3.6.59 \(\int \frac {e^{-2 e^2+2 e^{10}-2 x} (-54-18 x)}{x^7} \, dx\)

Optimal. Leaf size=21 \[ \frac {9 e^{-2 e^2+2 e^{10}-2 x}}{x^6} \]

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Rubi [A]  time = 0.06, antiderivative size = 23, normalized size of antiderivative = 1.10, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2197} \begin {gather*} \frac {9 e^{-2 x-2 e^2 \left (1-e^8\right )}}{x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-2*E^2 + 2*E^10 - 2*x)*(-54 - 18*x))/x^7,x]

[Out]

(9*E^(-2*E^2*(1 - E^8) - 2*x))/x^6

Rule 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {9 e^{-2 e^2 \left (1-e^8\right )-2 x}}{x^6}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 0.90 \begin {gather*} \frac {9 e^{-2 \left (e^2-e^{10}+x\right )}}{x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-2*E^2 + 2*E^10 - 2*x)*(-54 - 18*x))/x^7,x]

[Out]

9/(E^(2*(E^2 - E^10 + x))*x^6)

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fricas [A]  time = 0.59, size = 18, normalized size = 0.86 \begin {gather*} \frac {9 \, e^{\left (-2 \, x + 2 \, e^{10} - 2 \, e^{2}\right )}}{x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*x-54)*exp(exp(10))^2/x^7/exp(x+exp(2))^2,x, algorithm="fricas")

[Out]

9*e^(-2*x + 2*e^10 - 2*e^2)/x^6

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giac [A]  time = 0.57, size = 18, normalized size = 0.86 \begin {gather*} \frac {9 \, e^{\left (-2 \, x + 2 \, e^{10} - 2 \, e^{2}\right )}}{x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*x-54)*exp(exp(10))^2/x^7/exp(x+exp(2))^2,x, algorithm="giac")

[Out]

9*e^(-2*x + 2*e^10 - 2*e^2)/x^6

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maple [A]  time = 0.17, size = 18, normalized size = 0.86

