3.1119 \(\int \frac {e^{2 \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=51 \[ \frac {x}{3 c \sqrt {c-a^2 c x^2}}+\frac {2 (a x+1)}{3 a \left (c-a^2 c x^2\right )^{3/2}} \]

[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)

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Rubi [A]  time = 0.06, antiderivative size = 51, 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 = {6141, 653, 191} \[ \frac {x}{3 c \sqrt {c-a^2 c x^2}}+\frac {2 (a x+1)}{3 a \left (c-a^2 c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[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 6141

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[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]) && IGt
Q[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*}

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Mathematica [A]  time = 0.03, size = 63, normalized size = 1.24 \[ -\frac {(a x-2) \sqrt {a x+1} \sqrt {1-a^2 x^2}}{3 a c (1-a x)^{3/2} \sqrt {c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [A]  time = 0.79, size = 47, normalized size = 0.92 \[ -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x - 2\right )}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)

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giac [B]  time = 0.35, size = 148, normalized size = 2.90 \[ -\frac {{\left (a c - 3 \, \sqrt {-a^{2} c} \sqrt {c}\right )} \mathrm {sgn}\relax (x)}{3 \, {\left (a^{2} c^{\frac {5}{2}} - \sqrt {-a^{2} c} a c^{2}\right )}} + \frac {2 \, {\left (2 \, a^{2} c + 3 \, a \sqrt {c} {\left (\sqrt {-a^{2} c + \frac {c}{x^{2}}} - \frac {\sqrt {c}}{x}\right )} + 3 \, {\left (\sqrt {-a^{2} c + \frac {c}{x^{2}}} - \frac {\sqrt {c}}{x}\right )}^{2}\right )}}{3 \, {\left (a \sqrt {c} + \sqrt {-a^{2} c + \frac {c}{x^{2}}} - \frac {\sqrt {c}}{x}\right )}^{3} c \mathrm {sgn}\relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [A]  time = 0.03, size = 31, normalized size = 0.61 \[ -\frac {\left (a x -2\right ) \left (a x +1\right )^{2}}{3 \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.38, size = 196, normalized size = 3.84 \[ \frac {1}{3} \, a {\left (\frac {a}{\sqrt {-a^{2} c x^{2} + c} a^{4} c x + \sqrt {-a^{2} c x^{2} + c} a^{3} c} - \frac {a}{\sqrt {-a^{2} c x^{2} + c} a^{4} c x - \sqrt {-a^{2} c x^{2} + c} a^{3} c} - \frac {1}{\sqrt {-a^{2} c x^{2} + c} a^{3} c x + \sqrt {-a^{2} c x^{2} + c} a^{2} c} - \frac {1}{\sqrt {-a^{2} c x^{2} + c} a^{3} c x - \sqrt {-a^{2} c x^{2} + c} a^{2} c} + \frac {x}{\sqrt {-a^{2} c x^{2} + c} a c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.95, size = 33, normalized size = 0.65 \[ -\frac {\sqrt {c-a^2\,c\,x^2}\,\left (a\,x-2\right )}{3\,a\,c^2\,{\left (a\,x-1\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)

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