3.34.90 \(\int \frac {e^{e^5-x} (-1-x)+4 x^2+4 e^x x^2}{2 x^2} \, dx\)

Optimal. Leaf size=28 \[ \frac {e^{e^5-x}+3 x+4 x \left (e^x+x\right )}{2 x} \]

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Rubi [A]  time = 0.06, antiderivative size = 25, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 14, 2194, 2197} \begin {gather*} 2 x+2 e^x+\frac {e^{e^5-x}}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(E^5 - x)*(-1 - x) + 4*x^2 + 4*E^x*x^2)/(2*x^2),x]

[Out]

2*E^x + E^(E^5 - x)/(2*x) + 2*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 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 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

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 {1}{2} \int \frac {e^{e^5-x} (-1-x)+4 x^2+4 e^x x^2}{x^2} \, dx\\ &=\frac {1}{2} \int \left (4+4 e^x-\frac {e^{e^5-x} (1+x)}{x^2}\right ) \, dx\\ &=2 x-\frac {1}{2} \int \frac {e^{e^5-x} (1+x)}{x^2} \, dx+2 \int e^x \, dx\\ &=2 e^x+\frac {e^{e^5-x}}{2 x}+2 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 25, normalized size = 0.89 \begin {gather*} 2 e^x+\frac {e^{e^5-x}}{2 x}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^5 - x)*(-1 - x) + 4*x^2 + 4*E^x*x^2)/(2*x^2),x]

[Out]

2*E^x + E^(E^5 - x)/(2*x) + 2*x

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fricas [A]  time = 0.63, size = 40, normalized size = 1.43 \begin {gather*} \frac {{\left (4 \, x^{2} e^{\left (-x + e^{5}\right )} + 4 \, x e^{\left (e^{5}\right )} + e^{\left (-2 \, x + 2 \, e^{5}\right )}\right )} e^{\left (x - e^{5}\right )}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-x-1)*exp(exp(5)-x)+4*exp(x)*x^2+4*x^2)/x^2,x, algorithm="fricas")

[Out]

1/2*(4*x^2*e^(-x + e^5) + 4*x*e^(e^5) + e^(-2*x + 2*e^5))*e^(x - e^5)/x

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giac [A]  time = 0.73, size = 23, normalized size = 0.82 \begin {gather*} \frac {4 \, x^{2} + 4 \, x e^{x} + e^{\left (-x + e^{5}\right )}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-x-1)*exp(exp(5)-x)+4*exp(x)*x^2+4*x^2)/x^2,x, algorithm="giac")

[Out]

1/2*(4*x^2 + 4*x*e^x + e^(-x + e^5))/x

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maple [A]  time = 0.04, size = 21, normalized size = 0.75

method	result	size
risch	\(2 x +2 \,{\mathrm e}^{x}+\frac {{\mathrm e}^{{\mathrm e}^{5}-x}}{2 x}\)	\(21\)
norman	\(\frac {\left (2 x \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x^{2}+\frac {{\mathrm e}^{{\mathrm e}^{5}}}{2}\right ) {\mathrm e}^{-x}}{x}\)	\(29\)
default	\(2 x -\frac {{\mathrm e}^{{\mathrm e}^{5}} \left (-\frac {{\mathrm e}^{-x}}{x}+\expIntegralEi \left (1, x\right )\right )}{2}+\frac {{\mathrm e}^{{\mathrm e}^{5}} \expIntegralEi \left (1, x\right )}{2}+2 \,{\mathrm e}^{x}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((-x-1)*exp(exp(5)-x)+4*exp(x)*x^2+4*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [C]  time = 0.44, size = 25, normalized size = 0.89 \begin {gather*} -\frac {1}{2} \, {\rm Ei}\left (-x\right ) e^{\left (e^{5}\right )} + \frac {1}{2} \, e^{\left (e^{5}\right )} \Gamma \left (-1, x\right ) + 2 \, x + 2 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-x-1)*exp(exp(5)-x)+4*exp(x)*x^2+4*x^2)/x^2,x, algorithm="maxima")

[Out]

-1/2*Ei(-x)*e^(e^5) + 1/2*e^(e^5)*gamma(-1, x) + 2*x + 2*e^x

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mupad [B]  time = 2.07, size = 20, normalized size = 0.71 \begin {gather*} 2\,x+2\,{\mathrm {e}}^x+\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\mathrm {e}}^5}}{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + 2*exp(x) + (exp(-x)*exp(exp(5)))/(2*x)

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sympy [A]  time = 0.14, size = 20, normalized size = 0.71 \begin {gather*} 2 x + \frac {4 x e^{x} + e^{- x} e^{e^{5}}}{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-x-1)*exp(exp(5)-x)+4*exp(x)*x**2+4*x**2)/x**2,x)

[Out]

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

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