3.377 \(\int e^{\tanh ^{-1}(x)} \sqrt{1-x} x \, dx\)

Optimal. Leaf size=23 \[ \frac{2}{5} (x+1)^{5/2}-\frac{2}{3} (x+1)^{3/2} \]

[Out]

(-2*(1 + x)^(3/2))/3 + (2*(1 + x)^(5/2))/5

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Rubi [A]  time = 0.0393961, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6128, 26, 43} \[ \frac{2}{5} (x+1)^{5/2}-\frac{2}{3} (x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[x]*Sqrt[1 - x]*x,x]

[Out]

(-2*(1 + x)^(3/2))/3 + (2*(1 + x)^(5/2))/5

Rule 6128

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

Rule 26

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(j_))^(p_.), x_Symbol] :> Dist[(-(b^2/d))^m, Int[
u/(a - b*x^n)^m, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[j, 2*n] && EqQ[p, -m] && EqQ[b^2*c + a^2*d,
0] && GtQ[a, 0] && LtQ[d, 0]

Rule 43

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])

Rubi steps

\begin{align*} \int e^{\tanh ^{-1}(x)} \sqrt{1-x} x \, dx &=\int \frac{x \sqrt{1-x^2}}{\sqrt{1-x}} \, dx\\ &=\int x \sqrt{1+x} \, dx\\ &=\int \left (-\sqrt{1+x}+(1+x)^{3/2}\right ) \, dx\\ &=-\frac{2}{3} (1+x)^{3/2}+\frac{2}{5} (1+x)^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0069671, size = 16, normalized size = 0.7 \[ \frac{2}{15} (x+1)^{3/2} (3 x-2) \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[x]*Sqrt[1 - x]*x,x]

[Out]

(2*(1 + x)^(3/2)*(-2 + 3*x))/15

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Maple [A]  time = 0.031, size = 29, normalized size = 1.3 \begin{align*}{\frac{2\, \left ( 1+x \right ) ^{2} \left ( 3\,x-2 \right ) }{15}\sqrt{1-x}{\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(1+x)^2*(3*x-2)*(1-x)^(1/2)/(-x^2+1)^(1/2)

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Maxima [B]  time = 0.984432, size = 51, normalized size = 2.22 \begin{align*} \frac{2 \,{\left (3 \, x^{3} - x^{2} + 4 \, x + 8\right )}}{15 \, \sqrt{x + 1}} + \frac{2 \,{\left (x^{2} - x - 2\right )}}{3 \, \sqrt{x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*x^3 - x^2 + 4*x + 8)/sqrt(x + 1) + 2/3*(x^2 - x - 2)/sqrt(x + 1)

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Fricas [B]  time = 1.86095, size = 80, normalized size = 3.48 \begin{align*} -\frac{2 \,{\left (3 \, x^{2} + x - 2\right )} \sqrt{-x^{2} + 1} \sqrt{-x + 1}}{15 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(3*x^2 + x - 2)*sqrt(-x^2 + 1)*sqrt(-x + 1)/(x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18087, size = 27, normalized size = 1.17 \begin{align*} \frac{2}{5} \,{\left (x + 1\right )}^{\frac{5}{2}} - \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - \frac{4}{15} \, \sqrt{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(x + 1)^(5/2) - 2/3*(x + 1)^(3/2) - 4/15*sqrt(2)