3.42.55 \(\int \frac {10 x^2+10 x^2 \log (x)+(-8-10 x) \log ^2(x)+8 \log (x) \log (e^{.-\frac {2}{5}/x} x^2)+10 x \log ^2(e^{.-\frac {2}{5}/x} x^2)}{125 x^2} \, dx\)

Optimal. Leaf size=29 \[ \frac {2}{25} \log (x) \left (x+\left (-\log (x)+\log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )\right )^2\right ) \]

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Rubi [F]  time = 0.26, 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 x^2+10 x^2 \log (x)+(-8-10 x) \log ^2(x)+8 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )+10 x \log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(10*x^2 + 10*x^2*Log[x] + (-8 - 10*x)*Log[x]^2 + 8*Log[x]*Log[x^2/E^(2/(5*x))] + 10*x*Log[x^2/E^(2/(5*x))]
^2)/(125*x^2),x]

[Out]

-4/(625*x^2) - (8*(1 + 5*x)^2)/(625*x^2) + (8*Log[x])/25 + (16*Log[x])/(125*x) + (2*x*Log[x])/25 - (8*(1 + 5*x
)^2*Log[x])/(625*x^2) + (8*Log[x]^2)/(125*x) - (2*Log[x]^3)/75 - (8*Log[x^2/E^(2/(5*x))])/(125*x) - (8*Log[x]*
Log[x^2/E^(2/(5*x))])/(125*x) + (2*Defer[Int][Log[x^2/E^(2/(5*x))]^2/x, x])/25

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{125} \int \frac {10 x^2+10 x^2 \log (x)+(-8-10 x) \log ^2(x)+8 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )+10 x \log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{x^2} \, dx\\ &=\frac {1}{125} \int \left (-\frac {2 \left (-5 x^2-5 x^2 \log (x)+4 \log ^2(x)+5 x \log ^2(x)\right )}{x^2}+\frac {8 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{x^2}+\frac {10 \log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{x}\right ) \, dx\\ &=-\left (\frac {2}{125} \int \frac {-5 x^2-5 x^2 \log (x)+4 \log ^2(x)+5 x \log ^2(x)}{x^2} \, dx\right )+\frac {8}{125} \int \frac {\log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{x^2} \, dx+\frac {2}{25} \int \frac {\log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{x} \, dx\\ &=-\frac {8 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x}-\frac {2}{125} \int \left (-5-5 \log (x)+\frac {(4+5 x) \log ^2(x)}{x^2}\right ) \, dx-\frac {8}{125} \int \frac {2 (-1-5 x) \log (x)}{5 x^3} \, dx+\frac {8}{125} \int \frac {\log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{x^2} \, dx+\frac {2}{25} \int \frac {\log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{x} \, dx\\ &=\frac {2 x}{25}-\frac {8 \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x}-\frac {8 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x}-\frac {2}{125} \int \frac {(4+5 x) \log ^2(x)}{x^2} \, dx-\frac {16}{625} \int \frac {(-1-5 x) \log (x)}{x^3} \, dx+\frac {8}{125} \int \frac {2 (1+5 x)}{5 x^3} \, dx+\frac {2}{25} \int \log (x) \, dx+\frac {2}{25} \int \frac {\log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{x} \, dx\\ &=\frac {2}{25} x \log (x)-\frac {8 (1+5 x)^2 \log (x)}{625 x^2}-\frac {8 \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x}-\frac {8 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x}-\frac {2}{125} \int \left (\frac {4 \log ^2(x)}{x^2}+\frac {5 \log ^2(x)}{x}\right ) \, dx+\frac {16}{625} \int \frac {1+5 x}{x^3} \, dx+\frac {16}{625} \int \frac {(1+5 x)^2}{2 x^3} \, dx+\frac {2}{25} \int \frac {\log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{x} \, dx\\ &=-\frac {8 (1+5 x)^2}{625 x^2}+\frac {2}{25} x \log (x)-\frac {8 (1+5 x)^2 \log (x)}{625 x^2}-\frac {8 \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x}-\frac {8 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x}+\frac {8}{625} \int \frac {(1+5 x)^2}{x^3} \, dx-\frac {8}{125} \int \frac {\log ^2(x)}{x^2} \, dx-\frac {2}{25} \int \frac {\log ^2(x)}{x} \, dx+\frac {2}{25} \int \frac {\log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{x} \, dx\\ &=-\frac {8 (1+5 x)^2}{625 x^2}+\frac {2}{25} x \log (x)-\frac {8 (1+5 x)^2 \log (x)}{625 x^2}+\frac {8 \log ^2(x)}{125 x}-\frac {8 \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x}-\frac {8 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x}+\frac {8}{625} \int \left (\frac {1}{x^3}+\frac {10}{x^2}+\frac {25}{x}\right ) \, dx+\frac {2}{25} \int \frac {\log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{x} \, dx-\frac {2}{25} \operatorname {Subst}\left (\int x^2 \, dx,x,\log (x)\right )-\frac {16}{125} \int \frac {\log (x)}{x^2} \, dx\\ &=-\frac {4}{625 x^2}-\frac {8 (1+5 x)^2}{625 x^2}+\frac {8 \log (x)}{25}+\frac {16 \log (x)}{125 x}+\frac {2}{25} x \log (x)-\frac {8 (1+5 x)^2 \log (x)}{625 x^2}+\frac {8 \log ^2(x)}{125 x}-\frac {2 \log ^3(x)}{75}-\frac {8 \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x}-\frac {8 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x}+\frac {2}{25} \int \frac {\log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.08, size = 114, normalized size = 3.93 \begin {gather*} \frac {24 \log ^2(x)-10 x \log ^3(x)+\log ^2\left (x^2\right ) \left (6-10 x \log \left (x^2\right )+15 x \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )\right )+6 \log (x) \left (5 x \log ^2\left (x^2\right )-2 \log \left (x^2\right ) \left (2+5 x \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )\right )+5 x \left (x+\log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )\right )\right )}{375 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10*x^2 + 10*x^2*Log[x] + (-8 - 10*x)*Log[x]^2 + 8*Log[x]*Log[x^2/E^(2/(5*x))] + 10*x*Log[x^2/E^(2/(
5*x))]^2)/(125*x^2),x]

