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

Optimal result

Integrand size = 4, antiderivative size = 71 \[ \int e^{\cot ^{-1}(x)} \, dx=\left (\frac {4}{5}+\frac {8 i}{5}\right ) \left (\frac {-i+x}{x}\right )^{1+\frac {i}{2}} \left (\frac {i+x}{x}\right )^{-1-\frac {i}{2}} \operatorname {Hypergeometric2F1}\left (1+\frac {i}{2},2,2+\frac {i}{2},\frac {1-\frac {i}{x}}{1+\frac {i}{x}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5210, 133} \[ \int e^{\cot ^{-1}(x)} \, dx=\left (\frac {4}{5}+\frac {8 i}{5}\right ) \left (\frac {x-i}{x}\right )^{1+\frac {i}{2}} \left (\frac {x+i}{x}\right )^{-1-\frac {i}{2}} \operatorname {Hypergeometric2F1}\left (1+\frac {i}{2},2,2+\frac {i}{2},\frac {1-\frac {i}{x}}{1+\frac {i}{x}}\right ) \]

[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 133

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)/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2,
(-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) &&  !ILtQ[m, 0]

Rule 5210

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]

Rubi steps \begin{align*} \text {integral}& = -\text {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}} \operatorname {Hypergeometric2F1}\left (1+\frac {i}{2},2,2+\frac {i}{2},\frac {1-\frac {i}{x}}{1+\frac {i}{x}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00 \[ \int e^{\cot ^{-1}(x)} \, dx=e^{\cot ^{-1}(x)} x+i e^{\cot ^{-1}(x)} \operatorname {Hypergeometric2F1}\left (-\frac {i}{2},1,1-\frac {i}{2},e^{2 i \cot ^{-1}(x)}\right )+\left (\frac {2}{5}+\frac {i}{5}\right ) e^{(1+2 i) \cot ^{-1}(x)} \operatorname {Hypergeometric2F1}\left (1,1-\frac {i}{2},2-\frac {i}{2},e^{2 i \cot ^{-1}(x)}\right ) \]

[In]

Integrate[E^ArcCot[x],x]

[Out]

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

Maple [F]

\[\int {\mathrm e}^{\operatorname {arccot}\left (x \right )}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int e^{\cot ^{-1}(x)} \, dx=\int { e^{\operatorname {arccot}\left (x\right )} \,d x } \]

[In]

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

[Out]

integral(e^arccot(x), x)

Sympy [F]

\[ \int e^{\cot ^{-1}(x)} \, dx=\int e^{\operatorname {acot}{\left (x \right )}}\, dx \]

[In]

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

[Out]

Integral(exp(acot(x)), x)

Maxima [F]

\[ \int e^{\cot ^{-1}(x)} \, dx=\int { e^{\operatorname {arccot}\left (x\right )} \,d x } \]

[In]

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

[Out]

integrate(e^arccot(x), x)

Giac [F]

\[ \int e^{\cot ^{-1}(x)} \, dx=\int { e^{\operatorname {arccot}\left (x\right )} \,d x } \]

[In]

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

[Out]

integrate(e^arccot(x), x)

Mupad [F(-1)]

Timed out. \[ \int e^{\cot ^{-1}(x)} \, dx=\int {\mathrm {e}}^{\mathrm {acot}\left (x\right )} \,d x \]

[In]

int(exp(acot(x)),x)

[Out]

int(exp(acot(x)), x)