3.56.87 \(\int \frac {16 x-2 x^4+(8 x+2 x^4) \log (\frac {4+x^3}{x^2}) \log (\frac {3}{11} \log (\frac {4+x^3}{x^2}))+(-4-x^3) \log (\frac {4+x^3}{x^2}) \log ^3(\frac {3}{11} \log (\frac {4+x^3}{x^2}))}{(4+x^3) \log (\frac {4+x^3}{x^2}) \log ^3(\frac {3}{11} \log (\frac {4+x^3}{x^2}))} \, dx\)

Optimal. Leaf size=23 \[ -x+\frac {x^2}{\log ^2\left (\frac {3}{11} \log \left (\frac {4}{x^2}+x\right )\right )} \]

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Rubi [F]  time = 2.33, 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-2 x^4+\left (8 x+2 x^4\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log \left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )+\left (-4-x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )}{\left (4+x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(16*x - 2*x^4 + (8*x + 2*x^4)*Log[(4 + x^3)/x^2]*Log[(3*Log[(4 + x^3)/x^2])/11] + (-4 - x^3)*Log[(4 + x^3)
/x^2]*Log[(3*Log[(4 + x^3)/x^2])/11]^3)/((4 + x^3)*Log[(4 + x^3)/x^2]*Log[(3*Log[(4 + x^3)/x^2])/11]^3),x]

[Out]

-x - 2*Defer[Int][x/(Log[(4 + x^3)/x^2]*Log[(3*Log[(4 + x^3)/x^2])/11]^3), x] - 4*2^(1/3)*Defer[Int][1/((2^(2/
3) + x)*Log[(4 + x^3)/x^2]*Log[(3*Log[(4 + x^3)/x^2])/11]^3), x] - 4*(-1)^(2/3)*2^(1/3)*Defer[Int][1/((2^(2/3)
- (-1)^(1/3)*x)*Log[(4 + x^3)/x^2]*Log[(3*Log[(4 + x^3)/x^2])/11]^3), x] + 4*(-2)^(1/3)*Defer[Int][1/((2^(2/3
) + (-1)^(2/3)*x)*Log[(4 + x^3)/x^2]*Log[(3*Log[(4 + x^3)/x^2])/11]^3), x] + 2*Defer[Int][x/Log[(3*Log[(4 + x^
3)/x^2])/11]^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1-\frac {2 x \left (-8+x^3\right )}{\left (4+x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )}+\frac {2 x}{\log ^2\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )}\right ) \, dx\\ &=-x-2 \int \frac {x \left (-8+x^3\right )}{\left (4+x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx+2 \int \frac {x}{\log ^2\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx\\ &=-x-2 \int \left (\frac {x}{\log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )}+\frac {12 x}{\left (-4-x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )}\right ) \, dx+2 \int \frac {x}{\log ^2\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx\\ &=-x-2 \int \frac {x}{\log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx+2 \int \frac {x}{\log ^2\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx-24 \int \frac {x}{\left (-4-x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx\\ &=-x-2 \int \frac {x}{\log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx+2 \int \frac {x}{\log ^2\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx-24 \int \left (\frac {1}{3\ 2^{2/3} \left (2^{2/3}+x\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )}+\frac {\left (-\frac {1}{2}\right )^{2/3}}{3 \left (2^{2/3}-\sqrt [3]{-1} x\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )}-\frac {\sqrt [3]{-1}}{3\ 2^{2/3} \left (2^{2/3}+(-1)^{2/3} x\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )}\right ) \, dx\\ &=-x-2 \int \frac {x}{\log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx+2 \int \frac {x}{\log ^2\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx+\left (4 \sqrt [3]{-2}\right ) \int \frac {1}{\left (2^{2/3}+(-1)^{2/3} x\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx-\left (4 \sqrt [3]{2}\right ) \int \frac {1}{\left (2^{2/3}+x\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx-\left (4 (-1)^{2/3} \sqrt [3]{2}\right ) \int \frac {1}{\left (2^{2/3}-\sqrt [3]{-1} x\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.57, size = 23, normalized size = 1.00 \begin {gather*} -x+\frac {x^2}{\log ^2\left (\frac {3}{11} \log \left (\frac {4}{x^2}+x\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16*x - 2*x^4 + (8*x + 2*x^4)*Log[(4 + x^3)/x^2]*Log[(3*Log[(4 + x^3)/x^2])/11] + (-4 - x^3)*Log[(4
+ x^3)/x^2]*Log[(3*Log[(4 + x^3)/x^2])/11]^3)/((4 + x^3)*Log[(4 + x^3)/x^2]*Log[(3*Log[(4 + x^3)/x^2])/11]^3),
x]

