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

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

[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)

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Rubi [A]  time = 0.819183, antiderivative size = 156, normalized size of antiderivative = 1., 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 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]

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 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 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 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 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 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 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]

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 = 0.0630342, size = 108, normalized size = 0.69 \[ \frac{3 a^6 x^6-6 a^4 x^4-5 a^2 x^2+16 \log \left (1-a^2 x^2\right )+2 a x \left (9 a^6 x^6-21 a^4 x^4+7 a^2 x^2+21\right ) \tanh ^{-1}(a x)+21 \left (a^2 x^2-1\right )^3 \left (3 a^2 x^2+1\right ) \tanh ^{-1}(a x)^2}{504 a^4} \]

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)

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Maple [A]  time = 0.047, size = 239, normalized size = 1.5 \begin{align*}{\frac{{a}^{4}{x}^{8} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{8}}-{\frac{{a}^{2}{x}^{6} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{3}}+{\frac{{x}^{4} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{4}}+{\frac{{a}^{3}{x}^{7}{\it Artanh} \left ( ax \right ) }{28}}-{\frac{a{x}^{5}{\it Artanh} \left ( ax \right ) }{12}}+{\frac{{x}^{3}{\it Artanh} \left ( ax \right ) }{36\,a}}+{\frac{x{\it Artanh} \left ( ax \right ) }{12\,{a}^{3}}}+{\frac{{\it Artanh} \left ( ax \right ) \ln \left ( ax-1 \right ) }{24\,{a}^{4}}}-{\frac{{\it Artanh} \left ( ax \right ) \ln \left ( ax+1 \right ) }{24\,{a}^{4}}}+{\frac{ \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{96\,{a}^{4}}}-{\frac{\ln \left ( ax-1 \right ) }{48\,{a}^{4}}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{1}{48\,{a}^{4}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{\ln \left ( ax+1 \right ) }{48\,{a}^{4}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) }+{\frac{ \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{96\,{a}^{4}}}+{\frac{{x}^{6}{a}^{2}}{168}}-{\frac{{x}^{4}}{84}}-{\frac{5\,{x}^{2}}{504\,{a}^{2}}}+{\frac{2\,\ln \left ( ax-1 \right ) }{63\,{a}^{4}}}+{\frac{2\,\ln \left ( ax+1 \right ) }{63\,{a}^{4}}} \end{align*}

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)+1/48/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)+1/96/a^4*ln(a*x+1)^2+1/168*x^6*a^2-1/84*x^4-5/504*x^2/a^2+2/63/a^4*
ln(a*x-1)+2/63/a^4*ln(a*x+1)

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Maxima [A]  time = 0.959537, size = 230, normalized size = 1.47 \begin{align*} \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}} \end{align*}

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

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Fricas [A]  time = 1.88257, size = 300, normalized size = 1.92 \begin{align*} \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}} \end{align*}

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

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Sympy [A]  time = 5.25159, size = 153, normalized size = 0.98 \begin{align*} \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} \end{align*}

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))

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Giac [A]  time = 1.18168, size = 174, normalized size = 1.12 \begin{align*} \frac{1}{168} \, a^{2} x^{6} - \frac{1}{84} \, x^{4} + \frac{1}{96} \,{\left (3 \, a^{4} x^{8} - 8 \, a^{2} x^{6} + 6 \, x^{4} - \frac{1}{a^{4}}\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} + \frac{1}{504} \,{\left (9 \, a^{3} x^{7} - 21 \, a x^{5} + \frac{7 \, x^{3}}{a} + \frac{21 \, x}{a^{3}}\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) - \frac{5 \, x^{2}}{504 \, a^{2}} + \frac{2 \, \log \left (a^{2} x^{2} - 1\right )}{63 \, a^{4}} \end{align*}

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]

1/168*a^2*x^6 - 1/84*x^4 + 1/96*(3*a^4*x^8 - 8*a^2*x^6 + 6*x^4 - 1/a^4)*log(-(a*x + 1)/(a*x - 1))^2 + 1/504*(9
*a^3*x^7 - 21*a*x^5 + 7*x^3/a + 21*x/a^3)*log(-(a*x + 1)/(a*x - 1)) - 5/504*x^2/a^2 + 2/63*log(a^2*x^2 - 1)/a^
4