3.1 \(\int e^{\cot ^{-1}(x)} \, dx\)

Optimal. Leaf size=71 \[ \left (\frac{4}{5}+\frac{8 i}{5}\right ) \left (\frac{x-i}{x}\right )^{1+\frac{i}{2}} \left (\frac{x+i}{x}\right )^{-1-\frac{i}{2}} \, _2F_1\left (1+\frac{i}{2},2;2+\frac{i}{2};\frac{1-\frac{i}{x}}{1+\frac{i}{x}}\right ) \]

[Out]

((4/5 + (8*I)/5)*((-I + x)/x)^(1 + I/2)*Hypergeometric2F1[1 + I/2, 2, 2 + I/2, (1 - I/x)/(1 + I/x)])/((I + x)/
x)^(1 + I/2)

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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.

[In]

Int[E^ArcCot[x],x]

[Out]

((4/5 + (8*I)/5)*((-I + x)/x)^(1 + I/2)*Hypergeometric2F1[1 + I/2, 2, 2 + I/2, (1 - I/x)/(1 + I/x)])/((I + x)/
x)^(1 + I/2)

Rule 5102

Int[E^(ArcCot[(a_.)*(x_)]*(n_.)), x_Symbol] :> -Subst[Int[(1 - (I*x)/a)^((I*n)/2)/(x^2*(1 + (I*x)/a)^((I*n)/2)
), x], x, 1/x] /; FreeQ[{a, n}, x] &&  !IntegerQ[I*n]

Rule 131

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*c -
a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

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.

[In]

Integrate[E^ArcCot[x],x]

[Out]

E^ArcCot[x]*x + I*E^ArcCot[x]*Hypergeometric2F1[-I/2, 1, 1 - I/2, E^((2*I)*ArcCot[x])] + (2/5 + I/5)*E^((1 + 2
*I)*ArcCot[x])*Hypergeometric2F1[1, 1 - I/2, 2 - I/2, E^((2*I)*ArcCot[x])]

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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.

[In]

int(exp(arccot(x)),x)

[Out]

int(exp(arccot(x)),x)

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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.

[In]

integrate(exp(arccot(x)),x, algorithm="maxima")

[Out]

integrate(e^arccot(x), x)

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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.

[In]

integrate(exp(arccot(x)),x, algorithm="fricas")

[Out]

integral(e^arccot(x), x)

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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.

[In]

integrate(exp(acot(x)),x)

[Out]

Integral(exp(acot(x)), x)

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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.

[In]

integrate(exp(arccot(x)),x, algorithm="giac")

[Out]

integrate(e^arccot(x), x)