[Out]

-x + x^2/Log[(3*Log[4/x^2 + x])/11]^2

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fricas [A]  time = 0.58, size = 40, normalized size = 1.74 \begin {gather*} -\frac {x \log \left (\frac {3}{11} \, \log \left (\frac {x^{3} + 4}{x^{2}}\right )\right )^{2} - x^{2}}{\log \left (\frac {3}{11} \, \log \left (\frac {x^{3} + 4}{x^{2}}\right )\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-4)*log((x^3+4)/x^2)*log(3/11*log((x^3+4)/x^2))^3+(2*x^4+8*x)*log((x^3+4)/x^2)*log(3/11*log((x
^3+4)/x^2))-2*x^4+16*x)/(x^3+4)/log((x^3+4)/x^2)/log(3/11*log((x^3+4)/x^2))^3,x, algorithm="fricas")

[Out]

-(x*log(3/11*log((x^3 + 4)/x^2))^2 - x^2)/log(3/11*log((x^3 + 4)/x^2))^2

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giac [B]  time = 4.54, size = 1412, normalized size = 61.39 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-4)*log((x^3+4)/x^2)*log(3/11*log((x^3+4)/x^2))^3+(2*x^4+8*x)*log((x^3+4)/x^2)*log(3/11*log((x
^3+4)/x^2))-2*x^4+16*x)/(x^3+4)/log((x^3+4)/x^2)/log(3/11*log((x^3+4)/x^2))^3,x, algorithm="giac")

[Out]

-x + (2*x^5*log(11)*log(x^3 + 4)^2*log((x^3 + 4)/x^2) - 4*x^5*log(11)*log(x^3 + 4)*log(x^2)*log((x^3 + 4)/x^2)
+ 2*x^5*log(11)*log(x^2)^2*log((x^3 + 4)/x^2) - 2*x^5*log(x^3 + 4)^2*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/
x^2) + 4*x^5*log(x^3 + 4)*log(x^2)*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/x^2) - 2*x^5*log(x^2)^2*log(3*log((
x^3 + 4)/x^2))*log((x^3 + 4)/x^2) - 2*x^5*log(11)*log(x^3 + 4)*log((x^3 + 4)/x^2)^2 + 2*x^5*log(11)*log(x^2)*l
og((x^3 + 4)/x^2)^2 + 2*x^5*log(x^3 + 4)*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/x^2)^2 - 2*x^5*log(x^2)*log(3
*log((x^3 + 4)/x^2))*log((x^3 + 4)/x^2)^2 - x^5*log(11)*log(x^3 + 4)*log((x^3 + 4)/x^2) + x^5*log(11)*log(x^2)
*log((x^3 + 4)/x^2) + x^5*log(x^3 + 4)*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/x^2) - x^5*log(x^2)*log(3*log((
x^3 + 4)/x^2))*log((x^3 + 4)/x^2) + x^5*log(11)*log((x^3 + 4)/x^2)^2 - x^5*log(3*log((x^3 + 4)/x^2))*log((x^3
+ 4)/x^2)^2 + x^5*log(x^3 + 4)*log((x^3 + 4)/x^2) - x^5*log(x^2)*log((x^3 + 4)/x^2) + 8*x^2*log(11)*log(x^3 +
4)^2*log((x^3 + 4)/x^2) - 16*x^2*log(11)*log(x^3 + 4)*log(x^2)*log((x^3 + 4)/x^2) + 8*x^2*log(11)*log(x^2)^2*l
og((x^3 + 4)/x^2) - 8*x^2*log(x^3 + 4)^2*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/x^2) + 16*x^2*log(x^3 + 4)*lo
g(x^2)*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/x^2) - 8*x^2*log(x^2)^2*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)
/x^2) - 8*x^2*log(11)*log(x^3 + 4)*log((x^3 + 4)/x^2)^2 + 8*x^2*log(11)*log(x^2)*log((x^3 + 4)/x^2)^2 + 8*x^2*
log(x^3 + 4)*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/x^2)^2 - 8*x^2*log(x^2)*log(3*log((x^3 + 4)/x^2))*log((x^
3 + 4)/x^2)^2 + 8*x^2*log(11)*log(x^3 + 4)*log((x^3 + 4)/x^2) - 8*x^2*log(11)*log(x^2)*log((x^3 + 4)/x^2) - 8*
x^2*log(x^3 + 4)*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/x^2) + 8*x^2*log(x^2)*log(3*log((x^3 + 4)/x^2))*log((
x^3 + 4)/x^2) - 8*x^2*log(11)*log((x^3 + 4)/x^2)^2 + 8*x^2*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/x^2)^2 - 8*
x^2*log(x^3 + 4)*log((x^3 + 4)/x^2) + 8*x^2*log(x^2)*log((x^3 + 4)/x^2))/(x^3*log(11)^2*log(x^3 + 4)^2 - 2*x^3
*log(11)^2*log(x^3 + 4)*log(x^2) + x^3*log(11)^2*log(x^2)^2 - 2*x^3*log(11)*log(x^3 + 4)^2*log(3*log((x^3 + 4)
/x^2)) + 4*x^3*log(11)*log(x^3 + 4)*log(x^2)*log(3*log((x^3 + 4)/x^2)) - 2*x^3*log(11)*log(x^2)^2*log(3*log((x
^3 + 4)/x^2)) + x^3*log(x^3 + 4)^2*log(3*log((x^3 + 4)/x^2))^2 - 2*x^3*log(x^3 + 4)*log(x^2)*log(3*log((x^3 +
4)/x^2))^2 + x^3*log(x^2)^2*log(3*log((x^3 + 4)/x^2))^2 - 8*log(11)^2*log(x^3 + 4)^2 + 16*log(11)^2*log(x^3 +
4)*log(x^2) - 8*log(11)^2*log(x^2)^2 + 16*log(11)*log(x^3 + 4)^2*log(3*log((x^3 + 4)/x^2)) - 32*log(11)*log(x^
3 + 4)*log(x^2)*log(3*log((x^3 + 4)/x^2)) + 16*log(11)*log(x^2)^2*log(3*log((x^3 + 4)/x^2)) - 8*log(x^3 + 4)^2
*log(3*log((x^3 + 4)/x^2))^2 + 16*log(x^3 + 4)*log(x^2)*log(3*log((x^3 + 4)/x^2))^2 - 8*log(x^2)^2*log(3*log((
x^3 + 4)/x^2))^2)

