3.70.65 \(\int \frac {1}{72} e^{-x} (-1-480 e^x+4 e^{2 x}) \, dx\)

Optimal. Leaf size=20 \[ \frac {1}{18} \left (\frac {e^{-x}}{4}+e^x-120 x\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {12, 2282, 14} \begin {gather*} -\frac {20 x}{3}+\frac {e^{-x}}{72}+\frac {e^x}{18} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - 480*E^x + 4*E^(2*x))/(72*E^x),x]

[Out]

1/(72*E^x) + E^x/18 - (20*x)/3

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 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{72} \int e^{-x} \left (-1-480 e^x+4 e^{2 x}\right ) \, dx\\ &=\frac {1}{72} \operatorname {Subst}\left (\int \frac {-480-\frac {1}{x}+4 x}{x} \, dx,x,e^x\right )\\ &=\frac {1}{72} \operatorname {Subst}\left (\int \left (4-\frac {1}{x^2}-\frac {480}{x}\right ) \, dx,x,e^x\right )\\ &=\frac {e^{-x}}{72}+\frac {e^x}{18}-\frac {20 x}{3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.90 \begin {gather*} \frac {1}{72} \left (e^{-x}+4 e^x-480 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - 480*E^x + 4*E^(2*x))/(72*E^x),x]

[Out]

(E^(-x) + 4*E^x - 480*x)/72

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fricas [A]  time = 0.68, size = 19, normalized size = 0.95 \begin {gather*} -\frac {1}{72} \, {\left (480 \, x e^{x} - 4 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/72*(4*exp(x)^2-480*exp(x)-1)/exp(x),x, algorithm="fricas")

[Out]

-1/72*(480*x*e^x - 4*e^(2*x) - 1)*e^(-x)

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giac [A]  time = 0.41, size = 14, normalized size = 0.70 \begin {gather*} -\frac {20}{3} \, x + \frac {1}{72} \, e^{\left (-x\right )} + \frac {1}{18} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/72*(4*exp(x)^2-480*exp(x)-1)/exp(x),x, algorithm="giac")

[Out]

-20/3*x + 1/72*e^(-x) + 1/18*e^x

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maple [A]  time = 0.02, size = 15, normalized size = 0.75

method	result	size
risch	\(\frac {{\mathrm e}^{x}}{18}-\frac {20 x}{3}+\frac {{\mathrm e}^{-x}}{72}\)	\(15\)
derivativedivides	\(\frac {{\mathrm e}^{x}}{18}-\frac {20 \ln \left ({\mathrm e}^{x}\right )}{3}+\frac {{\mathrm e}^{-x}}{72}\)	\(17\)
default	\(\frac {{\mathrm e}^{x}}{18}-\frac {20 \ln \left ({\mathrm e}^{x}\right )}{3}+\frac {{\mathrm e}^{-x}}{72}\)	\(17\)
norman	\(\left (\frac {1}{72}+\frac {{\mathrm e}^{2 x}}{18}-\frac {20 \,{\mathrm e}^{x} x}{3}\right ) {\mathrm e}^{-x}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/72*(4*exp(x)^2-480*exp(x)-1)/exp(x),x,method=_RETURNVERBOSE)

[Out]

1/18*exp(x)-20/3*x+1/72*exp(-x)

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maxima [A]  time = 0.38, size = 14, normalized size = 0.70 \begin {gather*} -\frac {20}{3} \, x + \frac {1}{72} \, e^{\left (-x\right )} + \frac {1}{18} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/72*(4*exp(x)^2-480*exp(x)-1)/exp(x),x, algorithm="maxima")

[Out]

-20/3*x + 1/72*e^(-x) + 1/18*e^x

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mupad [B]  time = 4.16, size = 14, normalized size = 0.70 \begin {gather*} \frac {{\mathrm {e}}^{-x}}{72}-\frac {20\,x}{3}+\frac {{\mathrm {e}}^x}{18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-x)*((20*exp(x))/3 - exp(2*x)/18 + 1/72),x)

[Out]

exp(-x)/72 - (20*x)/3 + exp(x)/18

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sympy [A]  time = 0.09, size = 15, normalized size = 0.75 \begin {gather*} - \frac {20 x}{3} + \frac {e^{x}}{18} + \frac {e^{- x}}{72} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/72*(4*exp(x)**2-480*exp(x)-1)/exp(x),x)

[Out]

-20*x/3 + exp(x)/18 + exp(-x)/72

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