3.70.83 \(\int \frac {-123-4 e^{-4 x/41} x^2}{41 x^2} \, dx\)

Optimal. Leaf size=19 \[ e^{-4 x/41}-\frac {-3+x \log (3)}{x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 0.68, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 14, 2194} \begin {gather*} e^{-4 x/41}+\frac {3}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-123 - (4*x^2)/E^((4*x)/41))/(41*x^2),x]

[Out]

E^((-4*x)/41) + 3/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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{41} \int \frac {-123-4 e^{-4 x/41} x^2}{x^2} \, dx\\ &=\frac {1}{41} \int \left (-4 e^{-4 x/41}-\frac {123}{x^2}\right ) \, dx\\ &=\frac {3}{x}-\frac {4}{41} \int e^{-4 x/41} \, dx\\ &=e^{-4 x/41}+\frac {3}{x}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 13, normalized size = 0.68 \begin {gather*} e^{-4 x/41}+\frac {3}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-123 - (4*x^2)/E^((4*x)/41))/(41*x^2),x]

[Out]

E^((-4*x)/41) + 3/x

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 12, normalized size = 0.63 \begin {gather*} \frac {x e^{\left (-\frac {4}{41} \, x\right )} + 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/41*(-4*x^2*exp(-4/41*x)-123)/x^2,x, algorithm="fricas")

[Out]

(x*e^(-4/41*x) + 3)/x

________________________________________________________________________________________

giac [A]  time = 0.15, size = 12, normalized size = 0.63 \begin {gather*} \frac {x e^{\left (-\frac {4}{41} \, x\right )} + 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/41*(-4*x^2*exp(-4/41*x)-123)/x^2,x, algorithm="giac")

[Out]

(x*e^(-4/41*x) + 3)/x

________________________________________________________________________________________

maple [A]  time = 0.02, size = 11, normalized size = 0.58

method	result	size
derivativedivides	\(\frac {3}{x}+{\mathrm e}^{-\frac {4 x}{41}}\)	\(11\)
default	\(\frac {3}{x}+{\mathrm e}^{-\frac {4 x}{41}}\)	\(11\)
risch	\(\frac {3}{x}+{\mathrm e}^{-\frac {4 x}{41}}\)	\(11\)
norman	\(\frac {3+x \,{\mathrm e}^{-\frac {4 x}{41}}}{x}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/41*(-4*x^2*exp(-4/41*x)-123)/x^2,x,method=_RETURNVERBOSE)

[Out]

3/x+exp(-4/41*x)

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 10, normalized size = 0.53 \begin {gather*} \frac {3}{x} + e^{\left (-\frac {4}{41} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/41*(-4*x^2*exp(-4/41*x)-123)/x^2,x, algorithm="maxima")

[Out]

3/x + e^(-4/41*x)

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 10, normalized size = 0.53 \begin {gather*} {\mathrm {e}}^{-\frac {4\,x}{41}}+\frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((4*x^2*exp(-(4*x)/41))/41 + 3)/x^2,x)

[Out]

exp(-(4*x)/41) + 3/x

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 10, normalized size = 0.53 \begin {gather*} e^{- \frac {4 x}{41}} + \frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/41*(-4*x**2*exp(-4/41*x)-123)/x**2,x)

[Out]

exp(-4*x/41) + 3/x

________________________________________________________________________________________