3.520 \(\int \frac{\tanh ^{-1}(x)}{(a-a x^2)^{5/2}} \, dx\)

Optimal. Leaf size=83 \[ -\frac{2}{3 a^2 \sqrt{a-a x^2}}+\frac{2 x \tanh ^{-1}(x)}{3 a^2 \sqrt{a-a x^2}}-\frac{1}{9 a \left (a-a x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(x)}{3 a \left (a-a x^2\right )^{3/2}} \]

[Out]

-1/(9*a*(a - a*x^2)^(3/2)) - 2/(3*a^2*Sqrt[a - a*x^2]) + (x*ArcTanh[x])/(3*a*(a - a*x^2)^(3/2)) + (2*x*ArcTanh
[x])/(3*a^2*Sqrt[a - a*x^2])

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Rubi [A]  time = 0.0526752, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {5960, 5958} \[ -\frac{2}{3 a^2 \sqrt{a-a x^2}}+\frac{2 x \tanh ^{-1}(x)}{3 a^2 \sqrt{a-a x^2}}-\frac{1}{9 a \left (a-a x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(x)}{3 a \left (a-a x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[x]/(a - a*x^2)^(5/2),x]

[Out]

-1/(9*a*(a - a*x^2)^(3/2)) - 2/(3*a^2*Sqrt[a - a*x^2]) + (x*ArcTanh[x])/(3*a*(a - a*x^2)^(3/2)) + (2*x*ArcTanh
[x])/(3*a^2*Sqrt[a - a*x^2])

Rule 5960

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(b*(d + e*x^2)^(q + 1))
/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x] -
Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]))/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*
d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]

Rule 5958

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> -Simp[b/(c*d*Sqrt[d + e*x^2]
), x] + Simp[(x*(a + b*ArcTanh[c*x]))/(d*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0
]

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(x)}{\left (a-a x^2\right )^{5/2}} \, dx &=-\frac{1}{9 a \left (a-a x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(x)}{3 a \left (a-a x^2\right )^{3/2}}+\frac{2 \int \frac{\tanh ^{-1}(x)}{\left (a-a x^2\right )^{3/2}} \, dx}{3 a}\\ &=-\frac{1}{9 a \left (a-a x^2\right )^{3/2}}-\frac{2}{3 a^2 \sqrt{a-a x^2}}+\frac{x \tanh ^{-1}(x)}{3 a \left (a-a x^2\right )^{3/2}}+\frac{2 x \tanh ^{-1}(x)}{3 a^2 \sqrt{a-a x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0486671, size = 45, normalized size = 0.54 \[ -\frac{\sqrt{a-a x^2} \left (-6 x^2+\left (6 x^3-9 x\right ) \tanh ^{-1}(x)+7\right )}{9 a^3 \left (x^2-1\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[x]/(a - a*x^2)^(5/2),x]

[Out]

-(Sqrt[a - a*x^2]*(7 - 6*x^2 + (-9*x + 6*x^3)*ArcTanh[x]))/(9*a^3*(-1 + x^2)^2)

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Maple [A]  time = 0.237, size = 112, normalized size = 1.4 \begin{align*}{\frac{ \left ( 1+x \right ) \left ( -1+3\,{\it Artanh} \left ( x \right ) \right ) }{72\, \left ( -1+x \right ) ^{2}{a}^{3}}\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) a}}-{\frac{3\,{\it Artanh} \left ( x \right ) -3}{ \left ( -8+8\,x \right ){a}^{3}}\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) a}}-{\frac{3+3\,{\it Artanh} \left ( x \right ) }{ \left ( 8+8\,x \right ){a}^{3}}\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) a}}+{\frac{ \left ( -1+x \right ) \left ( 1+3\,{\it Artanh} \left ( x \right ) \right ) }{72\, \left ( 1+x \right ) ^{2}{a}^{3}}\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(x)/(-a*x^2+a)^(5/2),x)

[Out]

1/72*(1+x)*(-1+3*arctanh(x))*(-(-1+x)*(1+x)*a)^(1/2)/(-1+x)^2/a^3-3/8*(arctanh(x)-1)*(-(-1+x)*(1+x)*a)^(1/2)/(
-1+x)/a^3-3/8*(1+arctanh(x))*(-(-1+x)*(1+x)*a)^(1/2)/(1+x)/a^3+1/72*(-1+x)*(1+3*arctanh(x))*(-(-1+x)*(1+x)*a)^
(1/2)/(1+x)^2/a^3

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Maxima [A]  time = 0.975357, size = 90, normalized size = 1.08 \begin{align*} \frac{1}{3} \,{\left (\frac{2 \, x}{\sqrt{-a x^{2} + a} a^{2}} + \frac{x}{{\left (-a x^{2} + a\right )}^{\frac{3}{2}} a}\right )} \operatorname{artanh}\left (x\right ) - \frac{2}{3 \, \sqrt{-a x^{2} + a} a^{2}} - \frac{1}{9 \,{\left (-a x^{2} + a\right )}^{\frac{3}{2}} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x)/(-a*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

1/3*(2*x/(sqrt(-a*x^2 + a)*a^2) + x/((-a*x^2 + a)^(3/2)*a))*arctanh(x) - 2/3/(sqrt(-a*x^2 + a)*a^2) - 1/9/((-a
*x^2 + a)^(3/2)*a)

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Fricas [A]  time = 1.9577, size = 142, normalized size = 1.71 \begin{align*} \frac{\sqrt{-a x^{2} + a}{\left (12 \, x^{2} - 3 \,{\left (2 \, x^{3} - 3 \, x\right )} \log \left (-\frac{x + 1}{x - 1}\right ) - 14\right )}}{18 \,{\left (a^{3} x^{4} - 2 \, a^{3} x^{2} + a^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x)/(-a*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

1/18*sqrt(-a*x^2 + a)*(12*x^2 - 3*(2*x^3 - 3*x)*log(-(x + 1)/(x - 1)) - 14)/(a^3*x^4 - 2*a^3*x^2 + a^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atanh}{\left (x \right )}}{\left (- a \left (x - 1\right ) \left (x + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(x)/(-a*x**2+a)**(5/2),x)

[Out]

Integral(atanh(x)/(-a*(x - 1)*(x + 1))**(5/2), x)

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Giac [A]  time = 1.21212, size = 116, normalized size = 1.4 \begin{align*} -\frac{\sqrt{-a x^{2} + a} x{\left (\frac{2 \, x^{2}}{a} - \frac{3}{a}\right )} \log \left (-\frac{x + 1}{x - 1}\right )}{6 \,{\left (a x^{2} - a\right )}^{2}} - \frac{6 \, a x^{2} - 7 \, a}{9 \,{\left (a x^{2} - a\right )} \sqrt{-a x^{2} + a} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x)/(-a*x^2+a)^(5/2),x, algorithm="giac")

[Out]

-1/6*sqrt(-a*x^2 + a)*x*(2*x^2/a - 3/a)*log(-(x + 1)/(x - 1))/(a*x^2 - a)^2 - 1/9*(6*a*x^2 - 7*a)/((a*x^2 - a)
*sqrt(-a*x^2 + a)*a^2)