method	result	size
gosper	\(\frac {9 \,{\mathrm e}^{2 \,{\mathrm e}^{10}} {\mathrm e}^{-2 \,{\mathrm e}^{2}-2 x}}{x^{6}}\)	\(18\)
norman	\(\frac {9 \,{\mathrm e}^{2 \,{\mathrm e}^{10}} {\mathrm e}^{-2 \,{\mathrm e}^{2}-2 x}}{x^{6}}\)	\(18\)
risch	\(\frac {9 \,{\mathrm e}^{2 \,{\mathrm e}^{10}-2 \,{\mathrm e}^{2}-2 x}}{x^{6}}\)	\(19\)
derivativedivides	\(18 \,{\mathrm e}^{2 \,{\mathrm e}^{10}} \left ({\mathrm e}^{2} \left (-\frac {{\mathrm e}^{-2 \,{\mathrm e}^{2}-2 x} \left (4 \,{\mathrm e}^{10}-20 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{8}+40 \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{6}-40 \left (x +{\mathrm e}^{2}\right )^{3} {\mathrm e}^{4}+20 \left (x +{\mathrm e}^{2}\right )^{4} {\mathrm e}^{2}-4 \left (x +{\mathrm e}^{2}\right )^{5}+2 \,{\mathrm e}^{8}-8 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{6}+12 \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{4}-8 \left (x +{\mathrm e}^{2}\right )^{3} {\mathrm e}^{2}+2 \left (x +{\mathrm e}^{2}\right )^{4}+2 \,{\mathrm e}^{6}-6 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{4}+6 \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{2}-2 \left (x +{\mathrm e}^{2}\right )^{3}+3 \,{\mathrm e}^{4}-6 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{2}+3 \left (x +{\mathrm e}^{2}\right )^{2}-6 x +15\right )}{90 x^{6}}-\frac {4 \,{\mathrm e}^{-2 \,{\mathrm e}^{2}} \expIntegralEi \left (1, 2 x \right )}{45}\right )+\frac {{\mathrm e}^{-2 \,{\mathrm e}^{2}-2 x} \left (4 \,{\mathrm e}^{10}-20 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{8}+40 \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{6}-40 \left (x +{\mathrm e}^{2}\right )^{3} {\mathrm e}^{4}+20 \left (x +{\mathrm e}^{2}\right )^{4} {\mathrm e}^{2}-4 \left (x +{\mathrm e}^{2}\right )^{5}+2 \,{\mathrm e}^{8}-8 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{6}+12 \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{4}-8 \left (x +{\mathrm e}^{2}\right )^{3} {\mathrm e}^{2}+2 \left (x +{\mathrm e}^{2}\right )^{4}+2 \,{\mathrm e}^{6}-6 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{4}+6 \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{2}-2 \left (x +{\mathrm e}^{2}\right )^{3}+3 \,{\mathrm e}^{4}-6 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{2}+3 \left (x +{\mathrm e}^{2}\right )^{2}-6 x +15\right )}{30 x^{6}}+\frac {4 \,{\mathrm e}^{-2 \,{\mathrm e}^{2}} \expIntegralEi \left (1, 2 x \right )}{15}+\frac {{\mathrm e}^{-2 \,{\mathrm e}^{2}-2 x} \left (18 x -4 \,{\mathrm e}^{8}-10 \,{\mathrm e}^{10}-3 \,{\mathrm e}^{6}-3 \,{\mathrm e}^{4}+15 \,{\mathrm e}^{2}+4 \,{\mathrm e}^{12}-9 \left (x +{\mathrm e}^{2}\right )^{2}-58 \left (x +{\mathrm e}^{2}\right )^{4} {\mathrm e}^{2}+22 \left (x +{\mathrm e}^{2}\right )^{3} {\mathrm e}^{2}-15 \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{2}+12 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{2}-4 \left (x +{\mathrm e}^{2}\right )^{5} {\mathrm e}^{2}+52 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{8}-108 \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{6}+112 \left (x +{\mathrm e}^{2}\right )^{3} {\mathrm e}^{4}+18 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{6}-30 \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{4}+12 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{4}-20 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{10}+40 \,{\mathrm e}^{8} \left (x +{\mathrm e}^{2}\right )^{2}-40 \,{\mathrm e}^{6} \left (x +{\mathrm e}^{2}\right )^{3}+20 \,{\mathrm e}^{4} \left (x +{\mathrm e}^{2}\right )^{4}+12 \left (x +{\mathrm e}^{2}\right )^{5}-6 \left (x +{\mathrm e}^{2}\right )^{4}+6 \left (x +{\mathrm e}^{2}\right )^{3}\right )}{90 x^{6}}+\left (\frac {4 \,{\mathrm e}^{2}}{45}-\frac {4}{15}\right ) {\mathrm e}^{-2 \,{\mathrm e}^{2}} \expIntegralEi \left (1, 2 x \right )\right )\)	\(656\)
default	\(18 \,{\mathrm e}^{2 \,{\mathrm e}^{10}} \left ({\mathrm e}^{2} \left (-\frac {{\mathrm e}^{-2 \,{\mathrm e}^{2}-2 x} \left (4 \,{\mathrm e}^{10}-20 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{8}+40 \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{6}-40 \left (x +{\mathrm e}^{2}\right )^{3} {\mathrm e}^{4}+20 \left (x +{\mathrm e}^{2}\right )^{4} {\mathrm e}^{2}-4 \left (x +{\mathrm e}^{2}\right )^{5}+2 \,{\mathrm e}^{8}-8 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{6}+12 \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{4}-8 \left (x +{\mathrm e}^{2}\right )^{3} {\mathrm e}^{2}+2 \left (x +{\mathrm e}^{2}\right )^{4}+2 \,{\mathrm e}^{6}-6 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{4}+6 \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{2}-2 \left (x +{\mathrm e}^{2}\right )^{3}+3 \,{\mathrm e}^{4}-6 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{2}+3 \left (x +{\mathrm e}^{2}\right )^{2}-6 x +15\right )}{90 x^{6}}-\frac {4 \,{\mathrm e}^{-2 \,{\mathrm e}^{2}} \expIntegralEi \left (1, 2 x \right )}{45}\right )+\frac {{\mathrm e}^{-2 \,{\mathrm e}^{2}-2 x} \left (4 \,{\mathrm e}^{10}-20 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{8}+40 \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{6}-40 \left (x +{\mathrm e}^{2}\right )^{3} {\mathrm e}^{4}+20 \left (x +{\mathrm e}^{2}\right )^{4} {\mathrm e}^{2}-4 \left (x +{\mathrm e}^{2}\right )^{5}+2 \,{\mathrm e}^{8}-8 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{6}+12 \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{4}-8 \left (x +{\mathrm e}^{2}\right )^{3} {\mathrm e}^{2}+2 \left (x +{\mathrm e}^{2}\right )^{4}+2 \,{\mathrm e}^{6}-6 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{4}+6 \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{2}-2 \left (x +{\mathrm e}^{2}\right )^{3}+3 \,{\mathrm e}^{4}-6 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{2}+3 \left (x +{\mathrm e}^{2}\right )^{2}-6 x +15\right )}{30 x^{6}}+\frac {4 \,{\mathrm e}^{-2 \,{\mathrm e}^{2}} \expIntegralEi \left (1, 2 x \right )}{15}+\frac {{\mathrm e}^{-2 \,{\mathrm e}^{2}-2 x} \left (18 x -4 \,{\mathrm e}^{8}-10 \,{\mathrm e}^{10}-3 \,{\mathrm e}^{6}-3 \,{\mathrm e}^{4}+15 \,{\mathrm e}^{2}+4 \,{\mathrm e}^{12}-9 \left (x +{\mathrm e}^{2}\right )^{2}-58 \left (x +{\mathrm e}^{2}\right )^{4} {\mathrm e}^{2}+22 \left (x +{\mathrm e}^{2}\right )^{3} {\mathrm e}^{2}-15 \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{2}+12 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{2}-4 \left (x +{\mathrm e}^{2}\right )^{5} {\mathrm e}^{2}+52 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{8}-108 \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{6}+112 \left (x +{\mathrm e}^{2}\right )^{3} {\mathrm e}^{4}+18 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{6}-30 \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{4}+12 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{4}-20 \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{10}+40 \,{\mathrm e}^{8} \left (x +{\mathrm e}^{2}\right )^{2}-40 \,{\mathrm e}^{6} \left (x +{\mathrm e}^{2}\right )^{3}+20 \,{\mathrm e}^{4} \left (x +{\mathrm e}^{2}\right )^{4}+12 \left (x +{\mathrm e}^{2}\right )^{5}-6 \left (x +{\mathrm e}^{2}\right )^{4}+6 \left (x +{\mathrm e}^{2}\right )^{3}\right )}{90 x^{6}}+\left (\frac {4 \,{\mathrm e}^{2}}{45}-\frac {4}{15}\right ) {\mathrm e}^{-2 \,{\mathrm e}^{2}} \expIntegralEi \left (1, 2 x \right )\right )\)	\(656\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-18*x-54)*exp(exp(10))^2/x^7/exp(x+exp(2))^2,x,method=_RETURNVERBOSE)

