3.364 \(\int e^{\tanh ^{-1}(x)} (1+x) \, dx\)

Optimal. Leaf size=47 \[ -\frac{1}{2} \sqrt{1-x} (x+1)^{3/2}-\frac{3}{2} \sqrt{1-x} \sqrt{x+1}+\frac{3}{2} \sin ^{-1}(x) \]

[Out]

(-3*Sqrt[1 - x]*Sqrt[1 + x])/2 - (Sqrt[1 - x]*(1 + x)^(3/2))/2 + (3*ArcSin[x])/2

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Rubi [A]  time = 0.0174384, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6129, 50, 41, 216} \[ -\frac{1}{2} \sqrt{1-x} (x+1)^{3/2}-\frac{3}{2} \sqrt{1-x} \sqrt{x+1}+\frac{3}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[x]*(1 + x),x]

[Out]

(-3*Sqrt[1 - x]*Sqrt[1 + x])/2 - (Sqrt[1 - x]*(1 + x)^(3/2))/2 + (3*ArcSin[x])/2

Rule 6129

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])

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 216

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

Rubi steps

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

Mathematica [A]  time = 0.0119762, size = 37, normalized size = 0.79 \[ -\frac{1}{2} \sqrt{1-x^2} (x+4)-3 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcTanh[x]*(1 + x),x]

[Out]

-((4 + x)*Sqrt[1 - x^2])/2 - 3*ArcSin[Sqrt[1 - x]/Sqrt[2]]

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Maple [A]  time = 0.034, size = 29, normalized size = 0.6 \begin{align*} -{\frac{x}{2}\sqrt{-{x}^{2}+1}}+{\frac{3\,\arcsin \left ( x \right ) }{2}}-2\,\sqrt{-{x}^{2}+1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.43312, size = 38, normalized size = 0.81 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 1} x - 2 \, \sqrt{-x^{2} + 1} + \frac{3}{2} \, \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.79934, size = 86, normalized size = 1.83 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 1}{\left (x + 4\right )} - 3 \, \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.236391, size = 27, normalized size = 0.57 \begin{align*} - \frac{x \sqrt{1 - x^{2}}}{2} - 2 \sqrt{1 - x^{2}} + \frac{3 \operatorname{asin}{\left (x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12413, size = 26, normalized size = 0.55 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 1}{\left (x + 4\right )} + \frac{3}{2} \, \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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