3.49.60 \(\int e^{81 x+e^{100 x-4 x^3} x-12 e^{75 x-3 x^3} x+54 e^{50 x-2 x^3} x-108 e^{25 x-x^3} x} (81+e^{50 x-2 x^3} (54+2700 x-324 x^3)+e^{100 x-4 x^3} (1+100 x-12 x^3)+e^{75 x-3 x^3} (-12-900 x+108 x^3)+e^{25 x-x^3} (-108-2700 x+324 x^3)) \, dx\)

Optimal. Leaf size=25 \[ e^{\left (3-e^{-x+x \left (26-x^2\right )}\right )^4 x} \]

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Rubi [F]  time = 7.43, 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 \exp \left (81 x+e^{100 x-4 x^3} x-12 e^{75 x-3 x^3} x+54 e^{50 x-2 x^3} x-108 e^{25 x-x^3} x\right ) \left (81+e^{50 x-2 x^3} \left (54+2700 x-324 x^3\right )+e^{100 x-4 x^3} \left (1+100 x-12 x^3\right )+e^{75 x-3 x^3} \left (-12-900 x+108 x^3\right )+e^{25 x-x^3} \left (-108-2700 x+324 x^3\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(81*x + E^(100*x - 4*x^3)*x - 12*E^(75*x - 3*x^3)*x + 54*E^(50*x - 2*x^3)*x - 108*E^(25*x - x^3)*x)*(81
+ E^(50*x - 2*x^3)*(54 + 2700*x - 324*x^3) + E^(100*x - 4*x^3)*(1 + 100*x - 12*x^3) + E^(75*x - 3*x^3)*(-12 -
900*x + 108*x^3) + E^(25*x - x^3)*(-108 - 2700*x + 324*x^3)),x]

[Out]

