3.82.49 \(\int \frac {1}{45} e^{-4/x} (e^{4/x} (1+2 x)-2 x^2 \log (x)+(-4 x-3 x^2) \log ^2(x)) \, dx\)

Optimal. Leaf size=28 \[ \frac {25}{4}+\frac {1}{45} x \left (1+x-e^{-4/x} x^2 \log ^2(x)\right ) \]

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Rubi [F]  time = 0.64, 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 {1}{45} e^{-4/x} \left (e^{4/x} (1+2 x)-2 x^2 \log (x)+\left (-4 x-3 x^2\right ) \log ^2(x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(4/x)*(1 + 2*x) - 2*x^2*Log[x] + (-4*x - 3*x^2)*Log[x]^2)/(45*E^(4/x)),x]

[Out]

x/45 + (88*x)/(405*E^(4/x)) + x^2/45 - (2*x^2)/(81*E^(4/x)) + (2*x^3)/(405*E^(4/x)) + (352*ExpIntegralEi[-4/x]
)/405 + (256*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -4/x])/(135*x) - (64*(ExpIntegralE[1, 4/x] + ExpIntegralE
i[-4/x])*Log[x^(-1)])/135 - (32*Log[4/x]^2)/135 + (64*EulerGamma*Log[x])/135 - (16*x*Log[x])/(135*E^(4/x)) + (
4*x^2*Log[x])/(135*E^(4/x)) - (2*x^3*Log[x])/(135*E^(4/x)) - (64*ExpIntegralEi[-4/x]*Log[x])/135 - (4*Defer[In
t][(x*Log[x]^2)/E^(4/x), x])/45 - Defer[Int][(x^2*Log[x]^2)/E^(4/x), x]/15

