3.27.97 \(\int \frac {-16 x+256 \sqrt [5]{e} x-256 x^2-1024 x^3+(-48+768 \sqrt [5]{e}-768 x-3072 x^2) \log (2)+(-16 x+256 \sqrt [5]{e} x+1024 x^3+(768 x+6144 x^2) \log (2)) \log (x) \log (\log (x))}{(x+256 e^{2/5} x+32 x^2+384 x^3+2048 x^4+4096 x^5+\sqrt [5]{e} (-32 x-512 x^2-2048 x^3)) \log (2) \log (x)} \, dx\)

Optimal. Leaf size=31 \[ \frac {\left (3+\frac {x}{\log (2)}\right ) \log (\log (x))}{\sqrt [5]{e}-\left (\frac {1}{4}+2 x\right )^2} \]

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Rubi [F]  time = 1.83, 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 {-16 x+256 \sqrt [5]{e} x-256 x^2-1024 x^3+\left (-48+768 \sqrt [5]{e}-768 x-3072 x^2\right ) \log (2)+\left (-16 x+256 \sqrt [5]{e} x+1024 x^3+\left (768 x+6144 x^2\right ) \log (2)\right ) \log (x) \log (\log (x))}{\left (x+256 e^{2/5} x+32 x^2+384 x^3+2048 x^4+4096 x^5+\sqrt [5]{e} \left (-32 x-512 x^2-2048 x^3\right )\right ) \log (2) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-16*x + 256*E^(1/5)*x - 256*x^2 - 1024*x^3 + (-48 + 768*E^(1/5) - 768*x - 3072*x^2)*Log[2] + (-16*x + 256
*E^(1/5)*x + 1024*x^3 + (768*x + 6144*x^2)*Log[2])*Log[x]*Log[Log[x]])/((x + 256*E^(2/5)*x + 32*x^2 + 384*x^3
+ 2048*x^4 + 4096*x^5 + E^(1/5)*(-32*x - 512*x^2 - 2048*x^3))*Log[2]*Log[x]),x]

[Out]

