3.4.70 \(\int \frac {e^{\tanh ^{-1}(x)}}{(1+x)^2} \, dx\) [370]

Optimal. Leaf size=18 \[ -\frac {\sqrt {1-x}}{\sqrt {1+x}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6264, 37} \begin {gather*} -\frac {\sqrt {1-x}}{\sqrt {x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[x]/(1 + x)^2,x]

[Out]

-(Sqrt[1 - x]/Sqrt[1 + x])

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]

Rule 6264

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[u*(1 + d*(x/c))^
p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ
[p] || GtQ[c, 0])

Rubi steps

\begin {align*} \int \frac {e^{\tanh ^{-1}(x)}}{(1+x)^2} \, dx &=\int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx\\ &=-\frac {\sqrt {1-x}}{\sqrt {1+x}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1-x}}{\sqrt {1+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[x]/(1 + x)^2,x]

[Out]

-(Sqrt[1 - x]/Sqrt[1 + x])

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Maple [A]
time = 1.02, size = 22, normalized size = 1.22

method	result	size
gosper	\(\frac {x -1}{\sqrt {-x^{2}+1}}\)	\(14\)
risch	\(\frac {x -1}{\sqrt {-x^{2}+1}}\)	\(14\)
trager	\(-\frac {\sqrt {-x^{2}+1}}{1+x}\)	\(17\)
default	\(-\frac {\sqrt {-\left (1+x \right )^{2}+2+2 x}}{1+x}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+x)/(-x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.46, size = 16, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {-x^{2} + 1}}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)/(x + 1)

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Fricas [A]
time = 0.33, size = 19, normalized size = 1.06 \begin {gather*} -\frac {x + \sqrt {-x^{2} + 1} + 1}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + sqrt(-x^2 + 1) + 1)/(x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.42, size = 21, normalized size = 1.17 \begin {gather*} \frac {2}{\frac {\sqrt {-x^{2} + 1} - 1}{x} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/((sqrt(-x^2 + 1) - 1)/x - 1)

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Mupad [B]
time = 0.03, size = 13, normalized size = 0.72 \begin {gather*} \frac {x-1}{\sqrt {1-x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - x^2)^(1/2)*(x + 1)),x)

[Out]

(x - 1)/(1 - x^2)^(1/2)

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