3.69.8 \(\int \frac {10 x+2 x^2-10 x^3-2 x^4+(-60+30 x+23 x^2-10 x^3-x^4) \log (\frac {1}{10} (-20+10 x+x^2))}{(-20 x^4+10 x^5+x^6) \log ^2(\frac {1}{10} (-20+10 x+x^2))} \, dx\)

Optimal. Leaf size=24 \[ \frac {-\frac {1}{x}+x}{x^2 \log \left (-2+x+\frac {x^2}{10}\right )} \]

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Rubi [F]  time = 1.66, 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 {10 x+2 x^2-10 x^3-2 x^4+\left (-60+30 x+23 x^2-10 x^3-x^4\right ) \log \left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )}{\left (-20 x^4+10 x^5+x^6\right ) \log ^2\left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(10*x + 2*x^2 - 10*x^3 - 2*x^4 + (-60 + 30*x + 23*x^2 - 10*x^3 - x^4)*Log[(-20 + 10*x + x^2)/10])/((-20*x^
4 + 10*x^5 + x^6)*Log[(-20 + 10*x + x^2)/10]^2),x]

[Out]

(31*Defer[Int][1/((-10 + 6*Sqrt[5] - 2*x)*Log[-2 + x + x^2/10]^2), x])/(20*Sqrt[5]) - Defer[Int][1/(x^3*Log[-2
+ x + x^2/10]^2), x]/2 - (7*Defer[Int][1/(x^2*Log[-2 + x + x^2/10]^2), x])/20 + (3*Defer[Int][1/(x*Log[-2 + x
+ x^2/10]^2), x])/10 - ((3 - Sqrt[5])*Defer[Int][1/((10 - 6*Sqrt[5] + 2*x)*Log[-2 + x + x^2/10]^2), x])/10 +
(31*Defer[Int][1/((10 + 6*Sqrt[5] + 2*x)*Log[-2 + x + x^2/10]^2), x])/(20*Sqrt[5]) - ((3 + Sqrt[5])*Defer[Int]
[1/((10 + 6*Sqrt[5] + 2*x)*Log[-2 + x + x^2/10]^2), x])/10 + 3*Defer[Int][1/(x^4*Log[-2 + x + x^2/10]), x] - D
efer[Int][1/(x^2*Log[-2 + x + x^2/10]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {10 x+2 x^2-10 x^3-2 x^4+\left (-60+30 x+23 x^2-10 x^3-x^4\right ) \log \left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )}{x^4 \left (-20+10 x+x^2\right ) \log ^2\left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )} \, dx\\ &=\int \left (-\frac {2 \left (-5-x+5 x^2+x^3\right )}{x^3 \left (-20+10 x+x^2\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )}+\frac {3-x^2}{x^4 \log \left (-2+x+\frac {x^2}{10}\right )}\right ) \, dx\\ &=-\left (2 \int \frac {-5-x+5 x^2+x^3}{x^3 \left (-20+10 x+x^2\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx\right )+\int \frac {3-x^2}{x^4 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx\\ &=-\left (2 \int \left (\frac {1}{4 x^3 \log ^2\left (-2+x+\frac {x^2}{10}\right )}+\frac {7}{40 x^2 \log ^2\left (-2+x+\frac {x^2}{10}\right )}-\frac {3}{20 x \log ^2\left (-2+x+\frac {x^2}{10}\right )}+\frac {3 (31+2 x)}{40 \left (-20+10 x+x^2\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )}\right ) \, dx\right )+\int \left (\frac {3}{x^4 \log \left (-2+x+\frac {x^2}{10}\right )}-\frac {1}{x^2 \log \left (-2+x+\frac {x^2}{10}\right )}\right ) \, dx\\ &=-\left (\frac {3}{20} \int \frac {31+2 x}{\left (-20+10 x+x^2\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx\right )+\frac {3}{10} \int \frac {1}{x \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {7}{20} \int \frac {1}{x^2 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {1}{2} \int \frac {1}{x^3 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx+3 \int \frac {1}{x^4 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx-\int \frac {1}{x^2 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx\\ &=-\left (\frac {3}{20} \int \left (\frac {31}{\left (-20+10 x+x^2\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )}+\frac {2 x}{\left (-20+10 x+x^2\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )}\right ) \, dx\right )+\frac {3}{10} \int \frac {1}{x \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {7}{20} \int \frac {1}{x^2 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {1}{2} \int \frac {1}{x^3 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx+3 \int \frac {1}{x^4 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx-\int \frac {1}{x^2 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx\\ &=\frac {3}{10} \int \frac {1}{x \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {3}{10} \int \frac {x}{\left (-20+10 x+x^2\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {7}{20} \int \frac {1}{x^2 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {1}{2} \int \frac {1}{x^3 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx+3 \int \frac {1}{x^4 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {93}{20} \int \frac {1}{\left (-20+10 x+x^2\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\int \frac {1}{x^2 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx\\ &=-\left (\frac {3}{10} \int \left (\frac {1-\frac {\sqrt {5}}{3}}{\left (10-6 \sqrt {5}+2 x\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )}+\frac {1+\frac {\sqrt {5}}{3}}{\left (10+6 \sqrt {5}+2 x\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )}\right ) \, dx\right )+\frac {3}{10} \int \frac {1}{x \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {7}{20} \int \frac {1}{x^2 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {1}{2} \int \frac {1}{x^3 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx+3 \int \frac {1}{x^4 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {93}{20} \int \left (-\frac {1}{3 \sqrt {5} \left (-10+6 \sqrt {5}-2 x\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )}-\frac {1}{3 \sqrt {5} \left (10+6 \sqrt {5}+2 x\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )}\right ) \, dx-\int \frac {1}{x^2 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx\\ &=\frac {3}{10} \int \frac {1}{x \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {7}{20} \int \frac {1}{x^2 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {1}{2} \int \frac {1}{x^3 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx+3 \int \frac {1}{x^4 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx+\frac {31 \int \frac {1}{\left (-10+6 \sqrt {5}-2 x\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx}{20 \sqrt {5}}+\frac {31 \int \frac {1}{\left (10+6 \sqrt {5}+2 x\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx}{20 \sqrt {5}}-\frac {1}{10} \left (3-\sqrt {5}\right ) \int \frac {1}{\left (10-6 \sqrt {5}+2 x\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {1}{10} \left (3+\sqrt {5}\right ) \int \frac {1}{\left (10+6 \sqrt {5}+2 x\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\int \frac {1}{x^2 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.38, size = 25, normalized size = 1.04 \begin {gather*} -\frac {1-x^2}{x^3 \log \left (-2+x+\frac {x^2}{10}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10*x + 2*x^2 - 10*x^3 - 2*x^4 + (-60 + 30*x + 23*x^2 - 10*x^3 - x^4)*Log[(-20 + 10*x + x^2)/10])/((
-20*x^4 + 10*x^5 + x^6)*Log[(-20 + 10*x + x^2)/10]^2),x]

