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

Optimal. Leaf size=121 \[ -\frac{3}{8 a^3 \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac{3 \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}+\frac{3 x \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^4}{8 a^3}+\frac{3 \tanh ^{-1}(a x)^2}{8 a^3} \]

[Out]

-3/(8*a^3*(1 - a^2*x^2)) + (3*x*ArcTanh[a*x])/(4*a^2*(1 - a^2*x^2)) + (3*ArcTanh[a*x]^2)/(8*a^3) - (3*ArcTanh[
a*x]^2)/(4*a^3*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^3)/(2*a^2*(1 - a^2*x^2)) - ArcTanh[a*x]^4/(8*a^3)

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Rubi [A]  time = 0.133575, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6000, 5994, 5956, 261} \[ -\frac{3}{8 a^3 \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac{3 \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}+\frac{3 x \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^4}{8 a^3}+\frac{3 \tanh ^{-1}(a x)^2}{8 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-3/(8*a^3*(1 - a^2*x^2)) + (3*x*ArcTanh[a*x])/(4*a^2*(1 - a^2*x^2)) + (3*ArcTanh[a*x]^2)/(8*a^3) - (3*ArcTanh[
a*x]^2)/(4*a^3*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^3)/(2*a^2*(1 - a^2*x^2)) - ArcTanh[a*x]^4/(8*a^3)

Rule 6000

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^2)/((d_) + (e_.)*(x_)^2)^2, x_Symbol] :> -Simp[(a + b*ArcT
anh[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)), x] + (-Dist[(b*p)/(2*c), Int[(x*(a + b*ArcTanh[c*x])^(p - 1))/(d + e*
x^2)^2, x], x] + Simp[(x*(a + b*ArcTanh[c*x])^p)/(2*c^2*d*(d + e*x^2)), x]) /; FreeQ[{a, b, c, d, e}, x] && Eq
Q[c^2*d + e, 0] && GtQ[p, 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]

Rule 5956

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

Rule 261

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

Rubi steps

\begin{align*} \int \frac{x^2 \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx &=\frac{x \tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^4}{8 a^3}-\frac{3 \int \frac{x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{2 a}\\ &=-\frac{3 \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^4}{8 a^3}+\frac{3 \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{2 a^2}\\ &=\frac{3 x \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )}+\frac{3 \tanh ^{-1}(a x)^2}{8 a^3}-\frac{3 \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^4}{8 a^3}-\frac{3 \int \frac{x}{\left (1-a^2 x^2\right )^2} \, dx}{4 a}\\ &=-\frac{3}{8 a^3 \left (1-a^2 x^2\right )}+\frac{3 x \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )}+\frac{3 \tanh ^{-1}(a x)^2}{8 a^3}-\frac{3 \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^4}{8 a^3}\\ \end{align*}

Mathematica [A]  time = 0.0811536, size = 72, normalized size = 0.6 \[ \frac{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4+3 \left (a^2 x^2+1\right ) \tanh ^{-1}(a x)^2-4 a x \tanh ^{-1}(a x)^3-6 a x \tanh ^{-1}(a x)+3}{8 a^3 \left (a^2 x^2-1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3 - 6*a*x*ArcTanh[a*x] + 3*(1 + a^2*x^2)*ArcTanh[a*x]^2 - 4*a*x*ArcTanh[a*x]^3 + (1 - a^2*x^2)*ArcTanh[a*x]^4
)/(8*a^3*(-1 + a^2*x^2))

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Maple [C]  time = 0.442, size = 1771, normalized size = 14.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/16/a^3/(a*x-1)/(a*x+1)-1/4/a^3*arctanh(a*x)^3/(a*x-1)-1/4/a^3*arctanh(a*x)^3/(a*x+1)-1/8*I/a^3/(a*x-1)/(a*x+
1)*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*arctanh(a*x)^3*Pi+1/8*I/a^3/(a*x-1)/(a*x
+1)*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*arctanh(a*x)^3*Pi
-1/8*I/a^3/(a*x-1)/(a*x+1)*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*csgn(I/((a*x+1)^2/(-a^2*
x^2+1)+1))*arctanh(a*x)^3*Pi-1/4*I/a/(a*x-1)/(a*x+1)*arctanh(a*x)^3*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*Pi*x^
2+1/4*I/a/(a*x-1)/(a*x+1)*arctanh(a*x)^3*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*Pi*x^2+1/8*I/a/(a*x-1)/(a*x+1)*a
rctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*Pi*x^2+1/8*I/a/(a*x-1)/(a*x+1)*arctanh(a*x)^3*csgn(I*(a*x+1)^2/(
a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3*Pi*x^2-1/8/a/(a*x-1)/(a*x+1)*arctanh(a*x)^4*x^2+3/8/a/(a*x-1)/(a*x+1)
*arctanh(a*x)^2*x^2-3/4/a^2/(a*x-1)/(a*x+1)*arctanh(a*x)*x+1/4/a^3*arctanh(a*x)^3*ln(a*x-1)-1/4/a^3*arctanh(a*
x)^3*ln(a*x+1)+1/2/a^3*arctanh(a*x)^3*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*I/a^3/(a*x-1)/(a*x+1)*csgn(I*(a*x+1)^
2/(a^2*x^2-1))^2*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)^3*Pi+1/8*I/a^3/(a*x-1)/(a*x+1)*csgn(I*(a*x+1)
^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*ar
ctanh(a*x)^3*Pi+1/4*I/a/(a*x-1)/(a*x+1)*arctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*csgn(I*(a*x+1)/(-a^2*x^
2+1)^(1/2))*Pi*x^2+1/8*I/a/(a*x-1)/(a*x+1)*arctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)/(-a^2*x
^2+1)^(1/2))^2*Pi*x^2-1/8*I/a/(a*x-1)/(a*x+1)*arctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a
^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*Pi*x^2+1/8*I/a/(a*x-1)/(a*x+1)*arctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^
2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*Pi*x^2-1/8*I/a/(a*x-1)/(a*x+1)*arctanh(a
*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I/((a*x+1)^2
/(-a^2*x^2+1)+1))*Pi*x^2+3/16/a/(a*x-1)/(a*x+1)*x^2+1/8/a^3/(a*x-1)/(a*x+1)*arctanh(a*x)^4+3/8/a^3/(a*x-1)/(a*
x+1)*arctanh(a*x)^2-1/4*I/a/(a*x-1)/(a*x+1)*arctanh(a*x)^3*Pi*x^2-1/8*I/a^3/(a*x-1)/(a*x+1)*csgn(I*(a*x+1)^2/(
a^2*x^2-1))^3*arctanh(a*x)^3*Pi-1/8*I/a^3/(a*x-1)/(a*x+1)*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)
+1))^3*arctanh(a*x)^3*Pi+1/4*I/a^3/(a*x-1)/(a*x+1)*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*arctanh(a*x)^3*Pi-1/4*
I/a^3/(a*x-1)/(a*x+1)*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)^3*Pi+1/4*I/a^3/(a*x-1)/(a*x+1)*arctanh
(a*x)^3*Pi

