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

Optimal. Leaf size=156 \[ \frac {1}{8} a^4 x^8 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{24 a^4}+\frac {1}{28} a^3 x^7 \tanh ^{-1}(a x)+\frac {x \tanh ^{-1}(a x)}{12 a^3}+\frac {a^2 x^6}{168}-\frac {1}{3} a^2 x^6 \tanh ^{-1}(a x)^2-\frac {5 x^2}{504 a^2}+\frac {2 \log \left (1-a^2 x^2\right )}{63 a^4}-\frac {1}{12} a x^5 \tanh ^{-1}(a x)+\frac {1}{4} x^4 \tanh ^{-1}(a x)^2+\frac {x^3 \tanh ^{-1}(a x)}{36 a}-\frac {x^4}{84} \]

[Out]

-5/504*x^2/a^2-1/84*x^4+1/168*a^2*x^6+1/12*x*arctanh(a*x)/a^3+1/36*x^3*arctanh(a*x)/a-1/12*a*x^5*arctanh(a*x)+
1/28*a^3*x^7*arctanh(a*x)-1/24*arctanh(a*x)^2/a^4+1/4*x^4*arctanh(a*x)^2-1/3*a^2*x^6*arctanh(a*x)^2+1/8*a^4*x^
8*arctanh(a*x)^2+2/63*ln(-a^2*x^2+1)/a^4

________________________________________________________________________________________

Rubi [A]  time = 0.82, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 47, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6012, 5916, 5980, 266, 43, 5910, 260, 5948} \[ \frac {a^2 x^6}{168}-\frac {5 x^2}{504 a^2}+\frac {2 \log \left (1-a^2 x^2\right )}{63 a^4}+\frac {1}{8} a^4 x^8 \tanh ^{-1}(a x)^2+\frac {1}{28} a^3 x^7 \tanh ^{-1}(a x)-\frac {1}{3} a^2 x^6 \tanh ^{-1}(a x)^2+\frac {x \tanh ^{-1}(a x)}{12 a^3}-\frac {\tanh ^{-1}(a x)^2}{24 a^4}-\frac {1}{12} a x^5 \tanh ^{-1}(a x)+\frac {1}{4} x^4 \tanh ^{-1}(a x)^2+\frac {x^3 \tanh ^{-1}(a x)}{36 a}-\frac {x^4}{84} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*x^2)/(504*a^2) - x^4/84 + (a^2*x^6)/168 + (x*ArcTanh[a*x])/(12*a^3) + (x^3*ArcTanh[a*x])/(36*a) - (a*x^5*A
rcTanh[a*x])/12 + (a^3*x^7*ArcTanh[a*x])/28 - ArcTanh[a*x]^2/(24*a^4) + (x^4*ArcTanh[a*x]^2)/4 - (a^2*x^6*ArcT
anh[a*x]^2)/3 + (a^4*x^8*ArcTanh[a*x]^2)/8 + (2*Log[1 - a^2*x^2])/(63*a^4)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 266

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

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 5916

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcT
anh[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(1 -
c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -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 5980

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

Rule 6012

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Int[E
xpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[
c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[q, 1]