(16*Defer[Int][(-x - Log[8])/(x*(1 - 16*E^(1/5) + 16*x + 64*x^2)*Log[x]), x])/Log[2] - (32*Defer[Int][Log[Log[
x]]/(-16 + 64*E^(1/10) - 128*x), x])/(E^(1/10)*Log[2]) - (32*Defer[Int][Log[Log[x]]/(16 + 64*E^(1/10) + 128*x)
, x])/(E^(1/10)*Log[2]) - (32*(1 - 16*E^(1/5) - 8*Log[8])*Defer[Int][Log[Log[x]]/(-1 + 16*E^(1/5) - 16*x - 64*
x^2)^2, x])/Log[2] - (256*(1 - 24*Log[2])*Defer[Int][(x*Log[Log[x]])/(-1 + 16*E^(1/5) - 16*x - 64*x^2)^2, x])/
Log[2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-16 x+256 \sqrt [5]{e} x-256 x^2-1024 x^3+\left (-48+768 \sqrt [5]{e}-768 x-3072 x^2\right ) \log (2)+\left (-16 x+256 \sqrt [5]{e} x+1024 x^3+\left (768 x+6144 x^2\right ) \log (2)\right ) \log (x) \log (\log (x))}{\left (\left (1+256 e^{2/5}\right ) x+32 x^2+384 x^3+2048 x^4+4096 x^5+\sqrt [5]{e} \left (-32 x-512 x^2-2048 x^3\right )\right ) \log (2) \log (x)} \, dx\\ &=\int \frac {\left (-16+256 \sqrt [5]{e}\right ) x-256 x^2-1024 x^3+\left (-48+768 \sqrt [5]{e}-768 x-3072 x^2\right ) \log (2)+\left (-16 x+256 \sqrt [5]{e} x+1024 x^3+\left (768 x+6144 x^2\right ) \log (2)\right ) \log (x) \log (\log (x))}{\left (\left (1+256 e^{2/5}\right ) x+32 x^2+384 x^3+2048 x^4+4096 x^5+\sqrt [5]{e} \left (-32 x-512 x^2-2048 x^3\right )\right ) \log (2) \log (x)} \, dx\\ &=\frac {\int \frac {\left (-16+256 \sqrt [5]{e}\right ) x-256 x^2-1024 x^3+\left (-48+768 \sqrt [5]{e}-768 x-3072 x^2\right ) \log (2)+\left (-16 x+256 \sqrt [5]{e} x+1024 x^3+\left (768 x+6144 x^2\right ) \log (2)\right ) \log (x) \log (\log (x))}{\left (\left (1+256 e^{2/5}\right ) x+32 x^2+384 x^3+2048 x^4+4096 x^5+\sqrt [5]{e} \left (-32 x-512 x^2-2048 x^3\right )\right ) \log (x)} \, dx}{\log (2)}\\ &=\frac {\int \frac {16 \left (\left (16 \sqrt [5]{e}-(1+8 x)^2\right ) (x+\log (8))+x \left (16 \sqrt [5]{e}+(1+8 x) (-1+8 x+48 \log (2))\right ) \log (x) \log (\log (x))\right )}{x \left (1-16 \sqrt [5]{e}+16 x+64 x^2\right )^2 \log (x)} \, dx}{\log (2)}\\ &=\frac {16 \int \frac {\left (16 \sqrt [5]{e}-(1+8 x)^2\right ) (x+\log (8))+x \left (16 \sqrt [5]{e}+(1+8 x) (-1+8 x+48 \log (2))\right ) \log (x) \log (\log (x))}{x \left (1-16 \sqrt [5]{e}+16 x+64 x^2\right )^2 \log (x)} \, dx}{\log (2)}\\ &=\frac {16 \int \left (\frac {-x-\log (8)}{x \left (1-16 \sqrt [5]{e}+16 x+64 x^2\right ) \log (x)}+\frac {\left (-1+16 \sqrt [5]{e}+64 x^2+48 \log (2)+384 x \log (2)\right ) \log (\log (x))}{\left (-1+16 \sqrt [5]{e}-16 x-64 x^2\right )^2}\right ) \, dx}{\log (2)}\\ &=\frac {16 \int \frac {-x-\log (8)}{x \left (1-16 \sqrt [5]{e}+16 x+64 x^2\right ) \log (x)} \, dx}{\log (2)}+\frac {16 \int \frac {\left (-1+16 \sqrt [5]{e}+64 x^2+48 \log (2)+384 x \log (2)\right ) \log (\log (x))}{\left (-1+16 \sqrt [5]{e}-16 x-64 x^2\right )^2} \, dx}{\log (2)}\\ &=\frac {16 \int \frac {-x-\log (8)}{x \left (1-16 \sqrt [5]{e}+16 x+64 x^2\right ) \log (x)} \, dx}{\log (2)}+\frac {16 \int \left (-\frac {\log (\log (x))}{-1+16 \sqrt [5]{e}-16 x-64 x^2}+\frac {2 \left (-1+16 \sqrt [5]{e}-8 x (1-24 \log (2))+24 \log (2)\right ) \log (\log (x))}{\left (1-16 \sqrt [5]{e}+16 x+64 x^2\right )^2}\right ) \, dx}{\log (2)}\\ &=\frac {16 \int \frac {-x-\log (8)}{x \left (1-16 \sqrt [5]{e}+16 x+64 x^2\right ) \log (x)} \, dx}{\log (2)}-\frac {16 \int \frac {\log (\log (x))}{-1+16 \sqrt [5]{e}-16 x-64 x^2} \, dx}{\log (2)}+\frac {32 \int \frac {\left (-1+16 \sqrt [5]{e}-8 x (1-24 \log (2))+24 \log (2)\right ) \log (\log (x))}{\left (1-16 \sqrt [5]{e}+16 x+64 x^2\right )^2} \, dx}{\log (2)}\\ &=\frac {16 \int \frac {-x-\log (8)}{x \left (1-16 \sqrt [5]{e}+16 x+64 x^2\right ) \log (x)} \, dx}{\log (2)}-\frac {16 \int \left (\frac {2 \log (\log (x))}{\sqrt [10]{e} \left (-16+64 \sqrt [10]{e}-128 x\right )}+\frac {2 \log (\log (x))}{\sqrt [10]{e} \left (16+64 \sqrt [10]{e}+128 x\right )}\right ) \, dx}{\log (2)}+\frac {32 \int \left (\frac {8 x (-1+24 \log (2)) \log (\log (x))}{\left (-1+16 \sqrt [5]{e}-16 x-64 x^2\right )^2}-\frac {\left (1-8 \left (2 \sqrt [5]{e}+\log (8)\right )\right ) \log (\log (x))}{\left (-1+16 \sqrt [5]{e}-16 x-64 x^2\right )^2}\right ) \, dx}{\log (2)}\\ &=\frac {16 \int \frac {-x-\log (8)}{x \left (1-16 \sqrt [5]{e}+16 x+64 x^2\right ) \log (x)} \, dx}{\log (2)}-\frac {32 \int \frac {\log (\log (x))}{-16+64 \sqrt [10]{e}-128 x} \, dx}{\sqrt [10]{e} \log (2)}-\frac {32 \int \frac {\log (\log (x))}{16+64 \sqrt [10]{e}+128 x} \, dx}{\sqrt [10]{e} \log (2)}-\frac {(256 (1-24 \log (2))) \int \frac {x \log (\log (x))}{\left (-1+16 \sqrt [5]{e}-16 x-64 x^2\right )^2} \, dx}{\log (2)}-\frac {\left (32 \left (1-16 \sqrt [5]{e}-8 \log (8)\right )\right ) \int \frac {\log (\log (x))}{\left (-1+16 \sqrt [5]{e}-16 x-64 x^2\right )^2} \, dx}{\log (2)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.80, size = 34, normalized size = 1.10 \begin {gather*} \frac {16 (x+3 \log (2)) \log (\log (x))}{\left (-1+16 \sqrt [5]{e}-16 x-64 x^2\right ) \log (2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16*x + 256*E^(1/5)*x - 256*x^2 - 1024*x^3 + (-48 + 768*E^(1/5) - 768*x - 3072*x^2)*Log[2] + (-16*x
+ 256*E^(1/5)*x + 1024*x^3 + (768*x + 6144*x^2)*Log[2])*Log[x]*Log[Log[x]])/((x + 256*E^(2/5)*x + 32*x^2 + 38
4*x^3 + 2048*x^4 + 4096*x^5 + E^(1/5)*(-32*x - 512*x^2 - 2048*x^3))*Log[2]*Log[x]),x]