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (-x^{3}-4\right ) \ln \left (\frac {x^{3}+4}{x^{2}}\right ) \ln \left (\frac {3 \ln \left (\frac {x^{3}+4}{x^{2}}\right )}{11}\right )^{3}+\left (2 x^{4}+8 x \right ) \ln \left (\frac {x^{3}+4}{x^{2}}\right ) \ln \left (\frac {3 \ln \left (\frac {x^{3}+4}{x^{2}}\right )}{11}\right )-2 x^{4}+16 x}{\left (x^{3}+4\right ) \ln \left (\frac {x^{3}+4}{x^{2}}\right ) \ln \left (\frac {3 \ln \left (\frac {x^{3}+4}{x^{2}}\right )}{11}\right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^3-4)*ln((x^3+4)/x^2)*ln(3/11*ln((x^3+4)/x^2))^3+(2*x^4+8*x)*ln((x^3+4)/x^2)*ln(3/11*ln((x^3+4)/x^2))-
2*x^4+16*x)/(x^3+4)/ln((x^3+4)/x^2)/ln(3/11*ln((x^3+4)/x^2))^3,x)

[Out]

int(((-x^3-4)*ln((x^3+4)/x^2)*ln(3/11*ln((x^3+4)/x^2))^3+(2*x^4+8*x)*ln((x^3+4)/x^2)*ln(3/11*ln((x^3+4)/x^2))-
2*x^4+16*x)/(x^3+4)/ln((x^3+4)/x^2)/ln(3/11*ln((x^3+4)/x^2))^3,x)

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maxima [B]  time = 0.55, size = 114, normalized size = 4.96 \begin {gather*} \frac {2 \, x {\left (\log \left (11\right ) - \log \relax (3)\right )} \log \left (\log \left (x^{3} + 4\right ) - 2 \, \log \relax (x)\right ) - x \log \left (\log \left (x^{3} + 4\right ) - 2 \, \log \relax (x)\right )^{2} - {\left (\log \left (11\right )^{2} - 2 \, \log \left (11\right ) \log \relax (3) + \log \relax (3)^{2}\right )} x + x^{2}}{\log \left (11\right )^{2} - 2 \, \log \left (11\right ) \log \relax (3) + \log \relax (3)^{2} - 2 \, {\left (\log \left (11\right ) - \log \relax (3)\right )} \log \left (\log \left (x^{3} + 4\right ) - 2 \, \log \relax (x)\right ) + \log \left (\log \left (x^{3} + 4\right ) - 2 \, \log \relax (x)\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-4)*log((x^3+4)/x^2)*log(3/11*log((x^3+4)/x^2))^3+(2*x^4+8*x)*log((x^3+4)/x^2)*log(3/11*log((x
^3+4)/x^2))-2*x^4+16*x)/(x^3+4)/log((x^3+4)/x^2)/log(3/11*log((x^3+4)/x^2))^3,x, algorithm="maxima")

