3.55.75 \(\int \frac {2 x^2+(-20-5 x^2) \log (4+x^2) \log (\log (4+x^2))}{(4 x^6+x^8) \log (4+x^2)} \, dx\)

Optimal. Leaf size=13 \[ -2+\frac {\log \left (\log \left (4+x^2\right )\right )}{x^5} \]

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Rubi [F]  time = 0.33, 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+\left (-20-5 x^2\right ) \log \left (4+x^2\right ) \log \left (\log \left (4+x^2\right )\right )}{\left (4 x^6+x^8\right ) \log \left (4+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*x^2 + (-20 - 5*x^2)*Log[4 + x^2]*Log[Log[4 + x^2]])/((4*x^6 + x^8)*Log[4 + x^2]),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x^2+\left (-20-5 x^2\right ) \log \left (4+x^2\right ) \log \left (\log \left (4+x^2\right )\right )}{x^6 \left (4+x^2\right ) \log \left (4+x^2\right )} \, dx\\ &=\int \left (\frac {2}{x^4 \left (4+x^2\right ) \log \left (4+x^2\right )}-\frac {5 \log \left (\log \left (4+x^2\right )\right )}{x^6}\right ) \, dx\\ &=2 \int \frac {1}{x^4 \left (4+x^2\right ) \log \left (4+x^2\right )} \, dx-5 \int \frac {\log \left (\log \left (4+x^2\right )\right )}{x^6} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.77, size = 11, normalized size = 0.85 \begin {gather*} \frac {\log \left (\log \left (4+x^2\right )\right )}{x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^2 + (-20 - 5*x^2)*Log[4 + x^2]*Log[Log[4 + x^2]])/((4*x^6 + x^8)*Log[4 + x^2]),x]

[Out]

Log[Log[4 + x^2]]/x^5

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fricas [A]  time = 0.63, size = 11, normalized size = 0.85 \begin {gather*} \frac {\log \left (\log \left (x^{2} + 4\right )\right )}{x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^2-20)*log(x^2+4)*log(log(x^2+4))+2*x^2)/(x^8+4*x^6)/log(x^2+4),x, algorithm="fricas")

[Out]

log(log(x^2 + 4))/x^5

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giac [A]  time = 0.16, size = 11, normalized size = 0.85 \begin {gather*} \frac {\log \left (\log \left (x^{2} + 4\right )\right )}{x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^2-20)*log(x^2+4)*log(log(x^2+4))+2*x^2)/(x^8+4*x^6)/log(x^2+4),x, algorithm="giac")

[Out]

log(log(x^2 + 4))/x^5

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maple [A]  time = 0.20, size = 12, normalized size = 0.92

method	result	size
risch	\(\frac {\ln \left (\ln \left (x^{2}+4\right )\right )}{x^{5}}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-5*x^2-20)*ln(x^2+4)*ln(ln(x^2+4))+2*x^2)/(x^8+4*x^6)/ln(x^2+4),x,method=_RETURNVERBOSE)

[Out]

ln(ln(x^2+4))/x^5

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maxima [A]  time = 0.41, size = 11, normalized size = 0.85 \begin {gather*} \frac {\log \left (\log \left (x^{2} + 4\right )\right )}{x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^2-20)*log(x^2+4)*log(log(x^2+4))+2*x^2)/(x^8+4*x^6)/log(x^2+4),x, algorithm="maxima")

[Out]

log(log(x^2 + 4))/x^5

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mupad [B]  time = 3.81, size = 11, normalized size = 0.85 \begin {gather*} \frac {\ln \left (\ln \left (x^2+4\right )\right )}{x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2 - log(log(x^2 + 4))*log(x^2 + 4)*(5*x^2 + 20))/(log(x^2 + 4)*(4*x^6 + x^8)),x)

[Out]

log(log(x^2 + 4))/x^5

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sympy [A]  time = 0.32, size = 10, normalized size = 0.77 \begin {gather*} \frac {\log {\left (\log {\left (x^{2} + 4 \right )} \right )}}{x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x**2-20)*ln(x**2+4)*ln(ln(x**2+4))+2*x**2)/(x**8+4*x**6)/ln(x**2+4),x)

[Out]

log(log(x**2 + 4))/x**5

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