3.56.9 \(\int -81 e^{\frac {1}{2} (15-27 e^{2 x})+2 x} \, dx\)

Optimal. Leaf size=17 \[ 3 e^{\frac {3}{2} \left (5-9 e^{2 x}\right )} \]

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Rubi [A]  time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 2282, 2194} \begin {gather*} 3 e^{\frac {15}{2}-\frac {27 e^{2 x}}{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-81*E^((15 - 27*E^(2*x))/2 + 2*x),x]

[Out]

3*E^(15/2 - (27*E^(2*x))/2)

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} &=-\left (81 \int e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx\right )\\ &=-\left (\frac {81}{2} \operatorname {Subst}\left (\int e^{\frac {15}{2}-\frac {27 x}{2}} \, dx,x,e^{2 x}\right )\right )\\ &=3 e^{\frac {15}{2}-\frac {27 e^{2 x}}{2}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} 3 e^{\frac {15}{2}-\frac {27 e^{2 x}}{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-81*E^((15 - 27*E^(2*x))/2 + 2*x),x]

[Out]

3*E^(15/2 - (27*E^(2*x))/2)

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fricas [A]  time = 0.83, size = 11, normalized size = 0.65 \begin {gather*} 3 \, e^{\left (-\frac {27}{2} \, e^{\left (2 \, x\right )} + \frac {15}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-81*exp(x)^2*exp(-27/2*exp(x)^2+15/2),x, algorithm="fricas")

[Out]

3*e^(-27/2*e^(2*x) + 15/2)

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giac [A]  time = 0.23, size = 11, normalized size = 0.65 \begin {gather*} 3 \, e^{\left (-\frac {27}{2} \, e^{\left (2 \, x\right )} + \frac {15}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-81*exp(x)^2*exp(-27/2*exp(x)^2+15/2),x, algorithm="giac")

[Out]

3*e^(-27/2*e^(2*x) + 15/2)

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maple [A]  time = 0.02, size = 12, normalized size = 0.71

method	result	size
derivativedivides	\(3 \,{\mathrm e}^{-\frac {27 \,{\mathrm e}^{2 x}}{2}+\frac {15}{2}}\)	\(12\)
default	\(3 \,{\mathrm e}^{-\frac {27 \,{\mathrm e}^{2 x}}{2}+\frac {15}{2}}\)	\(12\)
norman	\(3 \,{\mathrm e}^{-\frac {27 \,{\mathrm e}^{2 x}}{2}+\frac {15}{2}}\)	\(12\)
risch	\(3 \,{\mathrm e}^{-\frac {27 \,{\mathrm e}^{2 x}}{2}+\frac {15}{2}}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-81*exp(x)^2*exp(-27/2*exp(x)^2+15/2),x,method=_RETURNVERBOSE)

[Out]

3*exp(-27/2*exp(x)^2+15/2)

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maxima [A]  time = 0.40, size = 11, normalized size = 0.65 \begin {gather*} 3 \, e^{\left (-\frac {27}{2} \, e^{\left (2 \, x\right )} + \frac {15}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-81*exp(x)^2*exp(-27/2*exp(x)^2+15/2),x, algorithm="maxima")

[Out]

3*e^(-27/2*e^(2*x) + 15/2)

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mupad [B]  time = 0.05, size = 11, normalized size = 0.65 \begin {gather*} 3\,{\mathrm {e}}^{-\frac {27\,{\mathrm {e}}^{2\,x}}{2}}\,{\mathrm {e}}^{15/2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-81*exp(2*x)*exp(15/2 - (27*exp(2*x))/2),x)

[Out]

3*exp(-(27*exp(2*x))/2)*exp(15/2)

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sympy [A]  time = 0.10, size = 14, normalized size = 0.82 \begin {gather*} 3 e^{\frac {15}{2} - \frac {27 e^{2 x}}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-81*exp(x)**2*exp(-27/2*exp(x)**2+15/2),x)

[Out]

3*exp(15/2 - 27*exp(2*x)/2)

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