3.163 \(\int \frac{e^{\coth ^{-1}(a x)}}{(c-a c x)^2} \, dx\)

Optimal. Leaf size=33 \[ -\frac{a^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{3 c^2 \left (a-\frac{1}{x}\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.0990949, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {6175, 6178, 651} \[ -\frac{a^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{3 c^2 \left (a-\frac{1}{x}\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCoth[a*x]/(c - a*c*x)^2,x]

[Out]

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

Rule 6175

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

Rule 6178

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> -Dist[c^n, Subst[Int[((c
+ d*x)^(p - n)*(1 - x^2/a^2)^(n/2))/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] &&
IntegerQ[(n - 1)/2] && IntegerQ[m] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1] || LtQ[-5, m, -1]) && In
tegerQ[2*p]

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

Mathematica [A]  time = 0.0488138, size = 34, normalized size = 1.03 \[ -\frac{x \sqrt{1-\frac{1}{a^2 x^2}} (a x+1)}{3 c^2 (a x-1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcCoth[a*x]/(c - a*c*x)^2,x]

[Out]

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

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Maple [A]  time = 0.045, size = 36, normalized size = 1.1 \begin{align*} -{\frac{ax+1}{ \left ( 3\,ax-3 \right ){c}^{2}a}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0126, size = 31, normalized size = 0.94 \begin{align*} -\frac{1}{3 \, a c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.50868, size = 122, normalized size = 3.7 \begin{align*} -\frac{{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19968, size = 47, normalized size = 1.42 \begin{align*} -\frac{a x + 1}{3 \,{\left (a x - 1\right )} a c^{2} \sqrt{\frac{a x - 1}{a x + 1}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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