3.22.85 \(\int \frac {x+7 x^2-4 x^4-7 x^5+(-8 x+8 x^4) \log (x-x^4)+(-2 x^2+2 x^5+(2 x-2 x^4) \log (x-x^4)) \log (-x+\log (x-x^4))}{x-x^4+(-1+x^3) \log (x-x^4)} \, dx\)

Optimal. Leaf size=22 \[ x \left (4 x-x \log \left (-x+\log \left (x-x^4\right )\right )\right ) \]

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Rubi [F]  time = 4.27, 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+7 x^2-4 x^4-7 x^5+\left (-8 x+8 x^4\right ) \log \left (x-x^4\right )+\left (-2 x^2+2 x^5+\left (2 x-2 x^4\right ) \log \left (x-x^4\right )\right ) \log \left (-x+\log \left (x-x^4\right )\right )}{x-x^4+\left (-1+x^3\right ) \log \left (x-x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x + 7*x^2 - 4*x^4 - 7*x^5 + (-8*x + 8*x^4)*Log[x - x^4] + (-2*x^2 + 2*x^5 + (2*x - 2*x^4)*Log[x - x^4])*L
og[-x + Log[x - x^4]])/(x - x^4 + (-1 + x^3)*Log[x - x^4]),x]

[Out]

((2*I)*Defer[Int][1/((-1 + I*Sqrt[3] - 2*x)*(x - Log[x - x^4])), x])/Sqrt[3] + Defer[Int][1/((-1 + x)*(x - Log
[x - x^4])), x] + 4*Defer[Int][x/(x - Log[x - x^4]), x] + 7*Defer[Int][x^2/(x - Log[x - x^4]), x] - ((3 + I*Sq
rt[3])*Defer[Int][1/((1 - I*Sqrt[3] + 2*x)*(x - Log[x - x^4])), x])/3 + ((2*I)*Defer[Int][1/((1 + I*Sqrt[3] +
2*x)*(x - Log[x - x^4])), x])/Sqrt[3] - ((3 - I*Sqrt[3])*Defer[Int][1/((1 + I*Sqrt[3] + 2*x)*(x - Log[x - x^4]
)), x])/3 + 8*Defer[Int][(x*Log[x*(1 - x^3)])/(-x + Log[x - x^4]), x] - 2*Defer[Int][x*Log[-x + Log[x - x^4]],
x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x+7 x^2-4 x^4-7 x^5+\left (-8 x+8 x^4\right ) \log \left (x-x^4\right )+\left (-2 x^2+2 x^5+\left (2 x-2 x^4\right ) \log \left (x-x^4\right )\right ) \log \left (-x+\log \left (x-x^4\right )\right )}{\left (1-x^3\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx\\ &=\int \left (-\frac {x}{(-1+x) \left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )}-\frac {7 x^2}{(-1+x) \left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )}+\frac {4 x^4}{(-1+x) \left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )}+\frac {7 x^5}{(-1+x) \left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )}+\frac {8 x \log \left (x \left (1-x^3\right )\right )}{-x+\log \left (x-x^4\right )}-2 x \log \left (-x+\log \left (x-x^4\right )\right )\right ) \, dx\\ &=-\left (2 \int x \log \left (-x+\log \left (x-x^4\right )\right ) \, dx\right )+4 \int \frac {x^4}{(-1+x) \left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx-7 \int \frac {x^2}{(-1+x) \left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx+7 \int \frac {x^5}{(-1+x) \left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx+8 \int \frac {x \log \left (x \left (1-x^3\right )\right )}{-x+\log \left (x-x^4\right )} \, dx-\int \frac {x}{(-1+x) \left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx\\ &=-\left (2 \int x \log \left (-x+\log \left (x-x^4\right )\right ) \, dx\right )+4 \int \left (\frac {1}{3 (-1+x) \left (x-\log \left (x-x^4\right )\right )}+\frac {x}{x-\log \left (x-x^4\right )}+\frac {1-x}{3 \left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )}\right ) \, dx-7 \int \left (\frac {1}{3 (-1+x) \left (x-\log \left (x-x^4\right )\right )}+\frac {1+2 x}{3 \left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )}\right ) \, dx+7 \int \left (\frac {1}{3 (-1+x) \left (x-\log \left (x-x^4\right )\right )}+\frac {x^2}{x-\log \left (x-x^4\right )}+\frac {1+2 x}{3 \left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )}\right ) \, dx+8 \int \frac {x \log \left (x \left (1-x^3\right )\right )}{-x+\log \left (x-x^4\right )} \, dx-\int \left (\frac {1}{3 (-1+x) \left (x-\log \left (x-x^4\right )\right )}+\frac {1-x}{3 \left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {1}{(-1+x) \left (x-\log \left (x-x^4\right )\right )} \, dx\right )-\frac {1}{3} \int \frac {1-x}{\left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx+\frac {4}{3} \int \frac {1}{(-1+x) \left (x-\log \left (x-x^4\right )\right )} \, dx+\frac {4}{3} \int \frac {1-x}{\left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx-2 \int x \log \left (-x+\log \left (x-x^4\right )\right ) \, dx+4 \int \frac {x}{x-\log \left (x-x^4\right )} \, dx+7 \int \frac {x^2}{x-\log \left (x-x^4\right )} \, dx+8 \int \frac {x \log \left (x \left (1-x^3\right )\right )}{-x+\log \left (x-x^4\right )} \, dx\\ &=-\left (\frac {1}{3} \int \left (\frac {1}{\left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )}-\frac {x}{\left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )}\right ) \, dx\right )-\frac {1}{3} \int \frac {1}{(-1+x) \left (x-\log \left (x-x^4\right )\right )} \, dx+\frac {4}{3} \int \left (\frac {1}{\left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )}-\frac {x}{\left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )}\right ) \, dx+\frac {4}{3} \int \frac {1}{(-1+x) \left (x-\log \left (x-x^4\right )\right )} \, dx-2 \int x \log \left (-x+\log \left (x-x^4\right )\right ) \, dx+4 \int \frac {x}{x-\log \left (x-x^4\right )} \, dx+7 \int \frac {x^2}{x-\log \left (x-x^4\right )} \, dx+8 \int \frac {x \log \left (x \left (1-x^3\right )\right )}{-x+\log \left (x-x^4\right )} \, dx\\ &=-\left (\frac {1}{3} \int \frac {1}{(-1+x) \left (x-\log \left (x-x^4\right )\right )} \, dx\right )-\frac {1}{3} \int \frac {1}{\left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx+\frac {1}{3} \int \frac {x}{\left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx+\frac {4}{3} \int \frac {1}{(-1+x) \left (x-\log \left (x-x^4\right )\right )} \, dx+\frac {4}{3} \int \frac {1}{\left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx-\frac {4}{3} \int \frac {x}{\left (1+x+x^2\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx-2 \int x \log \left (-x+\log \left (x-x^4\right )\right ) \, dx+4 \int \frac {x}{x-\log \left (x-x^4\right )} \, dx+7 \int \frac {x^2}{x-\log \left (x-x^4\right )} \, dx+8 \int \frac {x \log \left (x \left (1-x^3\right )\right )}{-x+\log \left (x-x^4\right )} \, dx\\ &=-\left (\frac {1}{3} \int \left (\frac {2 i}{\sqrt {3} \left (-1+i \sqrt {3}-2 x\right ) \left (x-\log \left (x-x^4\right )\right )}+\frac {2 i}{\sqrt {3} \left (1+i \sqrt {3}+2 x\right ) \left (x-\log \left (x-x^4\right )\right )}\right ) \, dx\right )+\frac {1}{3} \int \left (\frac {1+\frac {i}{\sqrt {3}}}{\left (1-i \sqrt {3}+2 x\right ) \left (x-\log \left (x-x^4\right )\right )}+\frac {1-\frac {i}{\sqrt {3}}}{\left (1+i \sqrt {3}+2 x\right ) \left (x-\log \left (x-x^4\right )\right )}\right ) \, dx-\frac {1}{3} \int \frac {1}{(-1+x) \left (x-\log \left (x-x^4\right )\right )} \, dx+\frac {4}{3} \int \left (\frac {2 i}{\sqrt {3} \left (-1+i \sqrt {3}-2 x\right ) \left (x-\log \left (x-x^4\right )\right )}+\frac {2 i}{\sqrt {3} \left (1+i \sqrt {3}+2 x\right ) \left (x-\log \left (x-x^4\right )\right )}\right ) \, dx-\frac {4}{3} \int \left (\frac {1+\frac {i}{\sqrt {3}}}{\left (1-i \sqrt {3}+2 x\right ) \left (x-\log \left (x-x^4\right )\right )}+\frac {1-\frac {i}{\sqrt {3}}}{\left (1+i \sqrt {3}+2 x\right ) \left (x-\log \left (x-x^4\right )\right )}\right ) \, dx+\frac {4}{3} \int \frac {1}{(-1+x) \left (x-\log \left (x-x^4\right )\right )} \, dx-2 \int x \log \left (-x+\log \left (x-x^4\right )\right ) \, dx+4 \int \frac {x}{x-\log \left (x-x^4\right )} \, dx+7 \int \frac {x^2}{x-\log \left (x-x^4\right )} \, dx+8 \int \frac {x \log \left (x \left (1-x^3\right )\right )}{-x+\log \left (x-x^4\right )} \, dx\\ &=-\left (\frac {1}{3} \int \frac {1}{(-1+x) \left (x-\log \left (x-x^4\right )\right )} \, dx\right )+\frac {4}{3} \int \frac {1}{(-1+x) \left (x-\log \left (x-x^4\right )\right )} \, dx-2 \int x \log \left (-x+\log \left (x-x^4\right )\right ) \, dx+4 \int \frac {x}{x-\log \left (x-x^4\right )} \, dx+7 \int \frac {x^2}{x-\log \left (x-x^4\right )} \, dx+8 \int \frac {x \log \left (x \left (1-x^3\right )\right )}{-x+\log \left (x-x^4\right )} \, dx-\frac {(2 i) \int \frac {1}{\left (-1+i \sqrt {3}-2 x\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx}{3 \sqrt {3}}-\frac {(2 i) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx}{3 \sqrt {3}}+\frac {(8 i) \int \frac {1}{\left (-1+i \sqrt {3}-2 x\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx}{3 \sqrt {3}}+\frac {(8 i) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx}{3 \sqrt {3}}+\frac {1}{9} \left (3-i \sqrt {3}\right ) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx-\frac {1}{9} \left (4 \left (3-i \sqrt {3}\right )\right ) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx+\frac {1}{9} \left (3+i \sqrt {3}\right ) \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx-\frac {1}{9} \left (4 \left (3+i \sqrt {3}\right )\right ) \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \left (x-\log \left (x-x^4\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 24, normalized size = 1.09 \begin {gather*} 4 x^2-x^2 \log \left (-x+\log \left (x-x^4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + 7*x^2 - 4*x^4 - 7*x^5 + (-8*x + 8*x^4)*Log[x - x^4] + (-2*x^2 + 2*x^5 + (2*x - 2*x^4)*Log[x - x
^4])*Log[-x + Log[x - x^4]])/(x - x^4 + (-1 + x^3)*Log[x - x^4]),x]

[Out]

4*x^2 - x^2*Log[-x + Log[x - x^4]]

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fricas [A]  time = 0.72, size = 24, normalized size = 1.09 \begin {gather*} -x^{2} \log \left (-x + \log \left (-x^{4} + x\right )\right ) + 4 \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^4+2*x)*log(-x^4+x)+2*x^5-2*x^2)*log(log(-x^4+x)-x)+(8*x^4-8*x)*log(-x^4+x)-7*x^5-4*x^4+7*x^2
+x)/((x^3-1)*log(-x^4+x)-x^4+x),x, algorithm="fricas")

[Out]

-x^2*log(-x + log(-x^4 + x)) + 4*x^2

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giac [A]  time = 0.38, size = 24, normalized size = 1.09 \begin {gather*} -x^{2} \log \left (-x + \log \left (-x^{4} + x\right )\right ) + 4 \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^4+2*x)*log(-x^4+x)+2*x^5-2*x^2)*log(log(-x^4+x)-x)+(8*x^4-8*x)*log(-x^4+x)-7*x^5-4*x^4+7*x^2
+x)/((x^3-1)*log(-x^4+x)-x^4+x),x, algorithm="giac")

[Out]

-x^2*log(-x + log(-x^4 + x)) + 4*x^2

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-2 x^{4}+2 x \right ) \ln \left (-x^{4}+x \right )+2 x^{5}-2 x^{2}\right ) \ln \left (\ln \left (-x^{4}+x \right )-x \right )+\left (8 x^{4}-8 x \right ) \ln \left (-x^{4}+x \right )-7 x^{5}-4 x^{4}+7 x^{2}+x}{\left (x^{3}-1\right ) \ln \left (-x^{4}+x \right )-x^{4}+x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x^4+2*x)*ln(-x^4+x)+2*x^5-2*x^2)*ln(ln(-x^4+x)-x)+(8*x^4-8*x)*ln(-x^4+x)-7*x^5-4*x^4+7*x^2+x)/((x^3-
1)*ln(-x^4+x)-x^4+x),x)

[Out]

int((((-2*x^4+2*x)*ln(-x^4+x)+2*x^5-2*x^2)*ln(ln(-x^4+x)-x)+(8*x^4-8*x)*ln(-x^4+x)-7*x^5-4*x^4+7*x^2+x)/((x^3-
1)*ln(-x^4+x)-x^4+x),x)

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maxima [A]  time = 0.59, size = 31, normalized size = 1.41 \begin {gather*} -x^{2} \log \left (-x + \log \left (x^{2} + x + 1\right ) + \log \relax (x) + \log \left (-x + 1\right )\right ) + 4 \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^4+2*x)*log(-x^4+x)+2*x^5-2*x^2)*log(log(-x^4+x)-x)+(8*x^4-8*x)*log(-x^4+x)-7*x^5-4*x^4+7*x^2
+x)/((x^3-1)*log(-x^4+x)-x^4+x),x, algorithm="maxima")

[Out]

-x^2*log(-x + log(x^2 + x + 1) + log(x) + log(-x + 1)) + 4*x^2

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mupad [B]  time = 1.38, size = 20, normalized size = 0.91 \begin {gather*} -x^2\,\left (\ln \left (\ln \left (x-x^4\right )-x\right )-4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - log(x - x^4)*(8*x - 8*x^4) + log(log(x - x^4) - x)*(log(x - x^4)*(2*x - 2*x^4) - 2*x^2 + 2*x^5) + 7*x
^2 - 4*x^4 - 7*x^5)/(x + log(x - x^4)*(x^3 - 1) - x^4),x)

[Out]

-x^2*(log(log(x - x^4) - x) - 4)

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sympy [A]  time = 0.83, size = 17, normalized size = 0.77 \begin {gather*} - x^{2} \log {\left (- x + \log {\left (- x^{4} + x \right )} \right )} + 4 x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x**4+2*x)*ln(-x**4+x)+2*x**5-2*x**2)*ln(ln(-x**4+x)-x)+(8*x**4-8*x)*ln(-x**4+x)-7*x**5-4*x**4+
7*x**2+x)/((x**3-1)*ln(-x**4+x)-x**4+x),x)

[Out]

-x**2*log(-x + log(-x**4 + x)) + 4*x**2

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