Integrand size = 11, antiderivative size = 50 \[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\frac {e^{\frac {e}{(c+d x)^2}} (c+d x)}{d}-\frac {\sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d} \] Output:
exp(e/(d*x+c)^2)*(d*x+c)/d-e^(1/2)*Pi^(1/2)*erfi(e^(1/2)/(d*x+c))/d
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00 \[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\frac {e^{\frac {e}{(c+d x)^2}} (c+d x)}{d}-\frac {\sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d} \] Input:
Integrate[E^(e/(c + d*x)^2),x]
Output:
(E^(e/(c + d*x)^2)*(c + d*x))/d - (Sqrt[e]*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x) ])/d
Time = 0.39 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2635, 2640, 2633}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int e^{\frac {e}{(c+d x)^2}} \, dx\) |
\(\Big \downarrow \) 2635 |
\(\displaystyle 2 e \int \frac {e^{\frac {e}{(c+d x)^2}}}{(c+d x)^2}dx+\frac {(c+d x) e^{\frac {e}{(c+d x)^2}}}{d}\) |
\(\Big \downarrow \) 2640 |
\(\displaystyle \frac {(c+d x) e^{\frac {e}{(c+d x)^2}}}{d}-\frac {2 e \int e^{\frac {e}{(c+d x)^2}}d\frac {1}{c+d x}}{d}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {(c+d x) e^{\frac {e}{(c+d x)^2}}}{d}-\frac {\sqrt {\pi } \sqrt {e} \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d}\) |
Input:
Int[E^(e/(c + d*x)^2),x]
Output:
(E^(e/(c + d*x)^2)*(c + d*x))/d - (Sqrt[e]*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x) ])/d
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]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(c + d*x)*(F^(a + b*(c + d*x)^n)/d), x] - Simp[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]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_ .), x_Symbol] :> Simp[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)]
Time = 0.08 (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 | \({\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x +\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c}{d}-\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{d \sqrt {-e}}\) | \(57\) |
Input:
int(exp(e/(d*x+c)^2),x,method=_RETURNVERBOSE)
Output:
-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)))
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.26 \[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\frac {\sqrt {\pi } d \sqrt {-\frac {e}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {e}{d^{2}}}}{d x + c}\right ) + {\left (d x + c\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{d} \] Input:
integrate(exp(e/(d*x+c)^2),x, algorithm="fricas")
Output:
(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
\[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\int e^{\frac {e}{\left (c + d x\right )^{2}}}\, dx \] Input:
integrate(exp(e/(d*x+c)**2),x)
Output:
Integral(exp(e/(c + d*x)**2), x)
\[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\int { e^{\left (\frac {e}{{\left (d x + c\right )}^{2}}\right )} \,d x } \] Input:
integrate(exp(e/(d*x+c)^2),x, algorithm="maxima")
Output:
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))
\[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\int { e^{\left (\frac {e}{{\left (d x + c\right )}^{2}}\right )} \,d x } \] Input:
integrate(exp(e/(d*x+c)^2),x, algorithm="giac")
Output:
integrate(e^(e/(d*x + c)^2), x)
Time = 0.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.86 \[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\frac {{\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^2}}\,\left (c+d\,x\right )}{d}-\frac {\sqrt {e}\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\sqrt {e}}{c+d\,x}\right )}{d} \] Input:
int(exp(e/(c + d*x)^2),x)
Output:
(exp(e/(c + d*x)^2)*(c + d*x))/d - (e^(1/2)*pi^(1/2)*erfi(e^(1/2)/(c + d*x )))/d
\[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\text {too large to display} \] Input:
int(exp(e/(d*x+c)^2),x)
Output:
( - 279*e**(e/(c**2 + 2*c*d*x + d**2*x**2))*c**9 - 2127*e**(e/(c**2 + 2*c* d*x + d**2*x**2))*c**8*d*x - 6972*e**(e/(c**2 + 2*c*d*x + d**2*x**2))*c**7 *d**2*x**2 + 174*e**(e/(c**2 + 2*c*d*x + d**2*x**2))*c**7*e - 12684*e**(e/ (c**2 + 2*c*d*x + d**2*x**2))*c**6*d**3*x**3 + 834*e**(e/(c**2 + 2*c*d*x + d**2*x**2))*c**6*d*e*x - 13650*e**(e/(c**2 + 2*c*d*x + d**2*x**2))*c**5*d **4*x**4 + 1350*e**(e/(c**2 + 2*c*d*x + d**2*x**2))*c**5*d**2*e*x**2 - 52* e**(e/(c**2 + 2*c*d*x + d**2*x**2))*c**5*e**2 - 8274*e**(e/(c**2 + 2*c*d*x + d**2*x**2))*c**4*d**5*x**5 + 330*e**(e/(c**2 + 2*c*d*x + d**2*x**2))*c* *4*d**3*e*x**3 - 68*e**(e/(c**2 + 2*c*d*x + d**2*x**2))*c**4*d*e**2*x - 19 32*e**(e/(c**2 + 2*c*d*x + d**2*x**2))*c**3*d**6*x**6 - 1590*e**(e/(c**2 + 2*c*d*x + d**2*x**2))*c**3*d**4*e*x**4 + 248*e**(e/(c**2 + 2*c*d*x + d**2 *x**2))*c**3*d**2*e**2*x**2 + 8*e**(e/(c**2 + 2*c*d*x + d**2*x**2))*c**3*e **3 + 708*e**(e/(c**2 + 2*c*d*x + d**2*x**2))*c**2*d**7*x**7 - 2106*e**(e/ (c**2 + 2*c*d*x + d**2*x**2))*c**2*d**5*e*x**5 + 632*e**(e/(c**2 + 2*c*d*x + d**2*x**2))*c**2*d**3*e**2*x**3 - 40*e**(e/(c**2 + 2*c*d*x + d**2*x**2) )*c**2*d*e**3*x + 561*e**(e/(c**2 + 2*c*d*x + d**2*x**2))*c*d**8*x**8 - 10 86*e**(e/(c**2 + 2*c*d*x + d**2*x**2))*c*d**6*e*x**6 + 508*e**(e/(c**2 + 2 *c*d*x + d**2*x**2))*c*d**4*e**2*x**4 - 104*e**(e/(c**2 + 2*c*d*x + d**2*x **2))*c*d**2*e**3*x**2 + 105*e**(e/(c**2 + 2*c*d*x + d**2*x**2))*d**9*x**9 - 210*e**(e/(c**2 + 2*c*d*x + d**2*x**2))*d**7*e*x**7 + 140*e**(e/(c**...