Optimal. Leaf size=50 \[ \frac {e^{\frac {e}{(c+d x)^2}} (c+d x)}{d}-\frac {\sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d} \]
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Rubi [A]
time = 0.03, 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 = {2237, 2242,
2235} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2237
Rule 2242
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) \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*}
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Mathematica [A]
time = 0.01, size = 50, normalized size = 1.00 \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 48, normalized size = 0.96
method | result | size |
derivativedivides | \(-\frac {-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}}{d}\) | \(48\) |
default | \(-\frac {-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}}{d}\) | \(48\) |
risch | \({\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x +\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c}{d}-\frac {e \sqrt {\pi }\, \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{d \sqrt {-e}}\) | \(57\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 64, normalized size = 1.28 \begin {gather*} -\frac {\sqrt {\pi } d \sqrt {\frac {1}{d^{2}}} \operatorname {erfi}\left (\frac {d \sqrt {\frac {1}{d^{2}}} e^{\frac {1}{2}}}{d x + c}\right ) e^{\frac {1}{2}} - {\left (d x + c\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{d} \end {gather*}
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 \begin {gather*} \int e^{\frac {e}{\left (c + d x\right )^{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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