[Out]

(24*Log[x]^2 - 10*x*Log[x]^3 + Log[x^2]^2*(6 - 10*x*Log[x^2] + 15*x*Log[x^2/E^(2/(5*x))]) + 6*Log[x]*(5*x*Log[
x^2]^2 - 2*Log[x^2]*(2 + 5*x*Log[x^2/E^(2/(5*x))]) + 5*x*(x + Log[x^2/E^(2/(5*x))]^2)))/(375*x)

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fricas [A]  time = 1.12, size = 32, normalized size = 1.10 \begin {gather*} \frac {2 \, {\left (25 \, x^{2} \log \relax (x)^{3} - 20 \, x \log \relax (x)^{2} + {\left (25 \, x^{3} + 4\right )} \log \relax (x)\right )}}{625 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/125*(10*x*log(x^2/exp(2/5/x))^2+8*log(x)*log(x^2/exp(2/5/x))+(-10*x-8)*log(x)^2+10*x^2*log(x)+10*x
^2)/x^2,x, algorithm="fricas")

[Out]

2/625*(25*x^2*log(x)^3 - 20*x*log(x)^2 + (25*x^3 + 4)*log(x))/x^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (5 \, x \log \left (x^{2} e^{\left (-\frac {2}{5 \, x}\right )}\right )^{2} + 5 \, x^{2} \log \relax (x) - {\left (5 \, x + 4\right )} \log \relax (x)^{2} + 5 \, x^{2} + 4 \, \log \left (x^{2} e^{\left (-\frac {2}{5 \, x}\right )}\right ) \log \relax (x)\right )}}{125 \, x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/125*(10*x*log(x^2/exp(2/5/x))^2+8*log(x)*log(x^2/exp(2/5/x))+(-10*x-8)*log(x)^2+10*x^2*log(x)+10*x
^2)/x^2,x, algorithm="giac")

