3.3.48 \(\int \frac {\sqrt {\tanh ^{-1}(a x)}}{1-a^2 x^2} \, dx\) [248]

Optimal. Leaf size=15 \[ \frac {2 \tanh ^{-1}(a x)^{3/2}}{3 a} \]

[Out]

2/3*arctanh(a*x)^(3/2)/a

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Rubi [A]
time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6095} \begin {gather*} \frac {2 \tanh ^{-1}(a x)^{3/2}}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6095

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

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} \frac {2 \tanh ^{-1}(a x)^{3/2}}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.72, size = 12, normalized size = 0.80

method	result	size
derivativedivides	\(\frac {2 \arctanh \left (a x \right )^{\frac {3}{2}}}{3 a}\)	\(12\)
default	\(\frac {2 \arctanh \left (a x \right )^{\frac {3}{2}}}{3 a}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(a*x)^(1/2)/(-a^2*x^2+1),x,method=_RETURNVERBOSE)

[Out]

2/3*arctanh(a*x)^(3/2)/a

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^(1/2)/(-a^2*x^2+1),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (11) = 22\).
time = 0.33, size = 25, normalized size = 1.67 \begin {gather*} \frac {\sqrt {2} \log \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {3}{2}}}{6 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^(1/2)/(-a^2*x^2+1),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.70, size = 14, normalized size = 0.93 \begin {gather*} \begin {cases} \frac {2 \operatorname {atanh}^{\frac {3}{2}}{\left (a x \right )}}{3 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*atanh(a*x)**(3/2)/(3*a), Ne(a, 0)), (0, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (11) = 22\).
time = 0.39, size = 25, normalized size = 1.67 \begin {gather*} \frac {\sqrt {2} \log \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {3}{2}}}{6 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^(1/2)/(-a^2*x^2+1),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.87, size = 11, normalized size = 0.73 \begin {gather*} \frac {2\,{\mathrm {atanh}\left (a\,x\right )}^{3/2}}{3\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-atanh(a*x)^(1/2)/(a^2*x^2 - 1),x)

[Out]

(2*atanh(a*x)^(3/2))/(3*a)

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