3.72.51 \(\int \frac {-5 x-5 x^2-5 x^4-5 x^5+(-5-5 x) \log (4)+(10 x+5 x^2+25 x^4+20 x^5+5 \log (4)) \log (-x)+(1+2 x+x^2) \log ^2(-x)}{(5+10 x+5 x^2) \log ^2(-x)} \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{5} \left (x+\frac {5 x \left (x+x^4+\log (4)\right )}{(1+x) \log (-x)}\right ) \]

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Rubi [F]  time = 0.24, 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 {-5 x-5 x^2-5 x^4-5 x^5+(-5-5 x) \log (4)+\left (10 x+5 x^2+25 x^4+20 x^5+5 \log (4)\right ) \log (-x)+\left (1+2 x+x^2\right ) \log ^2(-x)}{\left (5+10 x+5 x^2\right ) \log ^2(-x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5*x - 5*x^2 - 5*x^4 - 5*x^5 + (-5 - 5*x)*Log[4] + (10*x + 5*x^2 + 25*x^4 + 20*x^5 + 5*Log[4])*Log[-x] +
(1 + 2*x + x^2)*Log[-x]^2)/((5 + 10*x + 5*x^2)*Log[-x]^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-5 x-5 x^2-5 x^4-5 x^5+(-5-5 x) \log (4)+\left (10 x+5 x^2+25 x^4+20 x^5+5 \log (4)\right ) \log (-x)+\left (1+2 x+x^2\right ) \log ^2(-x)}{5 (1+x)^2 \log ^2(-x)} \, dx\\ &=\frac {1}{5} \int \frac {-5 x-5 x^2-5 x^4-5 x^5+(-5-5 x) \log (4)+\left (10 x+5 x^2+25 x^4+20 x^5+5 \log (4)\right ) \log (-x)+\left (1+2 x+x^2\right ) \log ^2(-x)}{(1+x)^2 \log ^2(-x)} \, dx\\ &=\frac {1}{5} \int \left (1-\frac {5 \left (x+x^4+\log (4)\right )}{(1+x) \log ^2(-x)}+\frac {5 \left (2 x+x^2+5 x^4+4 x^5+\log (4)\right )}{(1+x)^2 \log (-x)}\right ) \, dx\\ &=\frac {x}{5}-\int \frac {x+x^4+\log (4)}{(1+x) \log ^2(-x)} \, dx+\int \frac {2 x+x^2+5 x^4+4 x^5+\log (4)}{(1+x)^2 \log (-x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.68, size = 26, normalized size = 0.96 \begin {gather*} \frac {x}{5}+\frac {x \left (x+x^4+\log (4)\right )}{(1+x) \log (-x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5*x - 5*x^2 - 5*x^4 - 5*x^5 + (-5 - 5*x)*Log[4] + (10*x + 5*x^2 + 25*x^4 + 20*x^5 + 5*Log[4])*Log[
-x] + (1 + 2*x + x^2)*Log[-x]^2)/((5 + 10*x + 5*x^2)*Log[-x]^2),x]

[Out]

x/5 + (x*(x + x^4 + Log[4]))/((1 + x)*Log[-x])

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fricas [A]  time = 0.63, size = 39, normalized size = 1.44 \begin {gather*} \frac {5 \, x^{5} + 5 \, x^{2} + 10 \, x \log \relax (2) + {\left (x^{2} + x\right )} \log \left (-x\right )}{5 \, {\left (x + 1\right )} \log \left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+2*x+1)*log(-x)^2+(10*log(2)+20*x^5+25*x^4+5*x^2+10*x)*log(-x)+2*(-5*x-5)*log(2)-5*x^5-5*x^4-5*
x^2-5*x)/(5*x^2+10*x+5)/log(-x)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.19, size = 30, normalized size = 1.11 \begin {gather*} \frac {1}{5} \, x + \frac {x^{5} + x^{2} + 2 \, x \log \relax (2)}{x \log \left (-x\right ) + \log \left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+2*x+1)*log(-x)^2+(10*log(2)+20*x^5+25*x^4+5*x^2+10*x)*log(-x)+2*(-5*x-5)*log(2)-5*x^5-5*x^4-5*
x^2-5*x)/(5*x^2+10*x+5)/log(-x)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.15, size = 27, normalized size = 1.00

