3.30.23 \(\int \frac {10-20 \log (x)+5 \log (x) \log (\frac {5 x^2}{\log (x)})}{\log (x) \log ^3(\frac {5 x^2}{\log (x)})} \, dx\)

Optimal. Leaf size=15 \[ \frac {5 x}{\log ^2\left (\frac {5 x^2}{\log (x)}\right )} \]

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Rubi [F]  time = 0.30, 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 {10-20 \log (x)+5 \log (x) \log \left (\frac {5 x^2}{\log (x)}\right )}{\log (x) \log ^3\left (\frac {5 x^2}{\log (x)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(10 - 20*Log[x] + 5*Log[x]*Log[(5*x^2)/Log[x]])/(Log[x]*Log[(5*x^2)/Log[x]]^3),x]

[Out]

-20*Defer[Int][Log[(5*x^2)/Log[x]]^(-3), x] + 10*Defer[Int][1/(Log[x]*Log[(5*x^2)/Log[x]]^3), x] + 5*Defer[Int
][Log[(5*x^2)/Log[x]]^(-2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \left (2-4 \log (x)+\log (x) \log \left (\frac {5 x^2}{\log (x)}\right )\right )}{\log (x) \log ^3\left (\frac {5 x^2}{\log (x)}\right )} \, dx\\ &=5 \int \frac {2-4 \log (x)+\log (x) \log \left (\frac {5 x^2}{\log (x)}\right )}{\log (x) \log ^3\left (\frac {5 x^2}{\log (x)}\right )} \, dx\\ &=5 \int \left (-\frac {2 (-1+2 \log (x))}{\log (x) \log ^3\left (\frac {5 x^2}{\log (x)}\right )}+\frac {1}{\log ^2\left (\frac {5 x^2}{\log (x)}\right )}\right ) \, dx\\ &=5 \int \frac {1}{\log ^2\left (\frac {5 x^2}{\log (x)}\right )} \, dx-10 \int \frac {-1+2 \log (x)}{\log (x) \log ^3\left (\frac {5 x^2}{\log (x)}\right )} \, dx\\ &=5 \int \frac {1}{\log ^2\left (\frac {5 x^2}{\log (x)}\right )} \, dx-10 \int \left (\frac {2}{\log ^3\left (\frac {5 x^2}{\log (x)}\right )}-\frac {1}{\log (x) \log ^3\left (\frac {5 x^2}{\log (x)}\right )}\right ) \, dx\\ &=5 \int \frac {1}{\log ^2\left (\frac {5 x^2}{\log (x)}\right )} \, dx+10 \int \frac {1}{\log (x) \log ^3\left (\frac {5 x^2}{\log (x)}\right )} \, dx-20 \int \frac {1}{\log ^3\left (\frac {5 x^2}{\log (x)}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 15, normalized size = 1.00 \begin {gather*} \frac {5 x}{\log ^2\left (\frac {5 x^2}{\log (x)}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10 - 20*Log[x] + 5*Log[x]*Log[(5*x^2)/Log[x]])/(Log[x]*Log[(5*x^2)/Log[x]]^3),x]

[Out]

(5*x)/Log[(5*x^2)/Log[x]]^2

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fricas [A]  time = 0.68, size = 15, normalized size = 1.00 \begin {gather*} \frac {5 \, x}{\log \left (\frac {5 \, x^{2}}{\log \relax (x)}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*log(x)*log(5*x^2/log(x))-20*log(x)+10)/log(x)/log(5*x^2/log(x))^3,x, algorithm="fricas")

[Out]

5*x/log(5*x^2/log(x))^2

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giac [B]  time = 0.29, size = 102, normalized size = 6.80 \begin {gather*} \frac {5 \, {\left (2 \, x \log \relax (x) - x\right )}}{2 \, \log \relax (5)^{2} \log \relax (x) + 8 \, \log \relax (5) \log \relax (x)^{2} + 8 \, \log \relax (x)^{3} - 4 \, \log \relax (5) \log \relax (x) \log \left (\log \relax (x)\right ) - 8 \, \log \relax (x)^{2} \log \left (\log \relax (x)\right ) + 2 \, \log \relax (x) \log \left (\log \relax (x)\right )^{2} - \log \relax (5)^{2} - 4 \, \log \relax (5) \log \relax (x) - 4 \, \log \relax (x)^{2} + 2 \, \log \relax (5) \log \left (\log \relax (x)\right ) + 4 \, \log \relax (x) \log \left (\log \relax (x)\right ) - \log \left (\log \relax (x)\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*log(x)*log(5*x^2/log(x))-20*log(x)+10)/log(x)/log(5*x^2/log(x))^3,x, algorithm="giac")

[Out]

5*(2*x*log(x) - x)/(2*log(5)^2*log(x) + 8*log(5)*log(x)^2 + 8*log(x)^3 - 4*log(5)*log(x)*log(log(x)) - 8*log(x
)^2*log(log(x)) + 2*log(x)*log(log(x))^2 - log(5)^2 - 4*log(5)*log(x) - 4*log(x)^2 + 2*log(5)*log(log(x)) + 4*
log(x)*log(log(x)) - log(log(x))^2)

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maple [A]  time = 0.15, size = 16, normalized size = 1.07

method	result	size
norman	\(\frac {5 x}{\ln \left (\frac {5 x^{2}}{\ln \relax (x )}\right )^{2}}\)	\(16\)
risch	\(\frac {20 x}{\left (2 \ln \relax (5)+4 \ln \relax (x )-2 \ln \left (\ln \relax (x )\right )-i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \relax (x )}\right )+i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \relax (x )}\right )^{2}+i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \relax (x )}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i x^{2}}{\ln \relax (x )}\right )^{3}\right )^{2}}\)	\(165\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*ln(x)*ln(5*x^2/ln(x))-20*ln(x)+10)/ln(x)/ln(5*x^2/ln(x))^3,x,method=_RETURNVERBOSE)

[Out]

5*x/ln(5*x^2/ln(x))^2

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maxima [B]  time = 0.56, size = 39, normalized size = 2.60 \begin {gather*} \frac {5 \, x}{\log \relax (5)^{2} + 4 \, \log \relax (5) \log \relax (x) + 4 \, \log \relax (x)^{2} - 2 \, {\left (\log \relax (5) + 2 \, \log \relax (x)\right )} \log \left (\log \relax (x)\right ) + \log \left (\log \relax (x)\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*log(x)*log(5*x^2/log(x))-20*log(x)+10)/log(x)/log(5*x^2/log(x))^3,x, algorithm="maxima")

[Out]

5*x/(log(5)^2 + 4*log(5)*log(x) + 4*log(x)^2 - 2*(log(5) + 2*log(x))*log(log(x)) + log(log(x))^2)

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mupad [B]  time = 1.97, size = 15, normalized size = 1.00 \begin {gather*} \frac {5\,x}{{\ln \left (\frac {5\,x^2}{\ln \relax (x)}\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*log(x)*log((5*x^2)/log(x)) - 20*log(x) + 10)/(log(x)*log((5*x^2)/log(x))^3),x)

[Out]

(5*x)/log((5*x^2)/log(x))^2

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sympy [A]  time = 0.25, size = 14, normalized size = 0.93 \begin {gather*} \frac {5 x}{\log {\left (\frac {5 x^{2}}{\log {\relax (x )}} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*ln(x)*ln(5*x**2/ln(x))-20*ln(x)+10)/ln(x)/ln(5*x**2/ln(x))**3,x)

[Out]

5*x/log(5*x**2/log(x))**2

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