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 + 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])

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Rubi [A]  time = 0.117417, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, 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 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]

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 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]

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 = 0.0560338, 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)

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Maple [A]  time = 0.032, size = 32, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,ax-1 \right ) \left ( ax+1 \right ) ^{2}}{3\,{a}^{2}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 2.65501, size = 103, normalized size = 1.72 \begin{align*} \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 )}} \end{align*}

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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \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 \end{align*}

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)

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Giac [B]  time = 1.20668, size = 158, normalized size = 2.63 \begin{align*} -\frac{{\left (a c + 3 \, \sqrt{-a^{2} c} \sqrt{c}\right )} \mathrm{sgn}\left (x\right )}{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}\left (x\right )} \end{align*}

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))