3.52 \(\int \frac{\coth ^{-1}(x)}{(a-a x^2)^{7/2}} \, dx\)

Optimal. Leaf size=124 \[ -\frac{8}{15 a^3 \sqrt{a-a x^2}}-\frac{4}{45 a^2 \left (a-a x^2\right )^{3/2}}+\frac{8 x \coth ^{-1}(x)}{15 a^3 \sqrt{a-a x^2}}+\frac{4 x \coth ^{-1}(x)}{15 a^2 \left (a-a x^2\right )^{3/2}}-\frac{1}{25 a \left (a-a x^2\right )^{5/2}}+\frac{x \coth ^{-1}(x)}{5 a \left (a-a x^2\right )^{5/2}} \]

[Out]

-1/(25*a*(a - a*x^2)^(5/2)) - 4/(45*a^2*(a - a*x^2)^(3/2)) - 8/(15*a^3*Sqrt[a - a*x^2]) + (x*ArcCoth[x])/(5*a*
(a - a*x^2)^(5/2)) + (4*x*ArcCoth[x])/(15*a^2*(a - a*x^2)^(3/2)) + (8*x*ArcCoth[x])/(15*a^3*Sqrt[a - a*x^2])

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Rubi [A]  time = 0.0820825, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {5961, 5959} \[ -\frac{8}{15 a^3 \sqrt{a-a x^2}}-\frac{4}{45 a^2 \left (a-a x^2\right )^{3/2}}+\frac{8 x \coth ^{-1}(x)}{15 a^3 \sqrt{a-a x^2}}+\frac{4 x \coth ^{-1}(x)}{15 a^2 \left (a-a x^2\right )^{3/2}}-\frac{1}{25 a \left (a-a x^2\right )^{5/2}}+\frac{x \coth ^{-1}(x)}{5 a \left (a-a x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[x]/(a - a*x^2)^(7/2),x]

[Out]

-1/(25*a*(a - a*x^2)^(5/2)) - 4/(45*a^2*(a - a*x^2)^(3/2)) - 8/(15*a^3*Sqrt[a - a*x^2]) + (x*ArcCoth[x])/(5*a*
(a - a*x^2)^(5/2)) + (4*x*ArcCoth[x])/(15*a^2*(a - a*x^2)^(3/2)) + (8*x*ArcCoth[x])/(15*a^3*Sqrt[a - a*x^2])

Rule 5961

Int[((a_.) + ArcCoth[(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*ArcCoth[c*x]), x], x] -
Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[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 5959

Int[((a_.) + ArcCoth[(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*ArcCoth[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{\coth ^{-1}(x)}{\left (a-a x^2\right )^{7/2}} \, dx &=-\frac{1}{25 a \left (a-a x^2\right )^{5/2}}+\frac{x \coth ^{-1}(x)}{5 a \left (a-a x^2\right )^{5/2}}+\frac{4 \int \frac{\coth ^{-1}(x)}{\left (a-a x^2\right )^{5/2}} \, dx}{5 a}\\ &=-\frac{1}{25 a \left (a-a x^2\right )^{5/2}}-\frac{4}{45 a^2 \left (a-a x^2\right )^{3/2}}+\frac{x \coth ^{-1}(x)}{5 a \left (a-a x^2\right )^{5/2}}+\frac{4 x \coth ^{-1}(x)}{15 a^2 \left (a-a x^2\right )^{3/2}}+\frac{8 \int \frac{\coth ^{-1}(x)}{\left (a-a x^2\right )^{3/2}} \, dx}{15 a^2}\\ &=-\frac{1}{25 a \left (a-a x^2\right )^{5/2}}-\frac{4}{45 a^2 \left (a-a x^2\right )^{3/2}}-\frac{8}{15 a^3 \sqrt{a-a x^2}}+\frac{x \coth ^{-1}(x)}{5 a \left (a-a x^2\right )^{5/2}}+\frac{4 x \coth ^{-1}(x)}{15 a^2 \left (a-a x^2\right )^{3/2}}+\frac{8 x \coth ^{-1}(x)}{15 a^3 \sqrt{a-a x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0605613, size = 55, normalized size = 0.44 \[ \frac{\sqrt{a-a x^2} \left (120 x^4-260 x^2-15 \left (8 x^4-20 x^2+15\right ) x \coth ^{-1}(x)+149\right )}{225 a^4 \left (x^2-1\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[x]/(a - a*x^2)^(7/2),x]

[Out]

(Sqrt[a - a*x^2]*(149 - 260*x^2 + 120*x^4 - 15*x*(15 - 20*x^2 + 8*x^4)*ArcCoth[x]))/(225*a^4*(-1 + x^2)^3)

