3.43.95 \(\int \frac {-28+16 x+60000 x^4-10000 x^5+(24-12 x-30000 x^4+5000 x^5) \log (x)+(-4+2 x+3750 x^4-625 x^5) \log ^2(x)}{-16 x+10000 x^5+(12 x-5000 x^5) \log (x)+(-2 x+625 x^5) \log ^2(x)} \, dx\)

Optimal. Leaf size=27 \[ -x+\log \left (x \left (x-x \left (-1+625 x^4-\frac {4}{-4+\log (x)}\right )\right )\right ) \]

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Rubi [F]  time = 2.00, 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 {-28+16 x+60000 x^4-10000 x^5+\left (24-12 x-30000 x^4+5000 x^5\right ) \log (x)+\left (-4+2 x+3750 x^4-625 x^5\right ) \log ^2(x)}{-16 x+10000 x^5+\left (12 x-5000 x^5\right ) \log (x)+\left (-2 x+625 x^5\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-28 + 16*x + 60000*x^4 - 10000*x^5 + (24 - 12*x - 30000*x^4 + 5000*x^5)*Log[x] + (-4 + 2*x + 3750*x^4 - 6
25*x^5)*Log[x]^2)/(-16*x + 10000*x^5 + (12*x - 5000*x^5)*Log[x] + (-2*x + 625*x^5)*Log[x]^2),x]

[Out]

-x + 2*Log[x] + Log[2 - 625*x^4] - Log[4 - Log[x]] - 20*Defer[Int][1/((I*2^(1/4) - 5*x)*(4 - 2500*x^4 - 2*Log[
x] + 625*x^4*Log[x])), x] - 20*Defer[Int][1/((2^(1/4) - 5*x)*(4 - 2500*x^4 - 2*Log[x] + 625*x^4*Log[x])), x] -
2*Defer[Int][1/(x*(4 - 2500*x^4 - 2*Log[x] + 625*x^4*Log[x])), x] + 625*Defer[Int][x^3/(4 - 2500*x^4 - 2*Log[
x] + 625*x^4*Log[x]), x] + 20*Defer[Int][1/((I*2^(1/4) + 5*x)*(4 - 2500*x^4 - 2*Log[x] + 625*x^4*Log[x])), x]
+ 20*Defer[Int][1/((2^(1/4) + 5*x)*(4 - 2500*x^4 - 2*Log[x] + 625*x^4*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-4+2 x+3750 x^4-625 x^5}{x \left (-2+625 x^4\right )}-\frac {1}{x (-4+\log (x))}+\frac {4+7500 x^4+390625 x^8}{x \left (-2+625 x^4\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )}\right ) \, dx\\ &=\int \frac {-4+2 x+3750 x^4-625 x^5}{x \left (-2+625 x^4\right )} \, dx-\int \frac {1}{x (-4+\log (x))} \, dx+\int \frac {4+7500 x^4+390625 x^8}{x \left (-2+625 x^4\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx\\ &=\int \left (\frac {2-625 x^4}{-2+625 x^4}+\frac {-4+3750 x^4}{x \left (-2+625 x^4\right )}\right ) \, dx+\int \left (-\frac {2}{x \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )}+\frac {625 x^3}{4-2500 x^4-2 \log (x)+625 x^4 \log (x)}+\frac {10000 x^3}{\left (-2+625 x^4\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )}\right ) \, dx-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,-4+\log (x)\right )\\ &=-\log (4-\log (x))-2 \int \frac {1}{x \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx+625 \int \frac {x^3}{4-2500 x^4-2 \log (x)+625 x^4 \log (x)} \, dx+10000 \int \frac {x^3}{\left (-2+625 x^4\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx+\int \frac {2-625 x^4}{-2+625 x^4} \, dx+\int \frac {-4+3750 x^4}{x \left (-2+625 x^4\right )} \, dx\\ &=-\log (4-\log (x))+\frac {1}{4} \operatorname {Subst}\left (\int \frac {-4+3750 x}{x (-2+625 x)} \, dx,x,x^4\right )-2 \int \frac {1}{x \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx+625 \int \frac {x^3}{4-2500 x^4-2 \log (x)+625 x^4 \log (x)} \, dx+10000 \int \left (\frac {x}{2 \left (-25 \sqrt {2}+625 x^2\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )}+\frac {x}{2 \left (25 \sqrt {2}+625 x^2\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )}\right ) \, dx-\int 1 \, dx\\ &=-x-\log (4-\log (x))+\frac {1}{4} \operatorname {Subst}\left (\int \left (\frac {2}{x}+\frac {2500}{-2+625 x}\right ) \, dx,x,x^4\right )-2 \int \frac {1}{x \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx+625 \int \frac {x^3}{4-2500 x^4-2 \log (x)+625 x^4 \log (x)} \, dx+5000 \int \frac {x}{\left (-25 \sqrt {2}+625 x^2\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx+5000 \int \frac {x}{\left (25 \sqrt {2}+625 x^2\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx\\ &=-x+2 \log (x)+\log \left (2-625 x^4\right )-\log (4-\log (x))-2 \int \frac {1}{x \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx+625 \int \frac {x^3}{4-2500 x^4-2 \log (x)+625 x^4 \log (x)} \, dx+5000 \int \left (-\frac {1}{250 \left (i \sqrt [4]{2}-5 x\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )}+\frac {1}{250 \left (i \sqrt [4]{2}+5 x\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )}\right ) \, dx+5000 \int \left (-\frac {1}{250 \left (\sqrt [4]{2}-5 x\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )}+\frac {1}{250 \left (\sqrt [4]{2}+5 x\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )}\right ) \, dx\\ &=-x+2 \log (x)+\log \left (2-625 x^4\right )-\log (4-\log (x))-2 \int \frac {1}{x \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx-20 \int \frac {1}{\left (i \sqrt [4]{2}-5 x\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx-20 \int \frac {1}{\left (\sqrt [4]{2}-5 x\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx+20 \int \frac {1}{\left (i \sqrt [4]{2}+5 x\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx+20 \int \frac {1}{\left (\sqrt [4]{2}+5 x\right ) \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right )} \, dx+625 \int \frac {x^3}{4-2500 x^4-2 \log (x)+625 x^4 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 36, normalized size = 1.33 \begin {gather*} -x+2 \log (x)-\log (4-\log (x))+\log \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-28 + 16*x + 60000*x^4 - 10000*x^5 + (24 - 12*x - 30000*x^4 + 5000*x^5)*Log[x] + (-4 + 2*x + 3750*x
^4 - 625*x^5)*Log[x]^2)/(-16*x + 10000*x^5 + (12*x - 5000*x^5)*Log[x] + (-2*x + 625*x^5)*Log[x]^2),x]

