3.4.79 \(\int \frac {e^{\tanh ^{-1}(x)} x}{\sqrt {1+x}} \, dx\) [379]

Optimal. Leaf size=25 \[ -2 \sqrt {1-x}+\frac {2}{3} (1-x)^{3/2} \]

[Out]

2/3*(1-x)^(3/2)-2*(1-x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6264, 45} \begin {gather*} \frac {2}{3} (1-x)^{3/2}-2 \sqrt {1-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[x]*x)/Sqrt[1 + x],x]

[Out]

-2*Sqrt[1 - x] + (2*(1 - x)^(3/2))/3

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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}{\sqrt {1+x}} \, dx &=\int \frac {x}{\sqrt {1-x}} \, dx\\ &=\int \left (\frac {1}{\sqrt {1-x}}-\sqrt {1-x}\right ) \, dx\\ &=-2 \sqrt {1-x}+\frac {2}{3} (1-x)^{3/2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 16, normalized size = 0.64 \begin {gather*} -\frac {2}{3} \sqrt {1-x} (2+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcTanh[x]*x)/Sqrt[1 + x],x]

[Out]

(-2*Sqrt[1 - x]*(2 + x))/3

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Maple [A]
time = 0.99, size = 20, normalized size = 0.80

method	result	size
default	\(-\frac {2 \sqrt {-x^{2}+1}\, \left (x +2\right )}{3 \sqrt {1+x}}\)	\(20\)
gosper	\(\frac {2 \left (x -1\right ) \left (x +2\right ) \sqrt {1+x}}{3 \sqrt {-x^{2}+1}}\)	\(23\)
risch	\(\frac {2 \sqrt {\frac {-x^{2}+1}{1+x}}\, \sqrt {1+x}\, \left (x +2\right ) \left (x -1\right )}{3 \sqrt {-x^{2}+1}\, \sqrt {1-x}}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/(1+x)^(1/2)*(-x^2+1)^(1/2)*(x+2)

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Maxima [A]
time = 0.26, size = 15, normalized size = 0.60 \begin {gather*} \frac {2 \, {\left (x^{2} + x - 2\right )}}{3 \, \sqrt {-x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x^2 + x - 2)/sqrt(-x + 1)

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Fricas [A]
time = 0.36, size = 19, normalized size = 0.76 \begin {gather*} -\frac {2 \, \sqrt {-x^{2} + 1} {\left (x + 2\right )}}{3 \, \sqrt {x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(t_

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Mupad [B]
time = 0.02, size = 19, normalized size = 0.76 \begin {gather*} -\frac {2\,\sqrt {1-x^2}\,\left (x+2\right )}{3\,\sqrt {x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(1 - x^2)^(1/2)*(x + 2))/(3*(x + 1)^(1/2))

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