3.65.89 \(\int \frac {-3 x^2+e^{e^{4+e^2 (36+24 x+4 x^2)+e^4 (81+108 x+54 x^2+12 x^3+x^4)}} (-75+e^{4+e^2 (36+24 x+4 x^2)+e^4 (81+108 x+54 x^2+12 x^3+x^4)} (e^2 (600 x+200 x^2)+e^4 (2700 x+2700 x^2+900 x^3+100 x^4)))}{x^4} \, dx\)

Optimal. Leaf size=28 \[ \frac {3+\frac {25 e^{e^{\left (2+e^2 (3+x)^2\right )^2}}}{x^2}}{x} \]

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Rubi [F]  time = 6.67, 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 {-3 x^2+\exp \left (\exp \left (4+e^2 \left (36+24 x+4 x^2\right )+e^4 \left (81+108 x+54 x^2+12 x^3+x^4\right )\right )\right ) \left (-75+\exp \left (4+e^2 \left (36+24 x+4 x^2\right )+e^4 \left (81+108 x+54 x^2+12 x^3+x^4\right )\right ) \left (e^2 \left (600 x+200 x^2\right )+e^4 \left (2700 x+2700 x^2+900 x^3+100 x^4\right )\right )\right )}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3*x^2 + E^E^(4 + E^2*(36 + 24*x + 4*x^2) + E^4*(81 + 108*x + 54*x^2 + 12*x^3 + x^4))*(-75 + E^(4 + E^2*(
36 + 24*x + 4*x^2) + E^4*(81 + 108*x + 54*x^2 + 12*x^3 + x^4))*(E^2*(600*x + 200*x^2) + E^4*(2700*x + 2700*x^2
+ 900*x^3 + 100*x^4))))/x^4,x]

[Out]

