Optimal. Leaf size=50 \[ \frac {(c+d x) e^{\frac {e}{(c+d x)^2}}}{d}-\frac {\sqrt {\pi } \sqrt {e} \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d} \]
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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.
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Rule 2204
Rule 2206
Rule 2211
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.
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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.
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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.
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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.
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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.
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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.
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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.
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