Rubi steps

\begin {align*} \int x^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2 \, dx &=\int \left (x^3 \tanh ^{-1}(a x)^2-2 a^2 x^5 \tanh ^{-1}(a x)^2+a^4 x^7 \tanh ^{-1}(a x)^2\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int x^5 \tanh ^{-1}(a x)^2 \, dx\right )+a^4 \int x^7 \tanh ^{-1}(a x)^2 \, dx+\int x^3 \tanh ^{-1}(a x)^2 \, dx\\ &=\frac {1}{4} x^4 \tanh ^{-1}(a x)^2-\frac {1}{3} a^2 x^6 \tanh ^{-1}(a x)^2+\frac {1}{8} a^4 x^8 \tanh ^{-1}(a x)^2-\frac {1}{2} a \int \frac {x^4 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {x^6 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx-\frac {1}{4} a^5 \int \frac {x^8 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac {1}{4} x^4 \tanh ^{-1}(a x)^2-\frac {1}{3} a^2 x^6 \tanh ^{-1}(a x)^2+\frac {1}{8} a^4 x^8 \tanh ^{-1}(a x)^2+\frac {\int x^2 \tanh ^{-1}(a x) \, dx}{2 a}-\frac {\int \frac {x^2 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{2 a}-\frac {1}{3} (2 a) \int x^4 \tanh ^{-1}(a x) \, dx+\frac {1}{3} (2 a) \int \frac {x^4 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac {1}{4} a^3 \int x^6 \tanh ^{-1}(a x) \, dx-\frac {1}{4} a^3 \int \frac {x^6 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac {x^3 \tanh ^{-1}(a x)}{6 a}-\frac {2}{15} a x^5 \tanh ^{-1}(a x)+\frac {1}{28} a^3 x^7 \tanh ^{-1}(a x)+\frac {1}{4} x^4 \tanh ^{-1}(a x)^2-\frac {1}{3} a^2 x^6 \tanh ^{-1}(a x)^2+\frac {1}{8} a^4 x^8 \tanh ^{-1}(a x)^2-\frac {1}{6} \int \frac {x^3}{1-a^2 x^2} \, dx+\frac {\int \tanh ^{-1}(a x) \, dx}{2 a^3}-\frac {\int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{2 a^3}-\frac {2 \int x^2 \tanh ^{-1}(a x) \, dx}{3 a}+\frac {2 \int \frac {x^2 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a}+\frac {1}{4} a \int x^4 \tanh ^{-1}(a x) \, dx-\frac {1}{4} a \int \frac {x^4 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac {1}{15} \left (2 a^2\right ) \int \frac {x^5}{1-a^2 x^2} \, dx-\frac {1}{28} a^4 \int \frac {x^7}{1-a^2 x^2} \, dx\\ &=\frac {x \tanh ^{-1}(a x)}{2 a^3}-\frac {x^3 \tanh ^{-1}(a x)}{18 a}-\frac {1}{12} a x^5 \tanh ^{-1}(a x)+\frac {1}{28} a^3 x^7 \tanh ^{-1}(a x)-\frac {\tanh ^{-1}(a x)^2}{4 a^4}+\frac {1}{4} x^4 \tanh ^{-1}(a x)^2-\frac {1}{3} a^2 x^6 \tanh ^{-1}(a x)^2+\frac {1}{8} a^4 x^8 \tanh ^{-1}(a x)^2-\frac {1}{12} \operatorname {Subst}\left (\int \frac {x}{1-a^2 x} \, dx,x,x^2\right )+\frac {2}{9} \int \frac {x^3}{1-a^2 x^2} \, dx-\frac {2 \int \tanh ^{-1}(a x) \, dx}{3 a^3}+\frac {2 \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a^3}-\frac {\int \frac {x}{1-a^2 x^2} \, dx}{2 a^2}+\frac {\int x^2 \tanh ^{-1}(a x) \, dx}{4 a}-\frac {\int \frac {x^2 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{4 a}-\frac {1}{20} a^2 \int \frac {x^5}{1-a^2 x^2} \, dx+\frac {1}{15} a^2 \operatorname {Subst}\left (\int \frac {x^2}{1-a^2 x} \, dx,x,x^2\right )-\frac {1}{56} a^4 \operatorname {Subst}\left (\int \frac {x^3}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {x \tanh ^{-1}(a x)}{6 a^3}+\frac {x^3 \tanh ^{-1}(a x)}{36 a}-\frac {1}{12} a x^5 \tanh ^{-1}(a x)+\frac {1}{28} a^3 x^7 \tanh ^{-1}(a x)+\frac {\tanh ^{-1}(a x)^2}{12 a^4}+\frac {1}{4} x^4 \tanh ^{-1}(a x)^2-\frac {1}{3} a^2 x^6 \tanh ^{-1}(a x)^2+\frac {1}{8} a^4 x^8 \tanh ^{-1}(a x)^2+\frac {\log \left (1-a^2 x^2\right )}{4 a^4}-\frac {1}{12} \int \frac {x^3}{1-a^2 x^2} \, dx-\frac {1}{12} \operatorname {Subst}\left (\int \left (-\frac {1}{a^2}-\frac {1}{a^2 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac {1}{9} \operatorname {Subst}\left (\int \frac {x}{1-a^2 x} \, dx,x,x^2\right )+\frac {\int \tanh ^{-1}(a x) \, dx}{4 a^3}-\frac {\int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{4 a^3}+\frac {2 \int \frac {x}{1-a^2 x^2} \, dx}{3 a^2}-\frac {1}{40} a^2 \operatorname {Subst}\left (\int \frac {x^2}{1-a^2 x} \, dx,x,x^2\right )+\frac {1}{15} a^2 \operatorname {Subst}\left (\int \left (-\frac {1}{a^4}-\frac {x}{a^2}-\frac {1}{a^4 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {1}{56} a^4 \operatorname {Subst}\left (\int \left (-\frac {1}{a^6}-\frac {x}{a^4}-\frac {x^2}{a^2}-\frac {1}{a^6 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {29 x^2}{840 a^2}-\frac {41 x^4}{1680}+\frac {a^2 x^6}{168}+\frac {x \tanh ^{-1}(a x)}{12 a^3}+\frac {x^3 \tanh ^{-1}(a x)}{36 a}-\frac {1}{12} a x^5 \tanh ^{-1}(a x)+\frac {1}{28} a^3 x^7 \tanh ^{-1}(a x)-\frac {\tanh ^{-1}(a x)^2}{24 a^4}+\frac {1}{4} x^4 \tanh ^{-1}(a x)^2-\frac {1}{3} a^2 x^6 \tanh ^{-1}(a x)^2+\frac {1}{8} a^4 x^8 \tanh ^{-1}(a x)^2-\frac {41 \log \left (1-a^2 x^2\right )}{840 a^4}-\frac {1}{24} \operatorname {Subst}\left (\int \frac {x}{1-a^2 x} \, dx,x,x^2\right )+\frac {1}{9} \operatorname {Subst}\left (\int \left (-\frac {1}{a^2}-\frac {1}{a^2 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {\int \frac {x}{1-a^2 x^2} \, dx}{4 a^2}-\frac {1}{40} a^2 \operatorname {Subst}\left (\int \left (-\frac {1}{a^4}-\frac {x}{a^2}-\frac {1}{a^4 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {13 x^2}{252 a^2}-\frac {x^4}{84}+\frac {a^2 x^6}{168}+\frac {x \tanh ^{-1}(a x)}{12 a^3}+\frac {x^3 \tanh ^{-1}(a x)}{36 a}-\frac {1}{12} a x^5 \tanh ^{-1}(a x)+\frac {1}{28} a^3 x^7 \tanh ^{-1}(a x)-\frac {\tanh ^{-1}(a x)^2}{24 a^4}+\frac {1}{4} x^4 \tanh ^{-1}(a x)^2-\frac {1}{3} a^2 x^6 \tanh ^{-1}(a x)^2+\frac {1}{8} a^4 x^8 \tanh ^{-1}(a x)^2-\frac {5 \log \left (1-a^2 x^2\right )}{504 a^4}-\frac {1}{24} \operatorname {Subst}\left (\int \left (-\frac {1}{a^2}-\frac {1}{a^2 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {5 x^2}{504 a^2}-\frac {x^4}{84}+\frac {a^2 x^6}{168}+\frac {x \tanh ^{-1}(a x)}{12 a^3}+\frac {x^3 \tanh ^{-1}(a x)}{36 a}-\frac {1}{12} a x^5 \tanh ^{-1}(a x)+\frac {1}{28} a^3 x^7 \tanh ^{-1}(a x)-\frac {\tanh ^{-1}(a x)^2}{24 a^4}+\frac {1}{4} x^4 \tanh ^{-1}(a x)^2-\frac {1}{3} a^2 x^6 \tanh ^{-1}(a x)^2+\frac {1}{8} a^4 x^8 \tanh ^{-1}(a x)^2+\frac {2 \log \left (1-a^2 x^2\right )}{63 a^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 108, normalized size = 0.69 \[ \frac {3 a^6 x^6-6 a^4 x^4-5 a^2 x^2+16 \log \left (1-a^2 x^2\right )+21 \left (a^2 x^2-1\right )^3 \left (3 a^2 x^2+1\right ) \tanh ^{-1}(a x)^2+2 a x \left (9 a^6 x^6-21 a^4 x^4+7 a^2 x^2+21\right ) \tanh ^{-1}(a x)}{504 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*a^2*x^2 - 6*a^4*x^4 + 3*a^6*x^6 + 2*a*x*(21 + 7*a^2*x^2 - 21*a^4*x^4 + 9*a^6*x^6)*ArcTanh[a*x] + 21*(-1 +
a^2*x^2)^3*(1 + 3*a^2*x^2)*ArcTanh[a*x]^2 + 16*Log[1 - a^2*x^2])/(504*a^4)

