3.39.79 \(\int \frac {-x^4+(200 x+30 x^2-40 x^3-x^4+2 x^5-4 x^8) \log (x)+(40 x^3+x^4-4 x^5) \log (x) \log (e^{x^4} \log (x))+2 x^5 \log (x) \log ^2(e^{x^4} \log (x))}{(300-60 x^2+3 x^4) \log (x)+(60 x^2-6 x^4) \log (x) \log (e^{x^4} \log (x))+3 x^4 \log (x) \log ^2(e^{x^4} \log (x))} \, dx\)

Optimal. Leaf size=28 \[ \frac {1}{3} \left (x^2+\frac {x}{-1+\frac {10}{x^2}+\log \left (e^{x^4} \log (x)\right )}\right ) \]

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Rubi [F]  time = 1.01, 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 {-x^4+\left (200 x+30 x^2-40 x^3-x^4+2 x^5-4 x^8\right ) \log (x)+\left (40 x^3+x^4-4 x^5\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )}{\left (300-60 x^2+3 x^4\right ) \log (x)+\left (60 x^2-6 x^4\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+3 x^4 \log (x) \log ^2\left (e^{x^4} \log (x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-x^4 + (200*x + 30*x^2 - 40*x^3 - x^4 + 2*x^5 - 4*x^8)*Log[x] + (40*x^3 + x^4 - 4*x^5)*Log[x]*Log[E^x^4*L
og[x]] + 2*x^5*Log[x]*Log[E^x^4*Log[x]]^2)/((300 - 60*x^2 + 3*x^4)*Log[x] + (60*x^2 - 6*x^4)*Log[x]*Log[E^x^4*
Log[x]] + 3*x^4*Log[x]*Log[E^x^4*Log[x]]^2),x]

[Out]

x^2/3 + (20*Defer[Int][x^2/(10 - x^2 + x^2*Log[E^x^4*Log[x]])^2, x])/3 - (4*Defer[Int][x^8/(10 - x^2 + x^2*Log
[E^x^4*Log[x]])^2, x])/3 - Defer[Int][x^4/(Log[x]*(10 - x^2 + x^2*Log[E^x^4*Log[x]])^2), x]/3 + Defer[Int][x^2
/(10 - x^2 + x^2*Log[E^x^4*Log[x]]), x]/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x^4+x \log (x) \left (200+30 x-40 x^2-x^3+2 x^4-4 x^7+x^2 \left (40+x-4 x^2\right ) \log \left (e^{x^4} \log (x)\right )+2 x^4 \log ^2\left (e^{x^4} \log (x)\right )\right )}{3 \log (x) \left (10-x^2+x^2 \log \left (e^{x^4} \log (x)\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {-x^4+x \log (x) \left (200+30 x-40 x^2-x^3+2 x^4-4 x^7+x^2 \left (40+x-4 x^2\right ) \log \left (e^{x^4} \log (x)\right )+2 x^4 \log ^2\left (e^{x^4} \log (x)\right )\right )}{\log (x) \left (10-x^2+x^2 \log \left (e^{x^4} \log (x)\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \left (2 x-\frac {x^2 \left (x^2-20 \log (x)+4 x^6 \log (x)\right )}{\log (x) \left (10-x^2+x^2 \log \left (e^{x^4} \log (x)\right )\right )^2}+\frac {x^2}{10-x^2+x^2 \log \left (e^{x^4} \log (x)\right )}\right ) \, dx\\ &=\frac {x^2}{3}-\frac {1}{3} \int \frac {x^2 \left (x^2-20 \log (x)+4 x^6 \log (x)\right )}{\log (x) \left (10-x^2+x^2 \log \left (e^{x^4} \log (x)\right )\right )^2} \, dx+\frac {1}{3} \int \frac {x^2}{10-x^2+x^2 \log \left (e^{x^4} \log (x)\right )} \, dx\\ &=\frac {x^2}{3}+\frac {1}{3} \int \frac {x^2}{10-x^2+x^2 \log \left (e^{x^4} \log (x)\right )} \, dx-\frac {1}{3} \int \left (-\frac {20 x^2}{\left (10-x^2+x^2 \log \left (e^{x^4} \log (x)\right )\right )^2}+\frac {4 x^8}{\left (10-x^2+x^2 \log \left (e^{x^4} \log (x)\right )\right )^2}+\frac {x^4}{\log (x) \left (10-x^2+x^2 \log \left (e^{x^4} \log (x)\right )\right )^2}\right ) \, dx\\ &=\frac {x^2}{3}-\frac {1}{3} \int \frac {x^4}{\log (x) \left (10-x^2+x^2 \log \left (e^{x^4} \log (x)\right )\right )^2} \, dx+\frac {1}{3} \int \frac {x^2}{10-x^2+x^2 \log \left (e^{x^4} \log (x)\right )} \, dx-\frac {4}{3} \int \frac {x^8}{\left (10-x^2+x^2 \log \left (e^{x^4} \log (x)\right )\right )^2} \, dx+\frac {20}{3} \int \frac {x^2}{\left (10-x^2+x^2 \log \left (e^{x^4} \log (x)\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 50, normalized size = 1.79 \begin {gather*} \frac {x^2 \left (10+x-x^2+x^2 \log \left (e^{x^4} \log (x)\right )\right )}{3 \left (10-x^2+x^2 \log \left (e^{x^4} \log (x)\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^4 + (200*x + 30*x^2 - 40*x^3 - x^4 + 2*x^5 - 4*x^8)*Log[x] + (40*x^3 + x^4 - 4*x^5)*Log[x]*Log[E
^x^4*Log[x]] + 2*x^5*Log[x]*Log[E^x^4*Log[x]]^2)/((300 - 60*x^2 + 3*x^4)*Log[x] + (60*x^2 - 6*x^4)*Log[x]*Log[
E^x^4*Log[x]] + 3*x^4*Log[x]*Log[E^x^4*Log[x]]^2),x]

