3.205 \(\int \frac{e^{-\tanh ^{-1}(a x)}}{(c-a c x)^5} \, dx\)

Optimal. Leaf size=129 \[ \frac{2 \sqrt{1-a^2 x^2}}{35 a c^5 (1-a x)}+\frac{2 \sqrt{1-a^2 x^2}}{35 a c^5 (1-a x)^2}+\frac{3 \sqrt{1-a^2 x^2}}{35 a c^5 (1-a x)^3}+\frac{\sqrt{1-a^2 x^2}}{7 a c^5 (1-a x)^4} \]

[Out]

Sqrt[1 - a^2*x^2]/(7*a*c^5*(1 - a*x)^4) + (3*Sqrt[1 - a^2*x^2])/(35*a*c^5*(1 - a*x)^3) + (2*Sqrt[1 - a^2*x^2])
/(35*a*c^5*(1 - a*x)^2) + (2*Sqrt[1 - a^2*x^2])/(35*a*c^5*(1 - a*x))

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Rubi [A]  time = 0.0899088, antiderivative size = 129, 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, 651} \[ \frac{2 \sqrt{1-a^2 x^2}}{35 a c^5 (1-a x)}+\frac{2 \sqrt{1-a^2 x^2}}{35 a c^5 (1-a x)^2}+\frac{3 \sqrt{1-a^2 x^2}}{35 a c^5 (1-a x)^3}+\frac{\sqrt{1-a^2 x^2}}{7 a c^5 (1-a x)^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcTanh[a*x]*(c - a*c*x)^5),x]

[Out]

Sqrt[1 - a^2*x^2]/(7*a*c^5*(1 - a*x)^4) + (3*Sqrt[1 - a^2*x^2])/(35*a*c^5*(1 - a*x)^3) + (2*Sqrt[1 - a^2*x^2])
/(35*a*c^5*(1 - a*x)^2) + (2*Sqrt[1 - a^2*x^2])/(35*a*c^5*(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)^5} \, dx &=\frac{\int \frac{1}{(c-a c x)^4 \sqrt{1-a^2 x^2}} \, dx}{c}\\ &=\frac{\sqrt{1-a^2 x^2}}{7 a c^5 (1-a x)^4}+\frac{3 \int \frac{1}{(c-a c x)^3 \sqrt{1-a^2 x^2}} \, dx}{7 c^2}\\ &=\frac{\sqrt{1-a^2 x^2}}{7 a c^5 (1-a x)^4}+\frac{3 \sqrt{1-a^2 x^2}}{35 a c^5 (1-a x)^3}+\frac{6 \int \frac{1}{(c-a c x)^2 \sqrt{1-a^2 x^2}} \, dx}{35 c^3}\\ &=\frac{\sqrt{1-a^2 x^2}}{7 a c^5 (1-a x)^4}+\frac{3 \sqrt{1-a^2 x^2}}{35 a c^5 (1-a x)^3}+\frac{2 \sqrt{1-a^2 x^2}}{35 a c^5 (1-a x)^2}+\frac{2 \int \frac{1}{(c-a c x) \sqrt{1-a^2 x^2}} \, dx}{35 c^4}\\ &=\frac{\sqrt{1-a^2 x^2}}{7 a c^5 (1-a x)^4}+\frac{3 \sqrt{1-a^2 x^2}}{35 a c^5 (1-a x)^3}+\frac{2 \sqrt{1-a^2 x^2}}{35 a c^5 (1-a x)^2}+\frac{2 \sqrt{1-a^2 x^2}}{35 a c^5 (1-a x)}\\ \end{align*}

Mathematica [A]  time = 0.0243125, size = 51, normalized size = 0.4 \[ -\frac{\sqrt{a x+1} \left (2 a^3 x^3-8 a^2 x^2+13 a x-12\right )}{35 a c^5 (1-a x)^{7/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^ArcTanh[a*x]*(c - a*c*x)^5),x]

[Out]

