\(\int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx\) [221]

Optimal result

Integrand size = 18, antiderivative size = 28 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=\frac {a-\frac {1}{x}}{a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6310, 6313, 651} \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=\frac {a-\frac {1}{x}}{a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}} \]

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^2),x]

[Out]

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

Rule 651

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[((-a)*e + c*d*x)/(a*c*Sqrt[a + c*x^2]),
x] /; FreeQ[{a, c, d, e}, x]

Rule 6310

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 6313

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]

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2 (1+a x)} \]

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^2),x]

[Out]

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

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89

method	result	size
trager	\(\frac {\sqrt {-\frac {-a x +1}{a x +1}}}{a \,c^{2}}\)	\(25\)
gosper	\(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right )}{\left (a x -1\right ) a \,c^{2}}\)	\(35\)
default	\(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right )}{\left (a x -1\right ) a \,c^{2}}\)	\(35\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=\frac {\sqrt {\frac {a x - 1}{a x + 1}}}{a c^{2}} \]

[In]

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

[Out]

sqrt((a*x - 1)/(a*x + 1))/(a*c^2)

Sympy [F]

\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=\frac {\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}\right )\, dx + \int \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}\, dx}{c^{2}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=\frac {\sqrt {\frac {a x - 1}{a x + 1}}}{a c^{2}} \]

[In]

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

[Out]

sqrt((a*x - 1)/(a*x + 1))/(a*c^2)

Giac [F]

\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (a c x - c\right )}^{2}} \,d x } \]

[In]

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

[Out]

undef

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c^2} \]

[In]

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

[Out]

((a*x - 1)/(a*x + 1))^(1/2)/(a*c^2)