3.39.45 \(\int \frac {e^{-6-2 x} (120+105 x+18 x^2+4 e^{6+2 x} x^3+(-70-68 x-12 x^2) \log (x)+(10+11 x+2 x^2) \log ^2(x))}{3 x^3} \, dx\)

Optimal. Leaf size=35 \[ \frac {x}{3}+\frac {1}{3} (5+x) \left (3-\frac {e^{-6-2 x} (3-\log (x))^2}{x^2}\right ) \]

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Rubi [F]  time = 2.52, 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 {e^{-6-2 x} \left (120+105 x+18 x^2+4 e^{6+2 x} x^3+\left (-70-68 x-12 x^2\right ) \log (x)+\left (10+11 x+2 x^2\right ) \log ^2(x)\right )}{3 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-6 - 2*x)*(120 + 105*x + 18*x^2 + 4*E^(6 + 2*x)*x^3 + (-70 - 68*x - 12*x^2)*Log[x] + (10 + 11*x + 2*x^
2)*Log[x]^2))/(3*x^3),x]

[Out]

(-85*E^(-6 - 2*x))/(6*x^2) - (22*E^(-6 - 2*x))/(3*x) + (4*x)/3 - (26*ExpIntegralEi[-2*x])/(3*E^6) - (32*x*Hype
rgeometricPFQ[{1, 1, 1}, {2, 2, 2}, -2*x])/(3*E^6) + (16*EulerGamma*Log[x])/(3*E^6) + (35*E^(-6 - 2*x)*Log[x])
/(3*x^2) - (2*E^(-6 - 2*x)*Log[x])/(3*x) - (16*ExpIntegralEi[-2*x]*Log[x])/(3*E^6) + (16*(ExpIntegralE[1, 2*x]
+ ExpIntegralEi[-2*x])*Log[x])/(3*E^6) + (8*Log[2*x]^2)/(3*E^6) + (10*Defer[Int][(E^(-6 - 2*x)*Log[x]^2)/x^3,
x])/3 + (11*Defer[Int][(E^(-6 - 2*x)*Log[x]^2)/x^2, x])/3 + (2*Defer[Int][(E^(-6 - 2*x)*Log[x]^2)/x, x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {e^{-6-2 x} \left (120+105 x+18 x^2+4 e^{6+2 x} x^3+\left (-70-68 x-12 x^2\right ) \log (x)+\left (10+11 x+2 x^2\right ) \log ^2(x)\right )}{x^3} \, dx\\ &=\frac {1}{3} \int \left (4+\frac {e^{-6-2 x} (-3+\log (x)) \left (-40-35 x-6 x^2+10 \log (x)+11 x \log (x)+2 x^2 \log (x)\right )}{x^3}\right ) \, dx\\ &=\frac {4 x}{3}+\frac {1}{3} \int \frac {e^{-6-2 x} (-3+\log (x)) \left (-40-35 x-6 x^2+10 \log (x)+11 x \log (x)+2 x^2 \log (x)\right )}{x^3} \, dx\\ &=\frac {4 x}{3}+\frac {1}{3} \int \left (\frac {3 e^{-6-2 x} \left (40+35 x+6 x^2\right )}{x^3}-\frac {2 e^{-6-2 x} \left (35+34 x+6 x^2\right ) \log (x)}{x^3}+\frac {e^{-6-2 x} \left (10+11 x+2 x^2\right ) \log ^2(x)}{x^3}\right ) \, dx\\ &=\frac {4 x}{3}+\frac {1}{3} \int \frac {e^{-6-2 x} \left (10+11 x+2 x^2\right ) \log ^2(x)}{x^3} \, dx-\frac {2}{3} \int \frac {e^{-6-2 x} \left (35+34 x+6 x^2\right ) \log (x)}{x^3} \, dx+\int \frac {e^{-6-2 x} \left (40+35 x+6 x^2\right )}{x^3} \, dx\\ &=\frac {4 x}{3}+\frac {35 e^{-6-2 x} \log (x)}{3 x^2}-\frac {2 e^{-6-2 x} \log (x)}{3 x}-\frac {16 \text {Ei}(-2 x) \log (x)}{3 e^6}+\frac {1}{3} \int \left (\frac {10 e^{-6-2 x} \log ^2(x)}{x^3}+\frac {11 e^{-6-2 x} \log ^2(x)}{x^2}+\frac {2 e^{-6-2 x} \log ^2(x)}{x}\right ) \, dx+\frac {2}{3} \int \frac {e^{-6-2 x} \left (-35+2 x+16 e^{2 x} x^2 \text {Ei}(-2 x)\right )}{2 x^3} \, dx+\int \left (\frac {40 e^{-6-2 x}}{x^3}+\frac {35 e^{-6-2 x}}{x^2}+\frac {6 e^{-6-2 