3.1118 \(\int \frac {e^{2 \tanh ^{-1}(a x)} x}{(c-a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=60 \[ \frac {(a x+1)^2}{3 a^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {2 (a x+1)}{3 a^2 c \sqrt {c-a^2 c x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6151, 789, 637} \[ \frac {(a x+1)^2}{3 a^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {2 (a x+1)}{3 a^2 c \sqrt {c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 637

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 789

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

Rule 6151

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 38, normalized size = 0.63 \[ \frac {(2 a x-1) \sqrt {c-a^2 c x^2}}{3 a^2 c^2 (a x-1)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.68, size = 50, normalized size = 0.83 \[ \frac {\sqrt {-a^{2} c x^{2} + c} {\left (2 \, a x - 1\right )}}{3 \, {\left (a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x + a^{2} c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.31, size = 117, normalized size = 1.95 \[ -\frac {{\left (a c + 3 \, \sqrt {-a^{2} c} \sqrt {c}\right )} \mathrm {sgn}\relax (x)}{3 \, {\left (a^{3} c^{\frac {5}{2}} - \sqrt {-a^{2} c} a^{2} c^{2}\right )}} - \frac {2 \, {\left (a \sqrt {c} + 3 \, \sqrt {-a^{2} c + \frac {c}{x^{2}}} - \frac {3 \, \sqrt {c}}{x}\right )}}{3 \, {\left (a \sqrt {c} + \sqrt {-a^{2} c + \frac {c}{x^{2}}} - \frac {\sqrt {c}}{x}\right )}^{3} \sqrt {c} \mathrm {sgn}\relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.03, size = 32, normalized size = 0.53 \[ \frac {\left (2 a x -1\right ) \left (a x +1\right )^{2}}{3 \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.48, size = 218, normalized size = 3.63 \[ \frac {1}{3} \, a {\left (\frac {a}{\sqrt {-a^{2} c x^{2} + c} a^{5} c x + \sqrt {-a^{2} c x^{2} + c} a^{4} c} - \frac {a}{\sqrt {-a^{2} c x^{2} + c} a^{5} c x - \sqrt {-a^{2} c x^{2} + c} a^{4} c} - \frac {1}{\sqrt {-a^{2} c x^{2} + c} a^{4} c x + \sqrt {-a^{2} c x^{2} + c} a^{3} c} - \frac {1}{\sqrt {-a^{2} c x^{2} + c} a^{4} c x - \sqrt {-a^{2} c x^{2} + c} a^{3} c} - \frac {2 \, x}{\sqrt {-a^{2} c x^{2} + c} a^{2} c} - \frac {3}{\sqrt {-a^{2} c x^{2} + c} a^{3} c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.02, size = 34, normalized size = 0.57 \[ \frac {\sqrt {c-a^2\,c\,x^2}\,\left (2\,a\,x-1\right )}{3\,a^2\,c^2\,{\left (a\,x-1\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(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(a*x**2/(-a**3*c*x**3*sqrt(-a**2*c*x**2 + c) + a**2*c*x**2*sq
rt(-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)

________________________________________________________________________________________