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

Optimal. Leaf size=157 \[ -\frac {\log \left (1-a^2 x^2\right )}{2 a}+\frac {1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{2 a}+\frac {\text {Li}_3\left (1-\frac {2}{1-a x}\right )}{a}-\frac {2 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{a}+\frac {2}{3} x \tanh ^{-1}(a x)^3+\frac {2 \tanh ^{-1}(a x)^3}{3 a}-x \tanh ^{-1}(a x)-\frac {2 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a} \]

[Out]

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

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Rubi [A]  time = 0.19, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {5944, 5910, 5984, 5918, 5948, 6058, 6610, 260} \[ \frac {\text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{a}-\frac {2 \tanh ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}+\frac {1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{2 a}+\frac {2}{3} x \tanh ^{-1}(a x)^3+\frac {2 \tanh ^{-1}(a x)^3}{3 a}-x \tanh ^{-1}(a x)-\frac {2 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a} \]

Antiderivative was successfully verified.

[In]

Int[(1 - a^2*x^2)*ArcTanh[a*x]^3,x]

[Out]

-(x*ArcTanh[a*x]) + ((1 - a^2*x^2)*ArcTanh[a*x]^2)/(2*a) + (2*ArcTanh[a*x]^3)/(3*a) + (2*x*ArcTanh[a*x]^3)/3 +
(x*(1 - a^2*x^2)*ArcTanh[a*x]^3)/3 - (2*ArcTanh[a*x]^2*Log[2/(1 - a*x)])/a - Log[1 - a^2*x^2]/(2*a) - (2*ArcT
anh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/a + PolyLog[3, 1 - 2/(1 - a*x)]/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 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 ) \tanh ^{-1}(a x)^3 \, dx &=\frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{2 a}+\frac {1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {2}{3} \int \tanh ^{-1}(a x)^3 \, dx-\int \tanh ^{-1}(a x) \, dx\\ &=-x \tanh ^{-1}(a x)+\frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{2 a}+\frac {2}{3} x \tanh ^{-1}(a x)^3+\frac {1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+a \int \frac {x}{1-a^2 x^2} \, dx-(2 a) \int \frac {x \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=-x \tanh ^{-1}(a x)+\frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{2 a}+\frac {2 \tanh ^{-1}(a x)^3}{3 a}+\frac {2}{3} x \tanh ^{-1}(a x)^3+\frac {1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3-\frac {\log \left (1-a^2 x^2\right )}{2 a}-2 \int \frac {\tanh ^{-1}(a x)^2}{1-a x} \, dx\\ &=-x \tanh ^{-1}(a x)+\frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{2 a}+\frac {2 \tanh ^{-1}(a x)^3}{3 a}+\frac {2}{3} x \tanh ^{-1}(a x)^3+\frac {1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3-\frac {2 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}+4 \int \frac {\tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-x \tanh ^{-1}(a x)+\frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{2 a}+\frac {2 \tanh ^{-1}(a x)^3}{3 a}+\frac {2}{3} x \tanh ^{-1}(a x)^3+\frac {1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3-\frac {2 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-\frac {2 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a}+2 \int \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-x \tanh ^{-1}(a x)+\frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{2 a}+\frac {2 \tanh ^{-1}(a x)^3}{3 a}+\frac {2}{3} x \tanh ^{-1}(a x)^3+\frac {1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3-\frac {2 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-\frac {2 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a}+\frac {\text {Li}_3\left (1-\frac {2}{1-a x}\right )}{a}\\ \end {align*}

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Mathematica [A]  time = 0.30, size = 134, normalized size = 0.85 \[ -\frac {2 a^3 x^3 \tanh ^{-1}(a x)^3+3 \log \left (1-a^2 x^2\right )+3 a^2 x^2 \tanh ^{-1}(a x)^2-12 \tanh ^{-1}(a x) \text {Li}_2\left (-e^{-2 \tanh ^{-1}(a x)}\right )-6 \text {Li}_3\left (-e^{-2 \tanh ^{-1}(a x)}\right )-6 a x \tanh ^{-1}(a x)^3+4 \tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2+6 a x \tanh ^{-1}(a x)+12 \tanh ^{-1}(a x)^2 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )}{6 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/6*(6*a*x*ArcTanh[a*x] - 3*ArcTanh[a*x]^2 + 3*a^2*x^2*ArcTanh[a*x]^2 + 4*ArcTanh[a*x]^3 - 6*a*x*ArcTanh[a*x]
^3 + 2*a^3*x^3*ArcTanh[a*x]^3 + 12*ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] + 3*Log[1 - a^2*x^2] - 12*ArcTa
nh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])] - 6*PolyLog[3, -E^(-2*ArcTanh[a*x])])/a

