^2)])), x])/19 + 204*Sqrt[2/19]*Defer[Int][1/((-4 + 2*Sqrt[38] + 2*x)*(-2 - 4*x^2 + 3*x^2*Log[(4*(-8
+ x))/(-34 - 4*x + x^2)])), x] - (252*(19 - Sqrt[38])*Defer[Int][1/((-4 + 2*Sqrt[38] + 2*x)*(-2 - 4*x^2 + 3*x^
2*Log[(4*(-8 + x))/(-34 - 4*x + x^2)])), x])/19

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1088+8 x+48 x^2+194 x^3-48 x^4+3 x^5}{x \left (272-2 x-12 x^2+x^3\right ) \left (2+4 x^2-3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )} \, dx\\ &=\int \left (\frac {12}{-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )}+\frac {192}{(-8+x) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )}+\frac {4}{x \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )}-\frac {3 x}{-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )}-\frac {12 (34+21 x)}{\left (-34-4 x+x^2\right ) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )}\right ) \, dx\\ &=-\left (3 \int \frac {x}{-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )} \, dx\right )+4 \int \frac {1}{x \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )} \, dx+12 \int \frac {1}{-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )} \, dx-12 \int \frac {34+21 x}{\left (-34-4 x+x^2\right ) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )} \, dx+192 \int \frac {1}{(-8+x) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )} \, dx\\ &=-\left (3 \int \frac {x}{-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )} \, dx\right )+4 \int \frac {1}{x \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )} \, dx+12 \int \frac {1}{-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )} \, dx-12 \int \left (\frac {34}{\left (-34-4 x+x^2\right ) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )}+\frac {21 x}{\left (-34-4 x+x^2\right ) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )}\right ) \, dx+192 \int \frac {1}{(-8+x) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )} \, dx\\ &=-\left (3 \int \frac {x}{-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )} \, dx\right )+4 \int \frac {1}{x \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )} \, dx+12 \int \frac {1}{-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )} \, dx+192 \int \frac {1}{(-8+x) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )} \, dx-252 \int \frac {x}{\left (-34-4 x+x^2\right ) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )} \, dx-408 \int \frac {1}{\left (-34-4 x+x^2\right ) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )} \, dx\\ &=-\left (3 \int \frac {x}{-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )} \, dx\right )+4 \int \frac {1}{x \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )} \, dx+12 \int \frac {1}{-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )} \, dx+192 \int \frac {1}{(-8+x) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )} \, dx-252 \int \left (\frac {1+\sqrt {\frac {2}{19}}}{\left (-4-2 \sqrt {38}+2 x\right ) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )}+\frac {1-\sqrt {\frac {2}{19}}}{\left (-4+2 \sqrt {38}+2 x\right ) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )}\right ) \, dx-408 \int \left (-\frac {1}{\sqrt {38} \left (4+2 \sqrt {38}-2 x\right ) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )}-\frac {1}{\sqrt {38} \left (-4+2 \sqrt {38}+2 x\right ) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )}\right ) \, dx\\ &=-\left (3 \int \frac {x}{-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )} \, dx\right )+4 \int \frac {1}{x \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )} \, dx+12 \int \frac {1}{-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )} \, dx+192 \int \frac {1}{(-8+x) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )} \, dx+\left (204 \sqrt {\frac {2}{19}}\right ) \int \frac {1}{\left (4+2 \sqrt {38}-2 x\right ) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )} \, dx+\left (204 \sqrt {\frac {2}{19}}\right ) \int \frac {1}{\left (-4+2 \sqrt {38}+2 x\right ) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )} \, dx-\frac {1}{19} \left (252 \left (19-\sqrt {38}\right )\right ) \int \frac {1}{\left (-4+2 \sqrt {38}+2 x\right ) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )} \, dx-\frac {1}{19} \left (252 \left (19+\sqrt {38}\right )\right ) \int \frac {1}{\left (-4-2 \sqrt {38}+2 x\right ) \left (-2-4 x^2+3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.99, size = 34, normalized size = 1.26 \begin {gather*} -2 \log (x)+\log \left (2+4 x^2-3 x^2 \log \left (\frac {4 (-8+x)}{-34-4 x+x^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1088 - 8*x - 48*x^2 - 194*x^3 + 48*x^4 - 3*x^5)/(-544*x + 4*x^2 - 1064*x^3 + 6*x^4 + 48*x^5 - 4*x^6
+ (816*x^3 - 6*x^4 - 36*x^5 + 3*x^6)*Log[(-32 + 4*x)/(-34 - 4*x + x^2)]),x]

[Out]

-2*Log[x] + Log[2 + 4*x^2 - 3*x^2*Log[(4*(-8 + x))/(-34 - 4*x + x^2)]]

