Integrand size = 11, antiderivative size = 50 \[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\frac {e^{\frac {e}{(c+d x)^2}} (c+d x)}{d}-\frac {\sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d} \]
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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} \[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\frac {(c+d x) e^{\frac {e}{(c+d x)^2}}}{d}-\frac {\sqrt {\pi } \sqrt {e} \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d} \]
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Rule 2235
Rule 2237
Rule 2242
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*}
Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00 \[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\frac {e^{\frac {e}{(c+d x)^2}} (c+d x)}{d}-\frac {\sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d} \]
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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 | \({\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x +\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c}{d}-\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{d \sqrt {-e}}\) | \(57\) |
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Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.26 \[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\frac {\sqrt {\pi } d \sqrt {-\frac {e}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {e}{d^{2}}}}{d x + c}\right ) + {\left (d x + c\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{d} \]
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\[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\int e^{\frac {e}{\left (c + d x\right )^{2}}}\, dx \]
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\[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\int { e^{\left (\frac {e}{{\left (d x + c\right )}^{2}}\right )} \,d x } \]
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\[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\int { e^{\left (\frac {e}{{\left (d x + c\right )}^{2}}\right )} \,d x } \]
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Time = 0.38 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.86 \[ \int e^{\frac {e}{(c+d x)^2}} \, dx=\frac {{\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^2}}\,\left (c+d\,x\right )}{d}-\frac {\sqrt {e}\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\sqrt {e}}{c+d\,x}\right )}{d} \]
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