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

Optimal. Leaf size=188 \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2}+\frac{b^2 \text{PolyLog}\left (3,1-\frac{2}{c x+1}\right )}{2 c^2 d^2}+\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2 (c x+1)}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (c x+1)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 d^2}-\frac{\log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2}+\frac{b^2}{2 c^2 d^2 (c x+1)}-\frac{b^2 \tanh ^{-1}(c x)}{2 c^2 d^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.342551, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5940, 5928, 5926, 627, 44, 207, 5948, 5918, 6056, 6610} \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2}+\frac{b^2 \text{PolyLog}\left (3,1-\frac{2}{c x+1}\right )}{2 c^2 d^2}+\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2 (c x+1)}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (c x+1)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 d^2}-\frac{\log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2}+\frac{b^2}{2 c^2 d^2 (c x+1)}-\frac{b^2 \tanh ^{-1}(c x)}{2 c^2 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5940

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

Rule 5928

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1)*(
a + b*ArcTanh[c*x])^p)/(e*(q + 1)), x] - Dist[(b*c*p)/(e*(q + 1)), Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^(p
- 1), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] &
& NeQ[q, -1]

Rule 5926

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1)*(a + b
*ArcTanh[c*x]))/(e*(q + 1)), x] - Dist[(b*c)/(e*(q + 1)), Int[(d + e*x)^(q + 1)/(1 - c^2*x^2), x], x] /; FreeQ
[{a, b, c, d, e, q}, x] && NeQ[q, -1]

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

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

Mathematica [A]  time = 0.625016, size = 233, normalized size = 1.24 \[ \frac{2 a b \left (2 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )-\sinh \left (2 \tanh ^{-1}(c x)\right )+\cosh \left (2 \tanh ^{-1}(c x)\right )+2 \tanh ^{-1}(c x) \left (-2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-\sinh \left (2 \tanh ^{-1}(c x)\right )+\cosh \left (2 \tanh ^{-1}(c x)\right )\right )\right )+b^2 \left (4 \tanh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+2 \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )-4 \tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-2 \tanh ^{-1}(c x)^2 \sinh \left (2 \tanh ^{-1}(c x)\right )-2 \tanh ^{-1}(c x) \sinh \left (2 \tanh ^{-1}(c x)\right )-\sinh \left (2 \tanh ^{-1}(c x)\right )+2 \tanh ^{-1}(c x)^2 \cosh \left (2 \tanh ^{-1}(c x)\right )+2 \tanh ^{-1}(c x) \cosh \left (2 \tanh ^{-1}(c x)\right )+\cosh \left (2 \tanh ^{-1}(c x)\right )\right )+\frac{4 a^2}{c x+1}+4 a^2 \log (c x+1)}{4 c^2 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((4*a^2)/(1 + c*x) + 4*a^2*Log[1 + c*x] + 2*a*b*(Cosh[2*ArcTanh[c*x]] + 2*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 2
*ArcTanh[c*x]*(Cosh[2*ArcTanh[c*x]] - 2*Log[1 + E^(-2*ArcTanh[c*x])] - Sinh[2*ArcTanh[c*x]]) - Sinh[2*ArcTanh[
c*x]]) + b^2*(Cosh[2*ArcTanh[c*x]] + 2*ArcTanh[c*x]*Cosh[2*ArcTanh[c*x]] + 2*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x
]] - 4*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 4*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 2*PolyL
og[3, -E^(-2*ArcTanh[c*x])] - Sinh[2*ArcTanh[c*x]] - 2*ArcTanh[c*x]*Sinh[2*ArcTanh[c*x]] - 2*ArcTanh[c*x]^2*Si
nh[2*ArcTanh[c*x]]))/(4*c^2*d^2)

________________________________________________________________________________________

Maple [C]  time = 0.277, size = 1030, normalized size = 5.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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}{{\left (c d x + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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