3/x - 75*Defer[Int][E^E^(2 + 9*E^2 + 6*E^2*x + E^2*x^2)^2/x^4, x] + 300*(2 + 9*E^2)*Defer[Int][E^(6 + E^(2 + E
^2*(3 + x)^2)^2 + 4*E^2*(3 + x)^2 + E^4*(3 + x)^4)/x^3, x] + 100*(2 + 27*E^2)*Defer[Int][E^(6 + E^(2 + E^2*(3
+ x)^2)^2 + 4*E^2*(3 + x)^2 + E^4*(3 + x)^4)/x^2, x] + 900*Defer[Int][E^(8 + E^(2 + E^2*(3 + x)^2)^2 + 4*E^2*(
3 + x)^2 + E^4*(3 + x)^4)/x, x] + 100*Defer[Subst][Defer[Int][E^(8 + E^(2 + E^2*x^2)^2 + 4*E^2*x^2 + E^4*x^4),
x], x, 3 + x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {3 \left (25 e^{e^{\left (2+e^2 (3+x)^2\right )^2}}+x^2\right )}{x^4}+\frac {100 \exp \left (6+e^{\left (2+e^2 (3+x)^2\right )^2}+4 e^2 (3+x)^2+e^4 (3+x)^4\right ) (3+x) \left (2+9 e^2+6 e^2 x+e^2 x^2\right )}{x^3}\right ) \, dx\\ &=-\left (3 \int \frac {25 e^{e^{\left (2+e^2 (3+x)^2\right )^2}}+x^2}{x^4} \, dx\right )+100 \int \frac {\exp \left (6+e^{\left (2+e^2 (3+x)^2\right )^2}+4 e^2 (3+x)^2+e^4 (3+x)^4\right ) (3+x) \left (2+9 e^2+6 e^2 x+e^2 x^2\right )}{x^3} \, dx\\ &=-\left (3 \int \left (\frac {25 e^{e^{\left (2+9 e^2+6 e^2 x+e^2 x^2\right )^2}}}{x^4}+\frac {1}{x^2}\right ) \, dx\right )+100 \int \left (\exp \left (8+e^{\left (2+e^2 (3+x)^2\right )^2}+4 e^2 (3+x)^2+e^4 (3+x)^4\right )+\frac {3 \exp \left (6+e^{\left (2+e^2 (3+x)^2\right )^2}+4 e^2 (3+x)^2+e^4 (3+x)^4\right ) \left (2+9 e^2\right )}{x^3}+\frac {\exp \left (6+e^{\left (2+e^2 (3+x)^2\right )^2}+4 e^2 (3+x)^2+e^4 (3+x)^4\right ) \left (2+27 e^2\right )}{x^2}+\frac {9 \exp \left (8+e^{\left (2+e^2 (3+x)^2\right )^2}+4 e^2 (3+x)^2+e^4 (3+x)^4\right )}{x}\right ) \, dx\\ &=\frac {3}{x}-75 \int \frac {e^{e^{\left (2+9 e^2+6 e^2 x+e^2 x^2\right )^2}}}{x^4} \, dx+100 \int \exp \left (8+e^{\left (2+e^2 (3+x)^2\right )^2}+4 e^2 (3+x)^2+e^4 (3+x)^4\right ) \, dx+900 \int \frac {\exp \left (8+e^{\left (2+e^2 (3+x)^2\right )^2}+4 e^2 (3+x)^2+e^4 (3+x)^4\right )}{x} \, dx+\left (300 \left (2+9 e^2\right )\right ) \int \frac {\exp \left (6+e^{\left (2+e^2 (3+x)^2\right )^2}+4 e^2 (3+x)^2+e^4 (3+x)^4\right )}{x^3} \, dx+\left (100 \left (2+27 e^2\right )\right ) \int \frac {\exp \left (6+e^{\left (2+e^2 (3+x)^2\right )^2}+4 e^2 (3+x)^2+e^4 (3+x)^4\right )}{x^2} \, dx\\ &=\frac {3}{x}-75 \int \frac {e^{e^{\left (2+9 e^2+6 e^2 x+e^2 x^2\right )^2}}}{x^4} \, dx+100 \operatorname {Subst}\left (\int e^{8+e^{\left (2+e^2 x^2\right )^2}+4 e^2 x^2+e^4 x^4} \, dx,x,3+x\right )+900 \int \frac {\exp \left (8+e^{\left (2+e^2 (3+x)^2\right )^2}+4 e^2 (3+x)^2+e^4 (3+x)^4\right )}{x} \, dx+\left (300 \left (2+9 e^2\right )\right ) \int \frac {\exp \left (6+e^{\left (2+e^2 (3+x)^2\right )^2}+4 e^2 (3+x)^2+e^4 (3+x)^4\right )}{x^3} \, dx+\left (100 \left (2+27 e^2\right )\right ) \int \frac {\exp \left (6+e^{\left (2+e^2 (3+x)^2\right )^2}+4 e^2 (3+x)^2+e^4 (3+x)^4\right )}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 4.30, size = 29, normalized size = 1.04 \begin {gather*} \frac {25 e^{e^{\left (2+e^2 (3+x)^2\right )^2}}+3 x^2}{x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*x^2 + E^E^(4 + E^2*(36 + 24*x + 4*x^2) + E^4*(81 + 108*x + 54*x^2 + 12*x^3 + x^4))*(-75 + E^(4 +
E^2*(36 + 24*x + 4*x^2) + E^4*(81 + 108*x + 54*x^2 + 12*x^3 + x^4))*(E^2*(600*x + 200*x^2) + E^4*(2700*x + 27
00*x^2 + 900*x^3 + 100*x^4))))/x^4,x]

[Out]

(25*E^E^(2 + E^2*(3 + x)^2)^2 + 3*x^2)/x^3

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fricas [A]  time = 0.75, size = 49, normalized size = 1.75 \begin {gather*} \frac {3 \, x^{2} + 25 \, e^{\left (e^{\left ({\left (x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81\right )} e^{4} + 4 \, {\left (x^{2} + 6 \, x + 9\right )} e^{2} + 4\right )}\right )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((100*x^4+900*x^3+2700*x^2+2700*x)*exp(1)^4+(200*x^2+600*x)*exp(1)^2)*exp((x^4+12*x^3+54*x^2+108*x
+81)*exp(1)^4+(4*x^2+24*x+36)*exp(1)^2+4)-75)*exp(exp((x^4+12*x^3+54*x^2+108*x+81)*exp(1)^4+(4*x^2+24*x+36)*ex
p(1)^2+4))-3*x^2)/x^4,x, algorithm="fricas")

