3.1.51 \(\int \frac {\coth ^{-1}(x)}{(a-a x^2)^{5/2}} \, dx\) [51]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, 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 = {6108, 6106} \begin {gather*} -\frac {2}{3 a^2 \sqrt {a-a x^2}}+\frac {2 x \coth ^{-1}(x)}{3 a^2 \sqrt {a-a x^2}}-\frac {1}{9 a \left (a-a x^2\right )^{3/2}}+\frac {x \coth ^{-1}(x)}{3 a \left (a-a x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCoth[x]/(a - a*x^2)^(5/2),x]

[Out]

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

Rule 6106

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
]

Rule 6108

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]

Rubi steps

\begin {align*} \int \frac {\coth ^{-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 \coth ^{-1}(x)}{3 a \left (a-a x^2\right )^{3/2}}+\frac {2 \int \frac {\coth ^{-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 \coth ^{-1}(x)}{3 a \left (a-a x^2\right )^{3/2}}+\frac {2 x \coth ^{-1}(x)}{3 a^2 \sqrt {a-a x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 45, normalized size = 0.54 \begin {gather*} -\frac {\sqrt {a-a x^2} \left (7-6 x^2+\left (-9 x+6 x^3\right ) \coth ^{-1}(x)\right )}{9 a^3 \left (-1+x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.40, size = 112, normalized size = 1.35

method	result	size
risch	\(\frac {x \left (2 x^{2}-3\right ) \ln \left (1+x \right )}{6 a^{2} \left (x^{2}-1\right ) \sqrt {-a \left (x^{2}-1\right )}}-\frac {6 x^{3} \ln \left (-1+x \right )+12 x^{2}-9 \ln \left (-1+x \right ) x -14}{18 a^{2} \left (x^{2}-1\right ) \sqrt {-a \left (x^{2}-1\right )}}\)	\(81\)
default	\(\frac {\left (1+x \right ) \left (-1+3 \,\mathrm {arccoth}\left (x \right )\right ) \sqrt {-a \left (1+x \right ) \left (-1+x \right )}}{72 \left (-1+x \right )^{2} a^{3}}-\frac {3 \left (-1+\mathrm {arccoth}\left (x \right )\right ) \sqrt {-a \left (1+x \right ) \left (-1+x \right )}}{8 \left (-1+x \right ) a^{3}}-\frac {3 \left (1+\mathrm {arccoth}\left (x \right )\right ) \sqrt {-a \left (1+x \right ) \left (-1+x \right )}}{8 \left (1+x \right ) a^{3}}+\frac {\left (1+3 \,\mathrm {arccoth}\left (x \right )\right ) \left (-1+x \right ) \sqrt {-a \left (1+x \right ) \left (-1+x \right )}}{72 \left (1+x \right )^{2} a^{3}}\)	\(112\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(x)/(-a*x^2+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 67, normalized size = 0.81 \begin {gather*} \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 {arcoth}\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(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))*arccoth(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 = 0.39, size = 61, normalized size = 0.73 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acoth}{\left (x \right )}}{\left (- a \left (x - 1\right ) \left (x + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.42, size = 90, normalized size = 1.08 \begin {gather*} -\frac {\sqrt {-a x^{2} + a} x {\left (\frac {2 \, x^{2}}{a} - \frac {3}{a}\right )} \log \left (-\frac {\frac {1}{x} + 1}{\frac {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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(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(-(1/x + 1)/(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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {acoth}\left (x\right )}{{\left (a-a\,x^2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acoth(x)/(a - a*x^2)^(5/2),x)

[Out]

int(acoth(x)/(a - a*x^2)^(5/2), x)

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