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fricas [F]  time = 1.33, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 1.33, size = 829, normalized size = 5.28 \[ -\frac {a^{2} \arctanh \left (a x \right )^{3} x^{3}}{3}+x \arctanh \left (a x \right )^{3}-\frac {a \arctanh \left (a x \right )^{2} x^{2}}{2}+\frac {\arctanh \left (a x \right )^{2} \ln \left (a x -1\right )}{a}+\frac {\arctanh \left (a x \right )^{2} \ln \left (a x +1\right )}{a}-\frac {2 \arctanh \left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{a}+\frac {i \arctanh \left (a x \right )^{2} \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}\right )^{2} \pi }{2 a}+\frac {i \arctanh \left (a x \right )^{2} \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}\right ) \pi }{2 a}-\frac {i \arctanh \left (a x \right )^{2} \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}\right )^{3} \pi }{2 a}-\frac {i \arctanh \left (a x \right )^{2} \pi }{a}-\frac {i \arctanh \left (a x \right )^{2} \mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}\right )^{2} \pi }{2 a}-\frac {i \arctanh \left (a x \right )^{2} \mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{3} \pi }{a}+\frac {i \arctanh \left (a x \right )^{2} \mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{2} \pi }{a}-\frac {i \arctanh \left (a x \right )^{2} \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )^{2} \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \pi }{2 a}-\frac {i \arctanh \left (a x \right )^{2} \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \pi }{a}-\frac {i \arctanh \left (a x \right )^{2} \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \pi }{2 a}+\frac {2 \arctanh \left (a x \right )^{3}}{3 a}-\frac {2 \arctanh \left (a x \right )^{2} \ln \relax (2)}{a}-x \arctanh \left (a x \right )+\frac {\arctanh \left (a x \right )^{2}}{2 a}-\frac {\arctanh \left (a x \right )}{a}+\frac {\ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a}-\frac {2 \arctanh \left (a x \right ) \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a}+\frac {\polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a^2*arctanh(a*x)^3*x^3+x*arctanh(a*x)^3-1/2*a*arctanh(a*x)^2*x^2+1/a*arctanh(a*x)^2*ln(a*x-1)+1/a*arctanh
(a*x)^2*ln(a*x+1)-2/a*arctanh(a*x)^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*I/a*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a
^2*x^2-1))^3*Pi-I/a*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*Pi-1/2*I/a*arctanh(a*x)^2*csgn(I*(a*x+
1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*Pi-1/2*I/a*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*cs
gn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*Pi+I/a*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1
)))^2*Pi-I/a*arctanh(a*x)^2*Pi+1/2*I/a*arctanh(a*x)^2*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*Pi-1/2*I/a*arctanh(a*x)^2*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)
^2/(a^2*x^2-1))*Pi-I/a*arctanh(a*x)^2*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*Pi+1/
2*I/a*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*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)))*Pi+2/3*arctanh(a*x)^3/a-2/a*arctanh(a*x)^2*ln(2)-x*arctanh(a*x)+1/2*arctanh(a
*x)^2/a-arctanh(a*x)/a+1/a*ln(1+(a*x+1)^2/(-a^2*x^2+1))-2/a*arctanh(a*x)*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))+1/
a*polylog(3,-(a*x+1)^2/(-a^2*x^2+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 12 \, a x - 6 \, {\left (a^{3} x^{3} - 3 \, a x - 2\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{2}}{48 \, a} - \frac {{\left (\log \left (-a x + 1\right )^{3} - 3 \, \log \left (-a x + 1\right )^{2} + 6 \, \log \left (-a x + 1\right ) - 6\right )} {\left (a x - 1\right )}}{8 \, a} + \frac {4 \, {\left (9 \, \log \left (-a x + 1\right )^{3} - 9 \, \log \left (-a x + 1\right )^{2} + 6 \, \log \left (-a x + 1\right ) - 2\right )} {\left (a x - 1\right )}^{3} + 27 \, {\left (4 \, \log \left (-a x + 1\right )^{3} - 6 \, \log \left (-a x + 1\right )^{2} + 6 \, \log \left (-a x + 1\right ) - 3\right )} {\left (a x - 1\right )}^{2} + 108 \, {\left (\log \left (-a x + 1\right )^{3} - 3 \, \log \left (-a x + 1\right )^{2} + 6 \, \log \left (-a x + 1\right ) - 6\right )} {\left (a x - 1\right )}}{864 \, a} + \frac {1}{8} \, \int -\frac {3 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right )^{3} + {\left (2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 9 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right )^{2} - 12 \, a x - 6 \, {\left (a^{3} x^{3} - 3 \, a x - 2\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )}{3 \, {\left (a x - 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(2*a^3*x^3 - 3*a^2*x^2 - 12*a*x - 6*(a^3*x^3 - 3*a*x - 2)*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 + 1/864*(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(-1/3*(3*
(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1)^3 + (2*a^3*x^3 - 3*a^2*x^2 - 9*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*
x + 1)^2 - 12*a*x - 6*(a^3*x^3 - 3*a*x - 2)*log(a*x + 1))*log(-a*x + 1))/(a*x - 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int {\mathrm {atanh}\left (a\,x\right )}^3\,\left (a^2\,x^2-1\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-atanh(a*x)^3*(a^2*x^2 - 1),x)

[Out]

-int(atanh(a*x)^3*(a^2*x^2 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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