\(\int e^{\frac {e}{(c+d x)^2}} \, dx\) [412]

Optimal result

Integrand size = 11, antiderivative size = 50 \[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\frac {e^{\frac {e}{(c+d x)^2}} (c+d x)}{d}-\frac {\sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d} \]

[Out]

exp(e/(d*x+c)^2)*(d*x+c)/d-erfi(e^(1/2)/(d*x+c))*e^(1/2)*Pi^(1/2)/d

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2237, 2242, 2235} \[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\frac {(c+d x) e^{\frac {e}{(c+d x)^2}}}{d}-\frac {\sqrt {\pi } \sqrt {e} \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d} \]

[In]

Int[E^(e/(c + d*x)^2),x]

[Out]

(E^(e/(c + d*x)^2)*(c + d*x))/d - (Sqrt[e]*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2237

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(c + d*x)*(F^(a + b*(c + d*x)^n)/d), x]
- Dist[b*n*Log[F], Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n]
&& ILtQ[n, 0]

Rule 2242

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/(d*(m + 1))
, Subst[Int[F^(a + b*x^2), x], x, (c + d*x)^(m + 1)], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && EqQ[n, 2*(m + 1
)]

Rubi steps \begin{align*} \text {integral}& = \frac {e^{\frac {e}{(c+d x)^2}} (c+d x)}{d}+(2 e) \int \frac {e^{\frac {e}{(c+d x)^2}}}{(c+d x)^2} \, dx \\ & = \frac {e^{\frac {e}{(c+d x)^2}} (c+d x)}{d}-\frac {(2 e) \text {Subst}\left (\int e^{e x^2} \, dx,x,\frac {1}{c+d x}\right )}{d} \\ & = \frac {e^{\frac {e}{(c+d x)^2}} (c+d x)}{d}-\frac {\sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00 \[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\frac {e^{\frac {e}{(c+d x)^2}} (c+d x)}{d}-\frac {\sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d} \]

[In]

Integrate[E^(e/(c + d*x)^2),x]

[Out]

(E^(e/(c + d*x)^2)*(c + d*x))/d - (Sqrt[e]*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(-\frac {-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}}{d}\)	\(48\)
default	\(-\frac {-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}}{d}\)	\(48\)
risch	\({\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x +\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c}{d}-\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{d \sqrt {-e}}\)	\(57\)

[In]

int(exp(e/(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

-1/d*(-(d*x+c)*exp(e/(d*x+c)^2)+e*Pi^(1/2)/(-e)^(1/2)*erf((-e)^(1/2)/(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.26 \[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\frac {\sqrt {\pi } d \sqrt {-\frac {e}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {e}{d^{2}}}}{d x + c}\right ) + {\left (d x + c\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{d} \]

[In]

integrate(exp(e/(d*x+c)^2),x, algorithm="fricas")

[Out]

(sqrt(pi)*d*sqrt(-e/d^2)*erf(d*sqrt(-e/d^2)/(d*x + c)) + (d*x + c)*e^(e/(d^2*x^2 + 2*c*d*x + c^2)))/d

Sympy [F]

\[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\int e^{\frac {e}{\left (c + d x\right )^{2}}}\, dx \]

[In]

integrate(exp(e/(d*x+c)**2),x)

[Out]

Integral(exp(e/(c + d*x)**2), x)

Maxima [F]

\[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\int { e^{\left (\frac {e}{{\left (d x + c\right )}^{2}}\right )} \,d x } \]

[In]

integrate(exp(e/(d*x+c)^2),x, algorithm="maxima")

[Out]

2*d*e*integrate(x*e^(e/(d^2*x^2 + 2*c*d*x + c^2))/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x) + x*e^(e/(d^2*
x^2 + 2*c*d*x + c^2))

Giac [F]

\[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\int { e^{\left (\frac {e}{{\left (d x + c\right )}^{2}}\right )} \,d x } \]

[In]

integrate(exp(e/(d*x+c)^2),x, algorithm="giac")

[Out]

integrate(e^(e/(d*x + c)^2), x)

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.86 \[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\frac {{\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^2}}\,\left (c+d\,x\right )}{d}-\frac {\sqrt {e}\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\sqrt {e}}{c+d\,x}\right )}{d} \]

[In]

int(exp(e/(c + d*x)^2),x)

[Out]

(exp(e/(c + d*x)^2)*(c + d*x))/d - (e^(1/2)*pi^(1/2)*erfi(e^(1/2)/(c + d*x)))/d