### 3.324 $$\int e^{\coth ^{-1}(x)} \sqrt{1+x} \, dx$$

Optimal. Leaf size=70 $\frac{2 \sqrt{-\frac{1-x}{x}} \sqrt{x+1} x}{3 \sqrt{\frac{1}{x}+1}}+\frac{10 \sqrt{-\frac{1-x}{x}} \sqrt{x+1}}{3 \sqrt{\frac{1}{x}+1}}$

[Out]

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

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Rubi [A]  time = 0.0811716, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {6176, 6181, 78, 37} $\frac{2 \sqrt{-\frac{1-x}{x}} \sqrt{x+1} x}{3 \sqrt{\frac{1}{x}+1}}+\frac{10 \sqrt{-\frac{1-x}{x}} \sqrt{x+1}}{3 \sqrt{\frac{1}{x}+1}}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcCoth[x]*Sqrt[1 + x],x]

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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 e^{\coth ^{-1}(x)} \sqrt{1+x} \, dx &=\frac{\sqrt{1+x} \int e^{\coth ^{-1}(x)} \sqrt{1+\frac{1}{x}} \sqrt{x} \, dx}{\sqrt{1+\frac{1}{x}} \sqrt{x}}\\ &=-\frac{\left (\sqrt{\frac{1}{x}} \sqrt{1+x}\right ) \operatorname{Subst}\left (\int \frac{1+x}{\sqrt{1-x} x^{5/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1+\frac{1}{x}}}\\ &=\frac{2 \sqrt{-\frac{1-x}{x}} x \sqrt{1+x}}{3 \sqrt{1+\frac{1}{x}}}-\frac{\left (5 \sqrt{\frac{1}{x}} \sqrt{1+x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 \sqrt{1+\frac{1}{x}}}\\ &=\frac{10 \sqrt{-\frac{1-x}{x}} \sqrt{1+x}}{3 \sqrt{1+\frac{1}{x}}}+\frac{2 \sqrt{-\frac{1-x}{x}} x \sqrt{1+x}}{3 \sqrt{1+\frac{1}{x}}}\\ \end{align*}

Mathematica [A]  time = 0.0145598, size = 34, normalized size = 0.49 $\frac{2 \sqrt{\frac{x-1}{x}} \sqrt{x+1} (x+5)}{3 \sqrt{\frac{1}{x}+1}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^ArcCoth[x]*Sqrt[1 + x],x]

[Out]

(2*Sqrt[(-1 + x)/x]*Sqrt[1 + x]*(5 + x))/(3*Sqrt[1 + x^(-1)])

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Maple [A]  time = 0.058, size = 25, normalized size = 0.4 \begin{align*}{\frac{ \left ( -2+2\,x \right ) \left ( x+5 \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)*(1+x)^(1/2),x)

[Out]

2/3*(-1+x)*(x+5)/((-1+x)/(1+x))^(1/2)/(1+x)^(1/2)

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Maxima [A]  time = 1.05387, size = 20, normalized size = 0.29 \begin{align*} \frac{2 \,{\left (x^{2} + 4 \, x - 5\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)*(1+x)^(1/2),x, algorithm="maxima")

[Out]

2/3*(x^2 + 4*x - 5)/sqrt(x - 1)

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Fricas [A]  time = 1.61203, size = 63, normalized size = 0.9 \begin{align*} \frac{2}{3} \,{\left (x + 5\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)*(1+x)^(1/2),x, algorithm="fricas")

[Out]

2/3*(x + 5)*sqrt(x + 1)*sqrt((x - 1)/(x + 1))

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Sympy [A]  time = 23.2935, size = 39, normalized size = 0.56 \begin{align*} 2 \left (\begin{cases} 2 \sqrt{2} \left (\frac{\sqrt{2} \left (x - 1\right )^{\frac{3}{2}}}{12} + \frac{\sqrt{2} \sqrt{x - 1}}{2}\right ) & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*Piecewise((2*sqrt(2)*(sqrt(2)*(x - 1)**(3/2)/12 + sqrt(2)*sqrt(x - 1)/2), (x >= -1) & (x < 1)))

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Giac [C]  time = 1.13628, size = 27, normalized size = 0.39 \begin{align*} \frac{2}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} - \frac{8}{3} i \, \sqrt{2} + 4 \, \sqrt{x - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x - 1)^(3/2) - 8/3*I*sqrt(2) + 4*sqrt(x - 1)