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

Optimal. Leaf size=338 \[ -\frac {\left (1-a^2 x^2\right )^2}{140 a}-\frac {13 \left (1-a^2 x^2\right )}{210 a}-\frac {7 \log \left (1-a^2 x^2\right )}{15 a}+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}-\frac {13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac {24 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{35 a}-\frac {48 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^3+\frac {16 \tanh ^{-1}(a x)^3}{35 a}-\frac {14}{15} x \tanh ^{-1}(a x)-\frac {48 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{35 a} \]

[Out]

-13/210*(-a^2*x^2+1)/a-1/140*(-a^2*x^2+1)^2/a-14/15*x*arctanh(a*x)-13/105*x*(-a^2*x^2+1)*arctanh(a*x)-1/35*x*(
-a^2*x^2+1)^2*arctanh(a*x)+12/35*(-a^2*x^2+1)*arctanh(a*x)^2/a+9/70*(-a^2*x^2+1)^2*arctanh(a*x)^2/a+1/14*(-a^2
*x^2+1)^3*arctanh(a*x)^2/a+16/35*arctanh(a*x)^3/a+16/35*x*arctanh(a*x)^3+8/35*x*(-a^2*x^2+1)*arctanh(a*x)^3+6/
35*x*(-a^2*x^2+1)^2*arctanh(a*x)^3+1/7*x*(-a^2*x^2+1)^3*arctanh(a*x)^3-48/35*arctanh(a*x)^2*ln(2/(-a*x+1))/a-7
/15*ln(-a^2*x^2+1)/a-48/35*arctanh(a*x)*polylog(2,1-2/(-a*x+1))/a+24/35*polylog(3,1-2/(-a*x+1))/a

________________________________________________________________________________________

Rubi [A]  time = 0.33, antiderivative size = 338, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {5944, 5910, 5984, 5918, 5948, 6058, 6610, 260, 5942} \[ \frac {24 \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{35 a}-\frac {48 \tanh ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{35 a}-\frac {\left (1-a^2 x^2\right )^2}{140 a}-\frac {13 \left (1-a^2 x^2\right )}{210 a}-\frac {7 \log \left (1-a^2 x^2\right )}{15 a}+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}-\frac {13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac {16}{35} x \tanh ^{-1}(a x)^3+\frac {16 \tanh ^{-1}(a x)^3}{35 a}-\frac {14}{15} x \tanh ^{-1}(a x)-\frac {48 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{35 a} \]

Antiderivative was successfully verified.

[In]

Int[(1 - a^2*x^2)^3*ArcTanh[a*x]^3,x]

[Out]

(-13*(1 - a^2*x^2))/(210*a) - (1 - a^2*x^2)^2/(140*a) - (14*x*ArcTanh[a*x])/15 - (13*x*(1 - a^2*x^2)*ArcTanh[a
*x])/105 - (x*(1 - a^2*x^2)^2*ArcTanh[a*x])/35 + (12*(1 - a^2*x^2)*ArcTanh[a*x]^2)/(35*a) + (9*(1 - a^2*x^2)^2
*ArcTanh[a*x]^2)/(70*a) + ((1 - a^2*x^2)^3*ArcTanh[a*x]^2)/(14*a) + (16*ArcTanh[a*x]^3)/(35*a) + (16*x*ArcTanh
[a*x]^3)/35 + (8*x*(1 - a^2*x^2)*ArcTanh[a*x]^3)/35 + (6*x*(1 - a^2*x^2)^2*ArcTanh[a*x]^3)/35 + (x*(1 - a^2*x^
2)^3*ArcTanh[a*x]^3)/7 - (48*ArcTanh[a*x]^2*Log[2/(1 - a*x)])/(35*a) - (7*Log[1 - a^2*x^2])/(15*a) - (48*ArcTa
nh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/(35*a) + (24*PolyLog[3, 1 - 2/(1 - a*x)])/(35*a)

Rule 260

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

Rule 5910

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTanh[c*x])^p, x] - Dist[b*c*p, In
t[(x*(a + b*ArcTanh[c*x])^(p - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

Rule 5918

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

Rule 5942

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

Rule 5944

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

Rule 5948

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]