[Out]

(x^2*(10 + x - x^2 + x^2*Log[E^x^4*Log[x]]))/(3*(10 - x^2 + x^2*Log[E^x^4*Log[x]]))

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fricas [A]  time = 0.56, size = 49, normalized size = 1.75 \begin {gather*} \frac {x^{4} \log \left (e^{\left (x^{4}\right )} \log \relax (x)\right ) - x^{4} + x^{3} + 10 \, x^{2}}{3 \, {\left (x^{2} \log \left (e^{\left (x^{4}\right )} \log \relax (x)\right ) - x^{2} + 10\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5*log(x)*log(exp(x^4)*log(x))^2+(-4*x^5+x^4+40*x^3)*log(x)*log(exp(x^4)*log(x))+(-4*x^8+2*x^5-x
^4-40*x^3+30*x^2+200*x)*log(x)-x^4)/(3*x^4*log(x)*log(exp(x^4)*log(x))^2+(-6*x^4+60*x^2)*log(x)*log(exp(x^4)*l
og(x))+(3*x^4-60*x^2+300)*log(x)),x, algorithm="fricas")

[Out]

1/3*(x^4*log(e^(x^4)*log(x)) - x^4 + x^3 + 10*x^2)/(x^2*log(e^(x^4)*log(x)) - x^2 + 10)

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giac [A]  time = 0.14, size = 30, normalized size = 1.07 \begin {gather*} \frac {1}{3} \, x^{2} + \frac {x^{3}}{3 \, {\left (x^{6} + x^{2} \log \left (\log \relax (x)\right ) - x^{2} + 10\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5*log(x)*log(exp(x^4)*log(x))^2+(-4*x^5+x^4+40*x^3)*log(x)*log(exp(x^4)*log(x))+(-4*x^8+2*x^5-x
^4-40*x^3+30*x^2+200*x)*log(x)-x^4)/(3*x^4*log(x)*log(exp(x^4)*log(x))^2+(-6*x^4+60*x^2)*log(x)*log(exp(x^4)*l
og(x))+(3*x^4-60*x^2+300)*log(x)),x, algorithm="giac")

[Out]

1/3*x^2 + 1/3*x^3/(x^6 + x^2*log(log(x)) - x^2 + 10)