[Out]

-x + 2*Log[x] - Log[4 - Log[x]] + Log[4 - 2500*x^4 - 2*Log[x] + 625*x^4*Log[x]]

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fricas [A]  time = 0.51, size = 53, normalized size = 1.96 \begin {gather*} -x + \log \left (625 \, x^{4} - 2\right ) + 2 \, \log \relax (x) + \log \left (-\frac {2500 \, x^{4} - {\left (625 \, x^{4} - 2\right )} \log \relax (x) - 4}{625 \, x^{4} - 2}\right ) - \log \left (\log \relax (x) - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-625*x^5+3750*x^4+2*x-4)*log(x)^2+(5000*x^5-30000*x^4-12*x+24)*log(x)-10000*x^5+60000*x^4+16*x-28)
/((625*x^5-2*x)*log(x)^2+(-5000*x^5+12*x)*log(x)+10000*x^5-16*x),x, algorithm="fricas")

[Out]

-x + log(625*x^4 - 2) + 2*log(x) + log(-(2500*x^4 - (625*x^4 - 2)*log(x) - 4)/(625*x^4 - 2)) - log(log(x) - 4)

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giac [A]  time = 0.21, size = 34, normalized size = 1.26 \begin {gather*} -x + \log \left (625 \, x^{4} \log \relax (x) - 2500 \, x^{4} - 2 \, \log \relax (x) + 4\right ) + 2 \, \log \relax (x) - \log \left (\log \relax (x) - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-625*x^5+3750*x^4+2*x-4)*log(x)^2+(5000*x^5-30000*x^4-12*x+24)*log(x)-10000*x^5+60000*x^4+16*x-28)
/((625*x^5-2*x)*log(x)^2+(-5000*x^5+12*x)*log(x)+10000*x^5-16*x),x, algorithm="giac")

