3.225 \(\int \frac{e^{-3 \tanh ^{-1}(a x)}}{(c-a c x)^6} \, dx\)

Optimal. Leaf size=119 \[ \frac{8 x}{35 c^6 \sqrt{1-a^2 x^2}}+\frac{4}{35 a c^6 (1-a x) \sqrt{1-a^2 x^2}}+\frac{4}{35 a c^6 (1-a x)^2 \sqrt{1-a^2 x^2}}+\frac{1}{7 a c^6 (1-a x)^3 \sqrt{1-a^2 x^2}} \]

[Out]

(8*x)/(35*c^6*Sqrt[1 - a^2*x^2]) + 1/(7*a*c^6*(1 - a*x)^3*Sqrt[1 - a^2*x^2]) + 4/(35*a*c^6*(1 - a*x)^2*Sqrt[1
- a^2*x^2]) + 4/(35*a*c^6*(1 - a*x)*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.0762582, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6127, 659, 191} \[ \frac{8 x}{35 c^6 \sqrt{1-a^2 x^2}}+\frac{4}{35 a c^6 (1-a x) \sqrt{1-a^2 x^2}}+\frac{4}{35 a c^6 (1-a x)^2 \sqrt{1-a^2 x^2}}+\frac{1}{7 a c^6 (1-a x)^3 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^6),x]

[Out]

(8*x)/(35*c^6*Sqrt[1 - a^2*x^2]) + 1/(7*a*c^6*(1 - a*x)^3*Sqrt[1 - a^2*x^2]) + 4/(35*a*c^6*(1 - a*x)^2*Sqrt[1
- a^2*x^2]) + 4/(35*a*c^6*(1 - a*x)*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 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 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)}}{(c-a c x)^6} \, dx &=\frac{\int \frac{1}{(c-a c x)^3 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac{1}{7 a c^6 (1-a x)^3 \sqrt{1-a^2 x^2}}+\frac{4 \int \frac{1}{(c-a c x)^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{7 c^4}\\ &=\frac{1}{7 a c^6 (1-a x)^3 \sqrt{1-a^2 x^2}}+\frac{4}{35 a c^6 (1-a x)^2 \sqrt{1-a^2 x^2}}+\frac{12 \int \frac{1}{(c-a c x) \left (1-a^2 x^2\right )^{3/2}} \, dx}{35 c^5}\\ &=\frac{1}{7 a c^6 (1-a x)^3 \sqrt{1-a^2 x^2}}+\frac{4}{35 a c^6 (1-a x)^2 \sqrt{1-a^2 x^2}}+\frac{4}{35 a c^6 (1-a x) \sqrt{1-a^2 x^2}}+\frac{8 \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{35 c^6}\\ &=\frac{8 x}{35 c^6 \sqrt{1-a^2 x^2}}+\frac{1}{7 a c^6 (1-a x)^3 \sqrt{1-a^2 x^2}}+\frac{4}{35 a c^6 (1-a x)^2 \sqrt{1-a^2 x^2}}+\frac{4}{35 a c^6 (1-a x) \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0234367, size = 61, normalized size = 0.51 \[ \frac{8 a^4 x^4-24 a^3 x^3+20 a^2 x^2+4 a x-13}{35 a c^6 (a x-1)^3 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^6),x]

[Out]

(-13 + 4*a*x + 20*a^2*x^2 - 24*a^3*x^3 + 8*a^4*x^4)/(35*a*c^6*(-1 + a*x)^3*Sqrt[1 - a^2*x^2])

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Maple [A]  time = 0.031, size = 65, normalized size = 0.6 \begin{align*}{\frac{8\,{x}^{4}{a}^{4}-24\,{x}^{3}{a}^{3}+20\,{a}^{2}{x}^{2}+4\,ax-13}{35\, \left ( ax-1 \right ) ^{5}{c}^{6}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)^6,x)

[Out]

1/35*(-a^2*x^2+1)^(3/2)*(8*a^4*x^4-24*a^3*x^3+20*a^2*x^2+4*a*x-13)/(a*x-1)^5/c^6/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 )}^{6}{\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)^6,x, algorithm="maxima")

[Out]

integrate((-a^2*x^2 + 1)^(3/2)/((a*c*x - c)^6*(a*x + 1)^3), x)

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Fricas [A]  time = 1.6598, size = 308, normalized size = 2.59 \begin{align*} \frac{13 \, a^{5} x^{5} - 39 \, a^{4} x^{4} + 26 \, a^{3} x^{3} + 26 \, a^{2} x^{2} - 39 \, a x -{\left (8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 20 \, a^{2} x^{2} + 4 \, a x - 13\right )} \sqrt{-a^{2} x^{2} + 1} + 13}{35 \,{\left (a^{6} c^{6} x^{5} - 3 \, a^{5} c^{6} x^{4} + 2 \, a^{4} c^{6} x^{3} + 2 \, a^{3} c^{6} x^{2} - 3 \, a^{2} c^{6} x + a c^{6}\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)^6,x, algorithm="fricas")

[Out]

1/35*(13*a^5*x^5 - 39*a^4*x^4 + 26*a^3*x^3 + 26*a^2*x^2 - 39*a*x - (8*a^4*x^4 - 24*a^3*x^3 + 20*a^2*x^2 + 4*a*
x - 13)*sqrt(-a^2*x^2 + 1) + 13)/(a^6*c^6*x^5 - 3*a^5*c^6*x^4 + 2*a^4*c^6*x^3 + 2*a^3*c^6*x^2 - 3*a^2*c^6*x +
a*c^6)

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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^{9} x^{9} - 3 a^{8} x^{8} + 8 a^{6} x^{6} - 6 a^{5} x^{5} - 6 a^{4} x^{4} + 8 a^{3} x^{3} - 3 a x + 1}\, dx + \int - \frac{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{9} x^{9} - 3 a^{8} x^{8} + 8 a^{6} x^{6} - 6 a^{5} x^{5} - 6 a^{4} x^{4} + 8 a^{3} x^{3} - 3 a x + 1}\, dx}{c^{6}} \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)**6,x)

[Out]

(Integral(sqrt(-a**2*x**2 + 1)/(a**9*x**9 - 3*a**8*x**8 + 8*a**6*x**6 - 6*a**5*x**5 - 6*a**4*x**4 + 8*a**3*x**
3 - 3*a*x + 1), x) + Integral(-a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**9*x**9 - 3*a**8*x**8 + 8*a**6*x**6 - 6*a**5*
x**5 - 6*a**4*x**4 + 8*a**3*x**3 - 3*a*x + 1), x))/c**6

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Giac [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 )}^{6}{\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)^6,x, algorithm="giac")

[Out]

integrate((-a^2*x^2 + 1)^(3/2)/((a*c*x - c)^6*(a*x + 1)^3), x)