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]

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

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Rubi [A]  time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 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]

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
)]

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*}

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Mathematica [A]  time = 0.02, size = 50, normalized size = 1.00 \[ \frac {(c+d x) e^{\frac {e}{(c+d x)^2}}}{d}-\frac {\sqrt {\pi } \sqrt {e} \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d} \]

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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fricas [A]  time = 0.41, size = 63, normalized size = 1.26 \[ \frac {\sqrt {\pi } d \sqrt {-\frac {e}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {e}{d^{2}}}}{d x + c}\right ) + {\left (d x + c\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{d} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (\frac {e}{{\left (d x + c\right )}^{2}}\right )}\,{d x} \]

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)

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maple [A]  time = 0.02, size = 48, normalized size = 0.96 \[ -\frac {\frac {\sqrt {\pi }\, e \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(1/(d*x+c)^2*e),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ 2 \, d e \int \frac {x e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\,{d x} + x e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )} \]

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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mupad [B]  time = 3.69, size = 43, normalized size = 0.86 \[ \frac {{\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^2}}\,\left (c+d\,x\right )}{d}-\frac {\sqrt {e}\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\sqrt {e}}{c+d\,x}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\frac {e}{\left (c + d x\right )^{2}}}\, dx \]

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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