3.8.39 \(\int \frac {3+2 x^3+(-3+x^3) \log (\frac {3-x^3}{x})+(-3+x^3) \log (\frac {3-x^3}{x}) \log (\log (\frac {3-x^3}{x}))}{(-3+x^3) \log (\frac {3-x^3}{x})} \, dx\)

Optimal. Leaf size=17 \[ x+x \log \left (\log \left (\frac {3}{x}-x^2\right )\right ) \]

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Rubi [F]  time = 0.74, 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 {3+2 x^3+\left (-3+x^3\right ) \log \left (\frac {3-x^3}{x}\right )+\left (-3+x^3\right ) \log \left (\frac {3-x^3}{x}\right ) \log \left (\log \left (\frac {3-x^3}{x}\right )\right )}{\left (-3+x^3\right ) \log \left (\frac {3-x^3}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3 + 2*x^3 + (-3 + x^3)*Log[(3 - x^3)/x] + (-3 + x^3)*Log[(3 - x^3)/x]*Log[Log[(3 - x^3)/x]])/((-3 + x^3)*
Log[(3 - x^3)/x]),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {3+2 x^3-3 \log \left (\frac {3-x^3}{x}\right )+x^3 \log \left (\frac {3-x^3}{x}\right )}{\left (-3+x^3\right ) \log \left (\frac {3-x^3}{x}\right )}+\log \left (\log \left (\frac {3-x^3}{x}\right )\right )\right ) \, dx\\ &=\int \frac {3+2 x^3-3 \log \left (\frac {3-x^3}{x}\right )+x^3 \log \left (\frac {3-x^3}{x}\right )}{\left (-3+x^3\right ) \log \left (\frac {3-x^3}{x}\right )} \, dx+\int \log \left (\log \left (\frac {3-x^3}{x}\right )\right ) \, dx\\ &=\int \left (1+\frac {3+2 x^3}{\left (-3+x^3\right ) \log \left (\frac {3-x^3}{x}\right )}\right ) \, dx+\int \log \left (\log \left (\frac {3-x^3}{x}\right )\right ) \, dx\\ &=x+\int \frac {3+2 x^3}{\left (-3+x^3\right ) \log \left (\frac {3-x^3}{x}\right )} \, dx+\int \log \left (\log \left (\frac {3-x^3}{x}\right )\right ) \, dx\\ &=x+\int \left (\frac {2}{\log \left (\frac {3-x^3}{x}\right )}+\frac {9}{\left (-3+x^3\right ) \log \left (\frac {3-x^3}{x}\right )}\right ) \, dx+\int \log \left (\log \left (\frac {3-x^3}{x}\right )\right ) \, dx\\ &=x+2 \int \frac {1}{\log \left (\frac {3-x^3}{x}\right )} \, dx+9 \int \frac {1}{\left (-3+x^3\right ) \log \left (\frac {3-x^3}{x}\right )} \, dx+\int \log \left (\log \left (\frac {3-x^3}{x}\right )\right ) \, dx\\ &=x+2 \int \frac {1}{\log \left (\frac {3-x^3}{x}\right )} \, dx+9 \int \left (-\frac {1}{3\ 3^{2/3} \left (\sqrt [3]{3}-x\right ) \log \left (\frac {3-x^3}{x}\right )}-\frac {1}{3\ 3^{2/3} \left (\sqrt [3]{3}+\sqrt [3]{-1} x\right ) \log \left (\frac {3-x^3}{x}\right )}-\frac {1}{3\ 3^{2/3} \left (\sqrt [3]{3}-(-1)^{2/3} x\right ) \log \left (\frac {3-x^3}{x}\right )}\right ) \, dx+\int \log \left (\log \left (\frac {3-x^3}{x}\right )\right ) \, dx\\ &=x+2 \int \frac {1}{\log \left (\frac {3-x^3}{x}\right )} \, dx-\sqrt [3]{3} \int \frac {1}{\left (\sqrt [3]{3}-x\right ) \log \left (\frac {3-x^3}{x}\right )} \, dx-\sqrt [3]{3} \int \frac {1}{\left (\sqrt [3]{3}+\sqrt [3]{-1} x\right ) \log \left (\frac {3-x^3}{x}\right )} \, dx-\sqrt [3]{3} \int \frac {1}{\left (\sqrt [3]{3}-(-1)^{2/3} x\right ) \log \left (\frac {3-x^3}{x}\right )} \, dx+\int \log \left (\log \left (\frac {3-x^3}{x}\right )\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 17, normalized size = 1.00 \begin {gather*} x+x \log \left (\log \left (\frac {3-x^3}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 2*x^3 + (-3 + x^3)*Log[(3 - x^3)/x] + (-3 + x^3)*Log[(3 - x^3)/x]*Log[Log[(3 - x^3)/x]])/((-3 +
x^3)*Log[(3 - x^3)/x]),x]

