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]

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

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Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.54, size = 48, normalized size = 1.26 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x + 2\right )} + 2 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{2 \, a^{2}} \]

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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giac [A]  time = 1.11, size = 41, normalized size = 1.08 \[ -\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {x}{a} + \frac {2}{a^{2}}\right )} + \frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, a {\left | a \right |}} \]

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))

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maple [B]  time = 0.03, size = 67, normalized size = 1.76 \[ -\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a \sqrt {a^{2}}}-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{2}} \]

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^2*x^2+1)^(1/2)/a+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 = 0.42, size = 45, normalized size = 1.18 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} x}{2 \, a} + \frac {\arcsin \left (a x\right )}{2 \, a^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{2}} \]

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*x)/a^2 - sqrt(-a^2*x^2 + 1)/a^2

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mupad [B]  time = 0.80, size = 58, normalized size = 1.53 \[ \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {1}{\sqrt {-a^2}}-\frac {x\,\sqrt {-a^2}}{2\,a}\right )+\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a}}{\sqrt {-a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 3.16, size = 110, normalized size = 2.89 \[ a \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) + \begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases} \]

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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