3.27 \(\int x^2 (a+b \tanh ^{-1}(c x))^3 \, dx\)

Optimal. Leaf size=197 \[ -\frac{b^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3}+\frac{b^3 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )}{2 c^3}+\frac{a b^2 x}{c^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac{b \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3}+\frac{b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac{1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac{b^3 \log \left (1-c^2 x^2\right )}{2 c^3}+\frac{b^3 x \tanh ^{-1}(c x)}{c^2} \]

[Out]

(a*b^2*x)/c^2 + (b^3*x*ArcTanh[c*x])/c^2 - (b*(a + b*ArcTanh[c*x])^2)/(2*c^3) + (b*x^2*(a + b*ArcTanh[c*x])^2)
/(2*c) + (a + b*ArcTanh[c*x])^3/(3*c^3) + (x^3*(a + b*ArcTanh[c*x])^3)/3 - (b*(a + b*ArcTanh[c*x])^2*Log[2/(1
- c*x)])/c^3 + (b^3*Log[1 - c^2*x^2])/(2*c^3) - (b^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c^3 + (
b^3*PolyLog[3, 1 - 2/(1 - c*x)])/(2*c^3)

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Rubi [A]  time = 0.449067, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {5916, 5980, 5910, 260, 5948, 5984, 5918, 6058, 6610} \[ -\frac{b^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3}+\frac{b^3 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )}{2 c^3}+\frac{a b^2 x}{c^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac{b \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3}+\frac{b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac{1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac{b^3 \log \left (1-c^2 x^2\right )}{2 c^3}+\frac{b^3 x \tanh ^{-1}(c x)}{c^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcTanh[c*x])^3,x]

[Out]

(a*b^2*x)/c^2 + (b^3*x*ArcTanh[c*x])/c^2 - (b*(a + b*ArcTanh[c*x])^2)/(2*c^3) + (b*x^2*(a + b*ArcTanh[c*x])^2)
/(2*c) + (a + b*ArcTanh[c*x])^3/(3*c^3) + (x^3*(a + b*ArcTanh[c*x])^3)/3 - (b*(a + b*ArcTanh[c*x])^2*Log[2/(1
- c*x)])/c^3 + (b^3*Log[1 - c^2*x^2])/(2*c^3) - (b^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c^3 + (
b^3*PolyLog[3, 1 - 2/(1 - c*x)])/(2*c^3)

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 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]

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 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 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 x^2 \left (a+b \tanh ^{-1}(c x)\right )^3 \, dx &=\frac{1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-(b c) \int \frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac{b \int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c}-\frac{b \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx}{c}\\ &=\frac{b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}+\frac{1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-b^2 \int \frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac{b \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c x} \, dx}{c^2}\\ &=\frac{b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}+\frac{1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-c x}\right )}{c^3}+\frac{b^2 \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^2}-\frac{b^2 \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^2}+\frac{\left (2 b^2\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c^2}\\ &=\frac{a b^2 x}{c^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac{b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}+\frac{1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-c x}\right )}{c^3}-\frac{b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{c^3}+\frac{b^3 \int \tanh ^{-1}(c x) \, dx}{c^2}+\frac{b^3 \int \frac{\text{Li}_2\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c^2}\\ &=\frac{a b^2 x}{c^2}+\frac{b^3 x \tanh ^{-1}(c x)}{c^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac{b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}+\frac{1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-c x}\right )}{c^3}-\frac{b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{c^3}+\frac{b^3 \text{Li}_3\left (1-\frac{2}{1-c x}\right )}{2 c^3}-\frac{b^3 \int \frac{x}{1-c^2 x^2} \, dx}{c}\\ &=\frac{a b^2 x}{c^2}+\frac{b^3 x \tanh ^{-1}(c x)}{c^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac{b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}+\frac{1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-c x}\right )}{c^3}+\frac{b^3 \log \left (1-c^2 x^2\right )}{2 c^3}-\frac{b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{c^3}+\frac{b^3 \text{Li}_3\left (1-\frac{2}{1-c x}\right )}{2 c^3}\\ \end{align*}

