3.16.17 \(\int \frac {e^4 (-e^3-2 x^2)+e^{60 x^2} (3 x+120 e^3 x-120 x^3)+(-e^4 x^2+e^{60 x^2} (2 x-120 x^3)) \log (x)}{e^{120 x^2}-2 e^{4+60 x^2} x+e^8 x^2} \, dx\)

Optimal. Leaf size=31 \[ \frac {e^3-x^2 (1+\log (x))}{-e^{60 x^2}+e^4 x} \]

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Rubi [F]  time = 2.78, 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^4 \left (-e^3-2 x^2\right )+e^{60 x^2} \left (3 x+120 e^3 x-120 x^3\right )+\left (-e^4 x^2+e^{60 x^2} \left (2 x-120 x^3\right )\right ) \log (x)}{e^{120 x^2}-2 e^{4+60 x^2} x+e^8 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^4*(-E^3 - 2*x^2) + E^(60*x^2)*(3*x + 120*E^3*x - 120*x^3) + (-(E^4*x^2) + E^(60*x^2)*(2*x - 120*x^3))*L
og[x])/(E^(120*x^2) - 2*E^(4 + 60*x^2)*x + E^8*x^2),x]

[Out]

-(E^7*Defer[Int][(-E^(60*x^2) + E^4*x)^(-2), x]) + E^4*Defer[Int][x^2/(-E^(60*x^2) + E^4*x)^2, x] + 120*E^7*De
fer[Int][x^2/(-E^(60*x^2) + E^4*x)^2, x] + E^4*Log[x]*Defer[Int][x^2/(-E^(60*x^2) + E^4*x)^2, x] - 120*E^4*Def
er[Int][x^4/(-E^(60*x^2) + E^4*x)^2, x] - 120*E^4*Log[x]*Defer[Int][x^4/(-E^(60*x^2) + E^4*x)^2, x] - 3*(1 + 4
0*E^3)*Defer[Int][x/(-E^(60*x^2) + E^4*x), x] - 2*Log[x]*Defer[Int][x/(-E^(60*x^2) + E^4*x), x] + 120*Defer[In
t][x^3/(-E^(60*x^2) + E^4*x), x] + 120*Log[x]*Defer[Int][x^3/(-E^(60*x^2) + E^4*x), x] - E^4*Defer[Int][Defer[
Int][x^2/(E^(60*x^2) - E^4*x)^2, x]/x, x] + 120*E^4*Defer[Int][Defer[Int][x^4/(E^(60*x^2) - E^4*x)^2, x]/x, x]
+ 2*Defer[Int][Defer[Int][x/(-E^(60*x^2) + E^4*x), x]/x, x] - 120*Defer[Int][Defer[Int][x^3/(-E^(60*x^2) + E^
4*x), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^4 \left (-e^3-2 x^2\right )+e^{60 x^2} \left (3 x+120 e^3 x-120 x^3\right )+\left (-e^4 x^2+e^{60 x^2} \left (2 x-120 x^3\right )\right ) \log (x)}{\left (e^{60 x^2}-e^4 x\right )^2} \, dx\\ &=\int \left (\frac {x \left (3 \left (1+40 e^3\right )-120 x^2+2 \log (x)-120 x^2 \log (x)\right )}{e^{60 x^2}-e^4 x}+\frac {e^4 \left (-1+120 x^2\right ) \left (e^3-x^2-x^2 \log (x)\right )}{\left (-e^{60 x^2}+e^4 x\right )^2}\right ) \, dx\\ &=e^4 \int \frac {\left (-1+120 x^2\right ) \left (e^3-x^2-x^2 \log (x)\right )}{\left (-e^{60 x^2}+e^4 x\right )^2} \, dx+\int \frac {x \left (3 \left (1+40 e^3\right )-120 x^2+2 \log (x)-120 x^2 \log (x)\right )}{e^{60 x^2}-e^4 x} \, dx\\ &=e^4 \int \left (-\frac {e^3-x^2-x^2 \log (x)}{\left (e^{60 x^2}-e^4 x\right )^2}-\frac {120 x^2 \left (-e^3+x^2+x^2 \log (x)\right )}{\left (-e^{60 x^2}+e^4 x\right )^2}\right ) \, dx+\int \left (-\frac {3 \left (1+40 e^3\right ) x}{-e^{60 x^2}+e^4 x}+\frac {120 x^3}{-e^{60 x^2}+e^4 x}-\frac {2 x \log (x)}{-e^{60 x^2}+e^4 x}+\frac {120 x^3 \log (x)}{-e^{60 x^2}+e^4 x}\right ) \, dx\\ &=-\left (2 \int \frac {x \log (x)}{-e^{60 x^2}+e^4 x} \, dx\right )+120 \int \frac {x^3}{-e^{60 x^2}+e^4 x} \, dx+120 \int \frac {x^3 \log (x)}{-e^{60 x^2}+e^4 x} \, dx-e^4 \int \frac {e^3-x^2-x^2 \log (x)}{\left (e^{60 x^2}-e^4 x\right )^2} \, dx-\left (120 e^4\right ) \int \frac {x^2 \left (-e^3+x^2+x^2 \log (x)\right )}{\left (-e^{60 x^2}+e^4 x\right )^2} \, dx-\left (3 \left (1+40 e^3\right )\right ) \int \frac {x}{-e^{60 x^2}+e^4 x} \, dx\\ &=2 \int \frac {\int \frac {x}{-e^{60 x^2}+e^4 x} \, dx}{x} \, dx+120 \int \frac {x^3}{-e^{60 x^2}+e^4 x} \, dx-120 \int \frac {\int \frac {x^3}{-e^{60 x^2}+e^4 x} \, dx}{x} \, dx-e^4 \int \left (\frac {e^3}{\left (-e^{60 x^2}+e^4 x\right )^2}-\frac {x^2}{\left (-e^{60 x^2}+e^4 x\right )^2}-\frac {x^2 \log (x)}{\left (-e^{60 x^2}+e^4 x\right )^2}\right ) \, dx-\left (120 e^4\right ) \int \left (-\frac {e^3 x^2}{\left (-e^{60 x^2}+e^4 x\right )^2}+\frac {x^4}{\left (-e^{60 x^2}+e^4 x\right )^2}+\frac {x^4 \log (x)}{\left (-e^{60 x^2}+e^4 x\right )^2}\right ) \, dx-\left (3 \left (1+40 e^3\right )\right ) \int \frac {x}{-e^{60 x^2}+e^4 x} \, dx-(2 \log (x)) \int \frac {x}{-e^{60 x^2}+e^4 x} \, dx+(120 \log (x)) \int \frac {x^3}{-e^{60 x^2}+e^4 x} \, dx\\ &=2 \int \frac {\int \frac {x}{-e^{60 x^2}+e^4 x} \, dx}{x} \, dx+120 \int \frac {x^3}{-e^{60 x^2}+e^4 x} \, dx-120 \int \frac {\int \frac {x^3}{-e^{60 x^2}+e^4 