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

Optimal. Leaf size=13 \[ \frac{e^{\tanh ^{-1}(a x)}}{a c} \]

[Out]

E^ArcTanh[a*x]/(a*c)

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Rubi [A]  time = 0.0287232, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {6137} \[ \frac{e^{\tanh ^{-1}(a x)}}{a c} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/(c - a^2*c*x^2),x]

[Out]

E^ArcTanh[a*x]/(a*c)

Rule 6137

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcTanh[a*x])/(a*c*n), x] /; F
reeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2]

Rubi steps

\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=\frac{e^{\tanh ^{-1}(a x)}}{a c}\\ \end{align*}

Mathematica [A]  time = 0.0077084, size = 26, normalized size = 2. \[ \frac{\sqrt{a x+1}}{a c \sqrt{1-a x}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sqrt[1 + a*x]/(a*c*Sqrt[1 - a*x])

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Maple [A]  time = 0.03, size = 25, normalized size = 1.9 \begin{align*}{\frac{ax+1}{ac}{\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),x)

[Out]

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

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Maxima [B]  time = 1.4996, size = 220, normalized size = 16.92 \begin{align*} -\frac{a^{2} c{\left (\frac{\sqrt{-a^{2} x^{2} + 1} c}{\sqrt{a^{2} c^{2}} a^{2} c x + a^{2} c^{2}} + \frac{\sqrt{-a^{2} x^{2} + 1} c}{\sqrt{a^{2} c^{2}} a^{2} c x - a^{2} c^{2}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{3} c x + \sqrt{a^{2} c^{2}} a} + \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{3} c x - \sqrt{a^{2} c^{2}} a}\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)/(-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^2*c*x + a^2*c^2) + sqrt(-a^2*x^2 + 1)*c/(sqrt(a^2*c^2)*a^2*c
*x - a^2*c^2) - sqrt(-a^2*x^2 + 1)/(a^3*c*x + sqrt(a^2*c^2)*a) + sqrt(-a^2*x^2 + 1)/(a^3*c*x - sqrt(a^2*c^2)*a
))/sqrt(a^2*c^2)

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Fricas [A]  time = 1.53849, size = 65, normalized size = 5. \begin{align*} \frac{a x - \sqrt{-a^{2} x^{2} + 1} - 1}{a^{2} c x - a 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)/(-a^2*c*x^2+c),x, algorithm="fricas")

[Out]

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

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

[Out]

(Integral(a*x/(-a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-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 = 1.18228, size = 50, normalized size = 3.85 \begin{align*} \frac{2}{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)/(-a^2*c*x^2+c),x, algorithm="giac")

[Out]

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