Mathematica [A]  time = 0.520302, size = 250, normalized size = 1.27 \[ \frac{6 a b^2 \left (\text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+\left (c^3 x^3-1\right ) \tanh ^{-1}(c x)^2+\tanh ^{-1}(c x) \left (c^2 x^2-2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-1\right )+c x\right )+b^3 \left (6 \tanh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+3 \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )+3 \log \left (1-c^2 x^2\right )+2 c^3 x^3 \tanh ^{-1}(c x)^3+3 c^2 x^2 \tanh ^{-1}(c x)^2-2 \tanh ^{-1}(c x)^3-3 \tanh ^{-1}(c x)^2+6 c x \tanh ^{-1}(c x)-6 \tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+3 a^2 b c^2 x^2+3 a^2 b \log \left (1-c^2 x^2\right )+6 a^2 b c^3 x^3 \tanh ^{-1}(c x)+2 a^3 c^3 x^3}{6 c^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*(a + b*ArcTanh[c*x])^3,x]

[Out]

(3*a^2*b*c^2*x^2 + 2*a^3*c^3*x^3 + 6*a^2*b*c^3*x^3*ArcTanh[c*x] + 3*a^2*b*Log[1 - c^2*x^2] + 6*a*b^2*(c*x + (-
1 + c^3*x^3)*ArcTanh[c*x]^2 + ArcTanh[c*x]*(-1 + c^2*x^2 - 2*Log[1 + E^(-2*ArcTanh[c*x])]) + PolyLog[2, -E^(-2
*ArcTanh[c*x])]) + b^3*(6*c*x*ArcTanh[c*x] - 3*ArcTanh[c*x]^2 + 3*c^2*x^2*ArcTanh[c*x]^2 - 2*ArcTanh[c*x]^3 +
2*c^3*x^3*ArcTanh[c*x]^3 - 6*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 3*Log[1 - c^2*x^2] + 6*ArcTanh[c*x]
*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 3*PolyLog[3, -E^(-2*ArcTanh[c*x])]))/(6*c^3)

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Maple [C]  time = 0.477, size = 1177, normalized size = 6. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arctanh(c*x))^3,x)

[Out]

1/c^3*b^3*arctanh(c*x)+1/3/c^3*b^3*arctanh(c*x)^3-1/2/c^3*b^3*arctanh(c*x)^2+1/2/c^3*b^3*polylog(3,-(c*x+1)^2/
(-c^2*x^2+1))-1/c^3*b^3*ln((c*x+1)^2/(-c^2*x^2+1)+1)+1/3*x^3*b^3*arctanh(c*x)^3+a*b^2*x/c^2+b^3*x*arctanh(c*x)
/c^2+1/c*a*b^2*arctanh(c*x)*x^2+1/c^3*a*b^2*arctanh(c*x)*ln(c*x-1)+1/c^3*a*b^2*arctanh(c*x)*ln(c*x+1)-1/2/c^3*
a*b^2*ln(c*x-1)*ln(1/2+1/2*c*x)+1/2/c^3*a*b^2*ln(-1/2*c*x+1/2)*ln(c*x+1)-1/2/c^3*a*b^2*ln(-1/2*c*x+1/2)*ln(1/2
+1/2*c*x)-1/2*I/c^3*b^3*arctanh(c*x)^2*Pi+1/3*x^3*a^3+1/4*I/c^3*b^3*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1
))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))*Pi+1/2/c*a^2*b*
x^2+a^2*b*x^3*arctanh(c*x)+a*b^2*x^3*arctanh(c*x)^2+1/2/c^3*a*b^2*ln(c*x-1)-1/2/c^3*a*b^2*ln(c*x+1)+1/2/c^3*a^
2*b*ln(c*x-1)+1/2/c^3*a^2*b*ln(c*x+1)+1/4/c^3*a*b^2*ln(c*x-1)^2-1/4/c^3*a*b^2*ln(c*x+1)^2-1/c^3*b^3*arctanh(c*
x)^2*ln(2)-1/c^3*a*b^2*dilog(1/2+1/2*c*x)+1/2/c^3*b^3*arctanh(c*x)^2*ln(c*x-1)+1/2/c^3*b^3*arctanh(c*x)^2*ln(c
*x+1)-1/c^3*b^3*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))-1/c^3*b^3*arctanh(c*x)*polylog(2,-(c*x+1)^2/(-c^
2*x^2+1))+1/2/c*b^3*arctanh(c*x)^2*x^2-1/4*I/c^3*b^3*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*Pi-1/4*I/c
^3*b^3*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^3*Pi-1/2*I/c^3*b^3*arctanh(c*x)
^2*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))^3*Pi+1/2*I/c^3*b^3*arctanh(c*x)^2*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))^2*P
i-1/2*I/c^3*b^3*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*Pi+1/4*I/c^3
*b^3*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*P
i-1/4*I/c^3*b^3*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*Pi-1/4*I/c^3
*b^3*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+
1))*Pi

