\(\int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-\frac {c}{a x})^3} \, dx\) [435]

Optimal result

Integrand size = 22, antiderivative size = 45 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=-\frac {2}{a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {x}{c^3 \sqrt {1-\frac {1}{a^2 x^2}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6312, 277, 197} \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {x}{c^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {2}{a^2 c^3 x \sqrt {1-\frac {1}{a^2 x^2}}} \]

[In]

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

[Out]

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

Rule 197

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]

Rule 277

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

Rule 6312

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.73 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {-2+a^2 x^2}{a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91

method	result	size
trager	\(\frac {\left (a^{2} x^{2}-2\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{a \,c^{3} \left (a x -1\right )}\)	\(41\)
gosper	\(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a^{2} x^{2}-2\right ) \left (a x +1\right )}{a \left (a x -1\right )^{2} c^{3}}\)	\(44\)
default	\(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a^{3} x^{3}+a^{2} x^{2}-2 a x -2\right )}{a \left (a x -1\right )^{2} c^{3}}\)	\(50\)
risch	\(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{3}}-\frac {\sqrt {\frac {a x -1}{a x +1}}}{a \,c^{3} \left (a x -1\right )}\)	\(59\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {{\left (a^{2} x^{2} - 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3} x - a c^{3}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {a^{3} \left (\int \left (- \frac {x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{4} x^{4} - 2 a^{3} x^{3} + 2 a x - 1}\right )\, dx + \int \frac {a x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{4} x^{4} - 2 a^{3} x^{3} + 2 a x - 1}\, dx\right )}{c^{3}} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (41) = 82\).

Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.04 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=-\frac {1}{2} \, a {\left (\frac {\frac {5 \, {\left (a x - 1\right )}}{a x + 1} - 1}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - a^{2} c^{3} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3}}\right )} \]

[In]

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

[Out]

-1/2*a*((5*(a*x - 1)/(a*x + 1) - 1)/(a^2*c^3*((a*x - 1)/(a*x + 1))^(3/2) - a^2*c^3*sqrt((a*x - 1)/(a*x + 1)))
- sqrt((a*x - 1)/(a*x + 1))/(a^2*c^3))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {{\left (\frac {\sqrt {a^{2} x^{2} - 1}}{c^{3}} - \frac {1}{\sqrt {a^{2} x^{2} - 1} c^{3}}\right )} \mathrm {sgn}\left (a x + 1\right )}{a} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {a^2\,x^2-2}{\left (x\,a^2\,c^3+a\,c^3\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \]

[In]

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

[Out]

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