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

Optimal. Leaf size=28 \[ -\frac{1-a x}{a c^2 \sqrt{1-a^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0329932, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6127, 637} \[ -\frac{1-a x}{a c^2 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^2),x]

[Out]

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

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 637

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

Rubi steps

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

Mathematica [A]  time = 0.0104067, size = 27, normalized size = 0.96 \[ -\frac{\sqrt{1-a x}}{a c^2 \sqrt{a x+1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^2),x]

[Out]

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

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Maple [A]  time = 0.03, size = 34, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ( ax-1 \right ){c}^{2}a \left ( ax+1 \right ) ^{2}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a c x - c\right )}^{2}{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.65454, size = 72, normalized size = 2.57 \begin{align*} -\frac{a x + \sqrt{-a^{2} x^{2} + 1} + 1}{a^{2} c^{2} x + a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*x + sqrt(-a^2*x^2 + 1) + 1)/(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{\sqrt{- a^{2} x^{2} + 1}}{a^{5} x^{5} + a^{4} x^{4} - 2 a^{3} x^{3} - 2 a^{2} x^{2} + a x + 1}\, dx + \int - \frac{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{5} x^{5} + a^{4} x^{4} - 2 a^{3} x^{3} - 2 a^{2} x^{2} + a x + 1}\, dx}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(sqrt(-a**2*x**2 + 1)/(a**5*x**5 + a**4*x**4 - 2*a**3*x**3 - 2*a**2*x**2 + a*x + 1), x) + Integral(-a
**2*x**2*sqrt(-a**2*x**2 + 1)/(a**5*x**5 + a**4*x**4 - 2*a**3*x**3 - 2*a**2*x**2 + a*x + 1), x))/c**2

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Giac [C]  time = 1.22409, size = 93, normalized size = 3.32 \begin{align*} c^{2}{\left (\frac{i \, \mathrm{sgn}\left (\frac{1}{a c x - c}\right ) \mathrm{sgn}\left (a\right ) \mathrm{sgn}\left (c\right )}{a^{2} c^{4}} + \frac{\mathrm{sgn}\left (\frac{1}{a c x - c}\right ) \mathrm{sgn}\left (a\right ) \mathrm{sgn}\left (c\right )}{a^{2} c^{4} \sqrt{-\frac{2 \, c}{a c x - c} - 1}}\right )}{\left | a \right |} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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