Rule 5984

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

Rule 6058

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

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin {align*} \int \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3 \, dx &=\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac {1}{7} \int \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x) \, dx+\frac {6}{7} \int \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3 \, dx\\ &=-\frac {\left (1-a^2 x^2\right )^2}{140 a}-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac {9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac {4}{35} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx-\frac {9}{35} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx+\frac {24}{35} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3 \, dx\\ &=-\frac {13 \left (1-a^2 x^2\right )}{210 a}-\frac {\left (1-a^2 x^2\right )^2}{140 a}-\frac {13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac {12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac {9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac {8}{105} \int \tanh ^{-1}(a x) \, dx-\frac {6}{35} \int \tanh ^{-1}(a x) \, dx+\frac {16}{35} \int \tanh ^{-1}(a x)^3 \, dx-\frac {24}{35} \int \tanh ^{-1}(a x) \, dx\\ &=-\frac {13 \left (1-a^2 x^2\right )}{210 a}-\frac {\left (1-a^2 x^2\right )^2}{140 a}-\frac {14}{15} x \tanh ^{-1}(a x)-\frac {13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac {12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac {9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {16}{35} x \tanh ^{-1}(a x)^3+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3+\frac {1}{105} (8 a) \int \frac {x}{1-a^2 x^2} \, dx+\frac {1}{35} (6 a) \int \frac {x}{1-a^2 x^2} \, dx+\frac {1}{35} (24 a) \int \frac {x}{1-a^2 x^2} \, dx-\frac {1}{35} (48 a) \int \frac {x \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=-\frac {13 \left (1-a^2 x^2\right )}{210 a}-\frac {\left (1-a^2 x^2\right )^2}{140 a}-\frac {14}{15} x \tanh ^{-1}(a x)-\frac {13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac {12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac {9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {16 \tanh ^{-1}(a x)^3}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^3+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac {7 \log \left (1-a^2 x^2\right )}{15 a}-\frac {48}{35} \int \frac {\tanh ^{-1}(a x)^2}{1-a x} \, dx\\ &=-\frac {13 \left (1-a^2 x^2\right )}{210 a}-\frac {\left (1-a^2 x^2\right )^2}{140 a}-\frac {14}{15} x \tanh ^{-1}(a x)-\frac {13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac {12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac {9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {16 \tanh ^{-1}(a x)^3}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^3+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac {48 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{35 a}-\frac {7 \log \left (1-a^2 x^2\right )}{15 a}+\frac {96}{35} \int \frac {\tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {13 \left (1-a^2 x^2\right )}{210 a}-\frac {\left (1-a^2 x^2\right )^2}{140 a}-\frac {14}{15} x \tanh ^{-1}(a x)-\frac {13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac {12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac {9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {16 \tanh ^{-1}(a x)^3}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^3+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac {48 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{35 a}-\frac {7 \log \left (1-a^2 x^2\right )}{15 a}-\frac {48 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{35 a}+\frac {48}{35} \int \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {13 \left (1-a^2 x^2\right )}{210 a}-\frac {\left (1-a^2 x^2\right )^2}{140 a}-\frac {14}{15} x \tanh ^{-1}(a x)-\frac {13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac {12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac {9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {16 \tanh ^{-1}(a x)^3}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^3+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac {48 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{35 a}-\frac {7 \log \left (1-a^2 x^2\right )}{15 a}-\frac {48 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{35 a}+\frac {24 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{35 a}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.10, size = 231, normalized size = 0.68 \[ -\frac {60 a^7 x^7 \tanh ^{-1}(a x)^3+30 a^6 x^6 \tanh ^{-1}(a x)^2-252 a^5 x^5 \tanh ^{-1}(a x)^3+12 a^5 x^5 \tanh ^{-1}(a x)+3 a^4 x^4-144 a^4 x^4 \tanh ^{-1}(a x)^2+420 a^3 x^3 \tanh ^{-1}(a x)^3-76 a^3 x^3 \tanh ^{-1}(a x)-32 a^2 x^2+196 \log \left (1-a^2 x^2\right )+342 a^2 x^2 \tanh ^{-1}(a x)^2-576 \tanh ^{-1}(a x) \text {Li}_2\left (-e^{-2 \tanh ^{-1}(a x)}\right )-288 \text {Li}_3\left (-e^{-2 \tanh ^{-1}(a x)}\right )-420 a x \tanh ^{-1}(a x)^3+456 a x \tanh ^{-1}(a x)+192 \tanh ^{-1}(a x)^3-228 \tanh ^{-1}(a x)^2+576 \tanh ^{-1}(a x)^2 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )+29}{420 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/420*(29 - 32*a^2*x^2 + 3*a^4*x^4 + 456*a*x*ArcTanh[a*x] - 76*a^3*x^3*ArcTanh[a*x] + 12*a^5*x^5*ArcTanh[a*x]
- 228*ArcTanh[a*x]^2 + 342*a^2*x^2*ArcTanh[a*x]^2 - 144*a^4*x^4*ArcTanh[a*x]^2 + 30*a^6*x^6*ArcTanh[a*x]^2 +
192*ArcTanh[a*x]^3 - 420*a*x*ArcTanh[a*x]^3 + 420*a^3*x^3*ArcTanh[a*x]^3 - 252*a^5*x^5*ArcTanh[a*x]^3 + 60*a^7
*x^7*ArcTanh[a*x]^3 + 576*ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] + 196*Log[1 - a^2*x^2] - 576*ArcTanh[a*x
]*PolyLog[2, -E^(-2*ArcTanh[a*x])] - 288*PolyLog[3, -E^(-2*ArcTanh[a*x])])/a