[Out]

integrate(2/125*(5*x*log(x^2*e^(-2/5/x))^2 + 5*x^2*log(x) - (5*x + 4)*log(x)^2 + 5*x^2 + 4*log(x^2*e^(-2/5/x))
*log(x))/x^2, x)

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maple [B]  time = 0.21, size = 152, normalized size = 5.24

method	result	size
default	\(\frac {2 x \ln \relax (x )}{25}-\frac {8 \ln \relax (x )^{2}}{125 x}+\frac {2 \left (\ln \left (x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )-2 \ln \relax (x )+\ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )\right )^{2} \ln \relax (x )}{25}+\frac {2 \ln \relax (x ) \ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )^{2}}{25}-\frac {4 \left (\ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )-\frac {2}{5 x}\right ) \ln \relax (x )^{2}}{25}+\frac {4 \left (\ln \left (x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )-2 \ln \relax (x )+\ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )\right ) \ln \relax (x )^{2}}{25}+\frac {2 \ln \relax (x )^{3}}{25}-\frac {4 \left (\ln \left (x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )-2 \ln \relax (x )+\ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )\right ) \ln \relax (x ) \ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )}{25}\)	\(152\)
risch	\(\frac {2 \ln \relax (x )^{3}}{25}+\frac {2 x \ln \relax (x )}{25}+\frac {2 \pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{5}}{25}+\frac {\pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right )^{4} \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{2}}{25}-\frac {\pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i {\mathrm e}^{-\frac {2}{5 x}}\right )^{2} \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{4}}{50}+\frac {\pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i {\mathrm e}^{-\frac {2}{5 x}}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{5}}{25}-\frac {2 i \pi \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{3} \ln \relax (x )^{2}}{25}+\frac {\pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{5}}{25}-\frac {\pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{3}}{25}-\frac {\pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{4}}{50}-\frac {2 i \pi \mathrm {csgn}\left (i x^{2}\right )^{3} \ln \relax (x )^{2}}{25}-\frac {\pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x \right )^{4} \mathrm {csgn}\left (i x^{2}\right )^{2}}{50}+\frac {2 \pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{3}}{25}-\frac {3 \pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{4}}{25}-\frac {\pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right )^{6}}{50}-\frac {\pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{6}}{50}-\frac {2 i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-\frac {2}{5 x}}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right ) \ln \relax (x )^{2}}{25}+\frac {2 \pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i {\mathrm e}^{-\frac {2}{5 x}}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )}{25}-\frac {2 \pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-\frac {2}{5 x}}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{2}}{25}-\frac {\pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-\frac {2}{5 x}}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )}{25}+\frac {\pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-\frac {2}{5 x}}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{2}}{25}+\frac {\left (10 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-20 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+10 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+10 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-\frac {2}{5 x}}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )-10 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{2}-10 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-\frac {2}{5 x}}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{2}+10 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{3}-20 \ln \relax (x )^{2}\right ) \ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )}{125}+\frac {2 \ln \relax (x ) \ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )^{2}}{25}+\frac {\pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-\frac {2}{5 x}}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{3}}{25}+\frac {\pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-\frac {2}{5 x}}\right )^{2} \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{3}}{25}-\frac {2 i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) \ln \relax (x )^{2}}{25}+\frac {4 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} \ln \relax (x )^{2}}{25}+\frac {\pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i {\mathrm e}^{-\frac {2}{5 x}}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{2}}{25}-\frac {\pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-\frac {2}{5 x}}\right )^{2} \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{2}}{50}-\frac {\pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{3}}{25}-\frac {2 \pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{2}}{25}+\frac {2 \pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{3}}{25}-\frac {\pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right )^{4} \mathrm {csgn}\left (i {\mathrm e}^{-\frac {2}{5 x}}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )}{25}-\frac {2 \pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-\frac {2}{5 x}}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{4}}{25}+\frac {\pi ^{2} \ln \relax (x ) \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{2}}{25}+\frac {2 i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{2} \ln \relax (x )^{2}}{25}+\frac {2 i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-\frac {2}{5 x}}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )^{2} \ln \relax (x )^{2}}{25}\)	\(1286\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/125*(10*x*ln(x^2/exp(2/5/x))^2+8*ln(x)*ln(x^2/exp(2/5/x))+(-10*x-8)*ln(x)^2+10*x^2*ln(x)+10*x^2)/x^2,x,m
ethod=_RETURNVERBOSE)

