3.84.87 \(\int \frac {6+x^2+24 x^3+(3-2 x-48 x^2) \log (\frac {1}{x^2})+(1+24 x) \log ^2(\frac {1}{x^2})}{3 x^2-6 x \log (\frac {1}{x^2})+3 \log ^2(\frac {1}{x^2})} \, dx\)

Optimal. Leaf size=25 \[ -2+x-2 \left (\frac {1}{3}-2 x\right ) x+\frac {x}{-x+\log \left (\frac {1}{x^2}\right )} \]

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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 {6+x^2+24 x^3+\left (3-2 x-48 x^2\right ) \log \left (\frac {1}{x^2}\right )+(1+24 x) \log ^2\left (\frac {1}{x^2}\right )}{3 x^2-6 x \log \left (\frac {1}{x^2}\right )+3 \log ^2\left (\frac {1}{x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(6 + x^2 + 24*x^3 + (3 - 2*x - 48*x^2)*Log[x^(-2)] + (1 + 24*x)*Log[x^(-2)]^2)/(3*x^2 - 6*x*Log[x^(-2)] +
3*Log[x^(-2)]^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6+x^2+24 x^3+\left (3-2 x-48 x^2\right ) \log \left (\frac {1}{x^2}\right )+(1+24 x) \log ^2\left (\frac {1}{x^2}\right )}{3 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {6+x^2+24 x^3+\left (3-2 x-48 x^2\right ) \log \left (\frac {1}{x^2}\right )+(1+24 x) \log ^2\left (\frac {1}{x^2}\right )}{\left (x-\log \left (\frac {1}{x^2}\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \left (1+24 x+\frac {3 (2+x)}{\left (x-\log \left (\frac {1}{x^2}\right )\right )^2}-\frac {3}{x-\log \left (\frac {1}{x^2}\right )}\right ) \, dx\\ &=\frac {x}{3}+4 x^2+\int \frac {2+x}{\left (x-\log \left (\frac {1}{x^2}\right )\right )^2} \, dx-\int \frac {1}{x-\log \left (\frac {1}{x^2}\right )} \, dx\\ &=\frac {x}{3}+4 x^2+\int \left (\frac {2}{\left (x-\log \left (\frac {1}{x^2}\right )\right )^2}+\frac {x}{\left (x-\log \left (\frac {1}{x^2}\right )\right )^2}\right ) \, dx-\int \frac {1}{x-\log \left (\frac {1}{x^2}\right )} \, dx\\ &=\frac {x}{3}+4 x^2+2 \int \frac {1}{\left (x-\log \left (\frac {1}{x^2}\right )\right )^2} \, dx+\int \frac {x}{\left (x-\log \left (\frac {1}{x^2}\right )\right )^2} \, dx-\int \frac {1}{x-\log \left (\frac {1}{x^2}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 24, normalized size = 0.96 \begin {gather*} \frac {1}{3} \left (x+12 x^2+\frac {3 x}{-x+\log \left (\frac {1}{x^2}\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 + x^2 + 24*x^3 + (3 - 2*x - 48*x^2)*Log[x^(-2)] + (1 + 24*x)*Log[x^(-2)]^2)/(3*x^2 - 6*x*Log[x^(-
2)] + 3*Log[x^(-2)]^2),x]

[Out]

(x + 12*x^2 + (3*x)/(-x + Log[x^(-2)]))/3

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fricas [A]  time = 0.56, size = 37, normalized size = 1.48 \begin {gather*} \frac {12 \, x^{3} + x^{2} - {\left (12 \, x^{2} + x\right )} \log \left (\frac {1}{x^{2}}\right ) - 3 \, x}{3 \, {\left (x - \log \left (\frac {1}{x^{2}}\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+1)*log(1/x^2)^2+(-48*x^2-2*x+3)*log(1/x^2)+24*x^3+x^2+6)/(3*log(1/x^2)^2-6*x*log(1/x^2)+3*x^2
),x, algorithm="fricas")

[Out]

1/3*(12*x^3 + x^2 - (12*x^2 + x)*log(x^(-2)) - 3*x)/(x - log(x^(-2)))

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giac [A]  time = 0.22, size = 20, normalized size = 0.80 \begin {gather*} 4 \, x^{2} + \frac {1}{3} \, x - \frac {x}{x + \log \left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+1)*log(1/x^2)^2+(-48*x^2-2*x+3)*log(1/x^2)+24*x^3+x^2+6)/(3*log(1/x^2)^2-6*x*log(1/x^2)+3*x^2
),x, algorithm="giac")

[Out]

4*x^2 + 1/3*x - x/(x + log(x^2))

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maple [A]  time = 0.13, size = 23, normalized size = 0.92

method	result	size
risch	\(4 x^{2}+\frac {x}{3}-\frac {x}{x -\ln \left (\frac {1}{x^{2}}\right )}\)	\(23\)
norman	\(\frac {-x +\frac {x^{2}}{3}+4 x^{3}-\frac {x \ln \left (\frac {1}{x^{2}}\right )}{3}-4 x^{2} \ln \left (\frac {1}{x^{2}}\right )}{x -\ln \left (\frac {1}{x^{2}}\right )}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((24*x+1)*ln(1/x^2)^2+(-48*x^2-2*x+3)*ln(1/x^2)+24*x^3+x^2+6)/(3*ln(1/x^2)^2-6*x*ln(1/x^2)+3*x^2),x,method
=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.38, size = 33, normalized size = 1.32 \begin {gather*} \frac {12 \, x^{3} + x^{2} + 2 \, {\left (12 \, x^{2} + x\right )} \log \relax (x) - 3 \, x}{3 \, {\left (x + 2 \, \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+1)*log(1/x^2)^2+(-48*x^2-2*x+3)*log(1/x^2)+24*x^3+x^2+6)/(3*log(1/x^2)^2-6*x*log(1/x^2)+3*x^2
),x, algorithm="maxima")

[Out]

1/3*(12*x^3 + x^2 + 2*(12*x^2 + x)*log(x) - 3*x)/(x + 2*log(x))

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mupad [B]  time = 5.47, size = 25, normalized size = 1.00 \begin {gather*} \frac {x}{3}-\frac {\ln \left (\frac {1}{x^2}\right )}{x-\ln \left (\frac {1}{x^2}\right )}+4\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(1/x^2)^2*(24*x + 1) - log(1/x^2)*(2*x + 48*x^2 - 3) + x^2 + 24*x^3 + 6)/(3*log(1/x^2)^2 - 6*x*log(1/x
^2) + 3*x^2),x)

[Out]

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

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sympy [A]  time = 0.12, size = 17, normalized size = 0.68 \begin {gather*} 4 x^{2} + \frac {x}{3} + \frac {x}{- x + \log {\left (\frac {1}{x^{2}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+1)*ln(1/x**2)**2+(-48*x**2-2*x+3)*ln(1/x**2)+24*x**3+x**2+6)/(3*ln(1/x**2)**2-6*x*ln(1/x**2)+
3*x**2),x)

[Out]

4*x**2 + x/3 + x/(-x + log(x**(-2)))

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