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

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

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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 = {} $$\begin{array}{c}\int \frac{10x+2x^2-10x^3-2x^4+(-60+30x+23x^2-10x^3-x^4)\log \left(\frac{1}{10}(-20+10x+x^2)\right)}{(-20x^4+10x^5+x^6){\log }^2\left(\frac{1}{10}(-20+10x+x^2)\right)}dx\end{array}$$

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

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

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 $$\begin{array}{c}\frac{x^2-1}{x^3\log (\frac{1}{10}{x}^2+x-2)}\end{array}$$

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 $$\begin{array}{c}\frac{x^2-1}{x^3\log (\frac{1}{10}{x}^2+x-2)}\end{array}$$

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	$\frac{x^2-1}{\ln (\frac{1}{10}{x}^2+x-2)x^3}$	$21$
risch	$\frac{x^2-1}{\ln (\frac{1}{10}{x}^2+x-2)x^3}$	$21$

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 $$\begin{array}{c}-\frac{x^2-1}{x^3(\log (5)+\log (2))-x^3\log (x^2+10x-20)}\end{array}$$

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 $$\begin{array}{c}\frac{x^2-1}{x^3\ln (\frac{x^2}{10}+x-2)}\end{array}$$

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 $$\begin{array}{c}\frac{x^2-1}{x^3\log (\frac{x^2}{10}+x-2)}\end{array}$$

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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