3.61.47 \(\int \frac {1}{144} (143-e^x-576 e^{3+4 e^x+x}) \, dx\)

Optimal. Leaf size=27 \[ 1-e^{3+4 e^x}+\frac {1}{144} \left (-e^x-x\right )+x \]

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Rubi [A]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {12, 2194, 2282} \begin {gather*} \frac {143 x}{144}-e^{4 e^x+3}-\frac {e^x}{144} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(143 - E^x - 576*E^(3 + 4*E^x + x))/144,x]

[Out]

-E^(3 + 4*E^x) - E^x/144 + (143*x)/144

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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}{144} \int \left (143-e^x-576 e^{3+4 e^x+x}\right ) \, dx\\ &=\frac {143 x}{144}-\frac {\int e^x \, dx}{144}-4 \int e^{3+4 e^x+x} \, dx\\ &=-\frac {e^x}{144}+\frac {143 x}{144}-4 \operatorname {Subst}\left (\int e^{3+4 x} \, dx,x,e^x\right )\\ &=-e^{3+4 e^x}-\frac {e^x}{144}+\frac {143 x}{144}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 0.89 \begin {gather*} \frac {1}{144} \left (-144 e^{3+4 e^x}-e^x+143 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(143 - E^x - 576*E^(3 + 4*E^x + x))/144,x]

[Out]

(-144*E^(3 + 4*E^x) - E^x + 143*x)/144

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fricas [A]  time = 0.59, size = 28, normalized size = 1.04 \begin {gather*} \frac {1}{144} \, {\left (143 \, x e^{x} - e^{\left (2 \, x\right )} - 144 \, e^{\left (x + 4 \, e^{x} + 3\right )}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*exp(x)*exp(4*exp(x)+3)-1/144*exp(x)+143/144,x, algorithm="fricas")

[Out]

1/144*(143*x*e^x - e^(2*x) - 144*e^(x + 4*e^x + 3))*e^(-x)

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giac [A]  time = 0.13, size = 17, normalized size = 0.63 \begin {gather*} \frac {143}{144} \, x - \frac {1}{144} \, e^{x} - e^{\left (4 \, e^{x} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*exp(x)*exp(4*exp(x)+3)-1/144*exp(x)+143/144,x, algorithm="giac")

[Out]

143/144*x - 1/144*e^x - e^(4*e^x + 3)

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maple [A]  time = 0.12, size = 18, normalized size = 0.67

method	result	size
default	\(\frac {143 x}{144}-{\mathrm e}^{4 \,{\mathrm e}^{x}+3}-\frac {{\mathrm e}^{x}}{144}\)	\(18\)
norman	\(\frac {143 x}{144}-{\mathrm e}^{4 \,{\mathrm e}^{x}+3}-\frac {{\mathrm e}^{x}}{144}\)	\(18\)
risch	\(\frac {143 x}{144}-{\mathrm e}^{4 \,{\mathrm e}^{x}+3}-\frac {{\mathrm e}^{x}}{144}\)	\(18\)
derivativedivides	\(\frac {143 \ln \left (4 \,{\mathrm e}^{x}\right )}{144}-\frac {{\mathrm e}^{x}}{144}-\frac {1}{192}-{\mathrm e}^{4 \,{\mathrm e}^{x}+3}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-4*exp(x)*exp(4*exp(x)+3)-1/144*exp(x)+143/144,x,method=_RETURNVERBOSE)

[Out]

143/144*x-exp(4*exp(x)+3)-1/144*exp(x)

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maxima [A]  time = 0.34, size = 17, normalized size = 0.63 \begin {gather*} \frac {143}{144} \, x - \frac {1}{144} \, e^{x} - e^{\left (4 \, e^{x} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*exp(x)*exp(4*exp(x)+3)-1/144*exp(x)+143/144,x, algorithm="maxima")

[Out]

143/144*x - 1/144*e^x - e^(4*e^x + 3)

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mupad [B]  time = 0.08, size = 17, normalized size = 0.63 \begin {gather*} \frac {143\,x}{144}-\frac {{\mathrm {e}}^x}{144}-{\mathrm {e}}^3\,{\mathrm {e}}^{4\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(143/144 - 4*exp(4*exp(x) + 3)*exp(x) - exp(x)/144,x)

[Out]

(143*x)/144 - exp(x)/144 - exp(3)*exp(4*exp(x))

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sympy [A]  time = 0.11, size = 17, normalized size = 0.63 \begin {gather*} \frac {143 x}{144} - \frac {e^{x}}{144} - e^{4 e^{x} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*exp(x)*exp(4*exp(x)+3)-1/144*exp(x)+143/144,x)

[Out]

143*x/144 - exp(x)/144 - exp(4*exp(x) + 3)

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