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

Optimal. Leaf size=52 \[ \frac{2 x}{3 c^2 \sqrt{1-a^2 x^2}}+\frac{a x+1}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0396439, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6138, 639, 191} \[ \frac{2 x}{3 c^2 \sqrt{1-a^2 x^2}}+\frac{a x+1}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6138

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

Rule 639

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac{\int \frac{1+a x}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac{1+a x}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^2}\\ &=\frac{1+a x}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x}{3 c^2 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0131722, size = 45, normalized size = 0.87 \[ \frac{2 a^2 x^2-2 a x-1}{3 a c^2 (a x-1) \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.03, size = 42, normalized size = 0.8 \begin{align*}{\frac{2\,{a}^{2}{x}^{2}-2\,ax-1}{ \left ( 3\,ax-3 \right ){c}^{2}a}{\frac{1}{\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)/(-a^2*c*x^2+c)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.52428, size = 173, normalized size = 3.33 \begin{align*} \frac{a^{3} x^{3} - a^{2} x^{2} - a x -{\left (2 \, a^{2} x^{2} - 2 \, a x - 1\right )} \sqrt{-a^{2} x^{2} + 1} + 1}{3 \,{\left (a^{4} c^{2} x^{3} - a^{3} c^{2} x^{2} - a^{2} c^{2} x + a c^{2}\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)/(-a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a x}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{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)/(-a**2*c*x**2+c)**2,x)

[Out]

(Integral(a*x/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) +
Integral(1/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c*
*2

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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