3.331 \(\int \frac{e^{\coth ^{-1}(x)} x}{(1+x)^{3/2}} \, dx\)

Optimal. Leaf size=93 \[ \frac{2 \left (\frac{1}{x}+1\right )^{3/2} \sqrt{-\frac{1-x}{x}} x^2}{(x+1)^{3/2}}+\frac{\sqrt{2} \left (\frac{1}{x}+1\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{-\frac{1-x}{x}}}\right )}{\left (\frac{1}{x}\right )^{3/2} (x+1)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.116848, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {6176, 6181, 96, 93, 203} \[ \frac{2 \left (\frac{1}{x}+1\right )^{3/2} \sqrt{-\frac{1-x}{x}} x^2}{(x+1)^{3/2}}+\frac{\sqrt{2} \left (\frac{1}{x}+1\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{-\frac{1-x}{x}}}\right )}{\left (\frac{1}{x}\right )^{3/2} (x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcCoth[x]*x)/(1 + x)^(3/2),x]

[Out]

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

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 96

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] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0374601, size = 65, normalized size = 0.7 \[ \frac{\sqrt{\frac{1}{x}+1} x \left (2 \sqrt{\frac{x-1}{x}}-\sqrt{2} \sqrt{\frac{1}{x}} \tan ^{-1}\left (\frac{\sqrt{\frac{x-1}{x^2}} x}{\sqrt{2}}\right )\right )}{\sqrt{x+1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^ArcCoth[x]*x)/(1 + x)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.106, size = 47, normalized size = 0.5 \begin{align*} -{\sqrt{-1+x} \left ( \sqrt{2}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-1+x}} \right ) -2\,\sqrt{-1+x} \right ){\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)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{\frac{x - 1}{x + 1}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/2)*x/(1+x)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.49007, size = 138, normalized size = 1.48 \begin{align*} -\sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1} \sqrt{\frac{x - 1}{x + 1}}\right ) + 2 \, \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)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))**(1/2)*x/(1+x)**(3/2),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/2)*x/(1+x)^(3/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError