3.16.79 \(\int \frac {4 x^2+(2+2 x^2) \log (\frac {1}{1+x^2})}{(2+2 x^2+(1+x^2) \log (6)) \log ^2(\frac {1}{1+x^2})} \, dx\)

Optimal. Leaf size=19 \[ \frac {2 x}{(2+\log (6)) \log \left (\frac {1}{1+x^2}\right )} \]

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Rubi [F]  time = 0.46, 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 {4 x^2+\left (2+2 x^2\right ) \log \left (\frac {1}{1+x^2}\right )}{\left (2+2 x^2+\left (1+x^2\right ) \log (6)\right ) \log ^2\left (\frac {1}{1+x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*x^2 + (2 + 2*x^2)*Log[(1 + x^2)^(-1)])/((2 + 2*x^2 + (1 + x^2)*Log[6])*Log[(1 + x^2)^(-1)]^2),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 19, normalized size = 1.00 \begin {gather*} \frac {2 x}{(2+\log (6)) \log \left (\frac {1}{1+x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x^2 + (2 + 2*x^2)*Log[(1 + x^2)^(-1)])/((2 + 2*x^2 + (1 + x^2)*Log[6])*Log[(1 + x^2)^(-1)]^2),x]

[Out]

(2*x)/((2 + Log[6])*Log[(1 + x^2)^(-1)])

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fricas [A]  time = 0.82, size = 19, normalized size = 1.00 \begin {gather*} \frac {2 \, x}{{\left (\log \relax (6) + 2\right )} \log \left (\frac {1}{x^{2} + 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+2)*log(1/(x^2+1))+4*x^2)/((x^2+1)*log(6)+2*x^2+2)/log(1/(x^2+1))^2,x, algorithm="fricas")

[Out]

2*x/((log(6) + 2)*log(1/(x^2 + 1)))

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giac [A]  time = 0.27, size = 23, normalized size = 1.21 \begin {gather*} -\frac {2 \, x}{\log \relax (6) \log \left (x^{2} + 1\right ) + 2 \, \log \left (x^{2} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+2)*log(1/(x^2+1))+4*x^2)/((x^2+1)*log(6)+2*x^2+2)/log(1/(x^2+1))^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.26, size = 20, normalized size = 1.05

method	result	size
norman	\(\frac {2 x}{\left (\ln \relax (6)+2\right ) \ln \left (\frac {1}{x^{2}+1}\right )}\)	\(20\)
risch	\(\frac {2 x}{\left (\ln \relax (2)+\ln \relax (3)+2\right ) \ln \left (\frac {1}{x^{2}+1}\right )}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2+2)*ln(1/(x^2+1))+4*x^2)/((x^2+1)*ln(6)+2*x^2+2)/ln(1/(x^2+1))^2,x,method=_RETURNVERBOSE)

[Out]

2*x/(ln(6)+2)/ln(1/(x^2+1))

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maxima [A]  time = 1.19, size = 19, normalized size = 1.00 \begin {gather*} -\frac {2 \, x}{{\left (\log \relax (3) + \log \relax (2) + 2\right )} \log \left (x^{2} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+2)*log(1/(x^2+1))+4*x^2)/((x^2+1)*log(6)+2*x^2+2)/log(1/(x^2+1))^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.31, size = 17, normalized size = 0.89 \begin {gather*} -\frac {2\,x}{\ln \left (x^2+1\right )\,\left (\ln \relax (6)+2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2 + log(1/(x^2 + 1))*(2*x^2 + 2))/(log(1/(x^2 + 1))^2*(log(6)*(x^2 + 1) + 2*x^2 + 2)),x)

[Out]

-(2*x)/(log(x^2 + 1)*(log(6) + 2))

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sympy [A]  time = 0.12, size = 15, normalized size = 0.79 \begin {gather*} \frac {2 x}{\left (\log {\relax (6 )} + 2\right ) \log {\left (\frac {1}{x^{2} + 1} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2+2)*ln(1/(x**2+1))+4*x**2)/((x**2+1)*ln(6)+2*x**2+2)/ln(1/(x**2+1))**2,x)

[Out]

2*x/((log(6) + 2)*log(1/(x**2 + 1)))

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