Optimal. Leaf size=71 \[ \left (\frac{4}{5}+\frac{8 i}{5}\right ) \left (\frac{x-i}{x}\right )^{1+\frac{i}{2}} \left (\frac{x+i}{x}\right )^{-1-\frac{i}{2}} \, _2F_1\left (1+\frac{i}{2},2;2+\frac{i}{2};\frac{1-\frac{i}{x}}{1+\frac{i}{x}}\right ) \]
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Rubi [A] time = 0.014901, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5102, 131} \[ \left (\frac{4}{5}+\frac{8 i}{5}\right ) \left (\frac{x-i}{x}\right )^{1+\frac{i}{2}} \left (\frac{x+i}{x}\right )^{-1-\frac{i}{2}} \, _2F_1\left (1+\frac{i}{2},2;2+\frac{i}{2};\frac{1-\frac{i}{x}}{1+\frac{i}{x}}\right ) \]
Antiderivative was successfully verified.
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Rule 5102
Rule 131
Rubi steps
\begin{align*} \int e^{\cot ^{-1}(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{(1-i x)^{\frac{i}{2}} (1+i x)^{-\frac{i}{2}}}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\left (\frac{4}{5}+\frac{8 i}{5}\right ) \left (\frac{-i+x}{x}\right )^{1+\frac{i}{2}} \left (\frac{i+x}{x}\right )^{-1-\frac{i}{2}} \, _2F_1\left (1+\frac{i}{2},2;2+\frac{i}{2};\frac{1-\frac{i}{x}}{1+\frac{i}{x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0664228, size = 71, normalized size = 1. \[ i e^{\cot ^{-1}(x)} \, _2F_1\left (-\frac{i}{2},1;1-\frac{i}{2};e^{2 i \cot ^{-1}(x)}\right )+\left (\frac{2}{5}+\frac{i}{5}\right ) e^{(1+2 i) \cot ^{-1}(x)} \, _2F_1\left (1,1-\frac{i}{2};2-\frac{i}{2};e^{2 i \cot ^{-1}(x)}\right )+x e^{\cot ^{-1}(x)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.114, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{\rm arccot} \left (x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\operatorname{arccot}\left (x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (e^{\operatorname{arccot}\left (x\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\operatorname{acot}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\operatorname{arccot}\left (x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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