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fricas [A]  time = 0.61, size = 33, normalized size = 1.22 \begin {gather*} \log \left (\frac {3 \, x^{2} \log \left (\frac {4 \, {\left (x - 8\right )}}{x^{2} - 4 \, x - 34}\right ) - 4 \, x^{2} - 2}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^5+48*x^4-194*x^3-48*x^2-8*x+1088)/((3*x^6-36*x^5-6*x^4+816*x^3)*log((4*x-32)/(x^2-4*x-34))-4*x
^6+48*x^5+6*x^4-1064*x^3+4*x^2-544*x),x, algorithm="fricas")

[Out]

log((3*x^2*log(4*(x - 8)/(x^2 - 4*x - 34)) - 4*x^2 - 2)/x^2)

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giac [A]  time = 0.32, size = 34, normalized size = 1.26 \begin {gather*} \log \left (3 \, x^{2} \log \left (\frac {4 \, {\left (x - 8\right )}}{x^{2} - 4 \, x - 34}\right ) - 4 \, x^{2} - 2\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^5+48*x^4-194*x^3-48*x^2-8*x+1088)/((3*x^6-36*x^5-6*x^4+816*x^3)*log((4*x-32)/(x^2-4*x-34))-4*x
^6+48*x^5+6*x^4-1064*x^3+4*x^2-544*x),x, algorithm="giac")

[Out]

log(3*x^2*log(4*(x - 8)/(x^2 - 4*x - 34)) - 4*x^2 - 2) - 2*log(x)

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maple [A]  time = 0.07, size = 32, normalized size = 1.19

method	result	size
risch	\(\ln \left (\ln \left (\frac {4 x -32}{x^{2}-4 x -34}\right )-\frac {2 \left (2 x^{2}+1\right )}{3 x^{2}}\right )\)	\(32\)
norman	\(-2 \ln \relax (x )+\ln \left (3 x^{2} \ln \left (\frac {4 x -32}{x^{2}-4 x -34}\right )-4 x^{2}-2\right )\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*x^5+48*x^4-194*x^3-48*x^2-8*x+1088)/((3*x^6-36*x^5-6*x^4+816*x^3)*ln((4*x-32)/(x^2-4*x-34))-4*x^6+48*x
^5+6*x^4-1064*x^3+4*x^2-544*x),x,method=_RETURNVERBOSE)

[Out]

ln(ln((4*x-32)/(x^2-4*x-34))-2/3*(2*x^2+1)/x^2)

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maxima [A]  time = 0.76, size = 42, normalized size = 1.56 \begin {gather*} \log \left (-\frac {2 \, x^{2} {\left (3 \, \log \relax (2) - 2\right )} - 3 \, x^{2} \log \left (x^{2} - 4 \, x - 34\right ) + 3 \, x^{2} \log \left (x - 8\right ) - 2}{3 \, x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^5+48*x^4-194*x^3-48*x^2-8*x+1088)/((3*x^6-36*x^5-6*x^4+816*x^3)*log((4*x-32)/(x^2-4*x-34))-4*x
^6+48*x^5+6*x^4-1064*x^3+4*x^2-544*x),x, algorithm="maxima")

[Out]

log(-1/3*(2*x^2*(3*log(2) - 2) - 3*x^2*log(x^2 - 4*x - 34) + 3*x^2*log(x - 8) - 2)/x^2)

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mupad [B]  time = 0.70, size = 37, normalized size = 1.37 \begin {gather*} \ln \left (x^2\,\ln \left (-\frac {4\,x-32}{-x^2+4\,x+34}\right )-\frac {4\,x^2}{3}-\frac {2}{3}\right )+\ln \left (\frac {1}{x^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x + 48*x^2 + 194*x^3 - 48*x^4 + 3*x^5 - 1088)/(log(-(4*x - 32)/(4*x - x^2 + 34))*(816*x^3 - 6*x^4 - 36
*x^5 + 3*x^6) - 544*x + 4*x^2 - 1064*x^3 + 6*x^4 + 48*x^5 - 4*x^6),x)

[Out]

log(x^2*log(-(4*x - 32)/(4*x - x^2 + 34)) - (4*x^2)/3 - 2/3) + log(1/x^2)

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sympy [A]  time = 0.49, size = 29, normalized size = 1.07 \begin {gather*} \log {\left (\log {\left (\frac {4 x - 32}{x^{2} - 4 x - 34} \right )} + \frac {- 4 x^{2} - 2}{3 x^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x**5+48*x**4-194*x**3-48*x**2-8*x+1088)/((3*x**6-36*x**5-6*x**4+816*x**3)*ln((4*x-32)/(x**2-4*x-
34))-4*x**6+48*x**5+6*x**4-1064*x**3+4*x**2-544*x),x)

[Out]

log(log((4*x - 32)/(x**2 - 4*x - 34)) + (-4*x**2 - 2)/(3*x**2))

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