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

Optimal. Leaf size=73 \[ \frac {2 \sqrt {\frac {1}{x}+1} \sqrt {-\frac {1-x}{x}} x^2}{3 \sqrt {x+1}}+\frac {4 \sqrt {\frac {1}{x}+1} \sqrt {-\frac {1-x}{x}} x}{3 \sqrt {x+1}} \]

[Out]

4/3*x*(1/x+1)^(1/2)*((-1+x)/x)^(1/2)/(1+x)^(1/2)+2/3*x^2*(1/x+1)^(1/2)*((-1+x)/x)^(1/2)/(1+x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6176, 6181, 45, 37} \[ \frac {2 \sqrt {\frac {1}{x}+1} \sqrt {-\frac {1-x}{x}} x^2}{3 \sqrt {x+1}}+\frac {4 \sqrt {\frac {1}{x}+1} \sqrt {-\frac {1-x}{x}} x}{3 \sqrt {x+1}} \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcCoth[x]*x)/Sqrt[1 + x],x]

[Out]

(4*Sqrt[1 + x^(-1)]*Sqrt[-((1 - x)/x)]*x)/(3*Sqrt[1 + x]) + (2*Sqrt[1 + x^(-1)]*Sqrt[-((1 - x)/x)]*x^2)/(3*Sqr
t[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 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 6176

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 6181

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> -Dist[c^p*x^m*(1/x)^m, Sub
st[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*} \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1+x}} \, dx &=\frac {\left (\sqrt {1+\frac {1}{x}} \sqrt {x}\right ) \int \frac {e^{\coth ^{-1}(x)} \sqrt {x}}{\sqrt {1+\frac {1}{x}}} \, dx}{\sqrt {1+x}}\\ &=-\frac {\sqrt {1+\frac {1}{x}} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x^{5/2}} \, 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^2}{3 \sqrt {1+x}}-\frac {\left (2 \sqrt {1+\frac {1}{x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 \sqrt {\frac {1}{x}} \sqrt {1+x}}\\ &=\frac {4 \sqrt {1+\frac {1}{x}} \sqrt {-\frac {1-x}{x}} x}{3 \sqrt {1+x}}+\frac {2 \sqrt {1+\frac {1}{x}} \sqrt {-\frac {1-x}{x}} x^2}{3 \sqrt {1+x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 26, normalized size = 0.36 \[ \frac {2 \sqrt {1-\frac {1}{x^2}} x (x+2)}{3 \sqrt {x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcCoth[x]*x)/Sqrt[1 + x],x]

[Out]

(2*Sqrt[1 - x^(-2)]*x*(2 + x))/(3*Sqrt[1 + x])

________________________________________________________________________________________

fricas [A]  time = 0.63, size = 21, normalized size = 0.29 \[ \frac {2}{3} \, {\left (x + 2\right )} \sqrt {x + 1} \sqrt {\frac {x - 1}{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 2)*sqrt(x + 1)*sqrt((x - 1)/(x + 1))

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 0.04, size = 25, normalized size = 0.34 \[ \frac {2 \left (-1+x \right ) \left (x +2\right )}{3 \sqrt {\frac {-1+x}{1+x}}\, \sqrt {1+x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(-1+x)*(x+2)/((-1+x)/(1+x))^(1/2)/(1+x)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.00, size = 13, normalized size = 0.18 \[ \frac {2 \, {\left (x^{2} + x - 2\right )}}{3 \, \sqrt {x - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x^2 + x - 2)/sqrt(x - 1)

________________________________________________________________________________________

mupad [B]  time = 1.24, size = 21, normalized size = 0.29 \[ \frac {2\,\sqrt {\frac {x-1}{x+1}}\,\sqrt {x+1}\,\left (x+2\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*((x - 1)/(x + 1))^(1/2)*(x + 1)^(1/2)*(x + 2))/3

________________________________________________________________________________________

sympy [A]  time = 7.82, size = 48, normalized size = 0.66 \[ \begin {cases} \frac {2 x \sqrt {x - 1}}{3} + \frac {4 \sqrt {x - 1}}{3} & \text {for}\: \left |{x}\right | > 1 \\\frac {2 i x \sqrt {1 - x}}{3} + \frac {4 i \sqrt {1 - x}}{3} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*x*sqrt(x - 1)/3 + 4*sqrt(x - 1)/3, Abs(x) > 1), (2*I*x*sqrt(1 - x)/3 + 4*I*sqrt(1 - x)/3, True))

________________________________________________________________________________________