3.4.90 \(\int \frac {x \tanh ^{-1}(a x)}{(1-a^2 x^2)^{3/2}} \, dx\) [390]

Optimal. Leaf size=43 \[ -\frac {x}{a \sqrt {1-a^2 x^2}}+\frac {\tanh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6141, 197} \begin {gather*} \frac {\tanh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {x}{a \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 6141

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

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Mathematica [A]
time = 0.03, size = 27, normalized size = 0.63 \begin {gather*} \frac {-a x+\tanh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 1.07, size = 66, normalized size = 1.53

method	result	size
default	\(-\frac {\left (\arctanh \left (a x \right )-1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 \left (a x -1\right ) a^{2}}+\frac {\left (\arctanh \left (a x \right )+1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 \left (a x +1\right ) a^{2}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.25, size = 39, normalized size = 0.91 \begin {gather*} -\frac {x}{\sqrt {-a^{2} x^{2} + 1} a} + \frac {\operatorname {artanh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.38, size = 51, normalized size = 1.19 \begin {gather*} \frac {\sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x - \log \left (-\frac {a x + 1}{a x - 1}\right )\right )}}{2 \, {\left (a^{4} x^{2} - a^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.43, size = 61, normalized size = 1.42 \begin {gather*} \frac {\sqrt {-a^{2} x^{2} + 1} x}{{\left (a^{2} x^{2} - 1\right )} a} + \frac {\log \left (-\frac {a x + 1}{a x - 1}\right )}{2 \, \sqrt {-a^{2} x^{2} + 1} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x\,\mathrm {atanh}\left (a\,x\right )}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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