x}}{x}\right ) \, dx\\ &=\frac {4 x}{3}+\frac {35 e^{-6-2 x} \log (x)}{3 x^2}-\frac {2 e^{-6-2 x} \log (x)}{3 x}-\frac {16 \text {Ei}(-2 x) \log (x)}{3 e^6}+\frac {1}{3} \int \frac {e^{-6-2 x} \left (-35+2 x+16 e^{2 x} x^2 \text {Ei}(-2 x)\right )}{x^3} \, dx+\frac {2}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x} \, dx+\frac {10}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x^3} \, dx+\frac {11}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x^2} \, dx+6 \int \frac {e^{-6-2 x}}{x} \, dx+35 \int \frac {e^{-6-2 x}}{x^2} \, dx+40 \int \frac {e^{-6-2 x}}{x^3} \, dx\\ &=-\frac {20 e^{-6-2 x}}{x^2}-\frac {35 e^{-6-2 x}}{x}+\frac {4 x}{3}+\frac {6 \text {Ei}(-2 x)}{e^6}+\frac {35 e^{-6-2 x} \log (x)}{3 x^2}-\frac {2 e^{-6-2 x} \log (x)}{3 x}-\frac {16 \text {Ei}(-2 x) \log (x)}{3 e^6}+\frac {1}{3} \int \left (\frac {e^{-6-2 x} (-35+2 x)}{x^3}+\frac {16 \text {Ei}(-2 x)}{e^6 x}\right ) \, dx+\frac {2}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x} \, dx+\frac {10}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x^3} \, dx+\frac {11}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x^2} \, dx-40 \int \frac {e^{-6-2 x}}{x^2} \, dx-70 \int \frac {e^{-6-2 x}}{x} \, dx\\ &=-\frac {20 e^{-6-2 x}}{x^2}+\frac {5 e^{-6-2 x}}{x}+\frac {4 x}{3}-\frac {64 \text {Ei}(-2 x)}{e^6}+\frac {35 e^{-6-2 x} \log (x)}{3 x^2}-\frac {2 e^{-6-2 x} \log (x)}{3 x}-\frac {16 \text {Ei}(-2 x) \log (x)}{3 e^6}+\frac {1}{3} \int \frac {e^{-6-2 x} (-35+2 x)}{x^3} \, dx+\frac {2}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x} \, dx+\frac {10}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x^3} \, dx+\frac {11}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x^2} \, dx+80 \int \frac {e^{-6-2 x}}{x} \, dx+\frac {16 \int \frac {\text {Ei}(-2 x)}{x} \, dx}{3 e^6}\\ &=-\frac {20 e^{-6-2 x}}{x^2}+\frac {5 e^{-6-2 x}}{x}+\frac {4 x}{3}+\frac {16 \text {Ei}(-2 x)}{e^6}+\frac {35 e^{-6-2 x} \log (x)}{3 x^2}-\frac {2 e^{-6-2 x} \log (x)}{3 x}-\frac {16 \text {Ei}(-2 x) \log (x)}{3 e^6}+\frac {16 (E_1(2 x)+\text {Ei}(-2 x)) \log (x)}{3 e^6}+\frac {1}{3} \int \left (-\frac {35 e^{-6-2 x}}{x^3}+\frac {2 e^{-6-2 x}}{x^2}\right ) \, dx+\frac {2}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x} \, dx+\frac {10}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x^3} \, dx+\frac {11}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x^2} \, dx-\frac {16 \int \frac {E_1(2 x)}{x} \, dx}{3 e^6}\\ &=-\frac {20 e^{-6-2 x}}{x^2}+\frac {5 e^{-6-2 x}}{x}+\frac {4 x}{3}+\frac {16 \text {Ei}(-2 x)}{e^6}-\frac {32 x \, _3F_3(1,1,1;2,2,2;-2 x)}{3 e^6}+\frac {16 \gamma \log (x)}{3 e^6}+\frac {35 e^{-6-2 x} \log (x)}{3 x^2}-\frac {2 e^{-6-2 x} \log (x)}{3 x}-\frac {16 \text {Ei}(-2 x) \log (x)}{3 