\(\int -81 e^{\frac {1}{2} (15-27 e^{2 x})+2 x} \, dx\) [5509]

Optimal result

Integrand size = 21, antiderivative size = 17 \[ \int -81 e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx=3 e^{\frac {3}{2} \left (5-9 e^{2 x}\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 2320, 2225} \[ \int -81 e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx=3 e^{\frac {15}{2}-\frac {27 e^{2 x}}{2}} \]

[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 2225

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 2320

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{align*} \text {integral}& = -\left (81 \int e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx\right ) \\ & = -\left (\frac {81}{2} \text {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{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int -81 e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx=3 e^{\frac {15}{2}-\frac {27 e^{2 x}}{2}} \]

[In]

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

[Out]

3*E^(15/2 - (27*E^(2*x))/2)

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 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	\(3 \,{\mathrm e}^{-\frac {27 \,{\mathrm e}^{2 x}}{2}+\frac {15}{2}}\)	\(12\)
parallelrisch	\(3 \,{\mathrm e}^{-\frac {27 \,{\mathrm e}^{2 x}}{2}+\frac {15}{2}}\)	\(12\)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int -81 e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx=3 \, e^{\left (-\frac {27}{2} \, e^{\left (2 \, x\right )} + \frac {15}{2}\right )} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int -81 e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx=3 e^{\frac {15}{2} - \frac {27 e^{2 x}}{2}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int -81 e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx=3 \, e^{\left (-\frac {27}{2} \, e^{\left (2 \, x\right )} + \frac {15}{2}\right )} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int -81 e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx=3 \, e^{\left (-\frac {27}{2} \, e^{\left (2 \, x\right )} + \frac {15}{2}\right )} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int -81 e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx=3\,{\mathrm {e}}^{-\frac {27\,{\mathrm {e}}^{2\,x}}{2}}\,{\mathrm {e}}^{15/2} \]

[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)