3.182 \(\int \frac{e^{3 \tanh ^{-1}(a x)}}{c-a c x} \, dx\)

Optimal. Leaf size=74 \[ \frac{2 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^3}-\frac{2 \sqrt{1-a^2 x^2}}{a c (1-a x)}+\frac{\sin ^{-1}(a x)}{a c} \]

[Out]

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

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Rubi [A]  time = 0.0589498, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6127, 663, 216} \[ \frac{2 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^3}-\frac{2 \sqrt{1-a^2 x^2}}{a c (1-a x)}+\frac{\sin ^{-1}(a x)}{a c} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])/(c - a*c*x),x]

[Out]

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

Rule 6127

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

Rule 663

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(
e*(m + p + 1)), x] - Dist[(c*p)/(e^2*(m + p + 1)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1
, 0] && IntegerQ[2*p]

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^{3 \tanh ^{-1}(a x)}}{c-a c x} \, dx &=c^3 \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^4} \, dx\\ &=\frac{2 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^3}-c \int \frac{\sqrt{1-a^2 x^2}}{(c-a c x)^2} \, dx\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{a c (1-a x)}+\frac{2 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^3}+\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{c}\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{a c (1-a x)}+\frac{2 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^3}+\frac{\sin ^{-1}(a x)}{a c}\\ \end{align*}

Mathematica [C]  time = 0.0122167, size = 45, normalized size = 0.61 \[ \frac{4 \sqrt{2} \text{Hypergeometric2F1}\left (-\frac{3}{2},-\frac{3}{2},-\frac{1}{2},\frac{1}{2} (1-a x)\right )}{3 a c (1-a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a*x])/(c - a*c*x),x]

[Out]

(4*Sqrt[2]*Hypergeometric2F1[-3/2, -3/2, -1/2, (1 - a*x)/2])/(3*a*c*(1 - a*x)^(3/2))

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Maple [B]  time = 0.039, size = 146, normalized size = 2. \begin{align*} -8\,{\frac{x}{c\sqrt{-{a}^{2}{x}^{2}+1}}}+{\frac{1}{c}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-4\,{\frac{1}{ac\sqrt{-{a}^{2}{x}^{2}+1}}}-{\frac{8}{3\,{a}^{2}c} \left ( x-{a}^{-1} \right ) ^{-1}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}}+{\frac{16\,x}{3\,c}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/c*x/(-a^2*x^2+1)^(1/2)+1/c/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-4/c/a/(-a^2*x^2+1)^(1/2)-8/
3/c/a^2/(x-1/a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+16/3/c/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)*x

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.00288, size = 216, normalized size = 2.92 \begin{align*} -\frac{2 \,{\left (2 \, a^{2} x^{2} - 4 \, a x + 3 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) - 2 \, \sqrt{-a^{2} x^{2} + 1}{\left (2 \, a x - 1\right )} + 2\right )}}{3 \,{\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{3 a x}{- a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{2}}{- a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{3}}{- a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \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)**3/(-a**2*x**2+1)**(3/2)/(-a*c*x+c),x)

[Out]

-(Integral(3*a*x/(-a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1)
- sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**2/(-a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**
2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**3/(-a**3*x**3*sqrt(-a**2*x**2
+ 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a
**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 +
1)), x))/c

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Giac [A]  time = 1.23573, size = 107, normalized size = 1.45 \begin{align*} \frac{\arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{c{\left | a \right |}} + \frac{8 \,{\left (\frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - 1\right )}}{3 \, c{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{3}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c),x, algorithm="giac")

[Out]

arcsin(a*x)*sgn(a)/(c*abs(a)) + 8/3*(3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)/(c*((sqrt(-a^2*x^2 + 1)*ab
s(a) + a)/(a^2*x) - 1)^3*abs(a))