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

Optimal. Leaf size=50 \[ \frac{(c+d x) e^{\frac{e}{(c+d x)^2}}}{d}-\frac{\sqrt{\pi } \sqrt{e} \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.0399308, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2206, 2211, 2204} \[ \frac{(c+d x) e^{\frac{e}{(c+d x)^2}}}{d}-\frac{\sqrt{\pi } \sqrt{e} \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[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 2206

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 2211

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
)]

Rule 2204

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]

Rubi steps

\begin{align*} \int e^{\frac{e}{(c+d x)^2}} \, dx &=\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) \operatorname{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]  time = 0.0163764, size = 50, normalized size = 1. \[ \frac{(c+d x) e^{\frac{e}{(c+d x)^2}}}{d}-\frac{\sqrt{\pi } \sqrt{e} \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.005, size = 48, normalized size = 1. \begin{align*} -{\frac{1}{d} \left ( - \left ( dx+c \right ){{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{2}}}}}+{e\sqrt{\pi }{\it Erf} \left ({\frac{1}{dx+c}\sqrt{-e}} \right ){\frac{1}{\sqrt{-e}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 2 \, d e \int \frac{x e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\,{d x} + x e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Fricas [A]  time = 1.54464, size = 139, normalized size = 2.78 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\frac{e}{\left (c + d x\right )^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(e/(c + d*x)**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (\frac{e}{{\left (d x + c\right )}^{2}}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(e/(d*x + c)^2), x)