[Out]

(16*(x + 3*Log[2])*Log[Log[x]])/((-1 + 16*E^(1/5) - 16*x - 64*x^2)*Log[2])

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fricas [A]  time = 0.51, size = 31, normalized size = 1.00 \begin {gather*} -\frac {16 \, {\left (x + 3 \, \log \relax (2)\right )} \log \left (\log \relax (x)\right )}{{\left (64 \, x^{2} + 16 \, x - 16 \, e^{\frac {1}{5}} + 1\right )} \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6144*x^2+768*x)*log(2)+256*x*exp(1/5)+1024*x^3-16*x)*log(x)*log(log(x))+(768*exp(1/5)-3072*x^2-76
8*x-48)*log(2)+256*x*exp(1/5)-1024*x^3-256*x^2-16*x)/(256*x*exp(1/5)^2+(-2048*x^3-512*x^2-32*x)*exp(1/5)+4096*
x^5+2048*x^4+384*x^3+32*x^2+x)/log(2)/log(x),x, algorithm="fricas")

[Out]

-16*(x + 3*log(2))*log(log(x))/((64*x^2 + 16*x - 16*e^(1/5) + 1)*log(2))

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giac [A]  time = 0.39, size = 35, normalized size = 1.13 \begin {gather*} -\frac {16 \, {\left (x \log \left (\log \relax (x)\right ) + 3 \, \log \relax (2) \log \left (\log \relax (x)\right )\right )}}{{\left (64 \, x^{2} + 16 \, x - 16 \, e^{\frac {1}{5}} + 1\right )} \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6144*x^2+768*x)*log(2)+256*x*exp(1/5)+1024*x^3-16*x)*log(x)*log(log(x))+(768*exp(1/5)-3072*x^2-76
8*x-48)*log(2)+256*x*exp(1/5)-1024*x^3-256*x^2-16*x)/(256*x*exp(1/5)^2+(-2048*x^3-512*x^2-32*x)*exp(1/5)+4096*
x^5+2048*x^4+384*x^3+32*x^2+x)/log(2)/log(x),x, algorithm="giac")

