3.46.93 \(\int \frac {-6144 x^3+24 x^4+(-1572894 x^2+12288 x^3-24 x^4) \log (262149-2048 x+4 x^2)+(-262149+2048 x-4 x^2) \log ^2(262149-2048 x+4 x^2)}{(262149 x-2048 x^2+4 x^3) \log ^2(262149-2048 x+4 x^2)} \, dx\)

Optimal. Leaf size=27 \[ e^3-\frac {3 x^2}{\log \left (5+4 (256-x)^2\right )}-\log (x) \]

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Rubi [F]  time = 1.16, 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 {-6144 x^3+24 x^4+\left (-1572894 x^2+12288 x^3-24 x^4\right ) \log \left (262149-2048 x+4 x^2\right )+\left (-262149+2048 x-4 x^2\right ) \log ^2\left (262149-2048 x+4 x^2\right )}{\left (262149 x-2048 x^2+4 x^3\right ) \log ^2\left (262149-2048 x+4 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6144*x^3 + 24*x^4 + (-1572894*x^2 + 12288*x^3 - 24*x^4)*Log[262149 - 2048*x + 4*x^2] + (-262149 + 2048*x
- 4*x^2)*Log[262149 - 2048*x + 4*x^2]^2)/((262149*x - 2048*x^2 + 4*x^3)*Log[262149 - 2048*x + 4*x^2]^2),x]

[Out]

