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]
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Rubi [A] time = 0.0629938, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{(c+d x) e^{\frac{e}{(c+d x)^2}}}{d}-\frac{\sqrt{\pi } \sqrt{e} \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
[In] Int[E^(e/(c + d*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 7.34943, size = 39, normalized size = 0.78 \[ - \frac{\sqrt{\pi } \sqrt{e} \operatorname{erfi}{\left (\frac{\sqrt{e}}{c + d x} \right )}}{d} + \frac{\left (c + d x\right ) e^{\frac{e}{\left (c + d x\right )^{2}}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(exp(e/(d*x+c)**2),x)
[Out]
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Mathematica [A] time = 0.0168698, size = 50, normalized size = 1. \[ \frac{(c+d x) e^{\frac{e}{(c+d x)^2}}}{d}-\frac{\sqrt{\pi } \sqrt{e} \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
[In] Integrate[E^(e/(c + d*x)^2),x]
[Out]
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Maple [A] time = 0.007, size = 48, normalized size = 1. \[ -{\frac{1}{d} \left ( - \left ( dx+c \right ){{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{2}}}}}+{e\sqrt{\pi }{\it Erf} \left ({\frac{1}{dx+c}\sqrt{-e}} \right ){\frac{1}{\sqrt{-e}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(exp(e/(d*x+c)^2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ 2 \, d e \int \frac{x e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\,{d x} + x e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(e/(d*x + c)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.256635, size = 104, normalized size = 2.08 \[ -\frac{\sqrt{\pi } e \operatorname{erf}\left (\frac{d \sqrt{-\frac{e}{d^{2}}}}{d x + c}\right ) -{\left (d^{2} x + c d\right )} \sqrt{-\frac{e}{d^{2}}} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{d^{2} \sqrt{-\frac{e}{d^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(e/(d*x + c)^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int e^{\frac{e}{\left (c + d x\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(exp(e/(d*x+c)**2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int e^{\left (\frac{e}{{\left (d x + c\right )}^{2}}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(e/(d*x + c)^2),x, algorithm="giac")
[Out]