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

Optimal. Leaf size=65 \[ \frac{\sqrt{1-a^2 x^2}}{3 a c^3 (1-a x)}+\frac{\sqrt{1-a^2 x^2}}{3 a c^3 (1-a x)^2} \]

[Out]

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

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Rubi [A]  time = 0.0516709, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6127, 659, 651} \[ \frac{\sqrt{1-a^2 x^2}}{3 a c^3 (1-a x)}+\frac{\sqrt{1-a^2 x^2}}{3 a c^3 (1-a x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 659

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

Rule 651

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

Rubi steps

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

Mathematica [A]  time = 0.0168789, size = 34, normalized size = 0.52 \[ -\frac{(a x-2) \sqrt{a x+1}}{3 a c^3 (1-a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.032, size = 33, normalized size = 0.5 \begin{align*} -{\frac{ax-2}{3\, \left ( ax-1 \right ) ^{2}{c}^{3}a}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.61314, size = 130, normalized size = 2. \begin{align*} \frac{2 \, a^{2} x^{2} - 4 \, a x - \sqrt{-a^{2} x^{2} + 1}{\left (a x - 2\right )} + 2}{3 \,{\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^{4} x^{4} - 2 a^{3} x^{3} + 2 a x - 1}\, dx}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.23481, size = 123, normalized size = 1.89 \begin{align*} -\frac{2 \,{\left (\frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} - 2\right )}}{3 \, c^{3}{\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(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^3,x, algorithm="giac")

[Out]

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