[Out]

(2*x*(log(11) - log(3))*log(log(x^3 + 4) - 2*log(x)) - x*log(log(x^3 + 4) - 2*log(x))^2 - (log(11)^2 - 2*log(1
1)*log(3) + log(3)^2)*x + x^2)/(log(11)^2 - 2*log(11)*log(3) + log(3)^2 - 2*(log(11) - log(3))*log(log(x^3 + 4
) - 2*log(x)) + log(log(x^3 + 4) - 2*log(x))^2)

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mupad [B]  time = 4.12, size = 254, normalized size = 11.04 \begin {gather*} \frac {x^2}{{\ln \left (\frac {3\,\ln \left (\frac {1}{x^2}\right )}{11}+\frac {3\,\ln \left (x^3+4\right )}{11}\right )}^2}-x+\frac {4\,x^2\,\ln \left (\frac {1}{x^2}\right )}{x^3-8}+\frac {x^5\,\ln \left (\frac {1}{x^2}\right )}{x^3-8}+\frac {4\,x^2\,\ln \left (x^3+4\right )}{x^3-8}+\frac {x^5\,\ln \left (x^3+4\right )}{x^3-8}-\frac {256\,x^2\,\ln \left (\frac {1}{x^2}\right )}{x^9-24\,x^6+192\,x^3-512}+\frac {12\,x^8\,\ln \left (\frac {1}{x^2}\right )}{x^9-24\,x^6+192\,x^3-512}-\frac {x^{11}\,\ln \left (\frac {1}{x^2}\right )}{x^9-24\,x^6+192\,x^3-512}-\frac {256\,x^2\,\ln \left (x^3+4\right )}{x^9-24\,x^6+192\,x^3-512}+\frac {12\,x^8\,\ln \left (x^3+4\right )}{x^9-24\,x^6+192\,x^3-512}-\frac {x^{11}\,\ln \left (x^3+4\right )}{x^9-24\,x^6+192\,x^3-512} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x - 2*x^4 + log((3*log((x^3 + 4)/x^2))/11)*log((x^3 + 4)/x^2)*(8*x + 2*x^4) - log((3*log((x^3 + 4)/x^2
))/11)^3*log((x^3 + 4)/x^2)*(x^3 + 4))/(log((3*log((x^3 + 4)/x^2))/11)^3*log((x^3 + 4)/x^2)*(x^3 + 4)),x)

[Out]

x^2/log((3*log(1/x^2))/11 + (3*log(x^3 + 4))/11)^2 - x + (4*x^2*log(1/x^2))/(x^3 - 8) + (x^5*log(1/x^2))/(x^3
- 8) + (4*x^2*log(x^3 + 4))/(x^3 - 8) + (x^5*log(x^3 + 4))/(x^3 - 8) - (256*x^2*log(1/x^2))/(192*x^3 - 24*x^6
+ x^9 - 512) + (12*x^8*log(1/x^2))/(192*x^3 - 24*x^6 + x^9 - 512) - (x^11*log(1/x^2))/(192*x^3 - 24*x^6 + x^9
- 512) - (256*x^2*log(x^3 + 4))/(192*x^3 - 24*x^6 + x^9 - 512) + (12*x^8*log(x^3 + 4))/(192*x^3 - 24*x^6 + x^9
- 512) - (x^11*log(x^3 + 4))/(192*x^3 - 24*x^6 + x^9 - 512)

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sympy [A]  time = 0.46, size = 20, normalized size = 0.87 \begin {gather*} \frac {x^{2}}{\log {\left (\frac {3 \log {\left (\frac {x^{3} + 4}{x^{2}} \right )}}{11} \right )}^{2}} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**3-4)*ln((x**3+4)/x**2)*ln(3/11*ln((x**3+4)/x**2))**3+(2*x**4+8*x)*ln((x**3+4)/x**2)*ln(3/11*ln
((x**3+4)/x**2))-2*x**4+16*x)/(x**3+4)/ln((x**3+4)/x**2)/ln(3/11*ln((x**3+4)/x**2))**3,x)

[Out]

x**2/log(3*log((x**3 + 4)/x**2)/11)**2 - x

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