3.890 \(\int \frac{e^{\tanh ^{-1}(a x)} x}{c-a^2 c x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0639509, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6148, 778, 216} \[ \frac{a x+1}{a^2 c \sqrt{1-a^2 x^2}}-\frac{\sin ^{-1}(a x)}{a^2 c} \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*x)/(c - a^2*c*x^2),x]

[Out]

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

Rule 6148

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

Rule 778

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*(e*f + d*g) -
(c*d*f - a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[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 \frac{e^{\tanh ^{-1}(a x)} x}{c-a^2 c x^2} \, dx &=\frac{\int \frac{x (1+a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c}\\ &=\frac{1+a x}{a^2 c \sqrt{1-a^2 x^2}}-\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a c}\\ &=\frac{1+a x}{a^2 c \sqrt{1-a^2 x^2}}-\frac{\sin ^{-1}(a x)}{a^2 c}\\ \end{align*}

Mathematica [A]  time = 0.0293188, size = 45, normalized size = 1.15 \[ \frac{\frac{a x}{\sqrt{1-a^2 x^2}}+\frac{1}{\sqrt{1-a^2 x^2}}-\sin ^{-1}(a x)}{a^2 c} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcTanh[a*x]*x)/(c - a^2*c*x^2),x]

[Out]

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

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Maple [B]  time = 0.036, size = 79, normalized size = 2. \begin{align*} -{\frac{1}{ac}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{1}{{a}^{3}c}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-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/(-a^2*c*x^2+c),x)

[Out]

-1/c/a/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-1/c/a^3/(x-1/a)*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)

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Maxima [B]  time = 1.5419, size = 281, normalized size = 7.21 \begin{align*} -\frac{a^{2} c{\left (\frac{\sqrt{-a^{2} x^{2} + 1} c}{\sqrt{a^{2} c^{2}} a^{3} c x + a^{3} c^{2}} + \frac{\sqrt{-a^{2} x^{2} + 1} c}{\sqrt{a^{2} c^{2}} a^{3} c x - a^{3} c^{2}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{4} c x + \sqrt{a^{2} c^{2}} a^{2}} + \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{4} c x - \sqrt{a^{2} c^{2}} a^{2}} + \frac{2 \, \sqrt{a^{2} c^{2}} \arcsin \left (\frac{x}{c \sqrt{\frac{1}{a^{2} c^{2}}}}\right )}{a^{5} c^{3} \sqrt{\frac{1}{a^{2} c^{2}}}}\right )}}{2 \, \sqrt{a^{2} c^{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/(-a^2*c*x^2+c),x, algorithm="maxima")

[Out]

-1/2*a^2*c*(sqrt(-a^2*x^2 + 1)*c/(sqrt(a^2*c^2)*a^3*c*x + a^3*c^2) + sqrt(-a^2*x^2 + 1)*c/(sqrt(a^2*c^2)*a^3*c
*x - a^3*c^2) - sqrt(-a^2*x^2 + 1)/(a^4*c*x + sqrt(a^2*c^2)*a^2) + sqrt(-a^2*x^2 + 1)/(a^4*c*x - sqrt(a^2*c^2)
*a^2) + 2*sqrt(a^2*c^2)*arcsin(x/(c*sqrt(1/(a^2*c^2))))/(a^5*c^3*sqrt(1/(a^2*c^2))))/sqrt(a^2*c^2)

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Fricas [A]  time = 1.54584, size = 139, normalized size = 3.56 \begin{align*} \frac{a x + 2 \,{\left (a x - 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt{-a^{2} x^{2} + 1} - 1}{a^{3} c x - a^{2} c} \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/(-a^2*c*x^2+c),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x}{- a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{2}}{- a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c} \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/(-a**2*c*x**2+c),x)

[Out]

(Integral(x/(-a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**2/(-a**2*x**2*sqrt(-a
**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c

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Giac [A]  time = 1.20704, size = 80, normalized size = 2.05 \begin{align*} -\frac{\arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{a c{\left | a \right |}} + \frac{2}{a c{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}{\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/(-a^2*c*x^2+c),x, algorithm="giac")

[Out]

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