3.96 \(\int \frac{x^2 (a+b \tanh ^{-1}(c x))^2}{d+c d x} \, dx\)

Optimal. Leaf size=247 \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d}+\frac{b^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{c^3 d}+\frac{b^2 \text{PolyLog}\left (3,1-\frac{2}{c x+1}\right )}{2 c^3 d}+\frac{a b x}{c^2 d}-\frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d}-\frac{3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d}+\frac{2 b \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d}-\frac{\log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d}+\frac{b^2 \log \left (1-c^2 x^2\right )}{2 c^3 d}+\frac{b^2 x \tanh ^{-1}(c x)}{c^2 d} \]

[Out]

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

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Rubi [A]  time = 0.533158, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {5930, 5916, 5980, 5910, 260, 5948, 5984, 5918, 2402, 2315, 6056, 6610} \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d}+\frac{b^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{c^3 d}+\frac{b^2 \text{PolyLog}\left (3,1-\frac{2}{c x+1}\right )}{2 c^3 d}+\frac{a b x}{c^2 d}-\frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d}-\frac{3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d}+\frac{2 b \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d}-\frac{\log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d}+\frac{b^2 \log \left (1-c^2 x^2\right )}{2 c^3 d}+\frac{b^2 x \tanh ^{-1}(c x)}{c^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(a + b*ArcTanh[c*x])^2)/(d + c*d*x),x]

[Out]

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

Rule 5930

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[f/e,
Int[(f*x)^(m - 1)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[(d*f)/e, Int[((f*x)^(m - 1)*(a + b*ArcTanh[c*x])^p)/(
d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && GtQ[m, 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 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 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 6056

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[((a + b*ArcTa
nh[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 \frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d+c d x} \, dx &=-\frac{\int \frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{d+c d x} \, dx}{c}+\frac{\int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c d}\\ &=\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d}+\frac{\int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{d+c d x} \, dx}{c^2}-\frac{b \int \frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{d}-\frac{\int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c^2 d}\\ &=-\frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d}+\frac{b \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^2 d}-\frac{b \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^2 d}+\frac{(2 b) \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 d}+\frac{(2 b) \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c d}\\ &=\frac{a b x}{c^2 d}-\frac{3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d}-\frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d}+\frac{b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^3 d}+\frac{(2 b) \int \frac{a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{c^2 d}+\frac{b^2 \int \tanh ^{-1}(c x) \, dx}{c^2 d}-\frac{b^2 \int \frac{\text{Li}_2\left (1-\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c^2 d}\\ &=\frac{a b x}{c^2 d}+\frac{b^2 x \tanh ^{-1}(c x)}{c^2 d}-\frac{3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d}-\frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d}+\frac{2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{c^3 d}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d}+\frac{b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^3 d}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{2 c^3 d}-\frac{\left (2 b^2\right ) \int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c^2 d}-\frac{b^2 \int \frac{x}{1-c^2 x^2} \, dx}{c d}\\ &=\frac{a b x}{c^2 d}+\frac{b^2 x \tanh ^{-1}(c x)}{c^2 d}-\frac{3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d}-\frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d}+\frac{2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{c^3 d}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d}+\frac{b^2 \log \left (1-c^2 x^2\right )}{2 c^3 d}+\frac{b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^3 d}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{2 c^3 d}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c x}\right )}{c^3 d}\\ &=\frac{a b x}{c^2 d}+\frac{b^2 x \tanh ^{-1}(c x)}{c^2 d}-\frac{3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d}-\frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d}+\frac{2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{c^3 d}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d}+\frac{b^2 \log \left (1-c^2 x^2\right )}{2 c^3 d}+\frac{b^2 \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{c^3 d}+\frac{b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^3 d}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{2 c^3 d}\\ \end{align*}