81*Defer[Int][E^(((E^(25*x) - 3*E^x^3)^4*x)/E^(4*x^3)), x] + Defer[Int][E^(100*x + ((E^(25*x) - 3*E^x^3)^4*x)/
E^(4*x^3) - 4*x^3), x] - 12*Defer[Int][E^(75*x + ((E^(25*x) - 3*E^x^3)^4*x)/E^(4*x^3) - 3*x^3), x] - 108*Defer
[Int][E^(25*x + ((E^(25*x) - 3*E^x^3)^4*x)/E^(4*x^3) - x^3), x] + 54*Defer[Int][E^(((E^(25*x) - 3*E^x^3)^4*x)/
E^(4*x^3) - 4*x^3 + 2*x*(25 + x^2)), x] + 100*Defer[Int][E^(100*x + ((E^(25*x) - 3*E^x^3)^4*x)/E^(4*x^3) - 4*x
^3)*x, x] - 900*Defer[Int][E^(75*x + ((E^(25*x) - 3*E^x^3)^4*x)/E^(4*x^3) - 3*x^3)*x, x] - 2700*Defer[Int][E^(
25*x + ((E^(25*x) - 3*E^x^3)^4*x)/E^(4*x^3) - x^3)*x, x] + 2700*Defer[Int][E^(((E^(25*x) - 3*E^x^3)^4*x)/E^(4*
x^3) - 4*x^3 + 2*x*(25 + x^2))*x, x] - 12*Defer[Int][E^(100*x + ((E^(25*x) - 3*E^x^3)^4*x)/E^(4*x^3) - 4*x^3)*
x^3, x] + 108*Defer[Int][E^(75*x + ((E^(25*x) - 3*E^x^3)^4*x)/E^(4*x^3) - 3*x^3)*x^3, x] + 324*Defer[Int][E^(2
5*x + ((E^(25*x) - 3*E^x^3)^4*x)/E^(4*x^3) - x^3)*x^3, x] - 324*Defer[Int][E^(((E^(25*x) - 3*E^x^3)^4*x)/E^(4*
x^3) - 4*x^3 + 2*x*(25 + x^2))*x^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \exp \left (e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-4 x^3\right ) \left (e^{25 x}-3 e^{x^3}\right )^3 \left (-3 e^{x^3}-e^{25 x} \left (-1-100 x+12 x^3\right )\right ) \, dx\\ &=\int \left (81 e^{e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x}+108 \exp \left (25 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-x^3\right ) \left (-1-25 x+3 x^3\right )-54 \exp \left (e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-4 x^3+2 x \left (25+x^2\right )\right ) \left (-1-50 x+6 x^3\right )+12 \exp \left (75 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-3 x^3\right ) \left (-1-75 x+9 x^3\right )-\exp \left (100 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-4 x^3\right ) \left (-1-100 x+12 x^3\right )\right ) \, dx\\ &=12 \int \exp \left (75 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-3 x^3\right ) \left (-1-75 x+9 x^3\right ) \, dx-54 \int \exp \left (e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-4 x^3+2 x \left (25+x^2\right )\right ) \left (-1-50 x+6 x^3\right ) \, dx+81 \int e^{e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x} \, dx+108 \int \exp \left (25 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-x^3\right ) \left (-1-25 x+3 x^3\right ) \, dx-\int \exp \left (100 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-4 x^3\right ) \left (-1-100 x+12 x^3\right ) \, dx\\ &=12 \int \left (-\exp \left (75 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-3 x^3\right )-75 \exp \left (75 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-3 x^3\right ) x+9 \exp \left (75 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-3 x^3\right ) x^3\right ) \, dx-54 \int \left (-\exp \left (e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-4 x^3+2 x \left (25+x^2\right )\right )-50 \exp \left (e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-4 x^3+2 x \left (25+x^2\right )\right ) x+6 \exp \left (e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-4 x^3+2 x \left (25+x^2\right )\right ) x^3\right ) \, dx+81 \int e^{e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x} \, dx+108 \int \left (-\exp \left (25 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-x^3\right )-25 \exp \left (25 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-x^3\right ) x+3 \exp \left (25 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-x^3\right ) x^3\right ) \, dx-\int \left (-\exp \left (100 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-4 x^3\right )-100 \exp \left (100 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-4 x^3\right ) x+12 \exp \left (100 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-4 x^3\right ) x^3\right ) \, dx\\ &=-\left (12 \int \exp \left (75 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-3 x^3\right ) \, dx\right )-12 \int \exp \left (100 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-4 x^3\right ) x^3 \, dx+54 \int \exp \left (e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-4 x^3+2 x \left (25+x^2\right )\right ) \, dx+81 \int e^{e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x} \, dx+100 \int \exp \left (100 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-4 x^3\right ) x \, dx-108 \int \exp \left (25 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-x^3\right ) \, dx+108 \int \exp \left (75 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-3 x^3\right ) x^3 \, dx+324 \int \exp \left (25 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-x^3\right ) x^3 \, dx-324 \int \exp \left (e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-4 x^3+2 x \left (25+x^2\right )\right ) x^3 \, dx-900 \int \exp \left (75 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-3 x^3\right ) x \, dx-2700 \int \exp \left (25 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-x^3\right ) x \, dx+2700 \int \exp \left (e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-4 x^3+2 x \left (25+x^2\right )\right ) x \, dx+\int \exp \left (100 x+e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-4 x^3\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 2.62, size = 26, normalized size = 1.04 \begin {gather*} e^{e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(81*x + E^(100*x - 4*x^3)*x - 12*E^(75*x - 3*x^3)*x + 54*E^(50*x - 2*x^3)*x - 108*E^(25*x - x^3)*x
)*(81 + E^(50*x - 2*x^3)*(54 + 2700*x - 324*x^3) + E^(100*x - 4*x^3)*(1 + 100*x - 12*x^3) + E^(75*x - 3*x^3)*(
-12 - 900*x + 108*x^3) + E^(25*x - x^3)*(-108 - 2700*x + 324*x^3)),x]

[Out]

E^(((E^(25*x) - 3*E^x^3)^4*x)/E^(4*x^3))

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fricas [B]  time = 0.60, size = 56, normalized size = 2.24 \begin {gather*} e^{\left (-108 \, x e^{\left (-x^{3} + 25 \, x\right )} + 54 \, x e^{\left (-2 \, x^{3} + 50 \, x\right )} - 12 \, x e^{\left (-3 \, x^{3} + 75 \, x\right )} + x e^{\left (-4 \, x^{3} + 100 \, x\right )} + 81 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x^3+100*x+1)*exp(-x^3+25*x)^4+(108*x^3-900*x-12)*exp(-x^3+25*x)^3+(-324*x^3+2700*x+54)*exp(-x^
3+25*x)^2+(324*x^3-2700*x-108)*exp(-x^3+25*x)+81)*exp(x*exp(-x^3+25*x)^4-12*x*exp(-x^3+25*x)^3+54*x*exp(-x^3+2
5*x)^2-108*x*exp(-x^3+25*x)+81*x),x, algorithm="fricas")

