3.15.53 \(\int \frac {-540+36 x^2+24 x^3+108 \log (5)+(360 x-72 x \log (5)) \log (x^2)+(-60 x^2+12 x^2 \log (5)) \log ^2(x^2)}{9 x^2-6 x^3 \log (x^2)+x^4 \log ^2(x^2)} \, dx\)

Optimal. Leaf size=30 \[ 5-\frac {6 \left (x+2 \left (-5+x+\log (5)+\frac {x}{-\frac {3}{x}+\log \left (x^2\right )}\right )\right )}{x} \]

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Rubi [F]  time = 0.56, 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 {-540+36 x^2+24 x^3+108 \log (5)+(360 x-72 x \log (5)) \log \left (x^2\right )+\left (-60 x^2+12 x^2 \log (5)\right ) \log ^2\left (x^2\right )}{9 x^2-6 x^3 \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-540 + 36*x^2 + 24*x^3 + 108*Log[5] + (360*x - 72*x*Log[5])*Log[x^2] + (-60*x^2 + 12*x^2*Log[5])*Log[x^2]
^2)/(9*x^2 - 6*x^3*Log[x^2] + x^4*Log[x^2]^2),x]

[Out]

(12*(5 - Log[5]))/x + 36*Defer[Int][(-3 + x*Log[x^2])^(-2), x] + 24*Defer[Int][x/(-3 + x*Log[x^2])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {12 \left (3 x^2+2 x^3+9 (-5+\log (5))-6 x (-5+\log (5)) \log \left (x^2\right )+x^2 (-5+\log (5)) \log ^2\left (x^2\right )\right )}{x^2 \left (3-x \log \left (x^2\right )\right )^2} \, dx\\ &=12 \int \frac {3 x^2+2 x^3+9 (-5+\log (5))-6 x (-5+\log (5)) \log \left (x^2\right )+x^2 (-5+\log (5)) \log ^2\left (x^2\right )}{x^2 \left (3-x \log \left (x^2\right )\right )^2} \, dx\\ &=12 \int \left (\frac {-5+\log (5)}{x^2}+\frac {3+2 x}{\left (-3+x \log \left (x^2\right )\right )^2}\right ) \, dx\\ &=\frac {12 (5-\log (5))}{x}+12 \int \frac {3+2 x}{\left (-3+x \log \left (x^2\right )\right )^2} \, dx\\ &=\frac {12 (5-\log (5))}{x}+12 \int \left (\frac {3}{\left (-3+x \log \left (x^2\right )\right )^2}+\frac {2 x}{\left (-3+x \log \left (x^2\right )\right )^2}\right ) \, dx\\ &=\frac {12 (5-\log (5))}{x}+24 \int \frac {x}{\left (-3+x \log \left (x^2\right )\right )^2} \, dx+36 \int \frac {1}{\left (-3+x \log \left (x^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 26, normalized size = 0.87 \begin {gather*} 12 \left (\frac {5-\log (5)}{x}-\frac {x}{-3+x \log \left (x^2\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-540 + 36*x^2 + 24*x^3 + 108*Log[5] + (360*x - 72*x*Log[5])*Log[x^2] + (-60*x^2 + 12*x^2*Log[5])*Lo
g[x^2]^2)/(9*x^2 - 6*x^3*Log[x^2] + x^4*Log[x^2]^2),x]

[Out]

12*((5 - Log[5])/x - x/(-3 + x*Log[x^2]))

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fricas [A]  time = 0.66, size = 38, normalized size = 1.27 \begin {gather*} -\frac {12 \, {\left (x^{2} + {\left (x \log \relax (5) - 5 \, x\right )} \log \left (x^{2}\right ) - 3 \, \log \relax (5) + 15\right )}}{x^{2} \log \left (x^{2}\right ) - 3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^2*log(5)-60*x^2)*log(x^2)^2+(-72*x*log(5)+360*x)*log(x^2)+108*log(5)+24*x^3+36*x^2-540)/(x^4*
log(x^2)^2-6*x^3*log(x^2)+9*x^2),x, algorithm="fricas")

[Out]

