Optimal. Leaf size=20 \[ \frac{(x+1) e^{\tan ^{-1}(x)}}{2 \sqrt{x^2+1}} \]
[Out]
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Rubi [A] time = 0.0350961, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{(x+1) e^{\tan ^{-1}(x)}}{2 \sqrt{x^2+1}} \]
Antiderivative was successfully verified.
[In] Int[E^ArcTan[x]/(1 + x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 2.39228, size = 17, normalized size = 0.85 \[ \frac{\left (x + 1\right ) e^{\operatorname{atan}{\left (x \right )}}}{2 \sqrt{x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(exp(atan(x))/(x**2+1)**(3/2),x)
[Out]
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Mathematica [A] time = 0.064379, size = 20, normalized size = 1. \[ \frac{(x+1) e^{\tan ^{-1}(x)}}{2 \sqrt{x^2+1}} \]
Antiderivative was successfully verified.
[In] Integrate[E^ArcTan[x]/(1 + x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.006, size = 16, normalized size = 0.8 \[{\frac{{{\rm e}^{\arctan \left ( x \right ) }} \left ( 1+x \right ) }{2}{\frac{1}{\sqrt{{x}^{2}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(exp(arctan(x))/(x^2+1)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{\arctan \left (x\right )}}{{\left (x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^arctan(x)/(x^2 + 1)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227675, size = 20, normalized size = 1. \[ \frac{{\left (x + 1\right )} e^{\arctan \left (x\right )}}{2 \, \sqrt{x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^arctan(x)/(x^2 + 1)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(exp(atan(x))/(x**2+1)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{\arctan \left (x\right )}}{{\left (x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^arctan(x)/(x^2 + 1)^(3/2),x, algorithm="giac")
[Out]