e^6}+\frac {16 (E_1(2 x)+\text {Ei}(-2 x)) \log (x)}{3 e^6}+\frac {8 \log ^2(2 x)}{3 e^6}+\frac {2}{3} \int \frac {e^{-6-2 x}}{x^2} \, dx+\frac {2}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x} \, dx+\frac {10}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x^3} \, dx+\frac {11}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x^2} \, dx-\frac {35}{3} \int \frac {e^{-6-2 x}}{x^3} \, dx\\ &=-\frac {85 e^{-6-2 x}}{6 x^2}+\frac {13 e^{-6-2 x}}{3 x}+\frac {4 x}{3}+\frac {16 \text {Ei}(-2 x)}{e^6}-\frac {32 x \, _3F_3(1,1,1;2,2,2;-2 x)}{3 e^6}+\frac {16 \gamma \log (x)}{3 e^6}+\frac {35 e^{-6-2 x} \log (x)}{3 x^2}-\frac {2 e^{-6-2 x} \log (x)}{3 x}-\frac {16 \text {Ei}(-2 x) \log (x)}{3 e^6}+\frac {16 (E_1(2 x)+\text {Ei}(-2 x)) \log (x)}{3 e^6}+\frac {8 \log ^2(2 x)}{3 e^6}+\frac {2}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x} \, dx-\frac {4}{3} \int \frac {e^{-6-2 x}}{x} \, dx+\frac {10}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x^3} \, dx+\frac {11}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x^2} \, dx+\frac {35}{3} \int \frac {e^{-6-2 x}}{x^2} \, dx\\ &=-\frac {85 e^{-6-2 x}}{6 x^2}-\frac {22 e^{-6-2 x}}{3 x}+\frac {4 x}{3}+\frac {44 \text {Ei}(-2 x)}{3 e^6}-\frac {32 x \, _3F_3(1,1,1;2,2,2;-2 x)}{3 e^6}+\frac {16 \gamma \log (x)}{3 e^6}+\frac {35 e^{-6-2 x} \log (x)}{3 x^2}-\frac {2 e^{-6-2 x} \log (x)}{3 x}-\frac {16 \text {Ei}(-2 x) \log (x)}{3 e^6}+\frac {16 (E_1(2 x)+\text {Ei}(-2 x)) \log (x)}{3 e^6}+\frac {8 \log ^2(2 x)}{3 e^6}+\frac {2}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x} \, dx+\frac {10}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x^3} \, dx+\frac {11}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x^2} \, dx-\frac {70}{3} \int \frac {e^{-6-2 x}}{x} \, dx\\ &=-\frac {85 e^{-6-2 x}}{6 x^2}-\frac {22 e^{-6-2 x}}{3 x}+\frac {4 x}{3}-\frac {26 \text {Ei}(-2 x)}{3 e^6}-\frac {32 x \, _3F_3(1,1,1;2,2,2;-2 x)}{3 e^6}+\frac {16 \gamma \log (x)}{3 e^6}+\frac {35 e^{-6-2 x} \log (x)}{3 x^2}-\frac {2 e^{-6-2 x} \log (x)}{3 x}-\frac {16 \text {Ei}(-2 x) \log (x)}{3 e^6}+\frac {16 (E_1(2 x)+\text {Ei}(-2 x)) \log (x)}{3 e^6}+\frac {8 \log ^2(2 x)}{3 e^6}+\frac {2}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x} \, dx+\frac {10}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x^3} \, dx+\frac {11}{3} \int \frac {e^{-6-2 x} \log ^2(x)}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.95, size = 47, normalized size = 1.34 \begin {gather*} \frac {e^{-2 (3+x)} \left (-45-9 x+4 e^{6+2 x} x^3+6 (5+x) \log (x)-(5+x) \log ^2(x)\right )}{3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-6 - 2*x)*(120 + 105*x + 18*x^2 + 4*E^(6 + 2*x)*x^3 + (-70 - 68*x - 12*x^2)*Log[x] + (10 + 11*x
+ 2*x^2)*Log[x]^2))/(3*x^3),x]

