3.39.29 \(\int \frac {e^x (1-x)-3 x^2-\log ^2(4)}{x^2} \, dx\)

Optimal. Leaf size=30 \[ -\frac {e^x-x+3 \left (2 x+x^2\right )}{x}+\frac {\log ^2(4)}{x} \]

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Rubi [A]  time = 0.05, antiderivative size = 20, normalized size of antiderivative = 0.67, number of steps used = 5, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {14, 2197} \begin {gather*} -3 x-\frac {e^x}{x}+\frac {\log ^2(4)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x*(1 - x) - 3*x^2 - Log[4]^2)/x^2,x]

[Out]

-(E^x/x) - 3*x + Log[4]^2/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 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} &=\int \left (-\frac {e^x (-1+x)}{x^2}+\frac {-3 x^2-\log ^2(4)}{x^2}\right ) \, dx\\ &=-\int \frac {e^x (-1+x)}{x^2} \, dx+\int \frac {-3 x^2-\log ^2(4)}{x^2} \, dx\\ &=-\frac {e^x}{x}+\int \left (-3-\frac {\log ^2(4)}{x^2}\right ) \, dx\\ &=-\frac {e^x}{x}-3 x+\frac {\log ^2(4)}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 0.67 \begin {gather*} -\frac {e^x}{x}-3 x+\frac {\log ^2(4)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(1 - x) - 3*x^2 - Log[4]^2)/x^2,x]

[Out]

-(E^x/x) - 3*x + Log[4]^2/x

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fricas [A]  time = 1.07, size = 19, normalized size = 0.63 \begin {gather*} -\frac {3 \, x^{2} - 4 \, \log \relax (2)^{2} + e^{x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x+1)*exp(x)-4*log(2)^2-3*x^2)/x^2,x, algorithm="fricas")

[Out]

-(3*x^2 - 4*log(2)^2 + e^x)/x

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giac [A]  time = 0.13, size = 19, normalized size = 0.63 \begin {gather*} -\frac {3 \, x^{2} - 4 \, \log \relax (2)^{2} + e^{x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x+1)*exp(x)-4*log(2)^2-3*x^2)/x^2,x, algorithm="giac")

[Out]

-(3*x^2 - 4*log(2)^2 + e^x)/x

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maple [A]  time = 0.04, size = 21, normalized size = 0.70

method	result	size
default	\(-3 x -\frac {{\mathrm e}^{x}}{x}+\frac {4 \ln \relax (2)^{2}}{x}\)	\(21\)
norman	\(\frac {-3 x^{2}+4 \ln \relax (2)^{2}-{\mathrm e}^{x}}{x}\)	\(21\)
risch	\(-3 x -\frac {{\mathrm e}^{x}}{x}+\frac {4 \ln \relax (2)^{2}}{x}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1-x)*exp(x)-4*ln(2)^2-3*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [C]  time = 0.45, size = 22, normalized size = 0.73 \begin {gather*} -3 \, x + \frac {4 \, \log \relax (2)^{2}}{x} - {\rm Ei}\relax (x) + \Gamma \left (-1, -x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x+1)*exp(x)-4*log(2)^2-3*x^2)/x^2,x, algorithm="maxima")

[Out]

-3*x + 4*log(2)^2/x - Ei(x) + gamma(-1, -x)

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mupad [B]  time = 2.27, size = 18, normalized size = 0.60 \begin {gather*} -3\,x-\frac {{\mathrm {e}}^x-4\,{\ln \relax (2)}^2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(x - 1) + 4*log(2)^2 + 3*x^2)/x^2,x)

[Out]

- 3*x - (exp(x) - 4*log(2)^2)/x

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sympy [A]  time = 0.10, size = 15, normalized size = 0.50 \begin {gather*} - 3 x - \frac {e^{x}}{x} + \frac {4 \log {\relax (2 )}^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x+1)*exp(x)-4*ln(2)**2-3*x**2)/x**2,x)

[Out]

-3*x - exp(x)/x + 4*log(2)**2/x

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