[Out]

(3*x^2 + 25*e^(e^((x^4 + 12*x^3 + 54*x^2 + 108*x + 81)*e^4 + 4*(x^2 + 6*x + 9)*e^2 + 4)))/x^3

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {3 \, x^{2} - 25 \, {\left (4 \, {\left ({\left (x^{4} + 9 \, x^{3} + 27 \, x^{2} + 27 \, x\right )} e^{4} + 2 \, {\left (x^{2} + 3 \, x\right )} e^{2}\right )} e^{\left ({\left (x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81\right )} e^{4} + 4 \, {\left (x^{2} + 6 \, x + 9\right )} e^{2} + 4\right )} - 3\right )} e^{\left (e^{\left ({\left (x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81\right )} e^{4} + 4 \, {\left (x^{2} + 6 \, x + 9\right )} e^{2} + 4\right )}\right )}}{x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((100*x^4+900*x^3+2700*x^2+2700*x)*exp(1)^4+(200*x^2+600*x)*exp(1)^2)*exp((x^4+12*x^3+54*x^2+108*x
+81)*exp(1)^4+(4*x^2+24*x+36)*exp(1)^2+4)-75)*exp(exp((x^4+12*x^3+54*x^2+108*x+81)*exp(1)^4+(4*x^2+24*x+36)*ex
p(1)^2+4))-3*x^2)/x^4,x, algorithm="giac")

[Out]

integrate(-(3*x^2 - 25*(4*((x^4 + 9*x^3 + 27*x^2 + 27*x)*e^4 + 2*(x^2 + 3*x)*e^2)*e^((x^4 + 12*x^3 + 54*x^2 +
108*x + 81)*e^4 + 4*(x^2 + 6*x + 9)*e^2 + 4) - 3)*e^(e^((x^4 + 12*x^3 + 54*x^2 + 108*x + 81)*e^4 + 4*(x^2 + 6*
x + 9)*e^2 + 4)))/x^4, x)

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maple [A]  time = 0.22, size = 55, normalized size = 1.96

method	result	size
norman	\(\frac {3 x^{2}+25 \,{\mathrm e}^{{\mathrm e}^{\left (x^{4}+12 x^{3}+54 x^{2}+108 x +81\right ) {\mathrm e}^{4}+\left (4 x^{2}+24 x +36\right ) {\mathrm e}^{2}+4}}}{x^{3}}\)	\(55\)
risch	\(\frac {3}{x}+\frac {25 \,{\mathrm e}^{{\mathrm e}^{x^{4} {\mathrm e}^{4}+12 x^{3} {\mathrm e}^{4}+4 x^{2} {\mathrm e}^{2}+54 x^{2} {\mathrm e}^{4}+24 \,{\mathrm e}^{2} x +108 x \,{\mathrm e}^{4}+36 \,{\mathrm e}^{2}+81 \,{\mathrm e}^{4}+4}}}{x^{3}}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((100*x^4+900*x^3+2700*x^2+2700*x)*exp(1)^4+(200*x^2+600*x)*exp(1)^2)*exp((x^4+12*x^3+54*x^2+108*x+81)*e
xp(1)^4+(4*x^2+24*x+36)*exp(1)^2+4)-75)*exp(exp((x^4+12*x^3+54*x^2+108*x+81)*exp(1)^4+(4*x^2+24*x+36)*exp(1)^2
+4))-3*x^2)/x^4,x,method=_RETURNVERBOSE)

[Out]

