3.55.40 \(\int \frac {-2 x+2 x^2-x^4-2 x^5+33 x^6+30 x^9+(-2+4 x-4 x^3+65 x^4+60 x^7) \log (1-x)+(30 x^2+30 x^5) \log ^2(1-x)}{5 x^4+10 x^2 \log (1-x)+5 \log ^2(1-x)} \, dx\)

Optimal. Leaf size=33 \[ x \left (2 x+x^4\right ) \left (x+\frac {-1+x}{5 x \left (x^2+\log (1-x)\right )}\right ) \]

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Rubi [F]  time = 0.90, 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 {-2 x+2 x^2-x^4-2 x^5+33 x^6+30 x^9+\left (-2+4 x-4 x^3+65 x^4+60 x^7\right ) \log (1-x)+\left (30 x^2+30 x^5\right ) \log ^2(1-x)}{5 x^4+10 x^2 \log (1-x)+5 \log ^2(1-x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x + 2*x^2 - x^4 - 2*x^5 + 33*x^6 + 30*x^9 + (-2 + 4*x - 4*x^3 + 65*x^4 + 60*x^7)*Log[1 - x] + (30*x^2
+ 30*x^5)*Log[1 - x]^2)/(5*x^4 + 10*x^2*Log[1 - x] + 5*Log[1 - x]^2),x]

[Out]

2*x^3 + x^6 - (2*Defer[Int][x/(x^2 + Log[1 - x])^2, x])/5 + (4*Defer[Int][x^2/(x^2 + Log[1 - x])^2, x])/5 - (4
*Defer[Int][x^3/(x^2 + Log[1 - x])^2, x])/5 - Defer[Int][x^4/(x^2 + Log[1 - x])^2, x]/5 + (2*Defer[Int][x^5/(x
^2 + Log[1 - x])^2, x])/5 - (2*Defer[Int][x^6/(x^2 + Log[1 - x])^2, x])/5 - (2*Defer[Int][(x^2 + Log[1 - x])^(
-1), x])/5 + (4*Defer[Int][x/(x^2 + Log[1 - x]), x])/5 - (4*Defer[Int][x^3/(x^2 + Log[1 - x]), x])/5 + Defer[I
nt][x^4/(x^2 + Log[1 - x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 x+2 x^2-x^4-2 x^5+33 x^6+30 x^9+\left (-2+4 x-4 x^3+65 x^4+60 x^7\right ) \log (1-x)+\left (30 x^2+30 x^5\right ) \log ^2(1-x)}{5 \left (x^2+\log (1-x)\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {-2 x+2 x^2-x^4-2 x^5+33 x^6+30 x^9+\left (-2+4 x-4 x^3+65 x^4+60 x^7\right ) \log (1-x)+\left (30 x^2+30 x^5\right ) \log ^2(1-x)}{\left (x^2+\log (1-x)\right )^2} \, dx\\ &=\frac {1}{5} \int \left (30 x^2 \left (1+x^3\right )-\frac {x \left (2-4 x+4 x^2+x^3-2 x^4+2 x^5\right )}{\left (x^2+\log (1-x)\right )^2}+\frac {-2+4 x-4 x^3+5 x^4}{x^2+\log (1-x)}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {x \left (2-4 x+4 x^2+x^3-2 x^4+2 x^5\right )}{\left (x^2+\log (1-x)\right )^2} \, dx\right )+\frac {1}{5} \int \frac {-2+4 x-4 x^3+5 x^4}{x^2+\log (1-x)} \, dx+6 \int x^2 \left (1+x^3\right ) \, dx\\ &=-\left (\frac {1}{5} \int \left (\frac {2 x}{\left (x^2+\log (1-x)\right )^2}-\frac {4 x^2}{\left (x^2+\log (1-x)\right )^2}+\frac {4 x^3}{\left (x^2+\log (1-x)\right )^2}+\frac {x^4}{\left (x^2+\log (1-x)\right )^2}-\frac {2 x^5}{\left (x^2+\log (1-x)\right )^2}+\frac {2 x^6}{\left (x^2+\log (1-x)\right )^2}\right ) \, dx\right )+\frac {1}{5} \int \left (-\frac {2}{x^2+\log (1-x)}+\frac {4 x}{x^2+\log (1-x)}-\frac {4 x^3}{x^2+\log (1-x)}+\frac {5 x^4}{x^2+\log (1-x)}\right ) \, dx+6 \int \left (x^2+x^5\right ) \, dx\\ &=2 x^3+x^6-\frac {1}{5} \int \frac {x^4}{\left (x^2+\log (1-x)\right )^2} \, dx-\frac {2}{5} \int \frac {x}{\left (x^2+\log (1-x)\right )^2} \, dx+\frac {2}{5} \int \frac {x^5}{\left (x^2+\log (1-x)\right )^2} \, dx-\frac {2}{5} \int \frac {x^6}{\left (x^2+\log (1-x)\right )^2} \, dx-\frac {2}{5} \int \frac {1}{x^2+\log (1-x)} \, dx+\frac {4}{5} \int \frac {x^2}{\left (x^2+\log (1-x)\right )^2} \, dx-\frac {4}{5} \int \frac {x^3}{\left (x^2+\log (1-x)\right )^2} \, dx+\frac {4}{5} \int \frac {x}{x^2+\log (1-x)} \, dx-\frac {4}{5} \int \frac {x^3}{x^2+\log (1-x)} \, dx+\int \frac {x^4}{x^2+\log (1-x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 38, normalized size = 1.15 \begin {gather*} \frac {1}{5} \left (10 x^3+5 x^6+\frac {(-1+x) \left (2 x+x^4\right )}{x^2+\log (1-x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x + 2*x^2 - x^4 - 2*x^5 + 33*x^6 + 30*x^9 + (-2 + 4*x - 4*x^3 + 65*x^4 + 60*x^7)*Log[1 - x] + (3
0*x^2 + 30*x^5)*Log[1 - x]^2)/(5*x^4 + 10*x^2*Log[1 - x] + 5*Log[1 - x]^2),x]

