### 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*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])

________________________________________________________________________________________

Rubi [A]  time = 0.096523, antiderivative size = 73, normalized size of antiderivative = 1., 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 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]

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 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]

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 = 0.0147966, 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])

________________________________________________________________________________________

Maple [A]  time = 0.06, size = 25, normalized size = 0.3 \begin{align*}{\frac{ \left ( -2+2\,x \right ) \left ( x+2 \right ) }{3}{\frac{1}{\sqrt{{\frac{-1+x}{1+x}}}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

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 = 1.08373, size = 18, normalized size = 0.25 \begin{align*} \frac{2 \,{\left (x^{2} + x - 2\right )}}{3 \, \sqrt{x - 1}} \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.55459, size = 63, normalized size = 0.86 \begin{align*} \frac{2}{3} \,{\left (x + 2\right )} \sqrt{x + 1} \sqrt{\frac{x - 1}{x + 1}} \end{align*}

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))

________________________________________________________________________________________

Sympy [A]  time = 62.8377, size = 48, normalized size = 0.66 \begin{align*} \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} \end{align*}

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))

________________________________________________________________________________________

Giac [C]  time = 1.12459, size = 27, normalized size = 0.37 \begin{align*} \frac{2}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} - \frac{2}{3} i \, \sqrt{2} + 2 \, \sqrt{x - 1} \end{align*}

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]

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