x} \, dx}{x} \, dx+e^4 \int \frac {x^2}{\left (-e^{60 x^2}+e^4 x\right )^2} \, dx+e^4 \int \frac {x^2 \log (x)}{\left (-e^{60 x^2}+e^4 x\right )^2} \, dx-\left (120 e^4\right ) \int \frac {x^4}{\left (-e^{60 x^2}+e^4 x\right )^2} \, dx-\left (120 e^4\right ) \int \frac {x^4 \log (x)}{\left (-e^{60 x^2}+e^4 x\right )^2} \, dx-e^7 \int \frac {1}{\left (-e^{60 x^2}+e^4 x\right )^2} \, dx+\left (120 e^7\right ) \int \frac {x^2}{\left (-e^{60 x^2}+e^4 x\right )^2} \, dx-\left (3 \left (1+40 e^3\right )\right ) \int \frac {x}{-e^{60 x^2}+e^4 x} \, dx-(2 \log (x)) \int \frac {x}{-e^{60 x^2}+e^4 x} \, dx+(120 \log (x)) \int \frac {x^3}{-e^{60 x^2}+e^4 x} \, dx\\ &=2 \int \frac {\int \frac {x}{-e^{60 x^2}+e^4 x} \, dx}{x} \, dx+120 \int \frac {x^3}{-e^{60 x^2}+e^4 x} \, dx-120 \int \frac {\int \frac {x^3}{-e^{60 x^2}+e^4 x} \, dx}{x} \, dx+e^4 \int \frac {x^2}{\left (-e^{60 x^2}+e^4 x\right )^2} \, dx-e^4 \int \frac {\int \frac {x^2}{\left (e^{60 x^2}-e^4 x\right )^2} \, dx}{x} \, dx-\left (120 e^4\right ) \int \frac {x^4}{\left (-e^{60 x^2}+e^4 x\right )^2} \, dx+\left (120 e^4\right ) \int \frac {\int \frac {x^4}{\left (e^{60 x^2}-e^4 x\right )^2} \, dx}{x} \, dx-e^7 \int \frac {1}{\left (-e^{60 x^2}+e^4 x\right )^2} \, dx+\left (120 e^7\right ) \int \frac {x^2}{\left (-e^{60 x^2}+e^4 x\right )^2} \, dx-\left (3 \left (1+40 e^3\right )\right ) \int \frac {x}{-e^{60 x^2}+e^4 x} \, dx-(2 \log (x)) \int \frac {x}{-e^{60 x^2}+e^4 x} \, dx+(120 \log (x)) \int \frac {x^3}{-e^{60 x^2}+e^4 x} \, dx+\left (e^4 \log (x)\right ) \int \frac {x^2}{\left (-e^{60 x^2}+e^4 x\right )^2} \, dx-\left (120 e^4 \log (x)\right ) \int \frac {x^4}{\left (-e^{60 x^2}+e^4 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 34, normalized size = 1.10 \begin {gather*} -\frac {e^3-x^2-x^2 \log (x)}{e^{60 x^2}-e^4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^4*(-E^3 - 2*x^2) + E^(60*x^2)*(3*x + 120*E^3*x - 120*x^3) + (-(E^4*x^2) + E^(60*x^2)*(2*x - 120*x
^3))*Log[x])/(E^(120*x^2) - 2*E^(4 + 60*x^2)*x + E^8*x^2),x]

