\(\int \frac {e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx\) [577]

Optimal result

Integrand size = 22, antiderivative size = 18 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {e^{3 \coth ^{-1}(a x)}}{3 a c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6318} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {e^{3 \coth ^{-1}(a x)}}{3 a c} \]

[In]

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

[Out]

E^(3*ArcCoth[a*x])/(3*a*c)

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]

Rubi steps \begin{align*} \text {integral}& = \frac {e^{3 \coth ^{-1}(a x)}}{3 a c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {e^{3 \coth ^{-1}(a x)}}{3 a c} \]

[In]

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

[Out]

E^(3*ArcCoth[a*x])/(3*a*c)

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33

method	result	size
gosper	\(\frac {1}{3 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a c}\)	\(24\)
default	\(\frac {1}{3 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a c}\)	\(24\)
trager	\(\frac {\left (a x +1\right )^{2} \sqrt {-\frac {-a x +1}{a x +1}}}{3 a c \left (a x -1\right )^{2}}\)	\(40\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (23) = 46\).

Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.83 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {1}{3 \, a c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \]

[In]

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

[Out]

1/3/(a*c*((a*x - 1)/(a*x + 1))^(3/2))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (23) = 46\).

Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.72 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {2 \, {\left (3 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{2} x^{2} + 1\right )}}{3 \, {\left ({\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}^{3} a c} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {1}{3\,a\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \]

[In]

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

[Out]

1/(3*a*c*((a*x - 1)/(a*x + 1))^(3/2))