3.4.67 \(\int \frac {e^{\tanh ^{-1}(x)} x}{1+x} \, dx\) [367]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6264, 75} \begin {gather*} -\sqrt {1-x} \sqrt {x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[x]*x)/(1 + x),x]

[Out]

-(Sqrt[1 - x]*Sqrt[1 + x])

Rule 75

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

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)} x}{1+x} \, dx &=\int \frac {x}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\sqrt {1-x} \sqrt {1+x}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 13, normalized size = 0.72 \begin {gather*} -\sqrt {1-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcTanh[x]*x)/(1 + x),x]

[Out]

-Sqrt[1 - x^2]

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Maple [A]
time = 0.77, size = 12, normalized size = 0.67

method	result	size
derivativedivides	\(-\sqrt {-x^{2}+1}\)	\(12\)
default	\(-\sqrt {-x^{2}+1}\)	\(12\)
trager	\(-\sqrt {-x^{2}+1}\)	\(12\)
risch	\(\frac {x^{2}-1}{\sqrt {-x^{2}+1}}\)	\(16\)
gosper	\(\frac {\left (x -1\right ) \left (1+x \right )}{\sqrt {-x^{2}+1}}\)	\(17\)
meijerg	\(-\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-x^{2}+1}}{2 \sqrt {\pi }}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 0.25, size = 11, normalized size = 0.61 \begin {gather*} -\sqrt {-x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 0.35, size = 11, normalized size = 0.61 \begin {gather*} -\sqrt {-x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)

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Sympy [A]
time = 0.04, size = 8, normalized size = 0.44 \begin {gather*} - \sqrt {1 - x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(1 - x**2)

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Giac [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 0.41, size = 11, normalized size = 0.61 \begin {gather*} -\sqrt {-x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)

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Mupad [B]
time = 0.11, size = 11, normalized size = 0.61 \begin {gather*} -\sqrt {1-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(1 - x^2)^(1/2),x)

[Out]

-(1 - x^2)^(1/2)

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