[Out]

x + x*Log[Log[(3 - x^3)/x]]

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fricas [A]  time = 0.56, size = 16, normalized size = 0.94 \begin {gather*} x \log \left (\log \left (-\frac {x^{3} - 3}{x}\right )\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-3)*log((-x^3+3)/x)*log(log((-x^3+3)/x))+(x^3-3)*log((-x^3+3)/x)+2*x^3+3)/(x^3-3)/log((-x^3+3)/
x),x, algorithm="fricas")

[Out]

x*log(log(-(x^3 - 3)/x)) + x

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giac [A]  time = 0.53, size = 16, normalized size = 0.94 \begin {gather*} x \log \left (\log \left (-\frac {x^{3} - 3}{x}\right )\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-3)*log((-x^3+3)/x)*log(log((-x^3+3)/x))+(x^3-3)*log((-x^3+3)/x)+2*x^3+3)/(x^3-3)/log((-x^3+3)/
x),x, algorithm="giac")

[Out]

x*log(log(-(x^3 - 3)/x)) + x

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}-3\right ) \ln \left (\frac {-x^{3}+3}{x}\right ) \ln \left (\ln \left (\frac {-x^{3}+3}{x}\right )\right )+\left (x^{3}-3\right ) \ln \left (\frac {-x^{3}+3}{x}\right )+2 x^{3}+3}{\left (x^{3}-3\right ) \ln \left (\frac {-x^{3}+3}{x}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3-3)*ln((-x^3+3)/x)*ln(ln((-x^3+3)/x))+(x^3-3)*ln((-x^3+3)/x)+2*x^3+3)/(x^3-3)/ln((-x^3+3)/x),x)

[Out]

int(((x^3-3)*ln((-x^3+3)/x)*ln(ln((-x^3+3)/x))+(x^3-3)*ln((-x^3+3)/x)+2*x^3+3)/(x^3-3)/ln((-x^3+3)/x),x)

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maxima [A]  time = 0.64, size = 18, normalized size = 1.06 \begin {gather*} x \log \left (\log \left (-x^{3} + 3\right ) - \log \relax (x)\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-3)*log((-x^3+3)/x)*log(log((-x^3+3)/x))+(x^3-3)*log((-x^3+3)/x)+2*x^3+3)/(x^3-3)/log((-x^3+3)/
x),x, algorithm="maxima")

[Out]

x*log(log(-x^3 + 3) - log(x)) + x

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mupad [B]  time = 0.82, size = 16, normalized size = 0.94 \begin {gather*} x\,\left (\ln \left (\ln \left (-\frac {x^3-3}{x}\right )\right )+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(-(x^3 - 3)/x)*(x^3 - 3) + 2*x^3 + log(log(-(x^3 - 3)/x))*log(-(x^3 - 3)/x)*(x^3 - 3) + 3)/(log(-(x^3
- 3)/x)*(x^3 - 3)),x)

[Out]

x*(log(log(-(x^3 - 3)/x)) + 1)

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sympy [A]  time = 0.57, size = 12, normalized size = 0.71 \begin {gather*} x \log {\left (\log {\left (\frac {3 - x^{3}}{x} \right )} \right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**3-3)*ln((-x**3+3)/x)*ln(ln((-x**3+3)/x))+(x**3-3)*ln((-x**3+3)/x)+2*x**3+3)/(x**3-3)/ln((-x**3+
3)/x),x)

[Out]

x*log(log((3 - x**3)/x)) + x

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