method	result	size
risch	\(\frac {x \left (2 \ln \relax (2)+x^{4}+x \right )}{\left (x +1\right ) \ln \left (-x \right )}+\frac {x}{5}\)	\(27\)
norman	\(\frac {x^{2}+x^{5}-\frac {\ln \left (-x \right )}{5}+2 x \ln \relax (2)+\frac {\ln \left (-x \right ) x^{2}}{5}}{\left (x +1\right ) \ln \left (-x \right )}\)	\(40\)
derivativedivides	\(\frac {x}{5}-\frac {-10 x \ln \relax (2)-10 x^{2} \ln \relax (2)}{5 \left (-x -1\right )^{2} \ln \left (-x \right )}+\frac {x^{4}}{\ln \left (-x \right )}-\frac {x^{3}}{\ln \left (-x \right )}+\frac {x^{2}}{\ln \left (-x \right )}\)	\(64\)
default	\(\frac {x}{5}-\frac {-10 x \ln \relax (2)-10 x^{2} \ln \relax (2)}{5 \left (-x -1\right )^{2} \ln \left (-x \right )}+\frac {x^{4}}{\ln \left (-x \right )}-\frac {x^{3}}{\ln \left (-x \right )}+\frac {x^{2}}{\ln \left (-x \right )}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2+2*x+1)*ln(-x)^2+(10*ln(2)+20*x^5+25*x^4+5*x^2+10*x)*ln(-x)+2*(-5*x-5)*ln(2)-5*x^5-5*x^4-5*x^2-5*x)/(
5*x^2+10*x+5)/ln(-x)^2,x,method=_RETURNVERBOSE)

[Out]

x/(x+1)/ln(-x)*(2*ln(2)+x^4+x)+1/5*x

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maxima [A]  time = 0.47, size = 39, normalized size = 1.44 \begin {gather*} \frac {5 \, x^{5} + 5 \, x^{2} + 10 \, x \log \relax (2) + {\left (x^{2} + x\right )} \log \left (-x\right )}{5 \, {\left (x + 1\right )} \log \left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+2*x+1)*log(-x)^2+(10*log(2)+20*x^5+25*x^4+5*x^2+10*x)*log(-x)+2*(-5*x-5)*log(2)-5*x^5-5*x^4-5*
x^2-5*x)/(5*x^2+10*x+5)/log(-x)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.23, size = 31, normalized size = 1.15 \begin {gather*} \frac {x}{5}+\frac {x\,\left (5\,x^4+5\,x+10\,\ln \relax (2)\right )}{5\,\ln \left (-x\right )\,\left (x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*x + 2*log(2)*(5*x + 5) - log(-x)^2*(2*x + x^2 + 1) - log(-x)*(10*x + 10*log(2) + 5*x^2 + 25*x^4 + 20*x
^5) + 5*x^2 + 5*x^4 + 5*x^5)/(log(-x)^2*(10*x + 5*x^2 + 5)),x)

[Out]

x/5 + (x*(5*x + 10*log(2) + 5*x^4))/(5*log(-x)*(x + 1))

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sympy [A]  time = 0.17, size = 24, normalized size = 0.89 \begin {gather*} \frac {x}{5} + \frac {x^{5} + x^{2} + 2 x \log {\relax (2 )}}{\left (x + 1\right ) \log {\left (- x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2+2*x+1)*ln(-x)**2+(10*ln(2)+20*x**5+25*x**4+5*x**2+10*x)*ln(-x)+2*(-5*x-5)*ln(2)-5*x**5-5*x**4
-5*x**2-5*x)/(5*x**2+10*x+5)/ln(-x)**2,x)

[Out]

x/5 + (x**5 + x**2 + 2*x*log(2))/((x + 1)*log(-x))

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