[Out]

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

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fricas [A]  time = 0.57, size = 20, normalized size = 0.83 \begin {gather*} \frac {x^{2} - 1}{x^{3} \log \left (\frac {1}{10} \, x^{2} + x - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4-10*x^3+23*x^2+30*x-60)*log(1/10*x^2+x-2)-2*x^4-10*x^3+2*x^2+10*x)/(x^6+10*x^5-20*x^4)/log(1/1
0*x^2+x-2)^2,x, algorithm="fricas")

[Out]

(x^2 - 1)/(x^3*log(1/10*x^2 + x - 2))

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giac [A]  time = 0.18, size = 20, normalized size = 0.83 \begin {gather*} \frac {x^{2} - 1}{x^{3} \log \left (\frac {1}{10} \, x^{2} + x - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4-10*x^3+23*x^2+30*x-60)*log(1/10*x^2+x-2)-2*x^4-10*x^3+2*x^2+10*x)/(x^6+10*x^5-20*x^4)/log(1/1
0*x^2+x-2)^2,x, algorithm="giac")

[Out]

(x^2 - 1)/(x^3*log(1/10*x^2 + x - 2))

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maple [A]  time = 0.04, size = 21, normalized size = 0.88

method	result	size
norman	\(\frac {x^{2}-1}{\ln \left (\frac {1}{10} x^{2}+x -2\right ) x^{3}}\)	\(21\)
risch	\(\frac {x^{2}-1}{\ln \left (\frac {1}{10} x^{2}+x -2\right ) x^{3}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^4-10*x^3+23*x^2+30*x-60)*ln(1/10*x^2+x-2)-2*x^4-10*x^3+2*x^2+10*x)/(x^6+10*x^5-20*x^4)/ln(1/10*x^2+x-
2)^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.51, size = 33, normalized size = 1.38 \begin {gather*} -\frac {x^{2} - 1}{x^{3} {\left (\log \relax (5) + \log \relax (2)\right )} - x^{3} \log \left (x^{2} + 10 \, x - 20\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4-10*x^3+23*x^2+30*x-60)*log(1/10*x^2+x-2)-2*x^4-10*x^3+2*x^2+10*x)/(x^6+10*x^5-20*x^4)/log(1/1
0*x^2+x-2)^2,x, algorithm="maxima")

[Out]

-(x^2 - 1)/(x^3*(log(5) + log(2)) - x^3*log(x^2 + 10*x - 20))

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mupad [B]  time = 4.42, size = 20, normalized size = 0.83 \begin {gather*} \frac {x^2-1}{x^3\,\ln \left (\frac {x^2}{10}+x-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x + x^2/10 - 2)*(10*x^3 - 23*x^2 - 30*x + x^4 + 60) - 10*x - 2*x^2 + 10*x^3 + 2*x^4)/(log(x + x^2/10
- 2)^2*(10*x^5 - 20*x^4 + x^6)),x)

[Out]

(x^2 - 1)/(x^3*log(x + x^2/10 - 2))

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sympy [A]  time = 0.15, size = 17, normalized size = 0.71 \begin {gather*} \frac {x^{2} - 1}{x^{3} \log {\left (\frac {x^{2}}{10} + x - 2 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**4-10*x**3+23*x**2+30*x-60)*ln(1/10*x**2+x-2)-2*x**4-10*x**3+2*x**2+10*x)/(x**6+10*x**5-20*x**4
)/ln(1/10*x**2+x-2)**2,x)

[Out]

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

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