3.6.12 \(\int \frac {e^{\frac {1-6 x+x^2}{x}} (-16+16 x^2)}{x^2} \, dx\)

Optimal. Leaf size=10 \[ 16 e^{-6+\frac {1}{x}+x} \]

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Rubi [A]  time = 0.21, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6688, 12, 6706} \begin {gather*} 16 e^{x+\frac {1}{x}-6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((1 - 6*x + x^2)/x)*(-16 + 16*x^2))/x^2,x]

[Out]

16*E^(-6 + x^(-1) + x)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {16 e^{-6+\frac {1}{x}+x} \left (-1+x^2\right )}{x^2} \, dx\\ &=16 \int \frac {e^{-6+\frac {1}{x}+x} \left (-1+x^2\right )}{x^2} \, dx\\ &=16 e^{-6+\frac {1}{x}+x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 10, normalized size = 1.00 \begin {gather*} 16 e^{-6+\frac {1}{x}+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((1 - 6*x + x^2)/x)*(-16 + 16*x^2))/x^2,x]

[Out]

16*E^(-6 + x^(-1) + x)

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fricas [A]  time = 0.57, size = 15, normalized size = 1.50 \begin {gather*} 16 \, e^{\left (\frac {x^{2} - 6 \, x + 1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^2-16)*exp((x^2-6*x+1)/x)/x^2,x, algorithm="fricas")

[Out]

16*e^((x^2 - 6*x + 1)/x)

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giac [A]  time = 0.29, size = 9, normalized size = 0.90 \begin {gather*} 16 \, e^{\left (x + \frac {1}{x} - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^2-16)*exp((x^2-6*x+1)/x)/x^2,x, algorithm="giac")

[Out]

16*e^(x + 1/x - 6)

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maple [A]  time = 0.28, size = 16, normalized size = 1.60

method	result	size
gosper	\(16 \,{\mathrm e}^{\frac {x^{2}-6 x +1}{x}}\)	\(16\)
norman	\(16 \,{\mathrm e}^{\frac {x^{2}-6 x +1}{x}}\)	\(16\)
risch	\(16 \,{\mathrm e}^{\frac {x^{2}-6 x +1}{x}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x^2-16)*exp((x^2-6*x+1)/x)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.72, size = 9, normalized size = 0.90 \begin {gather*} 16 \, e^{\left (x + \frac {1}{x} - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^2-16)*exp((x^2-6*x+1)/x)/x^2,x, algorithm="maxima")

[Out]

16*e^(x + 1/x - 6)

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mupad [B]  time = 0.46, size = 10, normalized size = 1.00 \begin {gather*} 16\,{\mathrm {e}}^{1/x}\,{\mathrm {e}}^{-6}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((x^2 - 6*x + 1)/x)*(16*x^2 - 16))/x^2,x)

[Out]

16*exp(1/x)*exp(-6)*exp(x)

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sympy [A]  time = 0.11, size = 12, normalized size = 1.20 \begin {gather*} 16 e^{\frac {x^{2} - 6 x + 1}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x**2-16)*exp((x**2-6*x+1)/x)/x**2,x)

[Out]

16*exp((x**2 - 6*x + 1)/x)

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