[Out]

(-45 - 9*x + 4*E^(6 + 2*x)*x^3 + 6*(5 + x)*Log[x] - (5 + x)*Log[x]^2)/(3*E^(2*(3 + x))*x^2)

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fricas [A]  time = 0.68, size = 43, normalized size = 1.23 \begin {gather*} \frac {{\left (4 \, x^{3} e^{\left (2 \, x + 6\right )} - {\left (x + 5\right )} \log \relax (x)^{2} + 6 \, {\left (x + 5\right )} \log \relax (x) - 9 \, x - 45\right )} e^{\left (-2 \, x - 6\right )}}{3 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((2*x^2+11*x+10)*log(x)^2+(-12*x^2-68*x-70)*log(x)+4*x^3*exp(3+x)^2+18*x^2+105*x+120)/x^3/exp(3+
x)^2,x, algorithm="fricas")

[Out]

1/3*(4*x^3*e^(2*x + 6) - (x + 5)*log(x)^2 + 6*(x + 5)*log(x) - 9*x - 45)*e^(-2*x - 6)/x^2

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giac [B]  time = 0.18, size = 66, normalized size = 1.89 \begin {gather*} \frac {{\left (4 \, x^{3} e^{6} - x e^{\left (-2 \, x\right )} \log \relax (x)^{2} + 6 \, x e^{\left (-2 \, x\right )} \log \relax (x) - 5 \, e^{\left (-2 \, x\right )} \log \relax (x)^{2} - 9 \, x e^{\left (-2 \, x\right )} + 30 \, e^{\left (-2 \, x\right )} \log \relax (x) - 45 \, e^{\left (-2 \, x\right )}\right )} e^{\left (-6\right )}}{3 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((2*x^2+11*x+10)*log(x)^2+(-12*x^2-68*x-70)*log(x)+4*x^3*exp(3+x)^2+18*x^2+105*x+120)/x^3/exp(3+
x)^2,x, algorithm="giac")

[Out]

1/3*(4*x^3*e^6 - x*e^(-2*x)*log(x)^2 + 6*x*e^(-2*x)*log(x) - 5*e^(-2*x)*log(x)^2 - 9*x*e^(-2*x) + 30*e^(-2*x)*
log(x) - 45*e^(-2*x))*e^(-6)/x^2