Mathematica [A]  time = 0.503607, size = 260, normalized size = 1.05 \[ \frac{2 b \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right ) \left (a+b \tanh ^{-1}(c x)-b\right )+b^2 \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )+a^2 c^2 x^2-2 a^2 c x+2 a^2 \log (c x+1)-2 a b \log \left (1-c^2 x^2\right )+2 a b c^2 x^2 \tanh ^{-1}(c x)+2 a b c x-2 a b \tanh ^{-1}(c x)-4 a b c x \tanh ^{-1}(c x)-4 a b \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+b^2 \log \left (1-c^2 x^2\right )+b^2 c^2 x^2 \tanh ^{-1}(c x)^2+b^2 \tanh ^{-1}(c x)^2-2 b^2 c x \tanh ^{-1}(c x)^2+2 b^2 c x \tanh ^{-1}(c x)-2 b^2 \tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+4 b^2 \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )}{2 c^3 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*a^2*c*x + 2*a*b*c*x + a^2*c^2*x^2 - 2*a*b*ArcTanh[c*x] - 4*a*b*c*x*ArcTanh[c*x] + 2*b^2*c*x*ArcTanh[c*x] +
2*a*b*c^2*x^2*ArcTanh[c*x] + b^2*ArcTanh[c*x]^2 - 2*b^2*c*x*ArcTanh[c*x]^2 + b^2*c^2*x^2*ArcTanh[c*x]^2 - 4*a
*b*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] + 4*b^2*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 2*b^2*ArcTanh
[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 2*a^2*Log[1 + c*x] - 2*a*b*Log[1 - c^2*x^2] + b^2*Log[1 - c^2*x^2] + 2*
b*(a - b + b*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] + b^2*PolyLog[3, -E^(-2*ArcTanh[c*x])])/(2*c^3*d)

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Maple [C]  time = 0.623, size = 1192, normalized size = 4.8 \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))^2/(c*d*x+d),x)

[Out]

1/2*I/c^3*b^2/d*Pi*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*ar
ctanh(c*x)^2-I/c^3*b^2/d*Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2*arctanh(c*x)^2-
1/2*I/c^3*b^2/d*Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*arctanh(c*x)^2-1/2*I/c^3
*b^2/d*Pi*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctan
h(c*x)^2-2/c^3*b^2/d*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))+2/c^3*b^2/d*arctanh(c*x)*ln(1-I*(c*x+1)/(-c
^2*x^2+1)^(1/2))-1/c^3*b^2/d*arctanh(c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+1/c^3*b^2/d*arctanh(c*x)^2*ln(c*x
+1)+2/c^3*b^2/d*arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-1/c^3*b^2/d*arctanh(c*x)^2*ln(2)-1/c^3*a*b/d*d
ilog(1/2+1/2*c*x)+1/2/c*b^2/d*arctanh(c*x)^2*x^2-1/c^2*b^2/d*arctanh(c*x)^2*x-1/2*I/c^3*b^2/d*Pi*csgn(I*(c*x+1
)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^3*arctanh(c*x)^2-1/2*I/c^3*b^2/d*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^
3*arctanh(c*x)^2+1/2*I/c^3*b^2/d*Pi*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))*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))*arctanh(c*x)^2+a*b*x/c^2/d+b^2*x*arctanh(c*x)/c^2/d+1/c*a*b/d*
arctanh(c*x)*x^2-2/c^2*a*b/d*arctanh(c*x)*x+2/c^3*a*b/d*arctanh(c*x)*ln(c*x+1)+1/c^3*a*b/d*ln(-1/2*c*x+1/2)*ln
(c*x+1)-1/c^3*a*b/d*ln(-1/2*c*x+1/2)*ln(1/2+1/2*c*x)+1/c^3*a*b/d+1/2/c*a^2/d*x^2-1/c^2*a^2/d*x+1/c^3*a^2/d*ln(
c*x+1)+2/3/c^3*b^2/d*arctanh(c*x)^3+1/c^3*b^2/d*arctanh(c*x)-1/c^3*b^2/d*ln((c*x+1)^2/(-c^2*x^2+1)+1)-3/2/c^3*
b^2/d*arctanh(c*x)^2+2/c^3*b^2/d*dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))+2/c^3*b^2/d*dilog(1-I*(c*x+1)/(-c^2*x^2
+1)^(1/2))+1/2/c^3*b^2/d*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))-1/2/c^3*a*b/d*ln(c*x+1)^2-1/2/c^3*a*b/d*ln(c*x-1)-
3/2/c^3*a*b/d*ln(c*x+1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a^{2} x^{2}}{c x + 1}\, dx + \int \frac{b^{2} x^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{c x + 1}\, dx + \int \frac{2 a b x^{2} \operatorname{atanh}{\left (c x \right )}}{c x + 1}\, dx}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a**2*x**2/(c*x + 1), x) + Integral(b**2*x**2*atanh(c*x)**2/(c*x + 1), x) + Integral(2*a*b*x**2*atanh
(c*x)/(c*x + 1), x))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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