(3*x^2+25*exp(exp((x^4+12*x^3+54*x^2+108*x+81)*exp(1)^4+(4*x^2+24*x+36)*exp(1)^2+4)))/x^3

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maxima [B]  time = 0.47, size = 60, normalized size = 2.14 \begin {gather*} \frac {3}{x} + \frac {25 \, e^{\left (e^{\left (x^{4} e^{4} + 12 \, x^{3} e^{4} + 54 \, x^{2} e^{4} + 4 \, x^{2} e^{2} + 108 \, x e^{4} + 24 \, x e^{2} + 81 \, e^{4} + 36 \, e^{2} + 4\right )}\right )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((100*x^4+900*x^3+2700*x^2+2700*x)*exp(1)^4+(200*x^2+600*x)*exp(1)^2)*exp((x^4+12*x^3+54*x^2+108*x
+81)*exp(1)^4+(4*x^2+24*x+36)*exp(1)^2+4)-75)*exp(exp((x^4+12*x^3+54*x^2+108*x+81)*exp(1)^4+(4*x^2+24*x+36)*ex
p(1)^2+4))-3*x^2)/x^4,x, algorithm="maxima")

[Out]

3/x + 25*e^(e^(x^4*e^4 + 12*x^3*e^4 + 54*x^2*e^4 + 4*x^2*e^2 + 108*x*e^4 + 24*x*e^2 + 81*e^4 + 36*e^2 + 4))/x^
3

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mupad [B]  time = 4.33, size = 68, normalized size = 2.43 \begin {gather*} \frac {25\,{\mathrm {e}}^{{\mathrm {e}}^{x^4\,{\mathrm {e}}^4}\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^2}\,{\mathrm {e}}^{12\,x^3\,{\mathrm {e}}^4}\,{\mathrm {e}}^{54\,x^2\,{\mathrm {e}}^4}\,{\mathrm {e}}^{36\,{\mathrm {e}}^2}\,{\mathrm {e}}^{81\,{\mathrm {e}}^4}\,{\mathrm {e}}^4\,{\mathrm {e}}^{24\,x\,{\mathrm {e}}^2}\,{\mathrm {e}}^{108\,x\,{\mathrm {e}}^4}}}{x^3}+\frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(exp(2)*(24*x + 4*x^2 + 36) + exp(4)*(108*x + 54*x^2 + 12*x^3 + x^4 + 81) + 4))*(exp(exp(2)*(24*x
+ 4*x^2 + 36) + exp(4)*(108*x + 54*x^2 + 12*x^3 + x^4 + 81) + 4)*(exp(2)*(600*x + 200*x^2) + exp(4)*(2700*x +
2700*x^2 + 900*x^3 + 100*x^4)) - 75) - 3*x^2)/x^4,x)

[Out]

(25*exp(exp(x^4*exp(4))*exp(4*x^2*exp(2))*exp(12*x^3*exp(4))*exp(54*x^2*exp(4))*exp(36*exp(2))*exp(81*exp(4))*
exp(4)*exp(24*x*exp(2))*exp(108*x*exp(4))))/x^3 + 3/x

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sympy [B]  time = 0.46, size = 48, normalized size = 1.71 \begin {gather*} \frac {3}{x} + \frac {25 e^{e^{\left (4 x^{2} + 24 x + 36\right ) e^{2} + \left (x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81\right ) e^{4} + 4}}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((100*x**4+900*x**3+2700*x**2+2700*x)*exp(1)**4+(200*x**2+600*x)*exp(1)**2)*exp((x**4+12*x**3+54*x
**2+108*x+81)*exp(1)**4+(4*x**2+24*x+36)*exp(1)**2+4)-75)*exp(exp((x**4+12*x**3+54*x**2+108*x+81)*exp(1)**4+(4
*x**2+24*x+36)*exp(1)**2+4))-3*x**2)/x**4,x)

[Out]

3/x + 25*exp(exp((4*x**2 + 24*x + 36)*exp(2) + (x**4 + 12*x**3 + 54*x**2 + 108*x + 81)*exp(4) + 4))/x**3

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