3.23.28 \(\int \frac {16-12 x+e x-4 x^2+(12 x-e x+8 x^2) \log (x)}{80 x \log ^2(x)} \, dx\) [2228]

Optimal. Leaf size=24 \[ \frac {x \left (2-\frac {e}{4}+\frac {-4+x}{x}+x\right )}{20 \log (x)} \]

[Out]

1/20*x*(2-1/4*exp(1)+x+(x-4)/x)/ln(x)

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Rubi [F]
time = 0.14, 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 {16-12 x+e x-4 x^2+\left (12 x-e x+8 x^2\right ) \log (x)}{80 x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(16 - 12*x + E*x - 4*x^2 + (12*x - E*x + 8*x^2)*Log[x])/(80*x*Log[x]^2),x]

[Out]

ExpIntegralEi[2*Log[x]]/10 + ((12 - E)*LogIntegral[x])/80 + Defer[Int][(16 - (12 - E)*x - 4*x^2)/(x*Log[x]^2),
x]/80

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {16+(-12+e) x-4 x^2+\left (12 x-e x+8 x^2\right ) \log (x)}{80 x \log ^2(x)} \, dx\\ &=\frac {1}{80} \int \frac {16+(-12+e) x-4 x^2+\left (12 x-e x+8 x^2\right ) \log (x)}{x \log ^2(x)} \, dx\\ &=\frac {1}{80} \int \left (\frac {16-(12-e) x-4 x^2}{x \log ^2(x)}+\frac {12-e+8 x}{\log (x)}\right ) \, dx\\ &=\frac {1}{80} \int \frac {16-(12-e) x-4 x^2}{x \log ^2(x)} \, dx+\frac {1}{80} \int \frac {12-e+8 x}{\log (x)} \, dx\\ &=\frac {1}{80} \int \left (\frac {12 \left (1-\frac {e}{12}\right )}{\log (x)}+\frac {8 x}{\log (x)}\right ) \, dx+\frac {1}{80} \int \frac {16-(12-e) x-4 x^2}{x \log ^2(x)} \, dx\\ &=\frac {1}{80} \int \frac {16-(12-e) x-4 x^2}{x \log ^2(x)} \, dx+\frac {1}{10} \int \frac {x}{\log (x)} \, dx+\frac {1}{80} (12-e) \int \frac {1}{\log (x)} \, dx\\ &=\frac {1}{80} (12-e) \text {li}(x)+\frac {1}{80} \int \frac {16-(12-e) x-4 x^2}{x \log ^2(x)} \, dx+\frac {1}{10} \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=\frac {1}{10} \text {Ei}(2 \log (x))+\frac {1}{80} (12-e) \text {li}(x)+\frac {1}{80} \int \frac {16-(12-e) x-4 x^2}{x \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.04, size = 22, normalized size = 0.92 \begin {gather*} \frac {-16+12 x-e x+4 x^2}{80 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16 - 12*x + E*x - 4*x^2 + (12*x - E*x + 8*x^2)*Log[x])/(80*x*Log[x]^2),x]

[Out]

(-16 + 12*x - E*x + 4*x^2)/(80*Log[x])

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.04, size = 54, normalized size = 2.25

method	result	size
norman	\(\frac {-\frac {1}{5}+\left (-\frac {{\mathrm e}}{80}+\frac {3}{20}\right ) x +\frac {x^{2}}{20}}{\ln \left (x \right )}\)	\(21\)
risch	\(-\frac {x \,{\mathrm e}-4 x^{2}-12 x +16}{80 \ln \left (x \right )}\)	\(21\)
default	\(\frac {{\mathrm e} \expIntegral \left (1, -\ln \left (x \right )\right )}{80}+\frac {{\mathrm e} \left (-\frac {x}{\ln \left (x \right )}-\expIntegral \left (1, -\ln \left (x \right )\right )\right )}{80}+\frac {x^{2}}{20 \ln \left (x \right )}+\frac {3 x}{20 \ln \left (x \right )}-\frac {1}{5 \ln \left (x \right )}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/80*((-x*exp(1)+8*x^2+12*x)*ln(x)+x*exp(1)-4*x^2-12*x+16)/x/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

1/80*exp(1)*Ei(1,-ln(x))+1/80*exp(1)*(-x/ln(x)-Ei(1,-ln(x)))+1/20*x^2/ln(x)+3/20*x/ln(x)-1/5/ln(x)

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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.30, size = 52, normalized size = 2.17 \begin {gather*} -\frac {1}{80} \, {\rm Ei}\left (\log \left (x\right )\right ) e + \frac {1}{80} \, e \Gamma \left (-1, -\log \left (x\right )\right ) - \frac {1}{5 \, \log \left (x\right )} + \frac {1}{10} \, {\rm Ei}\left (2 \, \log \left (x\right )\right ) + \frac {3}{20} \, {\rm Ei}\left (\log \left (x\right )\right ) - \frac {3}{20} \, \Gamma \left (-1, -\log \left (x\right )\right ) - \frac {1}{10} \, \Gamma \left (-1, -2 \, \log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/80*((-x*exp(1)+8*x^2+12*x)*log(x)+x*exp(1)-4*x^2-12*x+16)/x/log(x)^2,x, algorithm="maxima")

[Out]

-1/80*Ei(log(x))*e + 1/80*e*gamma(-1, -log(x)) - 1/5/log(x) + 1/10*Ei(2*log(x)) + 3/20*Ei(log(x)) - 3/20*gamma
(-1, -log(x)) - 1/10*gamma(-1, -2*log(x))

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Fricas [A]
time = 0.39, size = 21, normalized size = 0.88 \begin {gather*} \frac {4 \, x^{2} - x e + 12 \, x - 16}{80 \, \log \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/80*((-x*exp(1)+8*x^2+12*x)*log(x)+x*exp(1)-4*x^2-12*x+16)/x/log(x)^2,x, algorithm="fricas")

[Out]

1/80*(4*x^2 - x*e + 12*x - 16)/log(x)

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Sympy [A]
time = 0.03, size = 19, normalized size = 0.79 \begin {gather*} \frac {4 x^{2} - e x + 12 x - 16}{80 \log {\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/80*((-x*exp(1)+8*x**2+12*x)*ln(x)+x*exp(1)-4*x**2-12*x+16)/x/ln(x)**2,x)

[Out]

(4*x**2 - E*x + 12*x - 16)/(80*log(x))

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Giac [A]
time = 0.40, size = 21, normalized size = 0.88 \begin {gather*} \frac {4 \, x^{2} - x e + 12 \, x - 16}{80 \, \log \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/80*((-x*exp(1)+8*x^2+12*x)*log(x)+x*exp(1)-4*x^2-12*x+16)/x/log(x)^2,x, algorithm="giac")

[Out]

1/80*(4*x^2 - x*e + 12*x - 16)/log(x)

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Mupad [B]
time = 1.27, size = 28, normalized size = 1.17 \begin {gather*} -\frac {-4\,x^4+\left (\mathrm {e}-12\right )\,x^3+16\,x^2}{80\,x^2\,\ln \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x*exp(1))/80 - (3*x)/20 - x^2/20 + (log(x)*(12*x - x*exp(1) + 8*x^2))/80 + 1/5)/(x*log(x)^2),x)

[Out]

-(16*x^2 - 4*x^4 + x^3*(exp(1) - 12))/(80*x^2*log(x))

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