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

Optimal. Leaf size=45 \[ -\frac{\sqrt{1-a^2 x^2}}{a c^3}-\frac{1}{a c^3 \sqrt{1-a^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.118959, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6131, 6128, 266, 43} \[ -\frac{\sqrt{1-a^2 x^2}}{a c^3}-\frac{1}{a c^3 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6131

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> Dist[d^p, Int[(u*(1 + (c*x)/d)
^p*E^(n*ArcTanh[a*x]))/x^p, x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]

Rule 6128

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[c^n,
Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c +
d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1, 0]) && IntegerQ[2*p]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.131466, size = 30, normalized size = 0.67 \[ \frac{a^2 x^2-2}{a c^3 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.034, size = 43, normalized size = 1. \begin{align*}{\frac{{a}^{2}{x}^{2}-2}{a \left ( ax-1 \right ) ^{2}{c}^{3} \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)/(c-c/a/x)^3,x)

[Out]

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

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Maxima [A]  time = 0.98244, size = 61, normalized size = 1.36 \begin{align*} -\frac{{\left (a^{2} x^{2} - 2\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{a^{3} c^{3} x^{2} - a c^{3}} \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)/(c-c/a/x)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.19505, size = 101, normalized size = 2.24 \begin{align*} -\frac{2 \, a^{2} x^{2} +{\left (a^{2} x^{2} - 2\right )} \sqrt{-a^{2} x^{2} + 1} - 2}{a^{3} c^{3} x^{2} - a c^{3}} \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)/(c-c/a/x)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 23.5224, size = 34, normalized size = 0.76 \begin{align*} - \frac{2 \left (\frac{\sqrt{- a^{2} x^{2} + 1}}{2 c^{3}} + \frac{1}{2 c^{3} \sqrt{- a^{2} x^{2} + 1}}\right )}{a} \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)/(c-c/a/x)**3,x)

[Out]

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

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Giac [A]  time = 1.17183, size = 45, normalized size = 1. \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} + \frac{1}{\sqrt{-a^{2} x^{2} + 1}}}{a c^{3}} \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)/(c-c/a/x)^3,x, algorithm="giac")

[Out]

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