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Maple [A]  time = 0.261, size = 176, normalized size = 1.4 \begin{align*} -{\frac{ \left ( 1+x \right ) ^{2} \left ( -1+5\,{\rm arccoth} \left (x\right ) \right ) }{800\, \left ( -1+x \right ) ^{3}{a}^{4}}\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) a}}+{\frac{ \left ( 5+5\,x \right ) \left ( -1+3\,{\rm arccoth} \left (x\right ) \right ) }{288\, \left ( -1+x \right ) ^{2}{a}^{4}}\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) a}}-{\frac{5\,{\rm arccoth} \left (x\right )-5}{ \left ( -16+16\,x \right ){a}^{4}}\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) a}}-{\frac{5\,{\rm arccoth} \left (x\right )+5}{ \left ( 16+16\,x \right ){a}^{4}}\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) a}}+{\frac{ \left ( 5+15\,{\rm arccoth} \left (x\right ) \right ) \left ( -1+x \right ) }{288\, \left ( 1+x \right ) ^{2}{a}^{4}}\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) a}}-{\frac{ \left ( 1+5\,{\rm arccoth} \left (x\right ) \right ) \left ( -1+x \right ) ^{2}}{800\, \left ( 1+x \right ) ^{3}{a}^{4}}\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(x)/(-a*x^2+a)^(7/2),x)

[Out]

-1/800*(1+x)^2*(-1+5*arccoth(x))*(-(-1+x)*(1+x)*a)^(1/2)/(-1+x)^3/a^4+5/288*(1+x)*(-1+3*arccoth(x))*(-(-1+x)*(
1+x)*a)^(1/2)/(-1+x)^2/a^4-5/16*(arccoth(x)-1)*(-(-1+x)*(1+x)*a)^(1/2)/(-1+x)/a^4-5/16*(arccoth(x)+1)*(-(-1+x)
*(1+x)*a)^(1/2)/(1+x)/a^4+5/288*(1+3*arccoth(x))*(-1+x)*(-(-1+x)*(1+x)*a)^(1/2)/(1+x)^2/a^4-1/800*(1+5*arccoth
(x))*(-1+x)^2*(-(-1+x)*(1+x)*a)^(1/2)/(1+x)^3/a^4

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Maxima [A]  time = 1.0259, size = 134, normalized size = 1.08 \begin{align*} \frac{1}{15} \,{\left (\frac{8 \, x}{\sqrt{-a x^{2} + a} a^{3}} + \frac{4 \, x}{{\left (-a x^{2} + a\right )}^{\frac{3}{2}} a^{2}} + \frac{3 \, x}{{\left (-a x^{2} + a\right )}^{\frac{5}{2}} a}\right )} \operatorname{arcoth}\left (x\right ) - \frac{8}{15 \, \sqrt{-a x^{2} + a} a^{3}} - \frac{4}{45 \,{\left (-a x^{2} + a\right )}^{\frac{3}{2}} a^{2}} - \frac{1}{25 \,{\left (-a x^{2} + a\right )}^{\frac{5}{2}} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(x)/(-a*x^2+a)^(7/2),x, algorithm="maxima")

[Out]

1/15*(8*x/(sqrt(-a*x^2 + a)*a^3) + 4*x/((-a*x^2 + a)^(3/2)*a^2) + 3*x/((-a*x^2 + a)^(5/2)*a))*arccoth(x) - 8/1
5/(sqrt(-a*x^2 + a)*a^3) - 4/45/((-a*x^2 + a)^(3/2)*a^2) - 1/25/((-a*x^2 + a)^(5/2)*a)

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Fricas [A]  time = 1.67926, size = 189, normalized size = 1.52 \begin{align*} \frac{{\left (240 \, x^{4} - 520 \, x^{2} - 15 \,{\left (8 \, x^{5} - 20 \, x^{3} + 15 \, x\right )} \log \left (\frac{x + 1}{x - 1}\right ) + 298\right )} \sqrt{-a x^{2} + a}}{450 \,{\left (a^{4} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{4} x^{2} - a^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(x)/(-a*x^2+a)^(7/2),x, algorithm="fricas")

[Out]

1/450*(240*x^4 - 520*x^2 - 15*(8*x^5 - 20*x^3 + 15*x)*log((x + 1)/(x - 1)) + 298)*sqrt(-a*x^2 + a)/(a^4*x^6 -
3*a^4*x^4 + 3*a^4*x^2 - a^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(x)/(-a*x**2+a)**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (x\right )}{{\left (-a x^{2} + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(x)/(-a*x^2+a)^(7/2),x, algorithm="giac")

[Out]

integrate(arccoth(x)/(-a*x^2 + a)^(7/2), x)