3.6.19 \(\int \frac {\tanh ^{-1}(x)}{(a-a x^2)^{3/2}} \, dx\) [519]

Optimal. Leaf size=37 \[ -\frac {1}{a \sqrt {a-a x^2}}+\frac {x \tanh ^{-1}(x)}{a \sqrt {a-a x^2}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {6105} \begin {gather*} \frac {x \tanh ^{-1}(x)}{a \sqrt {a-a x^2}}-\frac {1}{a \sqrt {a-a x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[x]/(a - a*x^2)^(3/2),x]

[Out]

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

Rule 6105

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

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Mathematica [A]
time = 0.03, size = 30, normalized size = 0.81 \begin {gather*} \frac {\sqrt {a-a x^2} \left (1-x \tanh ^{-1}(x)\right )}{a^2 \left (-1+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[x]/(a - a*x^2)^(3/2),x]

[Out]

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

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Maple [A]
time = 1.82, size = 52, normalized size = 1.41

method	result	size
risch	\(\frac {x \ln \left (1+x \right )}{2 a \sqrt {-a \left (x^{2}-1\right )}}-\frac {x \ln \left (1-x \right )+2}{2 a \sqrt {-a \left (x^{2}-1\right )}}\)	\(47\)
default	\(-\frac {\left (\arctanh \left (x \right )-1\right ) \sqrt {-\left (x -1\right ) \left (1+x \right ) a}}{2 a^{2} \left (x -1\right )}-\frac {\left (1+\arctanh \left (x \right )\right ) \sqrt {-\left (x -1\right ) \left (1+x \right ) a}}{2 a^{2} \left (1+x \right )}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(x)/(-a*x^2+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.47, size = 63, normalized size = 1.70 \begin {gather*} \frac {x \operatorname {artanh}\left (x\right )}{\sqrt {-a x^{2} + a} a} - \frac {\frac {\sqrt {-a x^{2} + a}}{a x + a} - \frac {\sqrt {-a x^{2} + a}}{a x - a}}{2 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x)/(-a*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 42, normalized size = 1.14 \begin {gather*} -\frac {\sqrt {-a x^{2} + a} {\left (x \log \left (-\frac {x + 1}{x - 1}\right ) - 2\right )}}{2 \, {\left (a^{2} x^{2} - a^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x)/(-a*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

-1/2*sqrt(-a*x^2 + a)*(x*log(-(x + 1)/(x - 1)) - 2)/(a^2*x^2 - a^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {atanh}{\left (x \right )}}{\left (- a \left (x - 1\right ) \left (x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(x)/(-a*x**2+a)**(3/2),x)

[Out]

Integral(atanh(x)/(-a*(x - 1)*(x + 1))**(3/2), x)

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Giac [A]
time = 0.41, size = 54, normalized size = 1.46 \begin {gather*} -\frac {\sqrt {-a x^{2} + a} x \log \left (-\frac {x + 1}{x - 1}\right )}{2 \, {\left (a x^{2} - a\right )} a} - \frac {1}{\sqrt {-a x^{2} + a} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x)/(-a*x^2+a)^(3/2),x, algorithm="giac")

[Out]

-1/2*sqrt(-a*x^2 + a)*x*log(-(x + 1)/(x - 1))/((a*x^2 - a)*a) - 1/(sqrt(-a*x^2 + a)*a)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\mathrm {atanh}\left (x\right )}{{\left (a-a\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(x)/(a - a*x^2)^(3/2),x)

[Out]

int(atanh(x)/(a - a*x^2)^(3/2), x)

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