3.1249 \(\int \frac {e^{-2 \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=52 \[ \frac {x}{3 c \sqrt {c-a^2 c x^2}}-\frac {2 (1-a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6142, 653, 191} \[ \frac {x}{3 c \sqrt {c-a^2 c x^2}}-\frac {2 (1-a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 191

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 653

Int[((d_) + (e_.)*(x_))^2*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)*(a + c*x^2)^(p + 1))/(c*(
p + 1)), x] - Dist[(e^2*(p + 2))/(c*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, p}, x] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && LtQ[p, -1]

Rule 6142

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

Rubi steps

\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=c \int \frac {(1-a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=-\frac {2 (1-a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}}+\frac {1}{3} \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (1-a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}}+\frac {x}{3 c \sqrt {c-a^2 c x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 63, normalized size = 1.21 \[ -\frac {\sqrt {1-a x} (a x+2) \sqrt {1-a^2 x^2}}{3 a c (a x+1)^{3/2} \sqrt {c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.58, size = 47, normalized size = 0.90 \[ -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 2\right )}}{3 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.98, size = 82, normalized size = 1.58 \[ \frac {\frac {2 \, \sqrt {-c} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)}{c^{2}} - \frac {3 \, c \sqrt {-c + \frac {2 \, c}{a x + 1}} + {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}}}{c^{3} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)}}{6 \, {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*sqrt(-c)*sgn(1/(a*x + 1))*sgn(a)/c^2 - (3*c*sqrt(-c + 2*c/(a*x + 1)) + (-c + 2*c/(a*x + 1))^(3/2))/(c^3
*sgn(1/(a*x + 1))*sgn(a)))/abs(a)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 31, normalized size = 0.60 \[ -\frac {\left (a x -1\right )^{2} \left (a x +2\right )}{3 \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.32, size = 60, normalized size = 1.15 \[ \frac {x}{3 \, \sqrt {-a^{2} c x^{2} + c} c} - \frac {2}{3 \, {\left (\sqrt {-a^{2} c x^{2} + c} a^{2} c x + \sqrt {-a^{2} c x^{2} + c} a c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.98, size = 33, normalized size = 0.63 \[ -\frac {\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+2\right )}{3\,a\,c^2\,{\left (a\,x+1\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {a x}{- a^{3} c x^{3} \sqrt {- a^{2} c x^{2} + c} - a^{2} c x^{2} \sqrt {- a^{2} c x^{2} + c} + a c x \sqrt {- a^{2} c x^{2} + c} + c \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \left (- \frac {1}{- a^{3} c x^{3} \sqrt {- a^{2} c x^{2} + c} - a^{2} c x^{2} \sqrt {- a^{2} c x^{2} + c} + a c x \sqrt {- a^{2} c x^{2} + c} + c \sqrt {- a^{2} c x^{2} + c}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________