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

Optimal. Leaf size=38 \[ \frac{\sin ^{-1}(a x)}{2 a^2}-\frac{(a x+2) \sqrt{1-a^2 x^2}}{2 a^2} \]

[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 = 0.0217802, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6124, 780, 216} \[ \frac{\sin ^{-1}(a x)}{2 a^2}-\frac{(a x+2) \sqrt{1-a^2 x^2}}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*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 = 0.0245613, size = 33, normalized size = 0.87 \[ \frac{\sin ^{-1}(a x)-(a x+2) \sqrt{1-a^2 x^2}}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a*x]*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 = 0.036, size = 67, normalized size = 1.8 \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}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]  time = 1.43191, size = 77, normalized size = 2.03 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} x}{2 \, a} + \frac{\arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{2 \, \sqrt{a^{2}} a} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.08853, size = 113, normalized size = 2.97 \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((a*x+1)/(-a^2*x^2+1)^(1/2)*x,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 [C]  time = 4.00632, size = 110, normalized size = 2.89 \begin{align*} a \left (\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left (a x \right )}}{2 a^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left (a x \right )}}{2 a^{3}} & \text{otherwise} \end{cases}\right ) + \begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a*
*2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) + Piecewise((x**2/2, Eq(a**2, 0))
, (-sqrt(-a**2*x**2 + 1)/a**2, True))

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Giac [A]  time = 1.21536, size = 55, normalized size = 1.45 \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((a*x+1)/(-a^2*x^2+1)^(1/2)*x,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))