[Out]

e^(-108*x*e^(-x^3 + 25*x) + 54*x*e^(-2*x^3 + 50*x) - 12*x*e^(-3*x^3 + 75*x) + x*e^(-4*x^3 + 100*x) + 81*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (108 \, {\left (3 \, x^{3} - 25 \, x - 1\right )} e^{\left (-x^{3} + 25 \, x\right )} - 54 \, {\left (6 \, x^{3} - 50 \, x - 1\right )} e^{\left (-2 \, x^{3} + 50 \, x\right )} + 12 \, {\left (9 \, x^{3} - 75 \, x - 1\right )} e^{\left (-3 \, x^{3} + 75 \, x\right )} - {\left (12 \, x^{3} - 100 \, x - 1\right )} e^{\left (-4 \, x^{3} + 100 \, x\right )} + 81\right )} e^{\left (-108 \, x e^{\left (-x^{3} + 25 \, x\right )} + 54 \, x e^{\left (-2 \, x^{3} + 50 \, x\right )} - 12 \, x e^{\left (-3 \, x^{3} + 75 \, x\right )} + x e^{\left (-4 \, x^{3} + 100 \, x\right )} + 81 \, x\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x^3+100*x+1)*exp(-x^3+25*x)^4+(108*x^3-900*x-12)*exp(-x^3+25*x)^3+(-324*x^3+2700*x+54)*exp(-x^
3+25*x)^2+(324*x^3-2700*x-108)*exp(-x^3+25*x)+81)*exp(x*exp(-x^3+25*x)^4-12*x*exp(-x^3+25*x)^3+54*x*exp(-x^3+2
5*x)^2-108*x*exp(-x^3+25*x)+81*x),x, algorithm="giac")

[Out]

integrate((108*(3*x^3 - 25*x - 1)*e^(-x^3 + 25*x) - 54*(6*x^3 - 50*x - 1)*e^(-2*x^3 + 50*x) + 12*(9*x^3 - 75*x
- 1)*e^(-3*x^3 + 75*x) - (12*x^3 - 100*x - 1)*e^(-4*x^3 + 100*x) + 81)*e^(-108*x*e^(-x^3 + 25*x) + 54*x*e^(-2
*x^3 + 50*x) - 12*x*e^(-3*x^3 + 75*x) + x*e^(-4*x^3 + 100*x) + 81*x), x)

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maple [B]  time = 0.15, size = 52, normalized size = 2.08

method	result	size
risch	\({\mathrm e}^{x \left ({\mathrm e}^{-4 x \left (x -5\right ) \left (5+x \right )}-12 \,{\mathrm e}^{-3 x \left (x -5\right ) \left (5+x \right )}+54 \,{\mathrm e}^{-2 x \left (x -5\right ) \left (5+x \right )}-108 \,{\mathrm e}^{-x \left (x -5\right ) \left (5+x \right )}+81\right )}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-12*x^3+100*x+1)*exp(-x^3+25*x)^4+(108*x^3-900*x-12)*exp(-x^3+25*x)^3+(-324*x^3+2700*x+54)*exp(-x^3+25*x
)^2+(324*x^3-2700*x-108)*exp(-x^3+25*x)+81)*exp(x*exp(-x^3+25*x)^4-12*x*exp(-x^3+25*x)^3+54*x*exp(-x^3+25*x)^2
-108*x*exp(-x^3+25*x)+81*x),x,method=_RETURNVERBOSE)

