3.38.60 \(\int \frac {96+e^4 (-8-2 x)+24 x+(96-8 e^4) \log (x)-10 \log ^2(x)}{(-96 x-24 x^2+e^4 (8 x+2 x^2)) \log (x)+(10 x+3 x^2) \log ^2(x)} \, dx\)

Optimal. Leaf size=29 \[ \log \left (\frac {3}{2}+\frac {5}{x}-\frac {\left (12-e^4\right ) (4+x)}{x \log (x)}\right ) \]

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Rubi [F]  time = 1.47, 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 {96+e^4 (-8-2 x)+24 x+\left (96-8 e^4\right ) \log (x)-10 \log ^2(x)}{\left (-96 x-24 x^2+e^4 \left (8 x+2 x^2\right )\right ) \log (x)+\left (10 x+3 x^2\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(96 + E^4*(-8 - 2*x) + 24*x + (96 - 8*E^4)*Log[x] - 10*Log[x]^2)/((-96*x - 24*x^2 + E^4*(8*x + 2*x^2))*Log
[x] + (10*x + 3*x^2)*Log[x]^2),x]

[Out]

-Log[x] + Log[10 + 3*x] - Log[Log[x]] + 3*Defer[Int][(2*(-12 + E^4)*(4 + x) + (10 + 3*x)*Log[x])^(-1), x] - 4*
(12 - E^4)*Defer[Int][1/((-10 - 3*x)*(2*(-12 + E^4)*(4 + x) + (10 + 3*x)*Log[x])), x] + 10*Defer[Int][1/(x*(2*
(-12 + E^4)*(4 + x) + (10 + 3*x)*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-96-e^4 (-8-2 x)-24 x-\left (96-8 e^4\right ) \log (x)+10 \log ^2(x)}{x \log (x) \left (96 \left (1-\frac {e^4}{12}\right )+24 \left (1-\frac {e^4}{12}\right ) x-10 \log (x)-3 x \log (x)\right )} \, dx\\ &=\int \frac {2 \left (-\left (\left (-12+e^4\right ) (4+x)\right )-4 \left (-12+e^4\right ) \log (x)-5 \log ^2(x)\right )}{x \log (x) \left (2 \left (-12+e^4\right ) (4+x)+(10+3 x) \log (x)\right )} \, dx\\ &=2 \int \frac {-\left (\left (-12+e^4\right ) (4+x)\right )-4 \left (-12+e^4\right ) \log (x)-5 \log ^2(x)}{x \log (x) \left (2 \left (-12+e^4\right ) (4+x)+(10+3 x) \log (x)\right )} \, dx\\ &=2 \int \left (-\frac {5}{x (10+3 x)}-\frac {1}{2 x \log (x)}+\frac {-10-3 x}{2 x \left (96 \left (1-\frac {e^4}{12}\right )+24 \left (1-\frac {e^4}{12}\right ) x-10 \log (x)-3 x \log (x)\right )}+\frac {2 \left (-12+e^4\right )}{(10+3 x) \left (96 \left (1-\frac {e^4}{12}\right )+24 \left (1-\frac {e^4}{12}\right ) x-10 \log (x)-3 x \log (x)\right )}\right ) \, dx\\ &=-\left (10 \int \frac {1}{x (10+3 x)} \, dx\right )-\left (4 \left (12-e^4\right )\right ) \int \frac {1}{(10+3 x) \left (96 \left (1-\frac {e^4}{12}\right )+24 \left (1-\frac {e^4}{12}\right ) x-10 \log (x)-3 x \log (x)\right )} \, dx-\int \frac {1}{x \log (x)} \, dx+\int \frac {-10-3 x}{x \left (96 \left (1-\frac {e^4}{12}\right )+24 \left (1-\frac {e^4}{12}\right ) x-10 \log (x)-3 x \log (x)\right )} \, dx\\ &=3 \int \frac {1}{10+3 x} \, dx-\left (4 \left (12-e^4\right )\right ) \int \frac {1}{(-10-3 x) \left (2 \left (-12+e^4\right ) (4+x)+(10+3 x) \log (x)\right )} \, dx-\int \frac {1}{x} \, dx+\int \frac {10+3 x}{x \left (2 \left (-12+e^4\right ) (4+x)+(10+3 x) \log (x)\right )} \, dx-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-\log (x)+\log (10+3 x)-\log (\log (x))-\left (4 \left (12-e^4\right )\right ) \int \frac {1}{(-10-3 x) \left (2 \left (-12+e^4\right ) (4+x)+(10+3 x) \log (x)\right )} \, dx+\int \left (\frac {3}{-96 \left (1-\frac {e^4}{12}\right )-24 \left (1-\frac {e^4}{12}\right ) x+10 \log (x)+3 x \log (x)}+\frac {10}{x \left (-96 \left (1-\frac {e^4}{12}\right )-24 \left (1-\frac {e^4}{12}\right ) x+10 \log (x)+3 x \log (x)\right )}\right ) \, dx\\ &=-\log (x)+\log (10+3 x)-\log (\log (x))+3 \int \frac {1}{-96 \left (1-\frac {e^4}{12}\right )-24 \left (1-\frac {e^4}{12}\right ) x+10 \log (x)+3 x \log (x)} \, dx+10 \int \frac {1}{x \left (-96 \left (1-\frac {e^4}{12}\right )-24 \left (1-\frac {e^4}{12}\right ) x+10 \log (x)+3 x \log (x)\right )} \, dx-\left (4 \left (12-e^4\right )\right ) \int \frac {1}{(-10-3 x) \left (2 \left (-12+e^4\right ) (4+x)+(10+3 x) \log (x)\right )} \, dx\\ &=-\log (x)+\log (10+3 x)-\log (\log (x))+3 \int \frac {1}{2 \left (-12+e^4\right ) (4+x)+(10+3 x) \log (x)} \, dx+10 \int \frac {1}{x \left (2 \left (-12+e^4\right ) (4+x)+(10+3 x) \log (x)\right )} \, dx-\left (4 \left (12-e^4\right )\right ) \int \frac {1}{(-10-3 x) \left (2 \left (-12+e^4\right ) (4+x)+(10+3 x) \log (x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 3.15, size = 70, normalized size = 2.41 \begin {gather*} -2 \left (\frac {\log (x)}{2}-\frac {1}{2} \log (10+3 x)+\frac {1}{2} \log (\log (x))\right )-2 \left (\frac {1}{2} \log (10+3 x)-\frac {1}{2} \log \left (96-8 e^4+24 x-2 e^4 x-10 \log (x)-3 x \log (x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(96 + E^4*(-8 - 2*x) + 24*x + (96 - 8*E^4)*Log[x] - 10*Log[x]^2)/((-96*x - 24*x^2 + E^4*(8*x + 2*x^2
))*Log[x] + (10*x + 3*x^2)*Log[x]^2),x]

