3.301 \(\int \frac{1}{(1-a^2 x^2)^2 \tanh ^{-1}(a x)^8} \, dx\)

Optimal. Leaf size=211 \[ -\frac{2 x}{315 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{x}{105 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac{x}{21 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac{2 \left (a^2 x^2+1\right )}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{a^2 x^2+1}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{a^2 x^2+1}{105 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac{1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}+\frac{4 \text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{315 a} \]

[Out]

-1/(7*a*(1 - a^2*x^2)*ArcTanh[a*x]^7) - x/(21*(1 - a^2*x^2)*ArcTanh[a*x]^6) - (1 + a^2*x^2)/(105*a*(1 - a^2*x^
2)*ArcTanh[a*x]^5) - x/(105*(1 - a^2*x^2)*ArcTanh[a*x]^4) - (1 + a^2*x^2)/(315*a*(1 - a^2*x^2)*ArcTanh[a*x]^3)
- (2*x)/(315*(1 - a^2*x^2)*ArcTanh[a*x]^2) - (2*(1 + a^2*x^2))/(315*a*(1 - a^2*x^2)*ArcTanh[a*x]) + (4*SinhIn
tegral[2*ArcTanh[a*x]])/(315*a)

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Rubi [A]  time = 0.255907, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {5966, 5996, 6034, 5448, 12, 3298} \[ -\frac{2 x}{315 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{x}{105 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac{x}{21 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac{2 \left (a^2 x^2+1\right )}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{a^2 x^2+1}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{a^2 x^2+1}{105 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac{1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}+\frac{4 \text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{315 a} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - a^2*x^2)^2*ArcTanh[a*x]^8),x]

[Out]

-1/(7*a*(1 - a^2*x^2)*ArcTanh[a*x]^7) - x/(21*(1 - a^2*x^2)*ArcTanh[a*x]^6) - (1 + a^2*x^2)/(105*a*(1 - a^2*x^
2)*ArcTanh[a*x]^5) - x/(105*(1 - a^2*x^2)*ArcTanh[a*x]^4) - (1 + a^2*x^2)/(315*a*(1 - a^2*x^2)*ArcTanh[a*x]^3)
- (2*x)/(315*(1 - a^2*x^2)*ArcTanh[a*x]^2) - (2*(1 + a^2*x^2))/(315*a*(1 - a^2*x^2)*ArcTanh[a*x]) + (4*SinhIn
tegral[2*ArcTanh[a*x]])/(315*a)

Rule 5966

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

Rule 5996

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*(x_))/((d_) + (e_.)*(x_)^2)^2, x_Symbol] :> Simp[(x*(a + b*ArcTa
nh[c*x])^(p + 1))/(b*c*d*(p + 1)*(d + e*x^2)), x] + (Dist[4/(b^2*(p + 1)*(p + 2)), Int[(x*(a + b*ArcTanh[c*x])
^(p + 2))/(d + e*x^2)^2, x], x] + Simp[((1 + c^2*x^2)*(a + b*ArcTanh[c*x])^(p + 2))/(b^2*e*(p + 1)*(p + 2)*(d
+ e*x^2)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1] && NeQ[p, -2]

Rule 6034

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[d^q/c^(
m + 1), Subst[Int[((a + b*x)^p*Sinh[x]^m)/Cosh[x]^(m + 2*(q + 1)), x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c,
d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q] || GtQ[d, 0])