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 133, normalized size = 0.85 \[ \frac {12 \, a^{6} x^{6} - 24 \, a^{4} x^{4} - 20 \, a^{2} x^{2} + 21 \, {\left (3 \, a^{8} x^{8} - 8 \, a^{6} x^{6} + 6 \, a^{4} x^{4} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, {\left (9 \, a^{7} x^{7} - 21 \, a^{5} x^{5} + 7 \, a^{3} x^{3} + 21 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + 64 \, \log \left (a^{2} x^{2} - 1\right )}{2016 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2016*(12*a^6*x^6 - 24*a^4*x^4 - 20*a^2*x^2 + 21*(3*a^8*x^8 - 8*a^6*x^6 + 6*a^4*x^4 - 1)*log(-(a*x + 1)/(a*x
- 1))^2 + 4*(9*a^7*x^7 - 21*a^5*x^5 + 7*a^3*x^3 + 21*a*x)*log(-(a*x + 1)/(a*x - 1)) + 64*log(a^2*x^2 - 1))/a^4

________________________________________________________________________________________

giac [B]  time = 0.23, size = 683, normalized size = 4.38 \[ \frac {2}{63} \, {\left (\frac {84 \, {\left (\frac {{\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} + \frac {{\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {{\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}}{\frac {{\left (a x + 1\right )}^{8} a^{5}}{{\left (a x - 1\right )}^{8}} - \frac {8 \, {\left (a x + 1\right )}^{7} a^{5}}{{\left (a x - 1\right )}^{7}} + \frac {28 \, {\left (a x + 1\right )}^{6} a^{5}}{{\left (a x - 1\right )}^{6}} - \frac {56 \, {\left (a x + 1\right )}^{5} a^{5}}{{\left (a x - 1\right )}^{5}} + \frac {70 \, {\left (a x + 1\right )}^{4} a^{5}}{{\left (a x - 1\right )}^{4}} - \frac {56 \, {\left (a x + 1\right )}^{3} a^{5}}{{\left (a x - 1\right )}^{3}} + \frac {28 \, {\left (a x + 1\right )}^{2} a^{5}}{{\left (a x - 1\right )}^{2}} - \frac {8 \, {\left (a x + 1\right )} a^{5}}{a x - 1} + a^{5}} + \frac {2 \, {\left (\frac {28 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} - \frac {7 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {21 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - \frac {7 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{\frac {{\left (a x + 1\right )}^{7} a^{5}}{{\left (a x - 1\right )}^{7}} - \frac {7 \, {\left (a x + 1\right )}^{6} a^{5}}{{\left (a x - 1\right )}^{6}} + \frac {21 \, {\left (a x + 1\right )}^{5} a^{5}}{{\left (a x - 1\right )}^{5}} - \frac {35 \, {\left (a x + 1\right )}^{4} a^{5}}{{\left (a x - 1\right )}^{4}} + \frac {35 \, {\left (a x + 1\right )}^{3} a^{5}}{{\left (a x - 1\right )}^{3}} - \frac {21 \, {\left (a x + 1\right )}^{2} a^{5}}{{\left (a x - 1\right )}^{2}} + \frac {7 \, {\left (a x + 1\right )} a^{5}}{a x - 1} - a^{5}} - \frac {\frac {2 \, {\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} - \frac {11 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {6 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} - \frac {11 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {2 \, {\left (a x + 1\right )}}{a x - 1}}{\frac {{\left (a x + 1\right )}^{6} a^{5}}{{\left (a x - 1\right )}^{6}} - \frac {6 \, {\left (a x + 1\right )}^{5} a^{5}}{{\left (a x - 1\right )}^{5}} + \frac {15 \, {\left (a x + 1\right )}^{4} a^{5}}{{\left (a x - 1\right )}^{4}} - \frac {20 \, {\left (a x + 1\right )}^{3} a^{5}}{{\left (a x - 1\right )}^{3}} + \frac {15 \, {\left (a x + 1\right )}^{2} a^{5}}{{\left (a x - 1\right )}^{2}} - \frac {6 \, {\left (a x + 1\right )} a^{5}}{a x - 1} + a^{5}} - \frac {2 \, \log \left (-\frac {a x + 1}{a x - 1} + 1\right )}{a^{5}} + \frac {2 \, \log \left (-\frac {a x + 1}{a x - 1}\right )}{a^{5}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/63*(84*((a*x + 1)^5/(a*x - 1)^5 + (a*x + 1)^4/(a*x - 1)^4 + (a*x + 1)^3/(a*x - 1)^3)*log(-(a*x + 1)/(a*x - 1
))^2/((a*x + 1)^8*a^5/(a*x - 1)^8 - 8*(a*x + 1)^7*a^5/(a*x - 1)^7 + 28*(a*x + 1)^6*a^5/(a*x - 1)^6 - 56*(a*x +
1)^5*a^5/(a*x - 1)^5 + 70*(a*x + 1)^4*a^5/(a*x - 1)^4 - 56*(a*x + 1)^3*a^5/(a*x - 1)^3 + 28*(a*x + 1)^2*a^5/(
a*x - 1)^2 - 8*(a*x + 1)*a^5/(a*x - 1) + a^5) + 2*(28*(a*x + 1)^4/(a*x - 1)^4 - 7*(a*x + 1)^3/(a*x - 1)^3 + 21
*(a*x + 1)^2/(a*x - 1)^2 - 7*(a*x + 1)/(a*x - 1) + 1)*log(-(a*x + 1)/(a*x - 1))/((a*x + 1)^7*a^5/(a*x - 1)^7 -
7*(a*x + 1)^6*a^5/(a*x - 1)^6 + 21*(a*x + 1)^5*a^5/(a*x - 1)^5 - 35*(a*x + 1)^4*a^5/(a*x - 1)^4 + 35*(a*x + 1
)^3*a^5/(a*x - 1)^3 - 21*(a*x + 1)^2*a^5/(a*x - 1)^2 + 7*(a*x + 1)*a^5/(a*x - 1) - a^5) - (2*(a*x + 1)^5/(a*x
- 1)^5 - 11*(a*x + 1)^4/(a*x - 1)^4 + 6*(a*x + 1)^3/(a*x - 1)^3 - 11*(a*x + 1)^2/(a*x - 1)^2 + 2*(a*x + 1)/(a*
x - 1))/((a*x + 1)^6*a^5/(a*x - 1)^6 - 6*(a*x + 1)^5*a^5/(a*x - 1)^5 + 15*(a*x + 1)^4*a^5/(a*x - 1)^4 - 20*(a*
x + 1)^3*a^5/(a*x - 1)^3 + 15*(a*x + 1)^2*a^5/(a*x - 1)^2 - 6*(a*x + 1)*a^5/(a*x - 1) + a^5) - 2*log(-(a*x + 1
)/(a*x - 1) + 1)/a^5 + 2*log(-(a*x + 1)/(a*x - 1))/a^5)*a

