3.1152 \(\int \frac{e^{3 \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx\)

Optimal. Leaf size=18 \[ \frac{e^{3 \tanh ^{-1}(a x)}}{3 a c} \]

[Out]

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

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Rubi [A]  time = 0.0305452, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {6137} \[ \frac{e^{3 \tanh ^{-1}(a x)}}{3 a c} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])/(c - a^2*c*x^2),x]

[Out]

E^(3*ArcTanh[a*x])/(3*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^{3 \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=\frac{e^{3 \tanh ^{-1}(a x)}}{3 a c}\\ \end{align*}

Mathematica [A]  time = 0.0111311, size = 29, normalized size = 1.61 \[ \frac{(a x+1)^{3/2}}{3 a c (1-a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a*x])/(c - a^2*c*x^2),x]

[Out]

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

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Maple [A]  time = 0.029, size = 28, normalized size = 1.6 \begin{align*}{\frac{ \left ( ax+1 \right ) ^{3}}{3\,ac} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \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^2*c*x^2+c),x)

[Out]

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

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Maxima [B]  time = 1.67419, size = 680, normalized size = 37.78 \begin{align*} \frac{a^{2} c{\left (\frac{\left (a^{2} c^{2}\right )^{\frac{3}{2}}}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{5} c^{3} x + \sqrt{-a^{2} x^{2} + 1} a^{5} c^{4}} - \frac{\left (a^{2} c^{2}\right )^{\frac{3}{2}}}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{5} c^{3} x - \sqrt{-a^{2} x^{2} + 1} a^{5} c^{4}} - \frac{4 \, c}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{2} c x + \sqrt{-a^{2} x^{2} + 1} a^{2} c^{2}} - \frac{4 \, c}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{2} c x - \sqrt{-a^{2} x^{2} + 1} a^{2} c^{2}} + \frac{16 \, x}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1}} + \frac{3 \, \sqrt{a^{2} c^{2}}}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{3} c x + \sqrt{-a^{2} x^{2} + 1} a^{3} c^{2}} - \frac{3 \, \sqrt{a^{2} c^{2}}}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{3} c x - \sqrt{-a^{2} x^{2} + 1} a^{3} c^{2}} - \frac{16}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a} - \frac{18 \, \sqrt{a^{2} c^{2}} x}{\sqrt{-a^{2} x^{2} + 1} a^{2} c^{2}} + \frac{6 \, \sqrt{a^{2} c^{2}}}{\sqrt{-a^{2} x^{2} + 1} a^{3} c^{2}} + \frac{4 \, \left (a^{2} c^{2}\right )^{\frac{3}{2}}}{\sqrt{-a^{2} x^{2} + 1} a^{5} c^{4}}\right )}}{6 \, \sqrt{a^{2} c^{2}}} \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^2*c*x^2+c),x, algorithm="maxima")

[Out]

1/6*a^2*c*((a^2*c^2)^(3/2)/(sqrt(a^2*c^2)*sqrt(-a^2*x^2 + 1)*a^5*c^3*x + sqrt(-a^2*x^2 + 1)*a^5*c^4) - (a^2*c^
2)^(3/2)/(sqrt(a^2*c^2)*sqrt(-a^2*x^2 + 1)*a^5*c^3*x - sqrt(-a^2*x^2 + 1)*a^5*c^4) - 4*c/(sqrt(a^2*c^2)*sqrt(-
a^2*x^2 + 1)*a^2*c*x + sqrt(-a^2*x^2 + 1)*a^2*c^2) - 4*c/(sqrt(a^2*c^2)*sqrt(-a^2*x^2 + 1)*a^2*c*x - sqrt(-a^2
*x^2 + 1)*a^2*c^2) + 16*x/(sqrt(a^2*c^2)*sqrt(-a^2*x^2 + 1)) + 3*sqrt(a^2*c^2)/(sqrt(a^2*c^2)*sqrt(-a^2*x^2 +
1)*a^3*c*x + sqrt(-a^2*x^2 + 1)*a^3*c^2) - 3*sqrt(a^2*c^2)/(sqrt(a^2*c^2)*sqrt(-a^2*x^2 + 1)*a^3*c*x - sqrt(-a
^2*x^2 + 1)*a^3*c^2) - 16/(sqrt(a^2*c^2)*sqrt(-a^2*x^2 + 1)*a) - 18*sqrt(a^2*c^2)*x/(sqrt(-a^2*x^2 + 1)*a^2*c^
2) + 6*sqrt(a^2*c^2)/(sqrt(-a^2*x^2 + 1)*a^3*c^2) + 4*(a^2*c^2)^(3/2)/(sqrt(-a^2*x^2 + 1)*a^5*c^4))/sqrt(a^2*c
^2)

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Fricas [B]  time = 2.526, size = 119, normalized size = 6.61 \begin{align*} \frac{a^{2} x^{2} - 2 \, a x + \sqrt{-a^{2} x^{2} + 1}{\left (a x + 1\right )} + 1}{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^2*c*x^2+c),x, algorithm="fricas")

[Out]

1/3*(a^2*x^2 - 2*a*x + sqrt(-a^2*x^2 + 1)*(a*x + 1) + 1)/(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^{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{3 a^{2} x^{2}}{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{a^{3} x^{3}}{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} \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**2*c*x**2+c),x)

[Out]

(Integral(3*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(3*a**2*x**2/(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(a**3*x**3/(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

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Giac [B]  time = 1.16098, size = 89, normalized size = 4.94 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + 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^2*c*x^2+c),x, algorithm="giac")

[Out]

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