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^8} \, dx &=-\frac{1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}+\frac{1}{7} (2 a) \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^7} \, dx\\ &=-\frac{1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}-\frac{x}{21 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac{1+a^2 x^2}{105 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}+\frac{1}{105} (4 a) \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^5} \, dx\\ &=-\frac{1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}-\frac{x}{21 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac{1+a^2 x^2}{105 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac{x}{105 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac{1+a^2 x^2}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}+\frac{1}{315} (4 a) \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx\\ &=-\frac{1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}-\frac{x}{21 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac{1+a^2 x^2}{105 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac{x}{105 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac{1+a^2 x^2}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{2 x}{315 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2 \left (1+a^2 x^2\right )}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{1}{315} (8 a) \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx\\ &=-\frac{1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}-\frac{x}{21 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac{1+a^2 x^2}{105 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac{x}{105 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac{1+a^2 x^2}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{2 x}{315 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2 \left (1+a^2 x^2\right )}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{8 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{315 a}\\ &=-\frac{1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}-\frac{x}{21 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac{1+a^2 x^2}{105 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac{x}{105 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac{1+a^2 x^2}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{2 x}{315 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2 \left (1+a^2 x^2\right )}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{8 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )}{315 a}\\ &=-\frac{1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}-\frac{x}{21 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac{1+a^2 x^2}{105 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac{x}{105 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac{1+a^2 x^2}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{2 x}{315 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2 \left (1+a^2 x^2\right )}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{4 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{315 a}\\ &=-\frac{1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}-\frac{x}{21 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac{1+a^2 x^2}{105 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac{x}{105 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac{1+a^2 x^2}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{2 x}{315 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2 \left (1+a^2 x^2\right )}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{4 \text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{315 a}\\ \end{align*}

Mathematica [A]  time = 0.125017, size = 128, normalized size = 0.61 \[ \frac{4 \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)^7 \text{Shi}\left (2 \tanh ^{-1}(a x)\right )+2 \left (a^2 x^2+1\right ) \tanh ^{-1}(a x)^6+\left (a^2 x^2+1\right ) \tanh ^{-1}(a x)^4+3 \left (a^2 x^2+1\right ) \tanh ^{-1}(a x)^2+2 a x \tanh ^{-1}(a x)^5+3 a x \tanh ^{-1}(a x)^3+15 a x \tanh ^{-1}(a x)+45}{315 a \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - a^2*x^2)^2*ArcTanh[a*x]^8),x]

[Out]

(45 + 15*a*x*ArcTanh[a*x] + 3*(1 + a^2*x^2)*ArcTanh[a*x]^2 + 3*a*x*ArcTanh[a*x]^3 + (1 + a^2*x^2)*ArcTanh[a*x]
^4 + 2*a*x*ArcTanh[a*x]^5 + 2*(1 + a^2*x^2)*ArcTanh[a*x]^6 + 4*(-1 + a^2*x^2)*ArcTanh[a*x]^7*SinhIntegral[2*Ar
cTanh[a*x]])/(315*a*(-1 + a^2*x^2)*ArcTanh[a*x]^7)

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Maple [A]  time = 0.066, size = 128, normalized size = 0.6 \begin{align*}{\frac{1}{a} \left ( -{\frac{1}{14\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{7}}}-{\frac{\cosh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{14\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{7}}}-{\frac{\sinh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{42\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{6}}}-{\frac{\cosh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{105\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{5}}}-{\frac{\sinh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{210\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{4}}}-{\frac{\cosh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{315\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}}-{\frac{\sinh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{315\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}-{\frac{2\,\cosh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{315\,{\it Artanh} \left ( ax \right ) }}+{\frac{4\,{\it Shi} \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{315}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-a^2*x^2+1)^2/arctanh(a*x)^8,x)

[Out]

1/a*(-1/14/arctanh(a*x)^7-1/14/arctanh(a*x)^7*cosh(2*arctanh(a*x))-1/42/arctanh(a*x)^6*sinh(2*arctanh(a*x))-1/
105/arctanh(a*x)^5*cosh(2*arctanh(a*x))-1/210/arctanh(a*x)^4*sinh(2*arctanh(a*x))-1/315/arctanh(a*x)^3*cosh(2*
arctanh(a*x))-1/315/arctanh(a*x)^2*sinh(2*arctanh(a*x))-2/315/arctanh(a*x)*cosh(2*arctanh(a*x))+4/315*Shi(2*ar
ctanh(a*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*x^2+1)^2/arctanh(a*x)^8,x, algorithm="maxima")

[Out]

