3.36.71 \(\int \frac {-93+138 x-32 x^2-10 x^3+3 x^4+(-63+70 x+2 x^2-6 x^3) \log (x)+(-11+8 x+3 x^2) \log ^2(x)}{9-6 x+x^2+(6-2 x) \log (x)+\log ^2(x)} \, dx\)

Optimal. Leaf size=23 \[ x \left (-3+4 (-2+x)+x^2-\frac {3}{-3+x-\log (x)}\right ) \]

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Rubi [F]  time = 0.28, 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 {-93+138 x-32 x^2-10 x^3+3 x^4+\left (-63+70 x+2 x^2-6 x^3\right ) \log (x)+\left (-11+8 x+3 x^2\right ) \log ^2(x)}{9-6 x+x^2+(6-2 x) \log (x)+\log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-93 + 138*x - 32*x^2 - 10*x^3 + 3*x^4 + (-63 + 70*x + 2*x^2 - 6*x^3)*Log[x] + (-11 + 8*x + 3*x^2)*Log[x]^
2)/(9 - 6*x + x^2 + (6 - 2*x)*Log[x] + Log[x]^2),x]

[Out]

-11*x + 4*x^2 + x^3 - 3*Defer[Int][(-3 + x - Log[x])^(-2), x] + 3*Defer[Int][x/(-3 + x - Log[x])^2, x] - 3*Def
er[Int][(-3 + x - Log[x])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-93+138 x-32 x^2-10 x^3+3 x^4+\left (-63+70 x+2 x^2-6 x^3\right ) \log (x)+\left (-11+8 x+3 x^2\right ) \log ^2(x)}{(3-x+\log (x))^2} \, dx\\ &=\int \left (-11+8 x+3 x^2+\frac {3 (-1+x)}{(-3+x-\log (x))^2}-\frac {3}{-3+x-\log (x)}\right ) \, dx\\ &=-11 x+4 x^2+x^3+3 \int \frac {-1+x}{(-3+x-\log (x))^2} \, dx-3 \int \frac {1}{-3+x-\log (x)} \, dx\\ &=-11 x+4 x^2+x^3+3 \int \left (-\frac {1}{(-3+x-\log (x))^2}+\frac {x}{(-3+x-\log (x))^2}\right ) \, dx-3 \int \frac {1}{-3+x-\log (x)} \, dx\\ &=-11 x+4 x^2+x^3-3 \int \frac {1}{(-3+x-\log (x))^2} \, dx+3 \int \frac {x}{(-3+x-\log (x))^2} \, dx-3 \int \frac {1}{-3+x-\log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 21, normalized size = 0.91 \begin {gather*} x \left (-11+4 x+x^2+\frac {3}{3-x+\log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-93 + 138*x - 32*x^2 - 10*x^3 + 3*x^4 + (-63 + 70*x + 2*x^2 - 6*x^3)*Log[x] + (-11 + 8*x + 3*x^2)*L
og[x]^2)/(9 - 6*x + x^2 + (6 - 2*x)*Log[x] + Log[x]^2),x]

[Out]

x*(-11 + 4*x + x^2 + 3/(3 - x + Log[x]))

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fricas [A]  time = 0.76, size = 41, normalized size = 1.78 \begin {gather*} \frac {x^{4} + x^{3} - 23 \, x^{2} - {\left (x^{3} + 4 \, x^{2} - 11 \, x\right )} \log \relax (x) + 30 \, x}{x - \log \relax (x) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+8*x-11)*log(x)^2+(-6*x^3+2*x^2+70*x-63)*log(x)+3*x^4-10*x^3-32*x^2+138*x-93)/(log(x)^2+(6-2*
x)*log(x)+x^2-6*x+9),x, algorithm="fricas")

[Out]

(x^4 + x^3 - 23*x^2 - (x^3 + 4*x^2 - 11*x)*log(x) + 30*x)/(x - log(x) - 3)

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giac [A]  time = 0.22, size = 24, normalized size = 1.04 \begin {gather*} x^{3} + 4 \, x^{2} - 11 \, x - \frac {3 \, x}{x - \log \relax (x) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+8*x-11)*log(x)^2+(-6*x^3+2*x^2+70*x-63)*log(x)+3*x^4-10*x^3-32*x^2+138*x-93)/(log(x)^2+(6-2*
x)*log(x)+x^2-6*x+9),x, algorithm="giac")

[Out]

x^3 + 4*x^2 - 11*x - 3*x/(x - log(x) - 3)

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maple [A]  time = 0.03, size = 25, normalized size = 1.09

method	result	size
risch	\(x^{3}+4 x^{2}-11 x -\frac {3 x}{x -\ln \relax (x )-3}\)	\(25\)
norman	\(\frac {x^{3}+x^{4}+30 \ln \relax (x )-23 x^{2}+11 x \ln \relax (x )-4 x^{2} \ln \relax (x )-x^{3} \ln \relax (x )+90}{x -\ln \relax (x )-3}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^2+8*x-11)*ln(x)^2+(-6*x^3+2*x^2+70*x-63)*ln(x)+3*x^4-10*x^3-32*x^2+138*x-93)/(ln(x)^2+(6-2*x)*ln(x)+
x^2-6*x+9),x,method=_RETURNVERBOSE)

[Out]

x^3+4*x^2-11*x-3*x/(x-ln(x)-3)

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maxima [A]  time = 0.39, size = 41, normalized size = 1.78 \begin {gather*} \frac {x^{4} + x^{3} - 23 \, x^{2} - {\left (x^{3} + 4 \, x^{2} - 11 \, x\right )} \log \relax (x) + 30 \, x}{x - \log \relax (x) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+8*x-11)*log(x)^2+(-6*x^3+2*x^2+70*x-63)*log(x)+3*x^4-10*x^3-32*x^2+138*x-93)/(log(x)^2+(6-2*
x)*log(x)+x^2-6*x+9),x, algorithm="maxima")

[Out]

(x^4 + x^3 - 23*x^2 - (x^3 + 4*x^2 - 11*x)*log(x) + 30*x)/(x - log(x) - 3)

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mupad [B]  time = 2.16, size = 28, normalized size = 1.22 \begin {gather*} \frac {3\,\ln \relax (x)+9}{\ln \relax (x)-x+3}-11\,x+4\,x^2+x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((138*x + log(x)^2*(8*x + 3*x^2 - 11) - 32*x^2 - 10*x^3 + 3*x^4 + log(x)*(70*x + 2*x^2 - 6*x^3 - 63) - 93)/
(log(x)^2 - 6*x - log(x)*(2*x - 6) + x^2 + 9),x)

[Out]

(3*log(x) + 9)/(log(x) - x + 3) - 11*x + 4*x^2 + x^3

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sympy [A]  time = 0.12, size = 20, normalized size = 0.87 \begin {gather*} x^{3} + 4 x^{2} - 11 x + \frac {3 x}{- x + \log {\relax (x )} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**2+8*x-11)*ln(x)**2+(-6*x**3+2*x**2+70*x-63)*ln(x)+3*x**4-10*x**3-32*x**2+138*x-93)/(ln(x)**2+
(6-2*x)*ln(x)+x**2-6*x+9),x)

[Out]

x**3 + 4*x**2 - 11*x + 3*x/(-x + log(x) + 3)

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