[Out]

(10*x^3 + 5*x^6 + ((-1 + x)*(2*x + x^4))/(x^2 + Log[1 - x]))/5

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fricas [A]  time = 0.63, size = 55, normalized size = 1.67 \begin {gather*} \frac {5 \, x^{8} + 11 \, x^{5} - x^{4} + 2 \, x^{2} + 5 \, {\left (x^{6} + 2 \, x^{3}\right )} \log \left (-x + 1\right ) - 2 \, x}{5 \, {\left (x^{2} + \log \left (-x + 1\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((30*x^5+30*x^2)*log(-x+1)^2+(60*x^7+65*x^4-4*x^3+4*x-2)*log(-x+1)+30*x^9+33*x^6-2*x^5-x^4+2*x^2-2*x
)/(5*log(-x+1)^2+10*x^2*log(-x+1)+5*x^4),x, algorithm="fricas")

[Out]

1/5*(5*x^8 + 11*x^5 - x^4 + 2*x^2 + 5*(x^6 + 2*x^3)*log(-x + 1) - 2*x)/(x^2 + log(-x + 1))

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giac [A]  time = 0.18, size = 40, normalized size = 1.21 \begin {gather*} x^{6} + 2 \, x^{3} + \frac {x^{5} - x^{4} + 2 \, x^{2} - 2 \, x}{5 \, {\left (x^{2} + \log \left (-x + 1\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((30*x^5+30*x^2)*log(-x+1)^2+(60*x^7+65*x^4-4*x^3+4*x-2)*log(-x+1)+30*x^9+33*x^6-2*x^5-x^4+2*x^2-2*x
)/(5*log(-x+1)^2+10*x^2*log(-x+1)+5*x^4),x, algorithm="giac")

[Out]

x^6 + 2*x^3 + 1/5*(x^5 - x^4 + 2*x^2 - 2*x)/(x^2 + log(-x + 1))

