3.2.72 \(\int -\frac {2400 e^{\frac {400-20 x^6}{x^6}}}{x^7} \, dx\)

Optimal. Leaf size=12 \[ e^{-20+\frac {400}{x^6}}+\log (25) \]

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Rubi [A]  time = 0.04, antiderivative size = 9, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 2225, 2209} \begin {gather*} e^{\frac {400}{x^6}-20} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2400*E^((400 - 20*x^6)/x^6))/x^7,x]

[Out]

E^(-20 + 400/x^6)

Rule 12

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

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2225

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && BinomialQ[v, x] &&  !(LinearMatchQ[u, x] && BinomialMatchQ[v, x])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (2400 \int \frac {e^{\frac {400-20 x^6}{x^6}}}{x^7} \, dx\right )\\ &=-\left (2400 \int \frac {e^{-20+\frac {400}{x^6}}}{x^7} \, dx\right )\\ &=e^{-20+\frac {400}{x^6}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 0.75 \begin {gather*} e^{-20+\frac {400}{x^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2400*E^((400 - 20*x^6)/x^6))/x^7,x]

[Out]

E^(-20 + 400/x^6)

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fricas [A]  time = 0.69, size = 11, normalized size = 0.92 \begin {gather*} e^{\left (-\frac {20 \, {\left (x^{6} - 20\right )}}{x^{6}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2400*exp((-20*x^6+400)/x^6)/x^7,x, algorithm="fricas")

[Out]

e^(-20*(x^6 - 20)/x^6)

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giac [A]  time = 0.37, size = 8, normalized size = 0.67 \begin {gather*} e^{\left (\frac {400}{x^{6}} - 20\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2400*exp((-20*x^6+400)/x^6)/x^7,x, algorithm="giac")

[Out]

e^(400/x^6 - 20)

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maple [A]  time = 0.03, size = 9, normalized size = 0.75

method	result	size
derivativedivides	\({\mathrm e}^{\frac {400}{x^{6}}-20}\)	\(9\)
default	\({\mathrm e}^{\frac {400}{x^{6}}-20}\)	\(9\)
gosper	\({\mathrm e}^{-\frac {20 \left (x^{6}-20\right )}{x^{6}}}\)	\(12\)
risch	\({\mathrm e}^{-\frac {20 \left (x^{6}-20\right )}{x^{6}}}\)	\(12\)
norman	\({\mathrm e}^{\frac {-20 x^{6}+400}{x^{6}}}\)	\(13\)
meijerg	\(-{\mathrm e}^{-20} \left (1-{\mathrm e}^{\frac {400}{x^{6}}}\right )\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-2400*exp((-20*x^6+400)/x^6)/x^7,x,method=_RETURNVERBOSE)

[Out]

exp(400/x^6-20)

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maxima [A]  time = 0.73, size = 8, normalized size = 0.67 \begin {gather*} e^{\left (\frac {400}{x^{6}} - 20\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2400*exp((-20*x^6+400)/x^6)/x^7,x, algorithm="maxima")

[Out]

e^(400/x^6 - 20)

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mupad [B]  time = 0.26, size = 9, normalized size = 0.75 \begin {gather*} {\mathrm {e}}^{-20}\,{\mathrm {e}}^{\frac {400}{x^6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2400*exp(-(20*x^6 - 400)/x^6))/x^7,x)

[Out]

exp(-20)*exp(400/x^6)

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sympy [A]  time = 0.09, size = 10, normalized size = 0.83 \begin {gather*} e^{\frac {400 - 20 x^{6}}{x^{6}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2400*exp((-20*x**6+400)/x**6)/x**7,x)

[Out]

exp((400 - 20*x**6)/x**6)

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