Optimal. Leaf size=20 \[ \frac {e^{\tan ^{-1}(x)} (1+x)}{2 \sqrt {1+x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {5177}
\begin {gather*} \frac {(x+1) e^{\text {ArcTan}(x)}}{2 \sqrt {x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 5177
Rubi steps
\begin {align*} \int \frac {e^{\tan ^{-1}(x)}}{\left (1+x^2\right )^{3/2}} \, dx &=\frac {e^{\tan ^{-1}(x)} (1+x)}{2 \sqrt {1+x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {e^{\tan ^{-1}(x)} (1+x)}{2 \sqrt {1+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 16, normalized size = 0.80
method | result | size |
gosper | \(\frac {{\mathrm e}^{\arctan \left (x \right )} \left (1+x \right )}{2 \sqrt {x^{2}+1}}\) | \(16\) |
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 = 0.90, size = 15, normalized size = 0.75 \begin {gather*} \frac {{\left (x + 1\right )} e^{\arctan \left (x\right )}}{2 \, \sqrt {x^{2} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 11.05, size = 31, normalized size = 1.55 \begin {gather*} \frac {x e^{\operatorname {atan}{\left (x \right )}}}{2 \sqrt {x^{2} + 1}} + \frac {e^{\operatorname {atan}{\left (x \right )}}}{2 \sqrt {x^{2} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 22, normalized size = 1.10 \begin {gather*} \frac {1}{2} \, {\left (\frac {x}{\sqrt {x^{2} + 1}} + \frac {1}{\sqrt {x^{2} + 1}}\right )} e^{\arctan \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {{\mathrm {e}}^{\mathrm {atan}\left (x\right )}}{{\left (x^2+1\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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