3.36 \(\int e^{-\tanh ^{-1}(a x)} x \, dx\)

Optimal. Leaf size=39 \[ -\frac{\sqrt{1-a^2 x^2} (2-a x)}{2 a^2}-\frac{\sin ^{-1}(a x)}{2 a^2} \]

[Out]

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

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Rubi [A]  time = 0.0231506, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6124, 780, 216} \[ -\frac{\sqrt{1-a^2 x^2} (2-a x)}{2 a^2}-\frac{\sin ^{-1}(a x)}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x/E^ArcTanh[a*x],x]

[Out]

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

Rule 6124

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*((1 + a*x)^((n + 1)/2)/((1 - a*x)^((n - 1)/
2)*Sqrt[1 - a^2*x^2])), x] /; FreeQ[{a, m}, x] && IntegerQ[(n - 1)/2]

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

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}(a x)} x \, dx &=\int \frac{x (1-a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{(2-a x) \sqrt{1-a^2 x^2}}{2 a^2}-\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{2 a}\\ &=-\frac{(2-a x) \sqrt{1-a^2 x^2}}{2 a^2}-\frac{\sin ^{-1}(a x)}{2 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0253898, size = 34, normalized size = 0.87 \[ \frac{(a x-2) \sqrt{1-a^2 x^2}-\sin ^{-1}(a x)}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x/E^ArcTanh[a*x],x]

[Out]

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

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Maple [B]  time = 0.04, size = 119, normalized size = 3.1 \begin{align*}{\frac{x}{2\,a}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{1}{2\,a}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{1}{{a}^{2}}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}-{\frac{1}{a}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.42151, size = 61, normalized size = 1.56 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1} x}{2 \, a} - \frac{\arcsin \left (a x\right )}{2 \, a^{2}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.9756, size = 112, normalized size = 2.87 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1}{\left (a x - 2\right )} + 2 \, \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right )}{2 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1623, size = 55, normalized size = 1.41 \begin{align*} \frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{x}{a} - \frac{2}{a^{2}}\right )} - \frac{\arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{2 \, a{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-a^2*x^2 + 1)*(x/a - 2/a^2) - 1/2*arcsin(a*x)*sgn(a)/(a*abs(a))