3.22.81 \(\int \frac {-28 x+4 x^4+6 x^4 \log (7-x^3)+(-14 x+14 x^3+2 x^4-2 x^6) \log ^2(7-x^3)}{14-2 x^3+(-7 x^2+x^5) \log ^2(7-x^3)} \, dx\)

Optimal. Leaf size=31 \[ -1-x^2+\log \left (-2+x^2 \log ^2\left (9-\frac {2 x+x^4}{x}\right )\right ) \]

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Rubi [F]  time = 5.89, 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 {-28 x+4 x^4+6 x^4 \log \left (7-x^3\right )+\left (-14 x+14 x^3+2 x^4-2 x^6\right ) \log ^2\left (7-x^3\right )}{14-2 x^3+\left (-7 x^2+x^5\right ) \log ^2\left (7-x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-28*x + 4*x^4 + 6*x^4*Log[7 - x^3] + (-14*x + 14*x^3 + 2*x^4 - 2*x^6)*Log[7 - x^3]^2)/(14 - 2*x^3 + (-7*x
^2 + x^5)*Log[7 - x^3]^2),x]

[Out]

-x^2 + 2*Log[x] + 4*Defer[Int][1/(x*(-2 + x^2*Log[7 - x^3]^2)), x] - 2*7^(2/3)*Defer[Int][Log[7 - x^3]/((7^(1/
3) - x)*(-2 + x^2*Log[7 - x^3]^2)), x] + 6*Defer[Int][(x*Log[7 - x^3])/(-2 + x^2*Log[7 - x^3]^2), x] - 2*(-7)^
(2/3)*Defer[Int][Log[7 - x^3]/((7^(1/3) + (-1)^(1/3)*x)*(-2 + x^2*Log[7 - x^3]^2)), x] + 2*(-1)^(1/3)*7^(2/3)*
Defer[Int][Log[7 - x^3]/((7^(1/3) - (-1)^(2/3)*x)*(-2 + x^2*Log[7 - x^3]^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-28 x+4 x^4+6 x^4 \log \left (7-x^3\right )+\left (-14 x+14 x^3+2 x^4-2 x^6\right ) \log ^2\left (7-x^3\right )}{\left (7-x^3\right ) \left (2-x^2 \log ^2\left (7-x^3\right )\right )} \, dx\\ &=\int \left (-\frac {2 \left (-1+x^2\right )}{x}+\frac {2 \left (-14+2 x^3+3 x^5 \log \left (7-x^3\right )\right )}{x \left (-7+x^3\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )}\right ) \, dx\\ &=-\left (2 \int \frac {-1+x^2}{x} \, dx\right )+2 \int \frac {-14+2 x^3+3 x^5 \log \left (7-x^3\right )}{x \left (-7+x^3\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx\\ &=-\left (2 \int \left (-\frac {1}{x}+x\right ) \, dx\right )+2 \int \left (\frac {14-2 x^3-3 x^5 \log \left (7-x^3\right )}{7 x \left (-2+x^2 \log ^2\left (7-x^3\right )\right )}+\frac {x^2 \left (-14+2 x^3+3 x^5 \log \left (7-x^3\right )\right )}{7 \left (-7+x^3\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )}\right ) \, dx\\ &=-x^2+2 \log (x)+\frac {2}{7} \int \frac {14-2 x^3-3 x^5 \log \left (7-x^3\right )}{x \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx+\frac {2}{7} \int \frac {x^2 \left (-14+2 x^3+3 x^5 \log \left (7-x^3\right )\right )}{\left (-7+x^3\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx\\ &=-x^2+2 \log (x)+\frac {2}{7} \int \left (\frac {14}{x \left (-2+x^2 \log ^2\left (7-x^3\right )\right )}-\frac {2 x^2}{-2+x^2 \log ^2\left (7-x^3\right )}-\frac {3 x^4 \log \left (7-x^3\right )}{-2+x^2 \log ^2\left (7-x^3\right )}\right ) \, dx+\frac {2}{7} \int \left (-\frac {14 x^2}{\left (-7+x^3\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )}+\frac {2 x^5}{\left (-7+x^3\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )}+\frac {3 x^7 \log \left (7-x^3\right )}{\left (-7+x^3\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )}\right ) \, dx\\ &=-x^2+2 \log (x)-\frac {4}{7} \int \frac {x^2}{-2+x^2 \log ^2\left (7-x^3\right )} \, dx+\frac {4}{7} \int \frac {x^5}{\left (-7+x^3\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx-\frac {6}{7} \int \frac {x^4 \log \left (7-x^3\right )}{-2+x^2 \log ^2\left (7-x^3\right )} \, dx+\frac {6}{7} \int \frac {x^7 \log \left (7-x^3\right )}{\left (-7+x^3\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx+4 \int \frac {1}{x \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx-4 \int \frac {x^2}{\left (-7+x^3\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx\\ &=-x^2+2 \log (x)-\frac {4}{7} \int \frac {x^2}{-2+x^2 \log ^2\left (7-x^3\right )} \, dx+\frac {4}{7} \int \left (\frac {x^2}{-2+x^2 \log ^2\left (7-x^3\right )}+\frac {7 x^2}{\left (-7+x^3\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )}\right ) \, dx-\frac {6}{7} \int \frac {x^4 \log \left (7-x^3\right )}{-2+x^2 \log ^2\left (7-x^3\right )} \, dx+\frac {6}{7} \int \left (\frac {7 x \log \left (7-x^3\right )}{-2+x^2 \log ^2\left (7-x^3\right )}+\frac {x^4 \log \left (7-x^3\right )}{-2+x^2 \log ^2\left (7-x^3\right )}+\frac {49 x \log \left (7-x^3\right )}{\left (-7+x^3\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )}\right ) \, dx+4 \int \frac {1}{x \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx-4 \int \left (-\frac {1}{3 \left (-\sqrt [3]{-7}-x\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )}-\frac {1}{3 \left (\sqrt [3]{7}-x\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )}-\frac {1}{3 \left ((-1)^{2/3} \sqrt [3]{7}-x\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )}\right ) \, dx\\ &=-x^2+2 \log (x)+\frac {4}{3} \int \frac {1}{\left (-\sqrt [3]{-7}-x\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx+\frac {4}{3} \int \frac {1}{\left (\sqrt [3]{7}-x\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx+\frac {4}{3} \int \frac {1}{\left ((-1)^{2/3} \sqrt [3]{7}-x\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx+4 \int \frac {1}{x \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx+4 \int \frac {x^2}{\left (-7+x^3\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx+6 \int \frac {x \log \left (7-x^3\right )}{-2+x^2 \log ^2\left (7-x^3\right )} \, dx+42 \int \frac {x \log \left (7-x^3\right )}{\left (-7+x^3\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx\\ &=-x^2+2 \log (x)+\frac {4}{3} \int \frac {1}{\left (-\sqrt [3]{-7}-x\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx+\frac {4}{3} \int \frac {1}{\left (\sqrt [3]{7}-x\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx+\frac {4}{3} \int \frac {1}{\left ((-1)^{2/3} \sqrt [3]{7}-x\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx+4 \int \frac {1}{x \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx+4 \int \left (-\frac {1}{3 \left (-\sqrt [3]{-7}-x\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )}-\frac {1}{3 \left (\sqrt [3]{7}-x\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )}-\frac {1}{3 \left ((-1)^{2/3} \sqrt [3]{7}-x\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )}\right ) \, dx+6 \int \frac {x \log \left (7-x^3\right )}{-2+x^2 \log ^2\left (7-x^3\right )} \, dx+42 \int \left (-\frac {\log \left (7-x^3\right )}{3 \sqrt [3]{7} \left (\sqrt [3]{7}-x\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )}-\frac {(-1)^{2/3} \log \left (7-x^3\right )}{3 \sqrt [3]{7} \left (\sqrt [3]{7}+\sqrt [3]{-1} x\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )}+\frac {\sqrt [3]{-\frac {1}{7}} \log \left (7-x^3\right )}{3 \left (\sqrt [3]{7}-(-1)^{2/3} x\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )}\right ) \, dx\\ &=-x^2+2 \log (x)+4 \int \frac {1}{x \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx+6 \int \frac {x \log \left (7-x^3\right )}{-2+x^2 \log ^2\left (7-x^3\right )} \, dx-\left (2 (-7)^{2/3}\right ) \int \frac {\log \left (7-x^3\right )}{\left (\sqrt [3]{7}+\sqrt [3]{-1} x\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx-\left (2\ 7^{2/3}\right ) \int \frac {\log \left (7-x^3\right )}{\left (\sqrt [3]{7}-x\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx+\left (2 \sqrt [3]{-1} 7^{2/3}\right ) \int \frac {\log \left (7-x^3\right )}{\left (\sqrt [3]{7}-(-1)^{2/3} x\right ) \left (-2+x^2 \log ^2\left (7-x^3\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.68, size = 32, normalized size = 1.03 \begin {gather*} 2 \left (-\frac {x^2}{2}+\frac {1}{2} \log \left (2-x^2 \log ^2\left (7-x^3\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-28*x + 4*x^4 + 6*x^4*Log[7 - x^3] + (-14*x + 14*x^3 + 2*x^4 - 2*x^6)*Log[7 - x^3]^2)/(14 - 2*x^3 +
(-7*x^2 + x^5)*Log[7 - x^3]^2),x]