________________________________________________________________________________________

fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*arctanh(a*x)^3, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 5.40, size = 932, normalized size = 2.76 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2*arctanh(a*x)^3*x^3-57/70*a*arctanh(a*x)^2*x^2-1/35*a^4*arctanh(a*x)*x^5-29/420/a+14/15/a*ln(1+(a*x+1)^2/(
-a^2*x^2+1))+24/35/a*polylog(3,-(a*x+1)^2/(-a^2*x^2+1))+8/105*a*x^2-14/15*arctanh(a*x)/a+24/35/a*arctanh(a*x)^
2*ln(a*x-1)+24/35/a*arctanh(a*x)^2*ln(a*x+1)-48/35/a*arctanh(a*x)^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-1/140*x^4*a
^3+19/35*arctanh(a*x)^2/a+16/35*arctanh(a*x)^3/a+x*arctanh(a*x)^3-38/35*x*arctanh(a*x)-48/35/a*arctanh(a*x)^2*
ln(2)-48/35/a*arctanh(a*x)*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))-12/35*I/a*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^
2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*arctanh(a*x)^2+12/35*I/a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^
2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*arctanh(a*x)^2-12/35*I/a*arctanh(a*x)^2*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^
2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2-24/35*I/a*arctanh(a*x)^2*Pi*csgn(I*(a*x+1)/(
-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2-24/35*I/a*Pi*arctanh(a*x)^2+12/35*I/a*Pi*csgn(I/(1+(a*x+1)^
2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*arctan
h(a*x)^2+3/5*a^4*arctanh(a*x)^3*x^5+19/105*a^2*arctanh(a*x)*x^3+12/35*a^3*arctanh(a*x)^2*x^4+24/35*I/a*Pi*csgn
(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*arctanh(a*x)^2-24/35*I/a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*arctanh(a*x)
^2-12/35*I/a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*arctanh(a*x)^2-12/35*I/a*Pi*csgn(I*
(a*x+1)^2/(a^2*x^2-1))^3*arctanh(a*x)^2-1/7*a^6*arctanh(a*x)^3*x^7-1/14*a^5*arctanh(a*x)^2*x^6