[Out]

-16*(x*log(log(x)) + 3*log(2)*log(log(x)))/((64*x^2 + 16*x - 16*e^(1/5) + 1)*log(2))

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maple [A]  time = 0.08, size = 32, normalized size = 1.03

method	result	size
risch	\(\frac {16 \left (3 \ln \relax (2)+x \right ) \ln \left (\ln \relax (x )\right )}{\ln \relax (2) \left (-64 x^{2}+16 \,{\mathrm e}^{\frac {1}{5}}-16 x -1\right )}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((6144*x^2+768*x)*ln(2)+256*x*exp(1/5)+1024*x^3-16*x)*ln(x)*ln(ln(x))+(768*exp(1/5)-3072*x^2-768*x-48)*ln
(2)+256*x*exp(1/5)-1024*x^3-256*x^2-16*x)/(256*x*exp(1/5)^2+(-2048*x^3-512*x^2-32*x)*exp(1/5)+4096*x^5+2048*x^
4+384*x^3+32*x^2+x)/ln(2)/ln(x),x,method=_RETURNVERBOSE)

[Out]

16/ln(2)*(3*ln(2)+x)/(-64*x^2+16*exp(1/5)-16*x-1)*ln(ln(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6144*x^2+768*x)*log(2)+256*x*exp(1/5)+1024*x^3-16*x)*log(x)*log(log(x))+(768*exp(1/5)-3072*x^2-76
8*x-48)*log(2)+256*x*exp(1/5)-1024*x^3-256*x^2-16*x)/(256*x*exp(1/5)^2+(-2048*x^3-512*x^2-32*x)*exp(1/5)+4096*
x^5+2048*x^4+384*x^3+32*x^2+x)/log(2)/log(x),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [B]  time = 2.54, size = 29, normalized size = 0.94 \begin {gather*} -\frac {16\,\ln \left (\ln \relax (x)\right )\,\left (x+\ln \relax (8)\right )}{\ln \relax (2)\,\left (64\,x^2+16\,x-16\,{\mathrm {e}}^{1/5}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(16*x - 256*x*exp(1/5) + 256*x^2 + 1024*x^3 + log(2)*(768*x - 768*exp(1/5) + 3072*x^2 + 48) - log(log(x))
*log(x)*(log(2)*(768*x + 6144*x^2) - 16*x + 256*x*exp(1/5) + 1024*x^3))/(log(2)*log(x)*(x + 256*x*exp(2/5) - e
xp(1/5)*(32*x + 512*x^2 + 2048*x^3) + 32*x^2 + 384*x^3 + 2048*x^4 + 4096*x^5)),x)

[Out]

-(16*log(log(x))*(x + log(8)))/(log(2)*(16*x - 16*exp(1/5) + 64*x^2 + 1))

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sympy [A]  time = 0.57, size = 42, normalized size = 1.35 \begin {gather*} \frac {\left (- 16 x - 48 \log {\relax (2 )}\right ) \log {\left (\log {\relax (x )} \right )}}{64 x^{2} \log {\relax (2 )} + 16 x \log {\relax (2 )} - 16 e^{\frac {1}{5}} \log {\relax (2 )} + \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6144*x**2+768*x)*ln(2)+256*x*exp(1/5)+1024*x**3-16*x)*ln(x)*ln(ln(x))+(768*exp(1/5)-3072*x**2-768
*x-48)*ln(2)+256*x*exp(1/5)-1024*x**3-256*x**2-16*x)/(256*x*exp(1/5)**2+(-2048*x**3-512*x**2-32*x)*exp(1/5)+40
96*x**5+2048*x**4+384*x**3+32*x**2+x)/ln(2)/ln(x),x)

[Out]

(-16*x - 48*log(2))*log(log(x))/(64*x**2*log(2) + 16*x*log(2) - 16*exp(1/5)*log(2) + log(2))

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