\(\int \frac {e^{\coth ^{-1}(x)} x}{1+x} \, dx\) [287]

Optimal result

Integrand size = 11, antiderivative size = 22 \[ \int \frac {e^{\coth ^{-1}(x)} x}{1+x} \, dx=\sqrt {1+\frac {1}{x}} \sqrt {\frac {-1+x}{x}} x \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6310, 6314, 97} \[ \int \frac {e^{\coth ^{-1}(x)} x}{1+x} \, dx=\sqrt {\frac {1}{x}+1} \sqrt {\frac {x-1}{x}} x \]

[In]

Int[(E^ArcCoth[x]*x)/(1 + x),x]

[Out]

Sqrt[1 + x^(-1)]*Sqrt[(-1 + x)/x]*x

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] /; FreeQ[{a, b, c, d,
e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 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 6314

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> Dist[-c^p, Subst[Int[(1 + d*(x/c))^p
*((1 + x/a)^(n/2)/(x^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])

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\coth ^{-1}(x)} x}{1+x} \, dx=x \sqrt {\frac {-1+x^2}{x^2}} \]

[In]

Integrate[(E^ArcCoth[x]*x)/(1 + x),x]

[Out]

x*Sqrt[(-1 + x^2)/x^2]

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73

method	result	size
gosper	\(\frac {x -1}{\sqrt {\frac {x -1}{1+x}}}\)	\(16\)
risch	\(\frac {x -1}{\sqrt {\frac {x -1}{1+x}}}\)	\(16\)
trager	\(\left (1+x \right ) \sqrt {-\frac {1-x}{1+x}}\)	\(19\)
default	\(\frac {\left (x -1\right ) \sqrt {x^{2}-1}}{\sqrt {\frac {x -1}{1+x}}\, \sqrt {\left (x -1\right ) \left (1+x \right )}}\)	\(32\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\coth ^{-1}(x)} x}{1+x} \, dx={\left (x + 1\right )} \sqrt {\frac {x - 1}{x + 1}} \]

[In]

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

[Out]

(x + 1)*sqrt((x - 1)/(x + 1))

Sympy [F]

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

[In]

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

[Out]

Integral(x/(sqrt((x - 1)/(x + 1))*(x + 1)), x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {e^{\coth ^{-1}(x)} x}{1+x} \, dx=-\frac {2 \, \sqrt {\frac {x - 1}{x + 1}}}{\frac {x - 1}{x + 1} - 1} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \frac {e^{\coth ^{-1}(x)} x}{1+x} \, dx=\frac {\sqrt {x^{2} - 1}}{\mathrm {sgn}\left (x + 1\right )} \]

[In]

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

[Out]

sqrt(x^2 - 1)/sgn(x + 1)

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {e^{\coth ^{-1}(x)} x}{1+x} \, dx=-\frac {2\,\sqrt {\frac {x-1}{x+1}}}{\frac {x-1}{x+1}-1} \]

[In]

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

[Out]

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