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19600*(150*a^7*x^7 - 175*a^6*x^6 - 672*a^5*x^5 + 840*a^4*x^4 + 1330*a^3*x^3 - 1995*a^2*x^2 - 3360*a*x - 210*
(5*a^7*x^7 - 21*a^5*x^5 + 35*a^3*x^3 - 35*a*x - 16)*log(a*x + 1))*log(-a*x + 1)^2/a - 1/8*(log(-a*x + 1)^3 - 3
*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 6)*(a*x - 1)/a + 1/691488000*(36000*(343*log(-a*x + 1)^3 - 147*log(-a*x +
1)^2 + 42*log(-a*x + 1) - 6)*(a*x - 1)^7 + 2401000*(36*log(-a*x + 1)^3 - 18*log(-a*x + 1)^2 + 6*log(-a*x + 1)
- 1)*(a*x - 1)^6 + 2074464*(125*log(-a*x + 1)^3 - 75*log(-a*x + 1)^2 + 30*log(-a*x + 1) - 6)*(a*x - 1)^5 + 13
505625*(32*log(-a*x + 1)^3 - 24*log(-a*x + 1)^2 + 12*log(-a*x + 1) - 3)*(a*x - 1)^4 + 48020000*(9*log(-a*x + 1
)^3 - 9*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 2)*(a*x - 1)^3 + 64827000*(4*log(-a*x + 1)^3 - 6*log(-a*x + 1)^2 +
6*log(-a*x + 1) - 3)*(a*x - 1)^2 + 86436000*(log(-a*x + 1)^3 - 3*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 6)*(a*x
- 1))/a - 1/480000*(288*(125*log(-a*x + 1)^3 - 75*log(-a*x + 1)^2 + 30*log(-a*x + 1) - 6)*(a*x - 1)^5 + 5625*(
32*log(-a*x + 1)^3 - 24*log(-a*x + 1)^2 + 12*log(-a*x + 1) - 3)*(a*x - 1)^4 + 40000*(9*log(-a*x + 1)^3 - 9*log
(-a*x + 1)^2 + 6*log(-a*x + 1) - 2)*(a*x - 1)^3 + 90000*(4*log(-a*x + 1)^3 - 6*log(-a*x + 1)^2 + 6*log(-a*x +
1) - 3)*(a*x - 1)^2 + 180000*(log(-a*x + 1)^3 - 3*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 6)*(a*x - 1))/a + 1/288*
(4*(9*log(-a*x + 1)^3 - 9*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 2)*(a*x - 1)^3 + 27*(4*log(-a*x + 1)^3 - 6*log(-
a*x + 1)^2 + 6*log(-a*x + 1) - 3)*(a*x - 1)^2 + 108*(log(-a*x + 1)^3 - 3*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 6
)*(a*x - 1))/a + 1/8*integrate(-1/1225*(1225*(a^7*x^7 - a^6*x^6 - 3*a^5*x^5 + 3*a^4*x^4 + 3*a^3*x^3 - 3*a^2*x^
2 - a*x + 1)*log(a*x + 1)^3 + (150*a^7*x^7 - 175*a^6*x^6 - 672*a^5*x^5 + 840*a^4*x^4 + 1330*a^3*x^3 - 1995*a^2
*x^2 - 3675*(a^7*x^7 - a^6*x^6 - 3*a^5*x^5 + 3*a^4*x^4 + 3*a^3*x^3 - 3*a^2*x^2 - a*x + 1)*log(a*x + 1)^2 - 336
0*a*x - 210*(5*a^7*x^7 - 21*a^5*x^5 + 35*a^3*x^3 - 35*a*x - 16)*log(a*x + 1))*log(-a*x + 1))/(a*x - 1), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-atanh(a*x)^3*(a^2*x^2 - 1)^3,x)

[Out]

-int(atanh(a*x)^3*(a^2*x^2 - 1)^3, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int 3 a^{2} x^{2} \operatorname {atanh}^{3}{\left (a x \right )}\, dx - \int \left (- 3 a^{4} x^{4} \operatorname {atanh}^{3}{\left (a x \right )}\right )\, dx - \int a^{6} x^{6} \operatorname {atanh}^{3}{\left (a x \right )}\, dx - \int \left (- \operatorname {atanh}^{3}{\left (a x \right )}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(3*a**2*x**2*atanh(a*x)**3, x) - Integral(-3*a**4*x**4*atanh(a*x)**3, x) - Integral(a**6*x**6*atanh(a
*x)**3, x) - Integral(-atanh(a*x)**3, x)

________________________________________________________________________________________