\(\int \frac {e^{\coth ^{-1}(x)}}{(1+x)^2} \, dx\) [292]

Optimal result

Integrand size = 10, antiderivative size = 21 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1+x)^2} \, dx=\frac {\sqrt {\frac {-1+x}{x}}}{\sqrt {1+\frac {1}{x}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6310, 6315, 37} \[ \int \frac {e^{\coth ^{-1}(x)}}{(1+x)^2} \, dx=\frac {\sqrt {\frac {x-1}{x}}}{\sqrt {\frac {1}{x}+1}} \]

[In]

Int[E^ArcCoth[x]/(1 + x)^2,x]

[Out]

Sqrt[(-1 + x)/x]/Sqrt[1 + x^(-1)]

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 6310

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

Rule 6315

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[-c^p, Subst[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, 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}& = \int \frac {e^{\coth ^{-1}(x)}}{\left (1+\frac {1}{x}\right )^2 x^2} \, dx \\ & = -\text {Subst}\left (\int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {\sqrt {\frac {-1+x}{x}}}{\sqrt {1+\frac {1}{x}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1+x)^2} \, dx=\frac {\sqrt {1-\frac {1}{x^2}} x}{1+x} \]

[In]

Integrate[E^ArcCoth[x]/(1 + x)^2,x]

[Out]

(Sqrt[1 - x^(-2)]*x)/(1 + x)

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.57

method	result	size
derivativedivides	\(\sqrt {\frac {x -1}{1+x}}\)	\(12\)
trager	\(\sqrt {-\frac {1-x}{1+x}}\)	\(15\)
gosper	\(\frac {x -1}{\sqrt {\frac {x -1}{1+x}}\, \left (1+x \right )}\)	\(21\)
risch	\(\frac {x -1}{\sqrt {\frac {x -1}{1+x}}\, \left (1+x \right )}\)	\(21\)
default	\(\frac {\sqrt {x^{2}-1}\, \left (x -1\right )}{\left (1+x \right ) \sqrt {\left (x -1\right ) \left (1+x \right )}\, \sqrt {\frac {x -1}{1+x}}}\)	\(37\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.52 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1+x)^2} \, dx=\sqrt {\frac {x - 1}{x + 1}} \]

[In]

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

[Out]

sqrt((x - 1)/(x + 1))

Sympy [A] (verification not implemented)

Time = 3.34 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.38 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1+x)^2} \, dx=\sqrt {\frac {x - 1}{x + 1}} \]

[In]

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

[Out]

sqrt((x - 1)/(x + 1))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.52 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1+x)^2} \, dx=\sqrt {\frac {x - 1}{x + 1}} \]

[In]

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

[Out]

sqrt((x - 1)/(x + 1))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1+x)^2} \, dx=\frac {2}{{\left (x - \sqrt {x^{2} - 1} + 1\right )} \mathrm {sgn}\left (x + 1\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.52 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1+x)^2} \, dx=\sqrt {1-\frac {2}{x+1}} \]

[In]

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

[Out]

(1 - 2/(x + 1))^(1/2)