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maple [B]  time = 0.08, size = 63, normalized size = 1.80

method	result	size
risch	\(-\frac {\left (5+x \right ) {\mathrm e}^{-2 x -6} \ln \relax (x )^{2}}{3 x^{2}}+\frac {2 \left (5+x \right ) {\mathrm e}^{-2 x -6} \ln \relax (x )}{x^{2}}+\frac {\left (4 x^{3} {\mathrm e}^{2 x +6}-9 x -45\right ) {\mathrm e}^{-2 x -6}}{3 x^{2}}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((2*x^2+11*x+10)*ln(x)^2+(-12*x^2-68*x-70)*ln(x)+4*x^3*exp(3+x)^2+18*x^2+105*x+120)/x^3/exp(3+x)^2,x,m
ethod=_RETURNVERBOSE)

[Out]

-1/3*(5+x)/x^2*exp(-2*x-6)*ln(x)^2+2*(5+x)/x^2*exp(-2*x-6)*ln(x)+1/3*(4*x^3*exp(2*x+6)-9*x-45)/x^2*exp(-2*x-6)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 6 \, {\rm Ei}\left (-2 \, x\right ) e^{\left (-6\right )} - 70 \, e^{\left (-6\right )} \Gamma \left (-1, 2 \, x\right ) - 160 \, e^{\left (-6\right )} \Gamma \left (-2, 2 \, x\right ) + \frac {4}{3} \, x - \frac {{\left ({\left (x + 5\right )} \log \relax (x)^{2} - 6 \, {\left (x + 5\right )} \log \relax (x)\right )} e^{\left (-2 \, x - 6\right )}}{3 \, x^{2}} - \frac {1}{3} \, \int \frac {6 \, {\left (x + 5\right )} e^{\left (-2 \, x - 6\right )}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((2*x^2+11*x+10)*log(x)^2+(-12*x^2-68*x-70)*log(x)+4*x^3*exp(3+x)^2+18*x^2+105*x+120)/x^3/exp(3+
x)^2,x, algorithm="maxima")

[Out]

6*Ei(-2*x)*e^(-6) - 70*e^(-6)*gamma(-1, 2*x) - 160*e^(-6)*gamma(-2, 2*x) + 4/3*x - 1/3*((x + 5)*log(x)^2 - 6*(
x + 5)*log(x))*e^(-2*x - 6)/x^2 - 1/3*integrate(6*(x + 5)*e^(-2*x - 6)/x^3, x)

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mupad [B]  time = 2.32, size = 49, normalized size = 1.40 \begin {gather*} \frac {4\,x}{3}-\frac {\frac {{\mathrm {e}}^{-2\,x-6}\,\left (5\,{\ln \relax (x)}^2-30\,\ln \relax (x)+45\right )}{3}+\frac {x\,{\mathrm {e}}^{-2\,x-6}\,\left ({\ln \relax (x)}^2-6\,\ln \relax (x)+9\right )}{3}}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(- 2*x - 6)*(35*x + (log(x)^2*(11*x + 2*x^2 + 10))/3 - (log(x)*(68*x + 12*x^2 + 70))/3 + (4*x^3*exp(2*
x + 6))/3 + 6*x^2 + 40))/x^3,x)

[Out]

(4*x)/3 - ((exp(- 2*x - 6)*(5*log(x)^2 - 30*log(x) + 45))/3 + (x*exp(- 2*x - 6)*(log(x)^2 - 6*log(x) + 9))/3)/
x^2

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sympy [A]  time = 0.34, size = 48, normalized size = 1.37 \begin {gather*} \frac {4 x}{3} + \frac {\left (- x \log {\relax (x )}^{2} + 6 x \log {\relax (x )} - 9 x - 5 \log {\relax (x )}^{2} + 30 \log {\relax (x )} - 45\right ) e^{- 2 x - 6}}{3 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((2*x**2+11*x+10)*ln(x)**2+(-12*x**2-68*x-70)*ln(x)+4*x**3*exp(3+x)**2+18*x**2+105*x+120)/x**3/e
xp(3+x)**2,x)

[Out]

4*x/3 + (-x*log(x)**2 + 6*x*log(x) - 9*x - 5*log(x)**2 + 30*log(x) - 45)*exp(-2*x - 6)/(3*x**2)

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