-12*(x^2 + (x*log(5) - 5*x)*log(x^2) - 3*log(5) + 15)/(x^2*log(x^2) - 3*x)

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giac [A]  time = 0.32, size = 23, normalized size = 0.77 \begin {gather*} -\frac {12 \, x}{x \log \left (x^{2}\right ) - 3} - \frac {12 \, {\left (\log \relax (5) - 5\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^2*log(5)-60*x^2)*log(x^2)^2+(-72*x*log(5)+360*x)*log(x^2)+108*log(5)+24*x^3+36*x^2-540)/(x^4*
log(x^2)^2-6*x^3*log(x^2)+9*x^2),x, algorithm="giac")

[Out]

-12*x/(x*log(x^2) - 3) - 12*(log(5) - 5)/x

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maple [A]  time = 0.08, size = 27, normalized size = 0.90

method	result	size
risch	\(\frac {60}{x}-\frac {12 \ln \relax (5)}{x}-\frac {12 x}{x \ln \left (x^{2}\right )-3}\)	\(27\)
default	\(-\frac {12 \ln \relax (5)}{x}+\frac {-180+60 x \ln \left (x^{2}\right )-12 x^{2}}{x \left (x \ln \left (x^{2}\right )-3\right )}\)	\(37\)
norman	\(\frac {\left (60-12 \ln \relax (5)\right ) x \ln \left (x^{2}\right )-12 x^{2}-180+36 \ln \relax (5)}{x \left (x \ln \left (x^{2}\right )-3\right )}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*x^2*ln(5)-60*x^2)*ln(x^2)^2+(-72*x*ln(5)+360*x)*ln(x^2)+108*ln(5)+24*x^3+36*x^2-540)/(x^4*ln(x^2)^2-6
*x^3*ln(x^2)+9*x^2),x,method=_RETURNVERBOSE)

[Out]

60/x-12*ln(5)/x-12*x/(x*ln(x^2)-3)

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maxima [A]  time = 0.70, size = 33, normalized size = 1.10 \begin {gather*} -\frac {12 \, {\left (2 \, x {\left (\log \relax (5) - 5\right )} \log \relax (x) + x^{2} - 3 \, \log \relax (5) + 15\right )}}{2 \, x^{2} \log \relax (x) - 3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^2*log(5)-60*x^2)*log(x^2)^2+(-72*x*log(5)+360*x)*log(x^2)+108*log(5)+24*x^3+36*x^2-540)/(x^4*
log(x^2)^2-6*x^3*log(x^2)+9*x^2),x, algorithm="maxima")

[Out]

-12*(2*x*(log(5) - 5)*log(x) + x^2 - 3*log(5) + 15)/(2*x^2*log(x) - 3*x)

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mupad [B]  time = 1.07, size = 47, normalized size = 1.57 \begin {gather*} -\frac {12\,x^3-x\,\left (36\,\ln \relax (5)-180\right )+x^2\,\ln \left (x^2\right )\,\left (12\,\ln \relax (5)-60\right )}{x^3\,\ln \left (x^2\right )-3\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((108*log(5) + log(x^2)*(360*x - 72*x*log(5)) + log(x^2)^2*(12*x^2*log(5) - 60*x^2) + 36*x^2 + 24*x^3 - 540
)/(9*x^2 - 6*x^3*log(x^2) + x^4*log(x^2)^2),x)

[Out]

-(12*x^3 - x*(36*log(5) - 180) + x^2*log(x^2)*(12*log(5) - 60))/(x^3*log(x^2) - 3*x^2)

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sympy [A]  time = 0.12, size = 20, normalized size = 0.67 \begin {gather*} - \frac {12 x}{x \log {\left (x^{2} \right )} - 3} - \frac {-60 + 12 \log {\relax (5 )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x**2*ln(5)-60*x**2)*ln(x**2)**2+(-72*x*ln(5)+360*x)*ln(x**2)+108*ln(5)+24*x**3+36*x**2-540)/(x*
*4*ln(x**2)**2-6*x**3*ln(x**2)+9*x**2),x)

[Out]

-12*x/(x*log(x**2) - 3) - (-60 + 12*log(5))/x

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