-Log[x] + 1536*Defer[Int][Log[262149 - 2048*x + 4*x^2]^(-2), x] - ((805321728*I)*Defer[Int][1/((2048 + (4*I)*S
qrt[5] - 8*x)*Log[262149 - 2048*x + 4*x^2]^2), x])/Sqrt[5] + 6*Defer[Int][x/Log[262149 - 2048*x + 4*x^2]^2, x]
+ (1572834*(5 - (512*I)*Sqrt[5])*Defer[Int][1/((-2048 - (4*I)*Sqrt[5] + 8*x)*Log[262149 - 2048*x + 4*x^2]^2),
x])/5 - ((805321728*I)*Defer[Int][1/((-2048 + (4*I)*Sqrt[5] + 8*x)*Log[262149 - 2048*x + 4*x^2]^2), x])/Sqrt[
5] + (1572834*(5 + (512*I)*Sqrt[5])*Defer[Int][1/((-2048 + (4*I)*Sqrt[5] + 8*x)*Log[262149 - 2048*x + 4*x^2]^2
), x])/5 - 6*Defer[Int][x/Log[262149 - 2048*x + 4*x^2], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6144 x^3+24 x^4+\left (-1572894 x^2+12288 x^3-24 x^4\right ) \log \left (262149-2048 x+4 x^2\right )+\left (-262149+2048 x-4 x^2\right ) \log ^2\left (262149-2048 x+4 x^2\right )}{x \left (262149-2048 x+4 x^2\right ) \log ^2\left (262149-2048 x+4 x^2\right )} \, dx\\ &=\int \left (-\frac {1}{x}+\frac {24 (-256+x) x^2}{\left (262149-2048 x+4 x^2\right ) \log ^2\left (262149-2048 x+4 x^2\right )}-\frac {6 x}{\log \left (262149-2048 x+4 x^2\right )}\right ) \, dx\\ &=-\log (x)-6 \int \frac {x}{\log \left (262149-2048 x+4 x^2\right )} \, dx+24 \int \frac {(-256+x) x^2}{\left (262149-2048 x+4 x^2\right ) \log ^2\left (262149-2048 x+4 x^2\right )} \, dx\\ &=-\log (x)-6 \int \frac {x}{\log \left (262149-2048 x+4 x^2\right )} \, dx+24 \int \left (\frac {64}{\log ^2\left (262149-2048 x+4 x^2\right )}+\frac {x}{4 \log ^2\left (262149-2048 x+4 x^2\right )}+\frac {-67110144+262139 x}{4 \left (262149-2048 x+4 x^2\right ) \log ^2\left (262149-2048 x+4 x^2\right )}\right ) \, dx\\ &=-\log (x)+6 \int \frac {x}{\log ^2\left (262149-2048 x+4 x^2\right )} \, dx+6 \int \frac {-67110144+262139 x}{\left (262149-2048 x+4 x^2\right ) \log ^2\left (262149-2048 x+4 x^2\right )} \, dx-6 \int \frac {x}{\log \left (262149-2048 x+4 x^2\right )} \, dx+1536 \int \frac {1}{\log ^2\left (262149-2048 x+4 x^2\right )} \, dx\\ &=-\log (x)+6 \int \left (-\frac {67110144}{\left (262149-2048 x+4 x^2\right ) \log ^2\left (262149-2048 x+4 x^2\right )}+\frac {262139 x}{\left (262149-2048 x+4 x^2\right ) \log ^2\left (262149-2048 x+4 x^2\right )}\right ) \, dx+6 \int \frac {x}{\log ^2\left (262149-2048 x+4 x^2\right )} \, dx-6 \int \frac {x}{\log \left (262149-2048 x+4 x^2\right )} \, dx+1536 \int \frac {1}{\log ^2\left (262149-2048 x+4 x^2\right )} \, dx\\ &=-\log (x)+6 \int \frac {x}{\log ^2\left (262149-2048 x+4 x^2\right )} \, dx-6 \int \frac {x}{\log \left (262149-2048 x+4 x^2\right )} \, dx+1536 \int \frac {1}{\log ^2\left (262149-2048 x+4 x^2\right )} \, dx+1572834 \int \frac {x}{\left (262149-2048 x+4 x^2\right ) \log ^2\left (262149-2048 x+4 x^2\right )} \, dx-402660864 \int \frac {1}{\left (262149-2048 x+4 x^2\right ) \log ^2\left (262149-2048 x+4 x^2\right )} \, dx\\ &=-\log (x)+6 \int \frac {x}{\log ^2\left (262149-2048 x+4 x^2\right )} \, dx-6 \int \frac {x}{\log \left (262149-2048 x+4 x^2\right )} \, dx+1536 \int \frac {1}{\log ^2\left (262149-2048 x+4 x^2\right )} \, dx+1572834 \int \left (\frac {1-\frac {512 i}{\sqrt {5}}}{\left (-2048-4 i \sqrt {5}+8 x\right ) \log ^2\left (262149-2048 x+4 x^2\right )}+\frac {1+\frac {512 i}{\sqrt {5}}}{\left (-2048+4 i \sqrt {5}+8 x\right ) \log ^2\left (262149-2048 x+4 x^2\right )}\right ) \, dx-402660864 \int \left (\frac {2 i}{\sqrt {5} \left (2048+4 i \sqrt {5}-8 x\right ) \log ^2\left (262149-2048 x+4 x^2\right )}+\frac {2 i}{\sqrt {5} \left (-2048+4 i \sqrt {5}+8 x\right ) \log ^2\left (262149-2048 x+4 x^2\right )}\right ) \, dx\\ &=-\log (x)+6 \int \frac {x}{\log ^2\left (262149-2048 x+4 x^2\right )} \, dx-6 \int \frac {x}{\log \left (262149-2048 x+4 x^2\right )} \, dx+1536 \int \frac {1}{\log ^2\left (262149-2048 x+4 x^2\right )} \, dx-\frac {(805321728 i) \int \frac {1}{\left (2048+4 i \sqrt {5}-8 x\right ) \log ^2\left (262149-2048 x+4 x^2\right )} \, dx}{\sqrt {5}}-\frac {(805321728 i) \int \frac {1}{\left (-2048+4 i \sqrt {5}+8 x\right ) \log ^2\left (262149-2048 x+4 x^2\right )} \, dx}{\sqrt {5}}+\frac {1}{5} \left (1572834 \left (5-512 i \sqrt {5}\right )\right ) \int \frac {1}{\left (-2048-4 i \sqrt {5}+8 x\right ) \log ^2\left (262149-2048 x+4 x^2\right )} \, dx+\frac {1}{5} \left (1572834 \left (5+512 i \sqrt {5}\right )\right ) \int \frac {1}{\left (-2048+4 i \sqrt {5}+8 x\right ) \log ^2\left (262149-2048 x+4 x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.30, size = 23, normalized size = 0.85 \begin {gather*} -\log (x)-\frac {3 x^2}{\log \left (262149-2048 x+4 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6144*x^3 + 24*x^4 + (-1572894*x^2 + 12288*x^3 - 24*x^4)*Log[262149 - 2048*x + 4*x^2] + (-262149 +
2048*x - 4*x^2)*Log[262149 - 2048*x + 4*x^2]^2)/((262149*x - 2048*x^2 + 4*x^3)*Log[262149 - 2048*x + 4*x^2]^2)
,x]

[Out]

-Log[x] - (3*x^2)/Log[262149 - 2048*x + 4*x^2]