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maple [A]  time = 0.05, size = 33, normalized size = 1.00

method	result	size
risch	\(x^{6}+2 x^{3}+\frac {\left (x^{3}+2\right ) x \left (x -1\right )}{5 x^{2}+5 \ln \left (1-x \right )}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((30*x^5+30*x^2)*ln(1-x)^2+(60*x^7+65*x^4-4*x^3+4*x-2)*ln(1-x)+30*x^9+33*x^6-2*x^5-x^4+2*x^2-2*x)/(5*ln(1-
x)^2+10*x^2*ln(1-x)+5*x^4),x,method=_RETURNVERBOSE)

[Out]

x^6+2*x^3+1/5*(x^3+2)*x*(x-1)/(x^2+ln(1-x))

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maxima [A]  time = 0.41, size = 55, normalized size = 1.67 \begin {gather*} \frac {5 \, x^{8} + 11 \, x^{5} - x^{4} + 2 \, x^{2} + 5 \, {\left (x^{6} + 2 \, x^{3}\right )} \log \left (-x + 1\right ) - 2 \, x}{5 \, {\left (x^{2} + \log \left (-x + 1\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((30*x^5+30*x^2)*log(-x+1)^2+(60*x^7+65*x^4-4*x^3+4*x-2)*log(-x+1)+30*x^9+33*x^6-2*x^5-x^4+2*x^2-2*x
)/(5*log(-x+1)^2+10*x^2*log(-x+1)+5*x^4),x, algorithm="maxima")

[Out]

1/5*(5*x^8 + 11*x^5 - x^4 + 2*x^2 + 5*(x^6 + 2*x^3)*log(-x + 1) - 2*x)/(x^2 + log(-x + 1))

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mupad [B]  time = 3.65, size = 124, normalized size = 3.76 \begin {gather*} \frac {5\,x^3}{2}-\frac {\frac {x}{8}-\frac {1}{40}}{x^2-x+\frac {1}{2}}-\frac {2\,x^2}{5}-\frac {x}{4}+x^6+\frac {\frac {\left (x-1\right )\,\left (-3\,x^6+2\,x^5+x^4-2\,x^2+2\,x\right )}{5\,\left (2\,x^2-2\,x+1\right )}-\frac {\ln \left (1-x\right )\,\left (x-1\right )\,\left (5\,x^4-4\,x^3+4\,x-2\right )}{5\,\left (2\,x^2-2\,x+1\right )}}{\ln \left (1-x\right )+x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(1 - x)^2*(30*x^2 + 30*x^5) - 2*x + 2*x^2 - x^4 - 2*x^5 + 33*x^6 + 30*x^9 + log(1 - x)*(4*x - 4*x^3 +
65*x^4 + 60*x^7 - 2))/(5*log(1 - x)^2 + 5*x^4 + 10*x^2*log(1 - x)),x)

[Out]

(5*x^3)/2 - (x/8 - 1/40)/(x^2 - x + 1/2) - (2*x^2)/5 - x/4 + x^6 + (((x - 1)*(2*x - 2*x^2 + x^4 + 2*x^5 - 3*x^
6))/(5*(2*x^2 - 2*x + 1)) - (log(1 - x)*(x - 1)*(4*x - 4*x^3 + 5*x^4 - 2))/(5*(2*x^2 - 2*x + 1)))/(log(1 - x)
+ x^2)

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sympy [A]  time = 0.14, size = 34, normalized size = 1.03 \begin {gather*} x^{6} + 2 x^{3} + \frac {x^{5} - x^{4} + 2 x^{2} - 2 x}{5 x^{2} + 5 \log {\left (1 - x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((30*x**5+30*x**2)*ln(-x+1)**2+(60*x**7+65*x**4-4*x**3+4*x-2)*ln(-x+1)+30*x**9+33*x**6-2*x**5-x**4+2
*x**2-2*x)/(5*ln(-x+1)**2+10*x**2*ln(-x+1)+5*x**4),x)

[Out]

x**6 + 2*x**3 + (x**5 - x**4 + 2*x**2 - 2*x)/(5*x**2 + 5*log(1 - x))

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