3.398 \(\int \frac {x \tanh ^{-1}(a x)^2}{(1-a^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=68 \[ \frac {2}{a^2 \sqrt {1-a^2 x^2}}+\frac {\tanh ^{-1}(a x)^2}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 x \tanh ^{-1}(a x)}{a \sqrt {1-a^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5994, 5958} \[ \frac {2}{a^2 \sqrt {1-a^2 x^2}}+\frac {\tanh ^{-1}(a x)^2}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 x \tanh ^{-1}(a x)}{a \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x*ArcTanh[a*x]^2)/(1 - a^2*x^2)^(3/2),x]

[Out]

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

Rule 5994

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

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 34, normalized size = 0.50 \[ \frac {\tanh ^{-1}(a x)^2-2 a x \tanh ^{-1}(a x)+2}{a^2 \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*ArcTanh[a*x]^2)/(1 - a^2*x^2)^(3/2),x]

[Out]

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

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fricas [A]  time = 0.51, size = 69, normalized size = 1.01 \[ \frac {\sqrt {-a^{2} x^{2} + 1} {\left (4 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) - \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 8\right )}}{4 \, {\left (a^{4} x^{2} - a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arctanh(a*x)^2/(-a^2*x^2+1)^(3/2),x, algorithm="fricas")

[Out]

1/4*sqrt(-a^2*x^2 + 1)*(4*a*x*log(-(a*x + 1)/(a*x - 1)) - log(-(a*x + 1)/(a*x - 1))^2 - 8)/(a^4*x^2 - a^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \operatorname {artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arctanh(a*x)^2/(-a^2*x^2+1)^(3/2),x, algorithm="giac")

[Out]

integrate(x*arctanh(a*x)^2/(-a^2*x^2 + 1)^(3/2), x)

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maple [A]  time = 0.32, size = 82, normalized size = 1.21 \[ -\frac {\left (\arctanh \left (a x \right )^{2}-2 \arctanh \left (a x \right )+2\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 a^{2} \left (a x -1\right )}+\frac {\left (\arctanh \left (a x \right )^{2}+2 \arctanh \left (a x \right )+2\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 a^{2} \left (a x +1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arctanh(a*x)^2/(-a^2*x^2+1)^(3/2),x)

[Out]

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

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maxima [A]  time = 0.31, size = 62, normalized size = 0.91 \[ -\frac {2 \, x \operatorname {artanh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} a} + \frac {\operatorname {artanh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} a^{2}} + \frac {2}{\sqrt {-a^{2} x^{2} + 1} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arctanh(a*x)^2/(-a^2*x^2+1)^(3/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,{\mathrm {atanh}\left (a\,x\right )}^2}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*atanh(a*x)^2)/(1 - a^2*x^2)^(3/2),x)

[Out]

int((x*atanh(a*x)^2)/(1 - a^2*x^2)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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