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maple [C]  time = 0.13, size = 140, normalized size = 5.00

method	result	size
risch	\(\frac {x^{2}}{3}+\frac {2 i x^{3}}{3 \left (\pi \,x^{2} \mathrm {csgn}\left (i {\mathrm e}^{x^{4}}\right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i {\mathrm e}^{x^{4}} \ln \relax (x )\right )-\pi \,x^{2} \mathrm {csgn}\left (i {\mathrm e}^{x^{4}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x^{4}} \ln \relax (x )\right )^{2}-\pi \,x^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i {\mathrm e}^{x^{4}} \ln \relax (x )\right )^{2}+\pi \,x^{2} \mathrm {csgn}\left (i {\mathrm e}^{x^{4}} \ln \relax (x )\right )^{3}+2 i x^{2} \ln \left (\ln \relax (x )\right )+2 i x^{2} \ln \left ({\mathrm e}^{x^{4}}\right )-2 i x^{2}+20 i\right )}\)	\(140\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^5*ln(x)*ln(exp(x^4)*ln(x))^2+(-4*x^5+x^4+40*x^3)*ln(x)*ln(exp(x^4)*ln(x))+(-4*x^8+2*x^5-x^4-40*x^3+30
*x^2+200*x)*ln(x)-x^4)/(3*x^4*ln(x)*ln(exp(x^4)*ln(x))^2+(-6*x^4+60*x^2)*ln(x)*ln(exp(x^4)*ln(x))+(3*x^4-60*x^
2+300)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

1/3*x^2+2/3*I*x^3/(Pi*x^2*csgn(I*exp(x^4))*csgn(I*ln(x))*csgn(I*exp(x^4)*ln(x))-Pi*x^2*csgn(I*exp(x^4))*csgn(I
*exp(x^4)*ln(x))^2-Pi*x^2*csgn(I*ln(x))*csgn(I*exp(x^4)*ln(x))^2+Pi*x^2*csgn(I*exp(x^4)*ln(x))^3+2*I*x^2*ln(ln
(x))+2*I*x^2*ln(exp(x^4))-2*I*x^2+20*I)

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maxima [A]  time = 0.39, size = 45, normalized size = 1.61 \begin {gather*} \frac {x^{8} + x^{4} \log \left (\log \relax (x)\right ) - x^{4} + x^{3} + 10 \, x^{2}}{3 \, {\left (x^{6} + x^{2} \log \left (\log \relax (x)\right ) - x^{2} + 10\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5*log(x)*log(exp(x^4)*log(x))^2+(-4*x^5+x^4+40*x^3)*log(x)*log(exp(x^4)*log(x))+(-4*x^8+2*x^5-x
^4-40*x^3+30*x^2+200*x)*log(x)-x^4)/(3*x^4*log(x)*log(exp(x^4)*log(x))^2+(-6*x^4+60*x^2)*log(x)*log(exp(x^4)*l
og(x))+(3*x^4-60*x^2+300)*log(x)),x, algorithm="maxima")

[Out]

1/3*(x^8 + x^4*log(log(x)) - x^4 + x^3 + 10*x^2)/(x^6 + x^2*log(log(x)) - x^2 + 10)

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mupad [B]  time = 2.47, size = 32, normalized size = 1.14 \begin {gather*} \frac {x^3}{3\,\left (x^2\,\ln \left (\ln \relax (x)\right )-x^2+x^6+10\right )}+\frac {x^2}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(200*x + 30*x^2 - 40*x^3 - x^4 + 2*x^5 - 4*x^8) - x^4 + 2*x^5*log(x)*log(exp(x^4)*log(x))^2 + log(
x)*log(exp(x^4)*log(x))*(40*x^3 + x^4 - 4*x^5))/(log(x)*(3*x^4 - 60*x^2 + 300) + 3*x^4*log(x)*log(exp(x^4)*log
(x))^2 + log(x)*log(exp(x^4)*log(x))*(60*x^2 - 6*x^4)),x)

[Out]

x^3/(3*(x^2*log(log(x)) - x^2 + x^6 + 10)) + x^2/3

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sympy [A]  time = 0.48, size = 29, normalized size = 1.04 \begin {gather*} \frac {x^{3}}{3 x^{2} \log {\left (e^{x^{4}} \log {\relax (x )} \right )} - 3 x^{2} + 30} + \frac {x^{2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**5*ln(x)*ln(exp(x**4)*ln(x))**2+(-4*x**5+x**4+40*x**3)*ln(x)*ln(exp(x**4)*ln(x))+(-4*x**8+2*x**
5-x**4-40*x**3+30*x**2+200*x)*ln(x)-x**4)/(3*x**4*ln(x)*ln(exp(x**4)*ln(x))**2+(-6*x**4+60*x**2)*ln(x)*ln(exp(
x**4)*ln(x))+(3*x**4-60*x**2+300)*ln(x)),x)

[Out]

x**3/(3*x**2*log(exp(x**4)*log(x)) - 3*x**2 + 30) + x**2/3

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