3.100.37 \(\int \frac {e^{6-x} (-12+e^5 (-12-6 x)-6 x)}{x^3} \, dx\)

Optimal. Leaf size=17 \[ \frac {6 e^{6-x} \left (1+e^5\right )}{x^2} \]

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Rubi [A]  time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 {6 \left (1+e^5\right ) e^{6-x}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(6 - x)*(-12 + E^5*(-12 - 6*x) - 6*x))/x^3,x]

[Out]

(6*E^(6 - x)*(1 + E^5))/x^2

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 {6 e^{6-x} \left (1+e^5\right )}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {6 e^{6-x} \left (1+e^5\right )}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(6 - x)*(-12 + E^5*(-12 - 6*x) - 6*x))/x^3,x]

[Out]

(6*E^(6 - x)*(1 + E^5))/x^2

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fricas [A]  time = 2.01, size = 15, normalized size = 0.88 \begin {gather*} \frac {6 \, {\left (e^{5} + 1\right )} e^{\left (-x + 6\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x-12)*exp(5)-6*x-12)*exp(-x+6)/x^3,x, algorithm="fricas")

[Out]

6*(e^5 + 1)*e^(-x + 6)/x^2

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giac [A]  time = 0.13, size = 18, normalized size = 1.06 \begin {gather*} \frac {6 \, {\left (e^{\left (-x + 11\right )} + e^{\left (-x + 6\right )}\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x-12)*exp(5)-6*x-12)*exp(-x+6)/x^3,x, algorithm="giac")

[Out]

6*(e^(-x + 11) + e^(-x + 6))/x^2

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maple [A]  time = 0.08, size = 16, normalized size = 0.94

method	result	size
gosper	\(\frac {6 \left ({\mathrm e}^{5}+1\right ) {\mathrm e}^{-x +6}}{x^{2}}\)	\(16\)
risch	\(\frac {6 \left ({\mathrm e}^{5}+1\right ) {\mathrm e}^{-x +6}}{x^{2}}\)	\(16\)
norman	\(\frac {\left (6 \,{\mathrm e}^{5}+6\right ) {\mathrm e}^{-x +6}}{x^{2}}\)	\(17\)
derivativedivides	\(\frac {6 \,{\mathrm e}^{-x +6}}{x^{2}}+48 \,{\mathrm e}^{5} \left (\frac {{\mathrm e}^{-x +6}}{2 x^{2}}-\frac {{\mathrm e}^{-x +6}}{2 x}+\frac {{\mathrm e}^{6} \expIntegralEi \left (1, x\right )}{2}\right )-6 \,{\mathrm e}^{5} \left (\frac {3 \,{\mathrm e}^{-x +6}}{x^{2}}-\frac {4 \,{\mathrm e}^{-x +6}}{x}+4 \,{\mathrm e}^{6} \expIntegralEi \left (1, x\right )\right )\)	\(81\)
default	\(\frac {6 \,{\mathrm e}^{-x +6}}{x^{2}}+48 \,{\mathrm e}^{5} \left (\frac {{\mathrm e}^{-x +6}}{2 x^{2}}-\frac {{\mathrm e}^{-x +6}}{2 x}+\frac {{\mathrm e}^{6} \expIntegralEi \left (1, x\right )}{2}\right )-6 \,{\mathrm e}^{5} \left (\frac {3 \,{\mathrm e}^{-x +6}}{x^{2}}-\frac {4 \,{\mathrm e}^{-x +6}}{x}+4 \,{\mathrm e}^{6} \expIntegralEi \left (1, x\right )\right )\)	\(81\)
meijerg	\(\left (-6 \,{\mathrm e}^{5}-6\right ) {\mathrm e}^{6} \left (-\frac {1}{x}+1+\frac {-2 x +2}{2 x}-\frac {{\mathrm e}^{-x}}{x}+\expIntegralEi \left (1, x\right )\right )-12 \,{\mathrm e}^{11} \left (-\frac {1}{2 x^{2}}+\frac {1}{x}-\frac {3}{4}+\frac {9 x^{2}-12 x +6}{12 x^{2}}-\frac {\left (-3 x +3\right ) {\mathrm e}^{-x}}{6 x^{2}}-\frac {\expIntegralEi \left (1, x\right )}{2}\right )-12 \,{\mathrm e}^{6} \left (-\frac {1}{2 x^{2}}+\frac {1}{x}-\frac {3}{4}+\frac {9 x^{2}-12 x +6}{12 x^{2}}-\frac {\left (-3 x +3\right ) {\mathrm e}^{-x}}{6 x^{2}}-\frac {\expIntegralEi \left (1, x\right )}{2}\right )\)	\(136\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x-12)*exp(5)-6*x-12)*exp(-x+6)/x^3,x,method=_RETURNVERBOSE)

[Out]

6/x^2*(exp(5)+1)*exp(-x+6)

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maxima [C]  time = 0.41, size = 29, normalized size = 1.71 \begin {gather*} 6 \, e^{11} \Gamma \left (-1, x\right ) + 6 \, e^{6} \Gamma \left (-1, x\right ) + 12 \, e^{11} \Gamma \left (-2, x\right ) + 12 \, e^{6} \Gamma \left (-2, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x-12)*exp(5)-6*x-12)*exp(-x+6)/x^3,x, algorithm="maxima")

[Out]

6*e^11*gamma(-1, x) + 6*e^6*gamma(-1, x) + 12*e^11*gamma(-2, x) + 12*e^6*gamma(-2, x)

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mupad [B]  time = 8.12, size = 15, normalized size = 0.88 \begin {gather*} \frac {6\,{\mathrm {e}}^{6-x}\,\left ({\mathrm {e}}^5+1\right )}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(6 - x)*(6*x + exp(5)*(6*x + 12) + 12))/x^3,x)

[Out]

(6*exp(6 - x)*(exp(5) + 1))/x^2

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sympy [A]  time = 0.14, size = 14, normalized size = 0.82 \begin {gather*} \frac {\left (6 + 6 e^{5}\right ) e^{6 - x}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x-12)*exp(5)-6*x-12)*exp(-x+6)/x**3,x)

[Out]

(6 + 6*exp(5))*exp(6 - x)/x**2

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