-16*a*integrate(-1/315*x/((a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1) - (a^4*x^4 - 2*a^2*x^2 + 1)*log(-a*x + 1)), x
) + 4/315*(2*a*x*log(a*x + 1)^5 + (a^2*x^2 + 1)*log(a*x + 1)^6 + (a^2*x^2 + 1)*log(-a*x + 1)^6 - 2*(a*x + 3*(a
^2*x^2 + 1)*log(a*x + 1))*log(-a*x + 1)^5 + 12*a*x*log(a*x + 1)^3 + 2*(a^2*x^2 + 1)*log(a*x + 1)^4 + (2*a^2*x^
2 + 10*a*x*log(a*x + 1) + 15*(a^2*x^2 + 1)*log(a*x + 1)^2 + 2)*log(-a*x + 1)^4 - 4*(5*a*x*log(a*x + 1)^2 + 5*(
a^2*x^2 + 1)*log(a*x + 1)^3 + 3*a*x + 2*(a^2*x^2 + 1)*log(a*x + 1))*log(-a*x + 1)^3 + 240*a*x*log(a*x + 1) + 2
4*(a^2*x^2 + 1)*log(a*x + 1)^2 + (20*a*x*log(a*x + 1)^3 + 15*(a^2*x^2 + 1)*log(a*x + 1)^4 + 24*a^2*x^2 + 36*a*
x*log(a*x + 1) + 12*(a^2*x^2 + 1)*log(a*x + 1)^2 + 24)*log(-a*x + 1)^2 - 2*(5*a*x*log(a*x + 1)^4 + 3*(a^2*x^2
+ 1)*log(a*x + 1)^5 + 18*a*x*log(a*x + 1)^2 + 4*(a^2*x^2 + 1)*log(a*x + 1)^3 + 120*a*x + 24*(a^2*x^2 + 1)*log(
a*x + 1))*log(-a*x + 1) + 1440)/((a^3*x^2 - a)*log(a*x + 1)^7 - 7*(a^3*x^2 - a)*log(a*x + 1)^6*log(-a*x + 1) +
21*(a^3*x^2 - a)*log(a*x + 1)^5*log(-a*x + 1)^2 - 35*(a^3*x^2 - a)*log(a*x + 1)^4*log(-a*x + 1)^3 + 35*(a^3*x
^2 - a)*log(a*x + 1)^3*log(-a*x + 1)^4 - 21*(a^3*x^2 - a)*log(a*x + 1)^2*log(-a*x + 1)^5 + 7*(a^3*x^2 - a)*log
(a*x + 1)*log(-a*x + 1)^6 - (a^3*x^2 - a)*log(-a*x + 1)^7)

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Fricas [A]  time = 2.3043, size = 591, normalized size = 2.8 \begin{align*} \frac{2 \,{\left ({\left ({\left (a^{2} x^{2} - 1\right )} \logintegral \left (-\frac{a x + 1}{a x - 1}\right ) -{\left (a^{2} x^{2} - 1\right )} \logintegral \left (-\frac{a x - 1}{a x + 1}\right )\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{7} + 4 \, a x \log \left (-\frac{a x + 1}{a x - 1}\right )^{5} + 2 \,{\left (a^{2} x^{2} + 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{6} + 24 \, a x \log \left (-\frac{a x + 1}{a x - 1}\right )^{3} + 4 \,{\left (a^{2} x^{2} + 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{4} + 480 \, a x \log \left (-\frac{a x + 1}{a x - 1}\right ) + 48 \,{\left (a^{2} x^{2} + 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} + 2880\right )}}{315 \,{\left (a^{3} x^{2} - a\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*x^2+1)^2/arctanh(a*x)^8,x, algorithm="fricas")

[Out]

2/315*(((a^2*x^2 - 1)*log_integral(-(a*x + 1)/(a*x - 1)) - (a^2*x^2 - 1)*log_integral(-(a*x - 1)/(a*x + 1)))*l
og(-(a*x + 1)/(a*x - 1))^7 + 4*a*x*log(-(a*x + 1)/(a*x - 1))^5 + 2*(a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1))^6 +
24*a*x*log(-(a*x + 1)/(a*x - 1))^3 + 4*(a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1))^4 + 480*a*x*log(-(a*x + 1)/(a*
x - 1)) + 48*(a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1))^2 + 2880)/((a^3*x^2 - a)*log(-(a*x + 1)/(a*x - 1))^7)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname{atanh}^{8}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a**2*x**2+1)**2/atanh(a*x)**8,x)

[Out]

Integral(1/((a*x - 1)**2*(a*x + 1)**2*atanh(a*x)**8), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname{artanh}\left (a x\right )^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*x^2+1)^2/arctanh(a*x)^8,x, algorithm="giac")

[Out]

integrate(1/((a^2*x^2 - 1)^2*arctanh(a*x)^8), x)