-(Sqrt[1 + a*x]*(-12 + 13*a*x - 8*a^2*x^2 + 2*a^3*x^3))/(35*a*c^5*(1 - a*x)^(7/2))

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Maple [A]  time = 0.03, size = 50, normalized size = 0.4 \begin{align*} -{\frac{2\,{x}^{3}{a}^{3}-8\,{a}^{2}{x}^{2}+13\,ax-12}{35\, \left ( ax-1 \right ) ^{4}{c}^{5}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)^5,x)

[Out]

-1/35*(-a^2*x^2+1)^(1/2)*(2*a^3*x^3-8*a^2*x^2+13*a*x-12)/(a*x-1)^4/c^5/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 )}^{5}{\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)^5,x, algorithm="maxima")

[Out]

-integrate(sqrt(-a^2*x^2 + 1)/((a*c*x - c)^5*(a*x + 1)), x)

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Fricas [A]  time = 1.59161, size = 251, normalized size = 1.95 \begin{align*} \frac{12 \, a^{4} x^{4} - 48 \, a^{3} x^{3} + 72 \, a^{2} x^{2} - 48 \, a x -{\left (2 \, a^{3} x^{3} - 8 \, a^{2} x^{2} + 13 \, a x - 12\right )} \sqrt{-a^{2} x^{2} + 1} + 12}{35 \,{\left (a^{5} c^{5} x^{4} - 4 \, a^{4} c^{5} x^{3} + 6 \, a^{3} c^{5} x^{2} - 4 \, a^{2} c^{5} x + a c^{5}\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)^5,x, algorithm="fricas")

[Out]

1/35*(12*a^4*x^4 - 48*a^3*x^3 + 72*a^2*x^2 - 48*a*x - (2*a^3*x^3 - 8*a^2*x^2 + 13*a*x - 12)*sqrt(-a^2*x^2 + 1)
+ 12)/(a^5*c^5*x^4 - 4*a^4*c^5*x^3 + 6*a^3*c^5*x^2 - 4*a^2*c^5*x + a*c^5)

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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^{6} x^{6} - 4 a^{5} x^{5} + 5 a^{4} x^{4} - 5 a^{2} x^{2} + 4 a x - 1}\, dx}{c^{5}} \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)**5,x)

[Out]

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

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Giac [C]  time = 1.25206, size = 221, normalized size = 1.71 \begin{align*} \frac{1}{280} \, c^{2}{\left (\frac{{\left (5 \,{\left (\frac{2 \, c}{a c x - c} + 1\right )}^{3} \sqrt{-\frac{2 \, c}{a c x - c} - 1} - 21 \,{\left (\frac{2 \, c}{a c x - c} + 1\right )}^{2} \sqrt{-\frac{2 \, c}{a c x - c} - 1} - 35 \,{\left (-\frac{2 \, c}{a c x - c} - 1\right )}^{\frac{3}{2}} - 35 \, \sqrt{-\frac{2 \, c}{a c x - c} - 1}\right )} \mathrm{sgn}\left (\frac{1}{a c x - c}\right ) \mathrm{sgn}\left (a\right ) \mathrm{sgn}\left (c\right )}{a^{2} c^{7}} + \frac{16 i \, \mathrm{sgn}\left (\frac{1}{a c x - c}\right ) \mathrm{sgn}\left (a\right ) \mathrm{sgn}\left (c\right )}{a^{2} c^{7}}\right )}{\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)^5,x, algorithm="giac")

[Out]

1/280*c^2*((5*(2*c/(a*c*x - c) + 1)^3*sqrt(-2*c/(a*c*x - c) - 1) - 21*(2*c/(a*c*x - c) + 1)^2*sqrt(-2*c/(a*c*x
- c) - 1) - 35*(-2*c/(a*c*x - c) - 1)^(3/2) - 35*sqrt(-2*c/(a*c*x - c) - 1))*sgn(1/(a*c*x - c))*sgn(a)*sgn(c)
/(a^2*c^7) + 16*I*sgn(1/(a*c*x - c))*sgn(a)*sgn(c)/(a^2*c^7))*abs(a)