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fricas [A]  time = 0.52, size = 35, normalized size = 1.30 \begin {gather*} -\frac {3 \, x^{2} + \log \left (4 \, x^{2} - 2048 \, x + 262149\right ) \log \relax (x)}{\log \left (4 \, x^{2} - 2048 \, x + 262149\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2+2048*x-262149)*log(4*x^2-2048*x+262149)^2+(-24*x^4+12288*x^3-1572894*x^2)*log(4*x^2-2048*x+
262149)+24*x^4-6144*x^3)/(4*x^3-2048*x^2+262149*x)/log(4*x^2-2048*x+262149)^2,x, algorithm="fricas")

[Out]

-(3*x^2 + log(4*x^2 - 2048*x + 262149)*log(x))/log(4*x^2 - 2048*x + 262149)

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giac [A]  time = 0.19, size = 23, normalized size = 0.85 \begin {gather*} -\frac {3 \, x^{2}}{\log \left (4 \, x^{2} - 2048 \, x + 262149\right )} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2+2048*x-262149)*log(4*x^2-2048*x+262149)^2+(-24*x^4+12288*x^3-1572894*x^2)*log(4*x^2-2048*x+
262149)+24*x^4-6144*x^3)/(4*x^3-2048*x^2+262149*x)/log(4*x^2-2048*x+262149)^2,x, algorithm="giac")

[Out]

-3*x^2/log(4*x^2 - 2048*x + 262149) - log(x)

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maple [A]  time = 0.04, size = 24, normalized size = 0.89

method	result	size
norman	\(-\frac {3 x^{2}}{\ln \left (4 x^{2}-2048 x +262149\right )}-\ln \relax (x )\)	\(24\)
risch	\(-\frac {3 x^{2}}{\ln \left (4 x^{2}-2048 x +262149\right )}-\ln \relax (x )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^2+2048*x-262149)*ln(4*x^2-2048*x+262149)^2+(-24*x^4+12288*x^3-1572894*x^2)*ln(4*x^2-2048*x+262149)+
24*x^4-6144*x^3)/(4*x^3-2048*x^2+262149*x)/ln(4*x^2-2048*x+262149)^2,x,method=_RETURNVERBOSE)

[Out]

-3*x^2/ln(4*x^2-2048*x+262149)-ln(x)

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maxima [A]  time = 0.40, size = 23, normalized size = 0.85 \begin {gather*} -\frac {3 \, x^{2}}{\log \left (4 \, x^{2} - 2048 \, x + 262149\right )} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2+2048*x-262149)*log(4*x^2-2048*x+262149)^2+(-24*x^4+12288*x^3-1572894*x^2)*log(4*x^2-2048*x+
262149)+24*x^4-6144*x^3)/(4*x^3-2048*x^2+262149*x)/log(4*x^2-2048*x+262149)^2,x, algorithm="maxima")

[Out]

-3*x^2/log(4*x^2 - 2048*x + 262149) - log(x)

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mupad [B]  time = 3.61, size = 72, normalized size = 2.67 \begin {gather*} 768\,x-\ln \relax (x)-\frac {960}{x-256}-\frac {3\,x^2-\frac {3\,x\,\ln \left (4\,x^2-2048\,x+262149\right )\,\left (4\,x^2-2048\,x+262149\right )}{4\,\left (x-256\right )}}{\ln \left (4\,x^2-2048\,x+262149\right )}-3\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(4*x^2 - 2048*x + 262149)^2*(4*x^2 - 2048*x + 262149) + log(4*x^2 - 2048*x + 262149)*(1572894*x^2 - 1
2288*x^3 + 24*x^4) + 6144*x^3 - 24*x^4)/(log(4*x^2 - 2048*x + 262149)^2*(262149*x - 2048*x^2 + 4*x^3)),x)

[Out]

768*x - log(x) - 960/(x - 256) - (3*x^2 - (3*x*log(4*x^2 - 2048*x + 262149)*(4*x^2 - 2048*x + 262149))/(4*(x -
256)))/log(4*x^2 - 2048*x + 262149) - 3*x^2

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sympy [A]  time = 0.15, size = 20, normalized size = 0.74 \begin {gather*} - \frac {3 x^{2}}{\log {\left (4 x^{2} - 2048 x + 262149 \right )}} - \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**2+2048*x-262149)*ln(4*x**2-2048*x+262149)**2+(-24*x**4+12288*x**3-1572894*x**2)*ln(4*x**2-20
48*x+262149)+24*x**4-6144*x**3)/(4*x**3-2048*x**2+262149*x)/ln(4*x**2-2048*x+262149)**2,x)

[Out]

-3*x**2/log(4*x**2 - 2048*x + 262149) - log(x)

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