[Out]

9*exp(exp(10))^2/x^6/exp(x+exp(2))^2

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maxima [C]  time = 0.58, size = 57, normalized size = 2.71 \begin {gather*} 576 \, e^{\left (2 \, {\left (e^{4} + 1\right )} {\left (e^{2} + 1\right )} {\left (e + 1\right )} {\left (e - 1\right )} e^{2}\right )} \Gamma \left (-5, 2 \, x\right ) + 3456 \, e^{\left (2 \, {\left (e^{4} + 1\right )} {\left (e^{2} + 1\right )} {\left (e + 1\right )} {\left (e - 1\right )} e^{2}\right )} \Gamma \left (-6, 2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*x-54)*exp(exp(10))^2/x^7/exp(x+exp(2))^2,x, algorithm="maxima")

[Out]

576*e^(2*(e^4 + 1)*(e^2 + 1)*(e + 1)*(e - 1)*e^2)*gamma(-5, 2*x) + 3456*e^(2*(e^4 + 1)*(e^2 + 1)*(e + 1)*(e -
1)*e^2)*gamma(-6, 2*x)

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mupad [B]  time = 0.08, size = 19, normalized size = 0.90 \begin {gather*} \frac {9\,{\mathrm {e}}^{-2\,{\mathrm {e}}^2}\,{\mathrm {e}}^{2\,{\mathrm {e}}^{10}}\,{\mathrm {e}}^{-2\,x}}{x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*exp(10))*exp(- 2*x - 2*exp(2))*(18*x + 54))/x^7,x)

[Out]

(9*exp(-2*exp(2))*exp(2*exp(10))*exp(-2*x))/x^6

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sympy [A]  time = 0.12, size = 22, normalized size = 1.05 \begin {gather*} \frac {9 e^{- 2 x - 2 e^{2}} e^{2 e^{10}}}{x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*x-54)*exp(exp(10))**2/x**7/exp(x+exp(2))**2,x)

[Out]

9*exp(-2*x - 2*exp(2))*exp(2*exp(10))/x**6

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