[Out]

-x + log(625*x^4*log(x) - 2500*x^4 - 2*log(x) + 4) + 2*log(x) - log(log(x) - 4)

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maple [A]  time = 0.05, size = 35, normalized size = 1.30

method	result	size
norman	\(2 \ln \relax (x )-x -\ln \left (\ln \relax (x )-4\right )+\ln \left (625 x^{4} \ln \relax (x )-2500 x^{4}-2 \ln \relax (x )+4\right )\)	\(35\)
risch	\(-x +2 \ln \relax (x )+\ln \left (625 x^{4}-2\right )+\ln \left (\ln \relax (x )-\frac {4 \left (625 x^{4}-1\right )}{625 x^{4}-2}\right )-\ln \left (\ln \relax (x )-4\right )\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-625*x^5+3750*x^4+2*x-4)*ln(x)^2+(5000*x^5-30000*x^4-12*x+24)*ln(x)-10000*x^5+60000*x^4+16*x-28)/((625*x
^5-2*x)*ln(x)^2+(-5000*x^5+12*x)*ln(x)+10000*x^5-16*x),x,method=_RETURNVERBOSE)

[Out]

2*ln(x)-x-ln(ln(x)-4)+ln(625*x^4*ln(x)-2500*x^4-2*ln(x)+4)

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maxima [A]  time = 0.49, size = 53, normalized size = 1.96 \begin {gather*} -x + \log \left (625 \, x^{4} - 2\right ) + 2 \, \log \relax (x) + \log \left (-\frac {2500 \, x^{4} - {\left (625 \, x^{4} - 2\right )} \log \relax (x) - 4}{625 \, x^{4} - 2}\right ) - \log \left (\log \relax (x) - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-625*x^5+3750*x^4+2*x-4)*log(x)^2+(5000*x^5-30000*x^4-12*x+24)*log(x)-10000*x^5+60000*x^4+16*x-28)
/((625*x^5-2*x)*log(x)^2+(-5000*x^5+12*x)*log(x)+10000*x^5-16*x),x, algorithm="maxima")

[Out]

-x + log(625*x^4 - 2) + 2*log(x) + log(-(2500*x^4 - (625*x^4 - 2)*log(x) - 4)/(625*x^4 - 2)) - log(log(x) - 4)

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mupad [B]  time = 3.58, size = 92, normalized size = 3.41 \begin {gather*} \ln \left (390625\,x^8+7500\,x^4+4\right )-x+\ln \left (\frac {4\,\ln \relax (x)-1250\,x^4\,\ln \relax (x)+5000\,x^4-8}{2\,x-625\,x^5}\right )-\ln \left (x^4-\frac {2}{625}\right )-\ln \left (\frac {\left (\ln \relax (x)-4\right )\,\left (390625\,x^8+7500\,x^4+4\right )}{x\,{\left (625\,x^4-2\right )}^2}\right )+2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(16*x + log(x)^2*(2*x + 3750*x^4 - 625*x^5 - 4) + 60000*x^4 - 10000*x^5 - log(x)*(12*x + 30000*x^4 - 5000
*x^5 - 24) - 28)/(16*x + log(x)^2*(2*x - 625*x^5) - log(x)*(12*x - 5000*x^5) - 10000*x^5),x)

[Out]

log(7500*x^4 + 390625*x^8 + 4) - x + log((4*log(x) - 1250*x^4*log(x) + 5000*x^4 - 8)/(2*x - 625*x^5)) - log(x^
4 - 2/625) - log(((log(x) - 4)*(7500*x^4 + 390625*x^8 + 4))/(x*(625*x^4 - 2)^2)) + 2*log(x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-625*x**5+3750*x**4+2*x-4)*ln(x)**2+(5000*x**5-30000*x**4-12*x+24)*ln(x)-10000*x**5+60000*x**4+16*
x-28)/((625*x**5-2*x)*ln(x)**2+(-5000*x**5+12*x)*ln(x)+10000*x**5-16*x),x)

[Out]

Exception raised: PolynomialError

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