[Out]

-((E^3 - x^2 - x^2*Log[x])/(E^(60*x^2) - E^4*x))

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fricas [A]  time = 0.62, size = 38, normalized size = 1.23 \begin {gather*} -\frac {x^{2} e^{4} \log \relax (x) + x^{2} e^{4} - e^{7}}{x e^{8} - e^{\left (60 \, x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-120*x^3+2*x)*exp(60*x^2)-x^2*exp(4))*log(x)+(120*x*exp(3)-120*x^3+3*x)*exp(60*x^2)+(-exp(3)-2*x^
2)*exp(4))/(exp(60*x^2)^2-2*x*exp(4)*exp(60*x^2)+x^2*exp(4)^2),x, algorithm="fricas")

[Out]

-(x^2*e^4*log(x) + x^2*e^4 - e^7)/(x*e^8 - e^(60*x^2 + 4))

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giac [B]  time = 0.20, size = 80, normalized size = 2.58 \begin {gather*} -\frac {x^{3} e^{4} \log \relax (x) + x^{3} e^{4} - x^{2} e^{\left (60 \, x^{2}\right )} \log \relax (x) - x^{2} e^{\left (60 \, x^{2}\right )} - x e^{7} + e^{\left (60 \, x^{2} + 3\right )}}{x^{2} e^{8} - 2 \, x e^{\left (60 \, x^{2} + 4\right )} + e^{\left (120 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-120*x^3+2*x)*exp(60*x^2)-x^2*exp(4))*log(x)+(120*x*exp(3)-120*x^3+3*x)*exp(60*x^2)+(-exp(3)-2*x^
2)*exp(4))/(exp(60*x^2)^2-2*x*exp(4)*exp(60*x^2)+x^2*exp(4)^2),x, algorithm="giac")

[Out]

-(x^3*e^4*log(x) + x^3*e^4 - x^2*e^(60*x^2)*log(x) - x^2*e^(60*x^2) - x*e^7 + e^(60*x^2 + 3))/(x^2*e^8 - 2*x*e
^(60*x^2 + 4) + e^(120*x^2))

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maple [A]  time = 0.15, size = 48, normalized size = 1.55

method	result	size
risch	\(-\frac {x^{2} \ln \relax (x )}{x \,{\mathrm e}^{4}-{\mathrm e}^{60 x^{2}}}+\frac {{\mathrm e}^{3}-x^{2}}{x \,{\mathrm e}^{4}-{\mathrm e}^{60 x^{2}}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-120*x^3+2*x)*exp(60*x^2)-x^2*exp(4))*ln(x)+(120*x*exp(3)-120*x^3+3*x)*exp(60*x^2)+(-exp(3)-2*x^2)*exp(
4))/(exp(60*x^2)^2-2*x*exp(4)*exp(60*x^2)+x^2*exp(4)^2),x,method=_RETURNVERBOSE)

[Out]

-x^2/(x*exp(4)-exp(60*x^2))*ln(x)+(exp(3)-x^2)/(x*exp(4)-exp(60*x^2))

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maxima [A]  time = 0.56, size = 31, normalized size = 1.00 \begin {gather*} -\frac {x^{2} \log \relax (x) + x^{2} - e^{3}}{x e^{4} - e^{\left (60 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-120*x^3+2*x)*exp(60*x^2)-x^2*exp(4))*log(x)+(120*x*exp(3)-120*x^3+3*x)*exp(60*x^2)+(-exp(3)-2*x^
2)*exp(4))/(exp(60*x^2)^2-2*x*exp(4)*exp(60*x^2)+x^2*exp(4)^2),x, algorithm="maxima")

[Out]

-(x^2*log(x) + x^2 - e^3)/(x*e^4 - e^(60*x^2))

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mupad [B]  time = 1.20, size = 29, normalized size = 0.94 \begin {gather*} \frac {x^2\,\ln \relax (x)-{\mathrm {e}}^3+x^2}{{\mathrm {e}}^{60\,x^2}-x\,{\mathrm {e}}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(60*x^2)*(3*x + 120*x*exp(3) - 120*x^3) + log(x)*(exp(60*x^2)*(2*x - 120*x^3) - x^2*exp(4)) - exp(4)*(
exp(3) + 2*x^2))/(exp(120*x^2) + x^2*exp(8) - 2*x*exp(4)*exp(60*x^2)),x)

[Out]

(x^2*log(x) - exp(3) + x^2)/(exp(60*x^2) - x*exp(4))

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sympy [A]  time = 0.28, size = 24, normalized size = 0.77 \begin {gather*} \frac {x^{2} \log {\relax (x )} + x^{2} - e^{3}}{- x e^{4} + e^{60 x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-120*x**3+2*x)*exp(60*x**2)-x**2*exp(4))*ln(x)+(120*x*exp(3)-120*x**3+3*x)*exp(60*x**2)+(-exp(3)-
2*x**2)*exp(4))/(exp(60*x**2)**2-2*x*exp(4)*exp(60*x**2)+x**2*exp(4)**2),x)

[Out]

(x**2*log(x) + x**2 - exp(3))/(-x*exp(4) + exp(60*x**2))

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