[Out]

-2*(Log[x]/2 - Log[10 + 3*x]/2 + Log[Log[x]]/2) - 2*(Log[10 + 3*x]/2 - Log[96 - 8*E^4 + 24*x - 2*E^4*x - 10*Lo
g[x] - 3*x*Log[x]]/2)

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fricas [A]  time = 0.72, size = 45, normalized size = 1.55 \begin {gather*} \log \left (3 \, x + 10\right ) - \log \relax (x) + \log \left (\frac {2 \, {\left (x + 4\right )} e^{4} + {\left (3 \, x + 10\right )} \log \relax (x) - 24 \, x - 96}{3 \, x + 10}\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*log(x)^2+(-8*exp(2)^2+96)*log(x)+(-2*x-8)*exp(2)^2+24*x+96)/((3*x^2+10*x)*log(x)^2+((2*x^2+8*x)
*exp(2)^2-24*x^2-96*x)*log(x)),x, algorithm="fricas")

[Out]

log(3*x + 10) - log(x) + log((2*(x + 4)*e^4 + (3*x + 10)*log(x) - 24*x - 96)/(3*x + 10)) - log(log(x))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*log(x)^2+(-8*exp(2)^2+96)*log(x)+(-2*x-8)*exp(2)^2+24*x+96)/((3*x^2+10*x)*log(x)^2+((2*x^2+8*x)
*exp(2)^2-24*x^2-96*x)*log(x)),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.08, size = 39, normalized size = 1.34

method	result	size
norman	\(-\ln \relax (x )-\ln \left (\ln \relax (x )\right )+\ln \left (2 x \,{\mathrm e}^{4}+8 \,{\mathrm e}^{4}+3 x \ln \relax (x )+10 \ln \relax (x )-24 x -96\right )\)	\(39\)
risch	\(-\ln \relax (x )+\ln \left (3 x +10\right )+\ln \left (\ln \relax (x )+\frac {2 x \,{\mathrm e}^{4}+8 \,{\mathrm e}^{4}-24 x -96}{3 x +10}\right )-\ln \left (\ln \relax (x )\right )\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-10*ln(x)^2+(-8*exp(2)^2+96)*ln(x)+(-2*x-8)*exp(2)^2+24*x+96)/((3*x^2+10*x)*ln(x)^2+((2*x^2+8*x)*exp(2)^2
-24*x^2-96*x)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

-ln(x)-ln(ln(x))+ln(2*x*exp(2)^2+8*exp(2)^2+3*x*ln(x)+10*ln(x)-24*x-96)

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maxima [A]  time = 0.55, size = 46, normalized size = 1.59 \begin {gather*} \log \left (3 \, x + 10\right ) - \log \relax (x) + \log \left (\frac {2 \, x {\left (e^{4} - 12\right )} + {\left (3 \, x + 10\right )} \log \relax (x) + 8 \, e^{4} - 96}{3 \, x + 10}\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*log(x)^2+(-8*exp(2)^2+96)*log(x)+(-2*x-8)*exp(2)^2+24*x+96)/((3*x^2+10*x)*log(x)^2+((2*x^2+8*x)
*exp(2)^2-24*x^2-96*x)*log(x)),x, algorithm="maxima")

[Out]

log(3*x + 10) - log(x) + log((2*x*(e^4 - 12) + (3*x + 10)*log(x) + 8*e^4 - 96)/(3*x + 10)) - log(log(x))

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mupad [B]  time = 2.36, size = 34, normalized size = 1.17 \begin {gather*} \ln \left (8\,{\mathrm {e}}^4-24\,x+10\,\ln \relax (x)+2\,x\,{\mathrm {e}}^4+3\,x\,\ln \relax (x)-96\right )-\ln \left (\ln \relax (x)\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(10*log(x)^2 - 24*x + log(x)*(8*exp(4) - 96) + exp(4)*(2*x + 8) - 96)/(log(x)^2*(10*x + 3*x^2) - log(x)*(
96*x - exp(4)*(8*x + 2*x^2) + 24*x^2)),x)

[Out]

log(8*exp(4) - 24*x + 10*log(x) + 2*x*exp(4) + 3*x*log(x) - 96) - log(log(x)) - 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((-10*ln(x)**2+(-8*exp(2)**2+96)*ln(x)+(-2*x-8)*exp(2)**2+24*x+96)/((3*x**2+10*x)*ln(x)**2+((2*x**2+8
*x)*exp(2)**2-24*x**2-96*x)*ln(x)),x)

[Out]

Exception raised: PolynomialError

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