\(\int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1+x}} \, dx\) [328]

Optimal result

Integrand size = 12, antiderivative size = 33 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1+x}} \, dx=\frac {2 \sqrt {1+\frac {1}{x}} \sqrt {-\frac {1-x}{x}} x}{\sqrt {1+x}} \]

[Out]

2*x*(1+1/x)^(1/2)*((-1+x)/x)^(1/2)/(1+x)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6311, 6316, 37} \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1+x}} \, dx=\frac {2 \sqrt {\frac {1}{x}+1} \sqrt {-\frac {1-x}{x}} x}{\sqrt {x+1}} \]

[In]

Int[E^ArcCoth[x]/Sqrt[1 + x],x]

[Out]

(2*Sqrt[1 + x^(-1)]*Sqrt[-((1 - x)/x)]*x)/Sqrt[1 + x]

Rule 37

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

Rule 6311

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^p/(x^p*(1 + c/(d
*x))^p), Int[u*x^p*(1 + c/(d*x))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^
2, 0] &&  !IntegerQ[n/2] &&  !IntegerQ[p]

Rule 6316

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> Dist[(-c^p)*x^m*(1/x)^m, S
ubst[Int[(1 + d*(x/c))^p*((1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2))), x], x, 1/x], x] /; FreeQ[{a, c, d, m,
n, p}, x] && EqQ[c^2 - a^2*d^2, 0] &&  !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) &&  !IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {1+\frac {1}{x}} \sqrt {x}\right ) \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1+\frac {1}{x}} \sqrt {x}} \, dx}{\sqrt {1+x}} \\ & = -\frac {\sqrt {1+\frac {1}{x}} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {\frac {1}{x}} \sqrt {1+x}} \\ & = \frac {2 \sqrt {1+\frac {1}{x}} \sqrt {-\frac {1-x}{x}} x}{\sqrt {1+x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.64 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1+x}} \, dx=\frac {2 \sqrt {1-\frac {1}{x^2}} x}{\sqrt {1+x}} \]

[In]

Integrate[E^ArcCoth[x]/Sqrt[1 + x],x]

[Out]

(2*Sqrt[1 - x^(-2)]*x)/Sqrt[1 + x]

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.67

method	result	size
gosper	\(\frac {2 x -2}{\sqrt {\frac {x -1}{1+x}}\, \sqrt {1+x}}\)	\(22\)
default	\(\frac {2 x -2}{\sqrt {\frac {x -1}{1+x}}\, \sqrt {1+x}}\)	\(22\)
risch	\(\frac {2 x -2}{\sqrt {\frac {x -1}{1+x}}\, \sqrt {1+x}}\)	\(22\)

[In]

int(1/((x-1)/(1+x))^(1/2)/(1+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*(x-1)/((x-1)/(1+x))^(1/2)/(1+x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.55 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1+x}} \, dx=2 \, \sqrt {x + 1} \sqrt {\frac {x - 1}{x + 1}} \]

[In]

integrate(1/((-1+x)/(1+x))^(1/2)/(1+x)^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(x + 1)*sqrt((x - 1)/(x + 1))

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.48 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.58 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1+x}} \, dx=\begin {cases} 2 \sqrt {x - 1} & \text {for}\: \left |{x}\right | > 1 \\2 i \sqrt {1 - x} & \text {otherwise} \end {cases} \]

[In]

integrate(1/((-1+x)/(1+x))**(1/2)/(1+x)**(1/2),x)

[Out]

Piecewise((2*sqrt(x - 1), Abs(x) > 1), (2*I*sqrt(1 - x), True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.21 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1+x}} \, dx=2 \, \sqrt {x - 1} \]

[In]

integrate(1/((-1+x)/(1+x))^(1/2)/(1+x)^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(x - 1)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1+x}} \, dx=-\frac {2 \, {\left (i \, \sqrt {2} - \sqrt {x - 1}\right )} \mathrm {sgn}\left (x\right )}{\mathrm {sgn}\left (x + 1\right )} \]

[In]

integrate(1/((-1+x)/(1+x))^(1/2)/(1+x)^(1/2),x, algorithm="giac")

[Out]

-2*(I*sqrt(2) - sqrt(x - 1))*sgn(x)/sgn(x + 1)

Mupad [B] (verification not implemented)

Time = 4.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.55 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1+x}} \, dx=2\,\sqrt {\frac {x-1}{x+1}}\,\sqrt {x+1} \]

[In]

int(1/(((x - 1)/(x + 1))^(1/2)*(x + 1)^(1/2)),x)

[Out]

2*((x - 1)/(x + 1))^(1/2)*(x + 1)^(1/2)