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a^{3} x^{3} + \frac{1}{2} \,{\left (2 \, x^{3} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{x^{2}}{c^{2}} + \frac{\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} a^{2} b - \frac{{\left (b^{3} c^{3} x^{3} - b^{3}\right )} \log \left (-c x + 1\right )^{3} - 3 \,{\left (2 \, a b^{2} c^{3} x^{3} + b^{3} c^{2} x^{2} +{\left (b^{3} c^{3} x^{3} + b^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )^{2}}{24 \, c^{3}} - \int -\frac{{\left (b^{3} c^{3} x^{3} - b^{3} c^{2} x^{2}\right )} \log \left (c x + 1\right )^{3} + 6 \,{\left (a b^{2} c^{3} x^{3} - a b^{2} c^{2} x^{2}\right )} \log \left (c x + 1\right )^{2} -{\left (4 \, a b^{2} c^{3} x^{3} + 2 \, b^{3} c^{2} x^{2} + 3 \,{\left (b^{3} c^{3} x^{3} - b^{3} c^{2} x^{2}\right )} \log \left (c x + 1\right )^{2} - 2 \,{\left (6 \, a b^{2} c^{2} x^{2} -{\left (6 \, a b^{2} c^{3} + b^{3} c^{3}\right )} x^{3} - b^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{8 \,{\left (c^{3} x - c^{2}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^3*x^3 + 1/2*(2*x^3*arctanh(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*a^2*b - 1/24*((b^3*c^3*x^3 - b^3)*
log(-c*x + 1)^3 - 3*(2*a*b^2*c^3*x^3 + b^3*c^2*x^2 + (b^3*c^3*x^3 + b^3)*log(c*x + 1))*log(-c*x + 1)^2)/c^3 -
integrate(-1/8*((b^3*c^3*x^3 - b^3*c^2*x^2)*log(c*x + 1)^3 + 6*(a*b^2*c^3*x^3 - a*b^2*c^2*x^2)*log(c*x + 1)^2
- (4*a*b^2*c^3*x^3 + 2*b^3*c^2*x^2 + 3*(b^3*c^3*x^3 - b^3*c^2*x^2)*log(c*x + 1)^2 - 2*(6*a*b^2*c^2*x^2 - (6*a*
b^2*c^3 + b^3*c^3)*x^3 - b^3)*log(c*x + 1))*log(-c*x + 1))/(c^3*x - c^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{3} x^{2} \operatorname{artanh}\left (c x\right )^{3} + 3 \, a b^{2} x^{2} \operatorname{artanh}\left (c x\right )^{2} + 3 \, a^{2} b x^{2} \operatorname{artanh}\left (c x\right ) + a^{3} x^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b^3*x^2*arctanh(c*x)^3 + 3*a*b^2*x^2*arctanh(c*x)^2 + 3*a^2*b*x^2*arctanh(c*x) + a^3*x^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (a + b \operatorname{atanh}{\left (c x \right )}\right )^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*atanh(c*x))**3,x)

[Out]

Integral(x**2*(a + b*atanh(c*x))**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}^{3} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arctanh(c*x) + a)^3*x^2, x)