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Maxima [B]  time = 1.03771, size = 628, normalized size = 5.19 \begin{align*} -\frac{1}{4} \,{\left (\frac{2 \, x}{a^{4} x^{2} - a^{2}} + \frac{\log \left (a x + 1\right )}{a^{3}} - \frac{\log \left (a x - 1\right )}{a^{3}}\right )} \operatorname{artanh}\left (a x\right )^{3} + \frac{3 \,{\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 2 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) +{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} + 4\right )} a \operatorname{artanh}\left (a x\right )^{2}}{16 \,{\left (a^{6} x^{2} - a^{4}\right )}} + \frac{1}{128} \,{\left (\frac{{\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{4} - 4 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} \log \left (a x - 1\right ) +{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{4} - 6 \,{\left (2 \, a^{2} x^{2} -{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 2\right )} \log \left (a x + 1\right )^{2} - 12 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 4 \,{\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} - 6 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 48\right )} a^{2}}{a^{8} x^{2} - a^{6}} - \frac{8 \,{\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} - 3 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) -{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} + 12 \, a x - 3 \,{\left (2 \, a^{2} x^{2} -{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 2\right )} \log \left (a x + 1\right ) + 6 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} a \operatorname{artanh}\left (a x\right )}{a^{7} x^{2} - a^{5}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*x/(a^4*x^2 - a^2) + log(a*x + 1)/a^3 - log(a*x - 1)/a^3)*arctanh(a*x)^3 + 3/16*((a^2*x^2 - 1)*log(a*x
+ 1)^2 - 2*(a^2*x^2 - 1)*log(a*x + 1)*log(a*x - 1) + (a^2*x^2 - 1)*log(a*x - 1)^2 + 4)*a*arctanh(a*x)^2/(a^6*x
^2 - a^4) + 1/128*(((a^2*x^2 - 1)*log(a*x + 1)^4 - 4*(a^2*x^2 - 1)*log(a*x + 1)^3*log(a*x - 1) + (a^2*x^2 - 1)
*log(a*x - 1)^4 - 6*(2*a^2*x^2 - (a^2*x^2 - 1)*log(a*x - 1)^2 - 2)*log(a*x + 1)^2 - 12*(a^2*x^2 - 1)*log(a*x -
1)^2 - 4*((a^2*x^2 - 1)*log(a*x - 1)^3 - 6*(a^2*x^2 - 1)*log(a*x - 1))*log(a*x + 1) + 48)*a^2/(a^8*x^2 - a^6)
- 8*((a^2*x^2 - 1)*log(a*x + 1)^3 - 3*(a^2*x^2 - 1)*log(a*x + 1)^2*log(a*x - 1) - (a^2*x^2 - 1)*log(a*x - 1)^
3 + 12*a*x - 3*(2*a^2*x^2 - (a^2*x^2 - 1)*log(a*x - 1)^2 - 2)*log(a*x + 1) + 6*(a^2*x^2 - 1)*log(a*x - 1))*a*a
rctanh(a*x)/(a^7*x^2 - a^5))*a

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Fricas [A]  time = 1.91611, size = 258, normalized size = 2.13 \begin{align*} -\frac{8 \, a x \log \left (-\frac{a x + 1}{a x - 1}\right )^{3} +{\left (a^{2} x^{2} - 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{4} + 48 \, a x \log \left (-\frac{a x + 1}{a x - 1}\right ) - 12 \,{\left (a^{2} x^{2} + 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} - 48}{128 \,{\left (a^{5} x^{2} - a^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128*(8*a*x*log(-(a*x + 1)/(a*x - 1))^3 + (a^2*x^2 - 1)*log(-(a*x + 1)/(a*x - 1))^4 + 48*a*x*log(-(a*x + 1)/
(a*x - 1)) - 12*(a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1))^2 - 48)/(a^5*x^2 - a^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \operatorname{artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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