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

Optimal. Leaf size=105 \[ -\frac{2}{3 a c^2 \sqrt{c-a^2 c x^2}}+\frac{2 x \tanh ^{-1}(a x)}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{1}{9 a c \left (c-a^2 c x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0656546, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {5960, 5958} \[ -\frac{2}{3 a c^2 \sqrt{c-a^2 c x^2}}+\frac{2 x \tanh ^{-1}(a x)}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{1}{9 a c \left (c-a^2 c x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0685026, size = 64, normalized size = 0.61 \[ -\frac{\sqrt{c-a^2 c x^2} \left (-6 a^2 x^2+\left (6 a^3 x^3-9 a x\right ) \tanh ^{-1}(a x)+7\right )}{9 a c^3 \left (a^2 x^2-1\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.256, size = 160, normalized size = 1.5 \begin{align*}{\frac{ \left ( ax+1 \right ) \left ( -1+3\,{\it Artanh} \left ( ax \right ) \right ) }{72\,a \left ( ax-1 \right ) ^{2}{c}^{3}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) c}}-{\frac{3\,{\it Artanh} \left ( ax \right ) -3}{8\,a \left ( ax-1 \right ){c}^{3}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) c}}-{\frac{3\,{\it Artanh} \left ( ax \right ) +3}{8\,a \left ( ax+1 \right ){c}^{3}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) c}}+{\frac{ \left ( ax-1 \right ) \left ( 1+3\,{\it Artanh} \left ( ax \right ) \right ) }{72\,a \left ( ax+1 \right ) ^{2}{c}^{3}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.989851, size = 122, normalized size = 1.16 \begin{align*} -\frac{1}{9} \, a{\left (\frac{6}{\sqrt{-a^{2} c x^{2} + c} a^{2} c^{2}} + \frac{1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} a^{2} c}\right )} + \frac{1}{3} \,{\left (\frac{2 \, x}{\sqrt{-a^{2} c x^{2} + c} c^{2}} + \frac{x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} c}\right )} \operatorname{artanh}\left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*a*(6/(sqrt(-a^2*c*x^2 + c)*a^2*c^2) + 1/((-a^2*c*x^2 + c)^(3/2)*a^2*c)) + 1/3*(2*x/(sqrt(-a^2*c*x^2 + c)*
c^2) + x/((-a^2*c*x^2 + c)^(3/2)*c))*arctanh(a*x)

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Fricas [A]  time = 1.5965, size = 180, normalized size = 1.71 \begin{align*} \frac{\sqrt{-a^{2} c x^{2} + c}{\left (12 \, a^{2} x^{2} - 3 \,{\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) - 14\right )}}{18 \,{\left (a^{5} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*sqrt(-a^2*c*x^2 + c)*(12*a^2*x^2 - 3*(2*a^3*x^3 - 3*a*x)*log(-(a*x + 1)/(a*x - 1)) - 14)/(a^5*c^3*x^4 - 2
*a^3*c^3*x^2 + a*c^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.29052, size = 150, normalized size = 1.43 \begin{align*} -\frac{\sqrt{-a^{2} c x^{2} + c}{\left (\frac{2 \, a^{2} x^{2}}{c} - \frac{3}{c}\right )} x \log \left (-\frac{a x + 1}{a x - 1}\right )}{6 \,{\left (a^{2} c x^{2} - c\right )}^{2}} - \frac{6 \, a^{2} c x^{2} - 7 \, c}{9 \,{\left (a^{2} c x^{2} - c\right )} \sqrt{-a^{2} c x^{2} + c} a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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