[Out]

exp(x*(exp(-4*x*(x-5)*(5+x))-12*exp(-3*x*(x-5)*(5+x))+54*exp(-2*x*(x-5)*(5+x))-108*exp(-x*(x-5)*(5+x))+81))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (108 \, {\left (3 \, x^{3} - 25 \, x - 1\right )} e^{\left (-x^{3} + 25 \, x\right )} - 54 \, {\left (6 \, x^{3} - 50 \, x - 1\right )} e^{\left (-2 \, x^{3} + 50 \, x\right )} + 12 \, {\left (9 \, x^{3} - 75 \, x - 1\right )} e^{\left (-3 \, x^{3} + 75 \, x\right )} - {\left (12 \, x^{3} - 100 \, x - 1\right )} e^{\left (-4 \, x^{3} + 100 \, x\right )} + 81\right )} e^{\left (-108 \, x e^{\left (-x^{3} + 25 \, x\right )} + 54 \, x e^{\left (-2 \, x^{3} + 50 \, x\right )} - 12 \, x e^{\left (-3 \, x^{3} + 75 \, x\right )} + x e^{\left (-4 \, x^{3} + 100 \, x\right )} + 81 \, x\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x^3+100*x+1)*exp(-x^3+25*x)^4+(108*x^3-900*x-12)*exp(-x^3+25*x)^3+(-324*x^3+2700*x+54)*exp(-x^
3+25*x)^2+(324*x^3-2700*x-108)*exp(-x^3+25*x)+81)*exp(x*exp(-x^3+25*x)^4-12*x*exp(-x^3+25*x)^3+54*x*exp(-x^3+2
5*x)^2-108*x*exp(-x^3+25*x)+81*x),x, algorithm="maxima")

[Out]

integrate((108*(3*x^3 - 25*x - 1)*e^(-x^3 + 25*x) - 54*(6*x^3 - 50*x - 1)*e^(-2*x^3 + 50*x) + 12*(9*x^3 - 75*x
- 1)*e^(-3*x^3 + 75*x) - (12*x^3 - 100*x - 1)*e^(-4*x^3 + 100*x) + 81)*e^(-108*x*e^(-x^3 + 25*x) + 54*x*e^(-2
*x^3 + 50*x) - 12*x*e^(-3*x^3 + 75*x) + x*e^(-4*x^3 + 100*x) + 81*x), x)

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mupad [B]  time = 3.88, size = 60, normalized size = 2.40 \begin {gather*} {\mathrm {e}}^{-12\,x\,{\mathrm {e}}^{75\,x}\,{\mathrm {e}}^{-3\,x^3}}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{100\,x}\,{\mathrm {e}}^{-4\,x^3}}\,{\mathrm {e}}^{54\,x\,{\mathrm {e}}^{50\,x}\,{\mathrm {e}}^{-2\,x^3}}\,{\mathrm {e}}^{-108\,x\,{\mathrm {e}}^{25\,x}\,{\mathrm {e}}^{-x^3}}\,{\mathrm {e}}^{81\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(81*x - 108*x*exp(25*x - x^3) + 54*x*exp(50*x - 2*x^3) - 12*x*exp(75*x - 3*x^3) + x*exp(100*x - 4*x^3))
*(exp(100*x - 4*x^3)*(100*x - 12*x^3 + 1) - exp(75*x - 3*x^3)*(900*x - 108*x^3 + 12) + exp(50*x - 2*x^3)*(2700
*x - 324*x^3 + 54) - exp(25*x - x^3)*(2700*x - 324*x^3 + 108) + 81),x)

[Out]

exp(-12*x*exp(75*x)*exp(-3*x^3))*exp(x*exp(100*x)*exp(-4*x^3))*exp(54*x*exp(50*x)*exp(-2*x^3))*exp(-108*x*exp(
25*x)*exp(-x^3))*exp(81*x)

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sympy [B]  time = 0.71, size = 54, normalized size = 2.16 \begin {gather*} e^{x e^{- 4 x^{3} + 100 x} - 12 x e^{- 3 x^{3} + 75 x} + 54 x e^{- 2 x^{3} + 50 x} - 108 x e^{- x^{3} + 25 x} + 81 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x**3+100*x+1)*exp(-x**3+25*x)**4+(108*x**3-900*x-12)*exp(-x**3+25*x)**3+(-324*x**3+2700*x+54)*
exp(-x**3+25*x)**2+(324*x**3-2700*x-108)*exp(-x**3+25*x)+81)*exp(x*exp(-x**3+25*x)**4-12*x*exp(-x**3+25*x)**3+
54*x*exp(-x**3+25*x)**2-108*x*exp(-x**3+25*x)+81*x),x)

[Out]

exp(x*exp(-4*x**3 + 100*x) - 12*x*exp(-3*x**3 + 75*x) + 54*x*exp(-2*x**3 + 50*x) - 108*x*exp(-x**3 + 25*x) + 8
1*x)

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