[Out]

2/25*x*ln(x)-8/125*ln(x)^2/x+2/25*(ln(x^2/exp(2/5/x))-2*ln(x)+ln(exp(2/5/x)))^2*ln(x)+2/25*ln(x)*ln(exp(2/5/x)
)^2-4/25*(ln(exp(2/5/x))-2/5/x)*ln(x)^2+4/25*(ln(x^2/exp(2/5/x))-2*ln(x)+ln(exp(2/5/x)))*ln(x)^2+2/25*ln(x)^3-
4/25*(ln(x^2/exp(2/5/x))-2*ln(x)+ln(exp(2/5/x)))*ln(x)*ln(exp(2/5/x))

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maxima [B]  time = 0.37, size = 96, normalized size = 3.31 \begin {gather*} -\frac {2}{75} \, \log \relax (x)^{3} + \frac {2}{25} \, x \log \relax (x) - \frac {8}{625} \, {\left (\frac {5 \, \log \left (x^{2} e^{\left (-\frac {2}{5 \, x}\right )}\right )}{x} + \frac {10 \, x + 1}{x^{2}}\right )} \log \relax (x) + \frac {8 \, {\left (\log \relax (x)^{2} + 2 \, \log \relax (x) + 2\right )}}{125 \, x} + \frac {4 \, {\left (50 \, x^{2} \log \relax (x)^{3} + 60 \, x \log \relax (x) + 60 \, x - 3\right )}}{1875 \, x^{2}} - \frac {4 \, {\left (20 \, x \log \relax (x) + 40 \, x - 1\right )}}{625 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/125*(10*x*log(x^2/exp(2/5/x))^2+8*log(x)*log(x^2/exp(2/5/x))+(-10*x-8)*log(x)^2+10*x^2*log(x)+10*x
^2)/x^2,x, algorithm="maxima")

[Out]

-2/75*log(x)^3 + 2/25*x*log(x) - 8/625*(5*log(x^2*e^(-2/5/x))/x + (10*x + 1)/x^2)*log(x) + 8/125*(log(x)^2 + 2
*log(x) + 2)/x + 4/1875*(50*x^2*log(x)^3 + 60*x*log(x) + 60*x - 3)/x^2 - 4/625*(20*x*log(x) + 40*x - 1)/x^2

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mupad [B]  time = 3.16, size = 57, normalized size = 1.97 \begin {gather*} \frac {2\,\ln \relax (x)\,\left (25\,x^3+25\,x^2\,{\ln \left (x^2\right )}^2-50\,x^2\,\ln \left (x^2\right )\,\ln \relax (x)+25\,x^2\,{\ln \relax (x)}^2-20\,x\,\ln \left (x^2\right )+20\,x\,\ln \relax (x)+4\right )}{625\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2*log(x))/25 + (2*x*log(x^2*exp(-2/(5*x)))^2)/25 + (8*log(x^2*exp(-2/(5*x)))*log(x))/125 + (2*x^2)/2
5 - (log(x)^2*(10*x + 8))/125)/x^2,x)

[Out]

(2*log(x)*(25*x^2*log(x)^2 - 20*x*log(x^2) + 20*x*log(x) + 25*x^3 + 25*x^2*log(x^2)^2 - 50*x^2*log(x^2)*log(x)
+ 4))/(625*x^2)

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sympy [A]  time = 0.33, size = 32, normalized size = 1.10 \begin {gather*} \frac {2 \log {\relax (x )}^{3}}{25} - \frac {8 \log {\relax (x )}^{2}}{125 x} + \frac {\left (50 x^{3} + 8\right ) \log {\relax (x )}}{625 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/125*(10*x*ln(x**2/exp(2/5/x))**2+8*ln(x)*ln(x**2/exp(2/5/x))+(-10*x-8)*ln(x)**2+10*x**2*ln(x)+10*x
**2)/x**2,x)

[Out]

2*log(x)**3/25 - 8*log(x)**2/(125*x) + (50*x**3 + 8)*log(x)/(625*x**2)

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