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+8/5*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))

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Rubi [A]  time = 0.01, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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]

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]

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*}

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Mathematica [A]  time = 0.07, size = 71, normalized size = 1.00 \[ i e^{\cot ^{-1}(x)} \, _2F_1\left (-\frac {i}{2},1;1-\frac {i}{2};e^{2 i \cot ^{-1}(x)}\right )+\left (\frac {2}{5}+\frac {i}{5}\right ) e^{(1+2 i) \cot ^{-1}(x)} \, _2F_1\left (1,1-\frac {i}{2};2-\frac {i}{2};e^{2 i \cot ^{-1}(x)}\right )+x e^{\cot ^{-1}(x)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCot[x],x]

[Out]

E^ArcCot[x]*x + I*E^ArcCot[x]*Hypergeometric2F1[-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])]

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fricas [F]  time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (e^{\operatorname {arccot}\relax (x)}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\operatorname {arccot}\relax (x)}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^arccot(x), x)

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{\mathrm {arccot}\relax (x )}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\operatorname {arccot}\relax (x)}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{\mathrm {acot}\relax (x)} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(exp(acot(x)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\operatorname {acot}{\relax (x )}}\, dx \]

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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