[Out]

2*(-1/2*x^2 + Log[2 - x^2*Log[7 - x^3]^2]/2)

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fricas [A]  time = 0.53, size = 31, normalized size = 1.00 \begin {gather*} -x^{2} + 2 \, \log \relax (x) + \log \left (\frac {x^{2} \log \left (-x^{3} + 7\right )^{2} - 2}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^6+2*x^4+14*x^3-14*x)*log(-x^3+7)^2+6*x^4*log(-x^3+7)+4*x^4-28*x)/((x^5-7*x^2)*log(-x^3+7)^2-2
*x^3+14),x, algorithm="fricas")

[Out]

-x^2 + 2*log(x) + log((x^2*log(-x^3 + 7)^2 - 2)/x^2)

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giac [A]  time = 0.46, size = 23, normalized size = 0.74 \begin {gather*} -x^{2} + \log \left (x^{2} \log \left (-x^{3} + 7\right )^{2} - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^6+2*x^4+14*x^3-14*x)*log(-x^3+7)^2+6*x^4*log(-x^3+7)+4*x^4-28*x)/((x^5-7*x^2)*log(-x^3+7)^2-2
*x^3+14),x, algorithm="giac")

[Out]

-x^2 + log(x^2*log(-x^3 + 7)^2 - 2)

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maple [A]  time = 0.05, size = 24, normalized size = 0.77

method	result	size
norman	\(-x^{2}+\ln \left (\ln \left (-x^{3}+7\right )^{2} x^{2}-2\right )\)	\(24\)
risch	\(-x^{2}+2 \ln \relax (x )+\ln \left (\ln \left (-x^{3}+7\right )^{2}-\frac {2}{x^{2}}\right )\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^6+2*x^4+14*x^3-14*x)*ln(-x^3+7)^2+6*x^4*ln(-x^3+7)+4*x^4-28*x)/((x^5-7*x^2)*ln(-x^3+7)^2-2*x^3+14),
x,method=_RETURNVERBOSE)

[Out]

-x^2+ln(ln(-x^3+7)^2*x^2-2)

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maxima [A]  time = 0.63, size = 31, normalized size = 1.00 \begin {gather*} -x^{2} + 2 \, \log \relax (x) + \log \left (\frac {x^{2} \log \left (-x^{3} + 7\right )^{2} - 2}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^6+2*x^4+14*x^3-14*x)*log(-x^3+7)^2+6*x^4*log(-x^3+7)+4*x^4-28*x)/((x^5-7*x^2)*log(-x^3+7)^2-2
*x^3+14),x, algorithm="maxima")

[Out]

-x^2 + 2*log(x) + log((x^2*log(-x^3 + 7)^2 - 2)/x^2)

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mupad [B]  time = 1.41, size = 31, normalized size = 1.00 \begin {gather*} \ln \left (\frac {x^2\,{\ln \left (7-x^3\right )}^2-2}{x^2}\right )+2\,\ln \relax (x)-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((28*x - 6*x^4*log(7 - x^3) + log(7 - x^3)^2*(14*x - 14*x^3 - 2*x^4 + 2*x^6) - 4*x^4)/(log(7 - x^3)^2*(7*x^
2 - x^5) + 2*x^3 - 14),x)

[Out]

log((x^2*log(7 - x^3)^2 - 2)/x^2) + 2*log(x) - x^2

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sympy [A]  time = 0.31, size = 22, normalized size = 0.71 \begin {gather*} - x^{2} + 2 \log {\relax (x )} + \log {\left (\log {\left (7 - x^{3} \right )}^{2} - \frac {2}{x^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**6+2*x**4+14*x**3-14*x)*ln(-x**3+7)**2+6*x**4*ln(-x**3+7)+4*x**4-28*x)/((x**5-7*x**2)*ln(-x**
3+7)**2-2*x**3+14),x)

[Out]

-x**2 + 2*log(x) + log(log(7 - x**3)**2 - 2/x**2)

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