3.17.14 \(\int \frac {-3 e^{3/x}+e^{-16+e^x-3 x^2-x^4-x^5} (e^x x^2-6 x^3-4 x^5-5 x^6)}{x^2} \, dx\)

Optimal. Leaf size=29 \[ e^{3/x}+e^{-16+e^x-x^2 \left (3+x^2 (1+x)\right )} \]

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Rubi [A]  time = 0.26, antiderivative size = 30, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {14, 2209, 6706} \begin {gather*} e^{-x^5-x^4-3 x^2+e^x-16}+e^{3/x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3*E^(3/x) + E^(-16 + E^x - 3*x^2 - x^4 - x^5)*(E^x*x^2 - 6*x^3 - 4*x^5 - 5*x^6))/x^2,x]

[Out]

E^(3/x) + E^(-16 + E^x - 3*x^2 - x^4 - x^5)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 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 \left (-\frac {3 e^{3/x}}{x^2}+e^{-16+e^x-3 x^2-x^4-x^5} \left (e^x-6 x-4 x^3-5 x^4\right )\right ) \, dx\\ &=-\left (3 \int \frac {e^{3/x}}{x^2} \, dx\right )+\int e^{-16+e^x-3 x^2-x^4-x^5} \left (e^x-6 x-4 x^3-5 x^4\right ) \, dx\\ &=e^{3/x}+e^{-16+e^x-3 x^2-x^4-x^5}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 30, normalized size = 1.03 \begin {gather*} e^{3/x}+e^{-16+e^x-3 x^2-x^4-x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*E^(3/x) + E^(-16 + E^x - 3*x^2 - x^4 - x^5)*(E^x*x^2 - 6*x^3 - 4*x^5 - 5*x^6))/x^2,x]

[Out]

E^(3/x) + E^(-16 + E^x - 3*x^2 - x^4 - x^5)

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fricas [A]  time = 0.77, size = 27, normalized size = 0.93 \begin {gather*} e^{\left (-x^{5} - x^{4} - 3 \, x^{2} + e^{x} - 16\right )} + e^{\frac {3}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)*x^2-5*x^6-4*x^5-6*x^3)*exp(exp(x)-x^5-x^4-3*x^2-16)-3*exp(3/x))/x^2,x, algorithm="fricas")

[Out]

e^(-x^5 - x^4 - 3*x^2 + e^x - 16) + e^(3/x)

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giac [A]  time = 0.49, size = 27, normalized size = 0.93 \begin {gather*} e^{\left (-x^{5} - x^{4} - 3 \, x^{2} + e^{x} - 16\right )} + e^{\frac {3}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)*x^2-5*x^6-4*x^5-6*x^3)*exp(exp(x)-x^5-x^4-3*x^2-16)-3*exp(3/x))/x^2,x, algorithm="giac")

[Out]

e^(-x^5 - x^4 - 3*x^2 + e^x - 16) + e^(3/x)

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maple [A]  time = 0.16, size = 28, normalized size = 0.97

method	result	size
risch	\({\mathrm e}^{\frac {3}{x}}+{\mathrm e}^{{\mathrm e}^{x}-x^{5}-x^{4}-3 x^{2}-16}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(x)*x^2-5*x^6-4*x^5-6*x^3)*exp(exp(x)-x^5-x^4-3*x^2-16)-3*exp(3/x))/x^2,x,method=_RETURNVERBOSE)

[Out]

exp(3/x)+exp(exp(x)-x^5-x^4-3*x^2-16)

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maxima [A]  time = 0.77, size = 27, normalized size = 0.93 \begin {gather*} e^{\left (-x^{5} - x^{4} - 3 \, x^{2} + e^{x} - 16\right )} + e^{\frac {3}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)*x^2-5*x^6-4*x^5-6*x^3)*exp(exp(x)-x^5-x^4-3*x^2-16)-3*exp(3/x))/x^2,x, algorithm="maxima")

[Out]

e^(-x^5 - x^4 - 3*x^2 + e^x - 16) + e^(3/x)

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mupad [B]  time = 1.13, size = 31, normalized size = 1.07 \begin {gather*} {\mathrm {e}}^{3/x}+{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{-16}\,{\mathrm {e}}^{-3\,x^2}\,{\mathrm {e}}^{-x^4}\,{\mathrm {e}}^{-x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*exp(3/x) + exp(exp(x) - 3*x^2 - x^4 - x^5 - 16)*(6*x^3 - x^2*exp(x) + 4*x^5 + 5*x^6))/x^2,x)

[Out]

exp(3/x) + exp(exp(x))*exp(-16)*exp(-3*x^2)*exp(-x^4)*exp(-x^5)

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sympy [A]  time = 0.43, size = 22, normalized size = 0.76 \begin {gather*} e^{\frac {3}{x}} + e^{- x^{5} - x^{4} - 3 x^{2} + e^{x} - 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)*x**2-5*x**6-4*x**5-6*x**3)*exp(exp(x)-x**5-x**4-3*x**2-16)-3*exp(3/x))/x**2,x)

[Out]

exp(3/x) + exp(-x**5 - x**4 - 3*x**2 + exp(x) - 16)

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