\(\int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a^2 c x^2)^2} \, dx\) [611]

Optimal result

Integrand size = 22, antiderivative size = 55 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {2 e^{-3 \coth ^{-1}(a x)}}{15 a c^2}-\frac {e^{-3 \coth ^{-1}(a x)} (3+2 a x)}{5 a c^2 \left (1-a^2 x^2\right )} \]

[Out]

2/15/a/c^2*((a*x-1)/(a*x+1))^(3/2)+1/5*(-2*a*x-3)/a/c^2*((a*x-1)/(a*x+1))^(3/2)/(-a^2*x^2+1)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6320, 6318} \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {2 e^{-3 \coth ^{-1}(a x)}}{15 a c^2}-\frac {(2 a x+3) e^{-3 \coth ^{-1}(a x)}}{5 a c^2 \left (1-a^2 x^2\right )} \]

[In]

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

[Out]

2/(15*a*c^2*E^(3*ArcCoth[a*x])) - (3 + 2*a*x)/(5*a*c^2*E^(3*ArcCoth[a*x])*(1 - a^2*x^2))

Rule 6318

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcCoth[a*x])/(a*c*n), x] /; F
reeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2]

Rule 6320

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(n + 2*a*(p + 1)*x)*(c + d*x^2
)^(p + 1)*(E^(n*ArcCoth[a*x])/(a*c*(n^2 - 4*(p + 1)^2))), x] - Dist[2*(p + 1)*((2*p + 3)/(c*(n^2 - 4*(p + 1)^2
))), Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !In
tegerQ[n/2] && LtQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] ||  !IntegerQ[n])

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-3 \coth ^{-1}(a x)} (3+2 a x)}{5 a c^2 \left (1-a^2 x^2\right )}-\frac {2 \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{5 c} \\ & = \frac {2 e^{-3 \coth ^{-1}(a x)}}{15 a c^2}-\frac {e^{-3 \coth ^{-1}(a x)} (3+2 a x)}{5 a c^2 \left (1-a^2 x^2\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (7+6 a x+2 a^2 x^2\right )}{15 c^2 (1+a x)^3} \]

[In]

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

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x*(7 + 6*a*x + 2*a^2*x^2))/(15*c^2*(1 + a*x)^3)

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85

method	result	size
trager	\(\frac {\left (2 a^{2} x^{2}+6 a x +7\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{15 a \,c^{2} \left (a x +1\right )^{2}}\)	\(47\)
gosper	\(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (2 a^{2} x^{2}+6 a x +7\right )}{15 \left (a^{2} x^{2}-1\right ) a \,c^{2}}\)	\(49\)
default	\(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (2 a^{2} x^{2}+6 a x +7\right )}{15 \left (a x -1\right ) c^{2} a \left (a x +1\right )}\)	\(52\)

[In]

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

[Out]

1/15/a/c^2*(2*a^2*x^2+6*a*x+7)/(a*x+1)^2*(-(-a*x+1)/(a*x+1))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.05 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {{\left (2 \, a^{2} x^{2} + 6 \, a x + 7\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} \]

[In]

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

[Out]

1/15*(2*a^2*x^2 + 6*a*x + 7)*sqrt((a*x - 1)/(a*x + 1))/(a^3*c^2*x^2 + 2*a^2*c^2*x + a*c^2)

Sympy [F]

\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{5} x^{5} + a^{4} x^{4} - 2 a^{3} x^{3} - 2 a^{2} x^{2} + a x + 1}\right )\, dx + \int \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{5} x^{5} + a^{4} x^{4} - 2 a^{3} x^{3} - 2 a^{2} x^{2} + a x + 1}\, dx}{c^{2}} \]

[In]

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

[Out]

(Integral(-sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**5*x**5 + a**4*x**4 - 2*a**3*x**3 - 2*a**2*x**2 + a*x + 1), x)
+ Integral(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**5*x**5 + a**4*x**4 - 2*a**3*x**3 - 2*a**2*x**2 + a*x + 1
), x))/c**2

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.09 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 10 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 15 \, \sqrt {\frac {a x - 1}{a x + 1}}}{60 \, a c^{2}} \]

[In]

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

[Out]

1/60*(3*((a*x - 1)/(a*x + 1))^(5/2) - 10*((a*x - 1)/(a*x + 1))^(3/2) + 15*sqrt((a*x - 1)/(a*x + 1)))/(a*c^2)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.18 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {4 \, {\left (10 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{2} x^{2} + 5 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x + 1\right )}}{15 \, {\left ({\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x + 1\right )}^{5} a c^{2}} \]

[In]

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

[Out]

-4/15*(10*(a + sqrt(a^2 - 1/x^2))^2*x^2 + 5*(a + sqrt(a^2 - 1/x^2))*x + 1)/(((a + sqrt(a^2 - 1/x^2))*x + 1)^5*
a*c^2)

Mupad [B] (verification not implemented)

Time = 3.85 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.09 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {15\,\sqrt {\frac {a\,x-1}{a\,x+1}}-10\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}+3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{60\,a\,c^2} \]

[In]

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

[Out]

(15*((a*x - 1)/(a*x + 1))^(1/2) - 10*((a*x - 1)/(a*x + 1))^(3/2) + 3*((a*x - 1)/(a*x + 1))^(5/2))/(60*a*c^2)