________________________________________________________________________________________

maple [A]  time = 0.06, size = 239, normalized size = 1.53 \[ \frac {a^{4} x^{8} \arctanh \left (a x \right )^{2}}{8}-\frac {a^{2} x^{6} \arctanh \left (a x \right )^{2}}{3}+\frac {x^{4} \arctanh \left (a x \right )^{2}}{4}+\frac {a^{3} x^{7} \arctanh \left (a x \right )}{28}-\frac {a \,x^{5} \arctanh \left (a x \right )}{12}+\frac {x^{3} \arctanh \left (a x \right )}{36 a}+\frac {x \arctanh \left (a x \right )}{12 a^{3}}+\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{24 a^{4}}-\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{24 a^{4}}+\frac {\ln \left (a x -1\right )^{2}}{96 a^{4}}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{48 a^{4}}+\frac {\ln \left (a x +1\right )^{2}}{96 a^{4}}+\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{48 a^{4}}-\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{48 a^{4}}+\frac {a^{2} x^{6}}{168}-\frac {x^{4}}{84}-\frac {5 x^{2}}{504 a^{2}}+\frac {2 \ln \left (a x -1\right )}{63 a^{4}}+\frac {2 \ln \left (a x +1\right )}{63 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a^4*x^8*arctanh(a*x)^2-1/3*a^2*x^6*arctanh(a*x)^2+1/4*x^4*arctanh(a*x)^2+1/28*a^3*x^7*arctanh(a*x)-1/12*a*
x^5*arctanh(a*x)+1/36*x^3*arctanh(a*x)/a+1/12*x*arctanh(a*x)/a^3+1/24/a^4*arctanh(a*x)*ln(a*x-1)-1/24/a^4*arct
anh(a*x)*ln(a*x+1)+1/96/a^4*ln(a*x-1)^2-1/48/a^4*ln(a*x-1)*ln(1/2+1/2*a*x)+1/96/a^4*ln(a*x+1)^2+1/48/a^4*ln(-1
/2*a*x+1/2)*ln(1/2+1/2*a*x)-1/48/a^4*ln(-1/2*a*x+1/2)*ln(a*x+1)+1/168*a^2*x^6-1/84*x^4-5/504*x^2/a^2+2/63/a^4*
ln(a*x-1)+2/63/a^4*ln(a*x+1)

________________________________________________________________________________________

maxima [A]  time = 0.32, size = 170, normalized size = 1.09 \[ \frac {1}{504} \, a {\left (\frac {2 \, {\left (9 \, a^{6} x^{7} - 21 \, a^{4} x^{5} + 7 \, a^{2} x^{3} + 21 \, x\right )}}{a^{4}} - \frac {21 \, \log \left (a x + 1\right )}{a^{5}} + \frac {21 \, \log \left (a x - 1\right )}{a^{5}}\right )} \operatorname {artanh}\left (a x\right ) + \frac {1}{24} \, {\left (3 \, a^{4} x^{8} - 8 \, a^{2} x^{6} + 6 \, x^{4}\right )} \operatorname {artanh}\left (a x\right )^{2} + \frac {12 \, a^{6} x^{6} - 24 \, a^{4} x^{4} - 20 \, a^{2} x^{2} - 2 \, {\left (21 \, \log \left (a x - 1\right ) - 32\right )} \log \left (a x + 1\right ) + 21 \, \log \left (a x + 1\right )^{2} + 21 \, \log \left (a x - 1\right )^{2} + 64 \, \log \left (a x - 1\right )}{2016 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/504*a*(2*(9*a^6*x^7 - 21*a^4*x^5 + 7*a^2*x^3 + 21*x)/a^4 - 21*log(a*x + 1)/a^5 + 21*log(a*x - 1)/a^5)*arctan
h(a*x) + 1/24*(3*a^4*x^8 - 8*a^2*x^6 + 6*x^4)*arctanh(a*x)^2 + 1/2016*(12*a^6*x^6 - 24*a^4*x^4 - 20*a^2*x^2 -
2*(21*log(a*x - 1) - 32)*log(a*x + 1) + 21*log(a*x + 1)^2 + 21*log(a*x - 1)^2 + 64*log(a*x - 1))/a^4