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{45} \int e^{-4/x} \left (e^{4/x} (1+2 x)-2 x^2 \log (x)+\left (-4 x-3 x^2\right ) \log ^2(x)\right ) \, dx\\ &=\frac {1}{45} \int \left (1+2 x-2 e^{-4/x} x^2 \log (x)-e^{-4/x} x (4+3 x) \log ^2(x)\right ) \, dx\\ &=\frac {x}{45}+\frac {x^2}{45}-\frac {1}{45} \int e^{-4/x} x (4+3 x) \log ^2(x) \, dx-\frac {2}{45} \int e^{-4/x} x^2 \log (x) \, dx\\ &=\frac {x}{45}+\frac {x^2}{45}-\frac {16}{135} e^{-4/x} x \log (x)+\frac {4}{135} e^{-4/x} x^2 \log (x)-\frac {2}{135} e^{-4/x} x^3 \log (x)-\frac {64}{135} \text {Ei}\left (-\frac {4}{x}\right ) \log (x)-\frac {1}{45} \int \left (4 e^{-4/x} x \log ^2(x)+3 e^{-4/x} x^2 \log ^2(x)\right ) \, dx+\frac {2}{45} \int \frac {1}{3} \left (e^{-4/x} \left (8-2 x+x^2\right )+\frac {32 \text {Ei}\left (-\frac {4}{x}\right )}{x}\right ) \, dx\\ &=\frac {x}{45}+\frac {x^2}{45}-\frac {16}{135} e^{-4/x} x \log (x)+\frac {4}{135} e^{-4/x} x^2 \log (x)-\frac {2}{135} e^{-4/x} x^3 \log (x)-\frac {64}{135} \text {Ei}\left (-\frac {4}{x}\right ) \log (x)+\frac {2}{135} \int \left (e^{-4/x} \left (8-2 x+x^2\right )+\frac {32 \text {Ei}\left (-\frac {4}{x}\right )}{x}\right ) \, dx-\frac {1}{15} \int e^{-4/x} x^2 \log ^2(x) \, dx-\frac {4}{45} \int e^{-4/x} x \log ^2(x) \, dx\\ &=\frac {x}{45}+\frac {x^2}{45}-\frac {16}{135} e^{-4/x} x \log (x)+\frac {4}{135} e^{-4/x} x^2 \log (x)-\frac {2}{135} e^{-4/x} x^3 \log (x)-\frac {64}{135} \text {Ei}\left (-\frac {4}{x}\right ) \log (x)+\frac {2}{135} \int e^{-4/x} \left (8-2 x+x^2\right ) \, dx-\frac {1}{15} \int e^{-4/x} x^2 \log ^2(x) \, dx-\frac {4}{45} \int e^{-4/x} x \log ^2(x) \, dx+\frac {64}{135} \int \frac {\text {Ei}\left (-\frac {4}{x}\right )}{x} \, dx\\ &=\frac {x}{45}+\frac {x^2}{45}-\frac {16}{135} e^{-4/x} x \log (x)+\frac {4}{135} e^{-4/x} x^2 \log (x)-\frac {2}{135} e^{-4/x} x^3 \log (x)-\frac {64}{135} \text {Ei}\left (-\frac {4}{x}\right ) \log (x)+\frac {2}{135} \int \left (8 e^{-4/x}-2 e^{-4/x} x+e^{-4/x} x^2\right ) \, dx-\frac {1}{15} \int e^{-4/x} x^2 \log ^2(x) \, dx-\frac {4}{45} \int e^{-4/x} x \log ^2(x) \, dx-\frac {64}{135} \operatorname {Subst}\left (\int \frac {\text {Ei}(-4 x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {x}{45}+\frac {x^2}{45}-\frac {64}{135} \left (E_1\left (\frac {4}{x}\right )+\text {Ei}\left (-\frac {4}{x}\right )\right ) \log \left (\frac {1}{x}\right )-\frac {16}{135} e^{-4/x} x \log (x)+\frac {4}{135} e^{-4/x} x^2 \log (x)-\frac {2}{135} e^{-4/x} x^3 \log (x)-\frac {64}{135} \text {Ei}\left (-\frac {4}{x}\right ) \log (x)+\frac {2}{135} \int e^{-4/x} x^2 \, dx-\frac {4}{135} \int e^{-4/x} x \, dx-\frac {1}{15} \int e^{-4/x} x^2 \log ^2(x) \, dx-\frac {4}{45} \int e^{-4/x} x \log ^2(x) \, dx+\frac {16}{135} \int e^{-4/x} \, dx+\frac {64}{135} \operatorname {Subst}\left (\int \frac {E_1(4 x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {x}{45}+\frac {16}{135} e^{-4/x} x+\frac {x^2}{45}-\frac {2}{135} e^{-4/x} x^2+\frac {2}{405} e^{-4/x} x^3+\frac {256 \, _3F_3\left (1,1,1;2,2,2;-\frac {4}{x}\right )}{135 x}-\frac {64}{135} \left (E_1\left (\frac {4}{x}\right )+\text {Ei}\left (-\frac {4}{x}\right )\right ) \log \left (\frac {1}{x}\right )-\frac {32}{135} \log ^2\left (\frac {4}{x}\right )+\frac {64}{135} \gamma \log (x)-\frac {16}{135} e^{-4/x} x \log (x)+\frac {4}{135} e^{-4/x} x^2 \log (x)-\frac {2}{135} e^{-4/x} x^3 \log (x)-\frac {64}{135} \text {Ei}\left (-\frac {4}{x}\right ) \log (x)-\frac {8}{405} \int e^{-4/x} x \, dx+\frac {8}{135} \int e^{-4/x} \, dx-\frac {1}{15} \int e^{-4/x} x^2 \log ^2(x) \, dx-\frac {4}{45} \int e^{-4/x} x \log ^2(x) \, dx-\frac {64}{135} \int \frac {e^{-4/x}}{x} \, dx\\ &=\frac {x}{45}+\frac {8}{45} e^{-4/x} x+\frac {x^2}{45}-\frac {2}{81} e^{-4/x} x^2+\frac {2}{405} e^{-4/x} x^3+\frac {64 \text {Ei}\left (-\frac {4}{x}\right )}{135}+\frac {256 \, _3F_3\left (1,1,1;2,2,2;-\frac {4}{x}\right )}{135 x}-\frac {64}{135} \left (E_1\left (\frac {4}{x}\right )+\text {Ei}\left (-\frac {4}{x}\right )\right ) \log \left (\frac {1}{x}\right )-\frac {32}{135} \log ^2\left (\frac {4}{x}\right )+\frac {64}{135} \gamma \log (x)-\frac {16}{135} e^{-4/x} x \log (x)+\frac {4}{135} e^{-4/x} x^2 \log (x)-\frac {2}{135} e^{-4/x} x^3 \log (x)-\frac {64}{135} \text {Ei}\left (-\frac {4}{x}\right ) \log (x)+\frac {16}{405} \int e^{-4/x} \, dx-\frac {1}{15} \int e^{-4/x} x^2 \log ^2(x) \, dx-\frac {4}{45} \int e^{-4/x} x \log ^2(x) \, dx-\frac {32}{135} \int \frac {e^{-4/x}}{x} \, dx\\ &=\frac {x}{45}+\frac {88}{405} e^{-4/x} x+\frac {x^2}{45}-\frac {2}{81} e^{-4/x} x^2+\frac {2}{405} e^{-4/x} x^3+\frac {32 \text {Ei}\left (-\frac {4}{x}\right )}{45}+\frac {256 \, _3F_3\left (1,1,1;2,2,2;-\frac {4}{x}\right )}{135 x}-\frac {64}{135} \left (E_1\left (\frac {4}{x}\right )+\text {Ei}\left (-\frac {4}{x}\right )\right ) \log \left (\frac {1}{x}\right )-\frac {32}{135} \log ^2\left (\frac {4}{x}\right )+\frac {64}{135} \gamma \log (x)-\frac {16}{135} e^{-4/x} x \log (x)+\frac {4}{135} e^{-4/x} x^2 \log (x)-\frac {2}{135} e^{-4/x} x^3 \log (x)-\frac {64}{135} \text {Ei}\left (-\frac {4}{x}\right ) \log (x)-\frac {1}{15} \int e^{-4/x} x^2 \log ^2(x) \, dx-\frac {4}{45} \int e^{-4/x} x \log ^2(x) \, dx-\frac {64}{405} \int \frac {e^{-4/x}}{x} \, dx\\ &=\frac {x}{45}+\frac {88}{405} e^{-4/x} x+\frac {x^2}{45}-\frac {2}{81} e^{-4/x} x^2+\frac {2}{405} e^{-4/x} x^3+\frac {352 \text {Ei}\left (-\frac {4}{x}\right )}{405}+\frac {256 \, _3F_3\left (1,1,1;2,2,2;-\frac {4}{x}\right )}{135 x}-\frac {64}{135} \left (E_1\left (\frac {4}{x}\right )+\text {Ei}\left (-\frac {4}{x}\right )\right ) \log \left (\frac {1}{x}\right )-\frac {32}{135} \log ^2\left (\frac {4}{x}\right )+\frac {64}{135} \gamma \log (x)-\frac {16}{135} e^{-4/x} x \log (x)+\frac {4}{135} e^{-4/x} x^2 \log (x)-\frac {2}{135} e^{-4/x} x^3 \log (x)-\frac {64}{135} \text {Ei}\left (-\frac {4}{x}\right ) \log (x)-\frac {1}{15} \int e^{-4/x} x^2 \log ^2(x) \, dx-\frac {4}{45} \int e^{-4/x} x \log ^2(x) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 25, normalized size = 0.89 \begin {gather*} \frac {1}{45} \left (x+x^2-e^{-4/x} x^3 \log ^2(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4/x)*(1 + 2*x) - 2*x^2*Log[x] + (-4*x - 3*x^2)*Log[x]^2)/(45*E^(4/x)),x]

