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

Optimal. Leaf size=22 \[ \tanh ^{-1}\left (\sqrt {\frac {1}{x}+1} \sqrt {\frac {x-1}{x}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6175, 6180, 92, 206} \[ \tanh ^{-1}\left (\sqrt {\frac {1}{x}+1} \sqrt {\frac {x-1}{x}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCoth[x]/(1 + x),x]

[Out]

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

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 206

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

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 6180

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

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.82 \[ \log \left (x \left (\sqrt {\frac {x^2-1}{x^2}}+1\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCoth[x]/(1 + x),x]

[Out]

Log[x*(1 + Sqrt[(-1 + x^2)/x^2])]

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fricas [A]  time = 0.61, size = 31, normalized size = 1.41 \[ \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.13, size = 32, normalized size = 1.45 \[ \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \log \left ({\left | \sqrt {\frac {x - 1}{x + 1}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sqrt((x - 1)/(x + 1)) + 1) - log(abs(sqrt((x - 1)/(x + 1)) - 1))

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maple [A]  time = 0.05, size = 35, normalized size = 1.59 \[ \frac {\left (-1+x \right ) \ln \left (x +\sqrt {x^{2}-1}\right )}{\sqrt {\frac {-1+x}{1+x}}\, \sqrt {\left (1+x \right ) \left (-1+x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.32, size = 31, normalized size = 1.41 \[ \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 14, normalized size = 0.64 \[ 2\,\mathrm {atanh}\left (\sqrt {\frac {x-1}{x+1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atanh(((x - 1)/(x + 1))^(1/2))

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sympy [A]  time = 4.03, size = 29, normalized size = 1.32 \[ - \log {\left (-1 + \frac {1}{\sqrt {1 - \frac {2}{x + 1}}} \right )} + \log {\left (1 + \frac {1}{\sqrt {1 - \frac {2}{x + 1}}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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