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

Optimal. Leaf size=22 \[ \sqrt {\frac {1}{x}+1} \sqrt {\frac {x-1}{x}} x \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6175, 6179, 95} \[ \sqrt {\frac {1}{x}+1} \sqrt {\frac {x-1}{x}} x \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcCoth[x]*x)/(1 + x),x]

[Out]

Sqrt[1 + x^(-1)]*Sqrt[(-1 + x)/x]*x

Rule 95

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 6175

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 6179

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*} \int \frac {e^{\coth ^{-1}(x)} x}{1+x} \, dx &=\int \frac {e^{\coth ^{-1}(x)}}{1+\frac {1}{x}} \, dx\\ &=-\operatorname {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*}

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Mathematica [A]  time = 0.03, size = 15, normalized size = 0.68 \[ x \sqrt {\frac {x^2-1}{x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcCoth[x]*x)/(1 + x),x]

[Out]

x*Sqrt[(-1 + x^2)/x^2]

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fricas [A]  time = 0.44, size = 15, normalized size = 0.68 \[ {\left (x + 1\right )} \sqrt {\frac {x - 1}{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 1)*sqrt((x - 1)/(x + 1))

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giac [A]  time = 0.14, size = 29, normalized size = 1.32 \[ -\frac {2}{\sqrt {\frac {x - 1}{x + 1}} - \frac {1}{\sqrt {\frac {x - 1}{x + 1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 16, normalized size = 0.73 \[ \frac {-1+x}{\sqrt {\frac {-1+x}{1+x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.31, size = 26, normalized size = 1.18 \[ -\frac {2 \, \sqrt {\frac {x - 1}{x + 1}}}{\frac {x - 1}{x + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.05, size = 26, normalized size = 1.18 \[ -\frac {2\,\sqrt {\frac {x-1}{x+1}}}{\frac {x-1}{x+1}-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {\frac {x - 1}{x + 1}} \left (x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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