[Out]

(x + x^2 - (x^3*Log[x]^2)/E^(4/x))/45

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fricas [A]  time = 0.56, size = 30, normalized size = 1.07 \begin {gather*} -\frac {1}{45} \, {\left (x^{3} \log \relax (x)^{2} - {\left (x^{2} + x\right )} e^{\frac {4}{x}}\right )} e^{\left (-\frac {4}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/45*((-3*x^2-4*x)*log(x)^2-2*x^2*log(x)+(2*x+1)*exp(4/x))/exp(4/x),x, algorithm="fricas")

[Out]

-1/45*(x^3*log(x)^2 - (x^2 + x)*e^(4/x))*e^(-4/x)

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giac [A]  time = 0.14, size = 24, normalized size = 0.86 \begin {gather*} -\frac {1}{45} \, x^{3} e^{\left (-\frac {4}{x}\right )} \log \relax (x)^{2} + \frac {1}{45} \, x^{2} + \frac {1}{45} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/45*((-3*x^2-4*x)*log(x)^2-2*x^2*log(x)+(2*x+1)*exp(4/x))/exp(4/x),x, algorithm="giac")

[Out]

-1/45*x^3*e^(-4/x)*log(x)^2 + 1/45*x^2 + 1/45*x

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maple [A]  time = 0.07, size = 25, normalized size = 0.89

method	result	size
risch	\(\frac {x}{45}-\frac {x^{3} \ln \relax (x )^{2} {\mathrm e}^{-\frac {4}{x}}}{45}+\frac {x^{2}}{45}\)	\(25\)
default	\(\frac {x}{45}-\frac {x^{3} \ln \relax (x )^{2} {\mathrm e}^{-\frac {4}{x}}}{45}+\frac {x^{2}}{45}\)	\(27\)
norman	\(\left (\frac {x \,{\mathrm e}^{\frac {4}{x}}}{45}+\frac {x^{2} {\mathrm e}^{\frac {4}{x}}}{45}-\frac {x^{3} \ln \relax (x )^{2}}{45}\right ) {\mathrm e}^{-\frac {4}{x}}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/45*((-3*x^2-4*x)*ln(x)^2-2*x^2*ln(x)+(2*x+1)*exp(4/x))/exp(4/x),x,method=_RETURNVERBOSE)

[Out]

1/45*x-1/45*x^3*ln(x)^2*exp(-4/x)+1/45*x^2

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maxima [A]  time = 0.41, size = 24, normalized size = 0.86 \begin {gather*} -\frac {1}{45} \, x^{3} e^{\left (-\frac {4}{x}\right )} \log \relax (x)^{2} + \frac {1}{45} \, x^{2} + \frac {1}{45} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/45*((-3*x^2-4*x)*log(x)^2-2*x^2*log(x)+(2*x+1)*exp(4/x))/exp(4/x),x, algorithm="maxima")

[Out]

-1/45*x^3*e^(-4/x)*log(x)^2 + 1/45*x^2 + 1/45*x

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mupad [B]  time = 7.75, size = 24, normalized size = 0.86 \begin {gather*} \frac {x}{45}+\frac {x^2}{45}-\frac {x^3\,{\mathrm {e}}^{-\frac {4}{x}}\,{\ln \relax (x)}^2}{45} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-4/x)*((log(x)^2*(4*x + 3*x^2))/45 + (2*x^2*log(x))/45 - (exp(4/x)*(2*x + 1))/45),x)

[Out]

x/45 + x^2/45 - (x^3*exp(-4/x)*log(x)^2)/45

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sympy [A]  time = 0.36, size = 22, normalized size = 0.79 \begin {gather*} - \frac {x^{3} e^{- \frac {4}{x}} \log {\relax (x )}^{2}}{45} + \frac {x^{2}}{45} + \frac {x}{45} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/45*((-3*x**2-4*x)*ln(x)**2-2*x**2*ln(x)+(2*x+1)*exp(4/x))/exp(4/x),x)

[Out]

-x**3*exp(-4/x)*log(x)**2/45 + x**2/45 + x/45

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