________________________________________________________________________________________

mupad [B]  time = 1.22, size = 221, normalized size = 1.42 \[ \frac {2\,\ln \left (a^2\,x^2-1\right )}{63\,a^4}-{\ln \left (1-a\,x\right )}^2\,\left (\frac {1}{96\,a^4}-\frac {x^4}{16}+\frac {a^2\,x^6}{12}-\frac {a^4\,x^8}{32}\right )-\frac {x^4}{84}-{\ln \left (a\,x+1\right )}^2\,\left (\frac {1}{96\,a^4}-\frac {x^4}{16}+\frac {a^2\,x^6}{12}-\frac {a^4\,x^8}{32}\right )-\ln \left (1-a\,x\right )\,\left (\frac {x}{24\,a^3}-\ln \left (a\,x+1\right )\,\left (\frac {1}{48\,a^4}-\frac {x^4}{8}+\frac {a^2\,x^6}{6}-\frac {a^4\,x^8}{16}\right )-\frac {a\,x^5}{24}+\frac {x^3}{72\,a}+\frac {a^3\,x^7}{56}\right )-\frac {5\,x^2}{504\,a^2}+\frac {a^2\,x^6}{168}+a\,\ln \left (a\,x+1\right )\,\left (\frac {x}{24\,a^4}-\frac {x^5}{24}+\frac {x^3}{72\,a^2}+\frac {a^2\,x^7}{56}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*log(a^2*x^2 - 1))/(63*a^4) - log(1 - a*x)^2*(1/(96*a^4) - x^4/16 + (a^2*x^6)/12 - (a^4*x^8)/32) - x^4/84 -
log(a*x + 1)^2*(1/(96*a^4) - x^4/16 + (a^2*x^6)/12 - (a^4*x^8)/32) - log(1 - a*x)*(x/(24*a^3) - log(a*x + 1)*(
1/(48*a^4) - x^4/8 + (a^2*x^6)/6 - (a^4*x^8)/16) - (a*x^5)/24 + x^3/(72*a) + (a^3*x^7)/56) - (5*x^2)/(504*a^2)
+ (a^2*x^6)/168 + a*log(a*x + 1)*(x/(24*a^4) - x^5/24 + x^3/(72*a^2) + (a^2*x^7)/56)

________________________________________________________________________________________

sympy [A]  time = 3.42, size = 153, normalized size = 0.98 \[ \begin {cases} \frac {a^{4} x^{8} \operatorname {atanh}^{2}{\left (a x \right )}}{8} + \frac {a^{3} x^{7} \operatorname {atanh}{\left (a x \right )}}{28} - \frac {a^{2} x^{6} \operatorname {atanh}^{2}{\left (a x \right )}}{3} + \frac {a^{2} x^{6}}{168} - \frac {a x^{5} \operatorname {atanh}{\left (a x \right )}}{12} + \frac {x^{4} \operatorname {atanh}^{2}{\left (a x \right )}}{4} - \frac {x^{4}}{84} + \frac {x^{3} \operatorname {atanh}{\left (a x \right )}}{36 a} - \frac {5 x^{2}}{504 a^{2}} + \frac {x \operatorname {atanh}{\left (a x \right )}}{12 a^{3}} + \frac {4 \log {\left (x - \frac {1}{a} \right )}}{63 a^{4}} - \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{24 a^{4}} + \frac {4 \operatorname {atanh}{\left (a x \right )}}{63 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**4*x**8*atanh(a*x)**2/8 + a**3*x**7*atanh(a*x)/28 - a**2*x**6*atanh(a*x)**2/3 + a**2*x**6/168 - a
*x**5*atanh(a*x)/12 + x**4*atanh(a*x)**2/4 - x**4/84 + x**3*atanh(a*x)/(36*a) - 5*x**2/(504*a**2) + x*atanh(a*
x)/(12*a**3) + 4*log(x - 1/a)/(63*a**4) - atanh(a*x)**2/(24*a**4) + 4*atanh(a*x)/(63*a**4), Ne(a, 0)), (0, Tru
e))

________________________________________________________________________________________