3.73 \(\int \frac{(d+c d x) (a+b \tanh ^{-1}(c x))^2}{x^2} \, dx\)
Optimal. Leaf size=201 \[ -b c d \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b c d \text{PolyLog}\left (2,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )+b^2 (-c) d \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )+\frac{1}{2} b^2 c d \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 c d \text{PolyLog}\left (3,\frac{2}{1-c x}-1\right )+c d \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c d \tanh ^{-1}\left (1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2+2 b c d \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right ) \]
[Out]
c*d*(a + b*ArcTanh[c*x])^2 - (d*(a + b*ArcTanh[c*x])^2)/x + 2*c*d*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*
x)] + 2*b*c*d*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - b*c*d*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x
)] + b*c*d*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] - b^2*c*d*PolyLog[2, -1 + 2/(1 + c*x)] + (b^2*c*d
*PolyLog[3, 1 - 2/(1 - c*x)])/2 - (b^2*c*d*PolyLog[3, -1 + 2/(1 - c*x)])/2
________________________________________________________________________________________
Rubi [A] time = 0.484035, antiderivative size = 201, normalized size of antiderivative
= 1., number of steps used = 12, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\)
= 0.5, Rules used = {5940, 5916, 5988, 5932, 2447, 5914, 6052, 5948, 6058, 6610}
\[ -b c d \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b c d \text{PolyLog}\left (2,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )+b^2 (-c) d \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )+\frac{1}{2} b^2 c d \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 c d \text{PolyLog}\left (3,\frac{2}{1-c x}-1\right )+c d \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c d \tanh ^{-1}\left (1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2+2 b c d \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right ) \]
Antiderivative was successfully verified.
[In]
Int[((d + c*d*x)*(a + b*ArcTanh[c*x])^2)/x^2,x]
[Out]
c*d*(a + b*ArcTanh[c*x])^2 - (d*(a + b*ArcTanh[c*x])^2)/x + 2*c*d*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*
x)] + 2*b*c*d*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - b*c*d*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x
)] + b*c*d*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] - b^2*c*d*PolyLog[2, -1 + 2/(1 + c*x)] + (b^2*c*d
*PolyLog[3, 1 - 2/(1 - c*x)])/2 - (b^2*c*d*PolyLog[3, -1 + 2/(1 - c*x)])/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 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 5988
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(a + b*ArcTanh[c
*x])^(p + 1)/(b*d*(p + 1)), x] + Dist[1/d, Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
Rule 5932
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[((a + b*ArcTanh[c*
x])^p*Log[2 - 2/(1 + (e*x)/d)])/d, x] - Dist[(b*c*p)/d, Int[((a + b*ArcTanh[c*x])^(p - 1)*Log[2 - 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 2447
Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]
Rule 5914
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + b*ArcTanh[c*x])^p*ArcTanh[1 - 2/(1
- c*x)], x] - Dist[2*b*c*p, Int[((a + b*ArcTanh[c*x])^(p - 1)*ArcTanh[1 - 2/(1 - c*x)])/(1 - c^2*x^2), x], x]
/; FreeQ[{a, b, c}, x] && IGtQ[p, 1]
Rule 6052
Int[(ArcTanh[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[
(Log[1 + u]*(a + b*ArcTanh[c*x])^p)/(d + e*x^2), x], x] - Dist[1/2, Int[(Log[1 - u]*(a + b*ArcTanh[c*x])^p)/(d
+ e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x
))^2, 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 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 \frac{(d+c d x) \left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx &=\int \left (\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{x^2}+\frac{c d \left (a+b \tanh ^{-1}(c x)\right )^2}{x}\right ) \, dx\\ &=d \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx+(c d) \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx\\ &=-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c d \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+(2 b c d) \int \frac{a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx-\left (4 b c^2 d\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=c d \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c d \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+(2 b c d) \int \frac{a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx+\left (2 b c^2 d\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-\left (2 b c^2 d\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=c d \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c d \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+2 b c d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )-b c d \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b c d \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )+\left (b^2 c^2 d\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (b^2 c^2 d\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (2 b^2 c^2 d\right ) \int \frac{\log \left (2-\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=c d \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c d \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+2 b c d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )-b c d \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b c d \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )-b^2 c d \text{Li}_2\left (-1+\frac{2}{1+c x}\right )+\frac{1}{2} b^2 c d \text{Li}_3\left (1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 c d \text{Li}_3\left (-1+\frac{2}{1-c x}\right )\\ \end{align*}
Mathematica [C] time = 0.515846, size = 249, normalized size = 1.24 \[ -\frac{d \left (a b c x (\text{PolyLog}(2,-c x)-\text{PolyLog}(2,c x))+b^2 \left (c x \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+\tanh ^{-1}(c x) \left ((1-c x) \tanh ^{-1}(c x)-2 c x \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )\right )\right )-b^2 c x \left (\tanh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+\tanh ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \tanh ^{-1}(c x)}\right )+\frac{1}{2} \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(c x)}\right )-\frac{2}{3} \tanh ^{-1}(c x)^3-\tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+\tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )+\frac{i \pi ^3}{24}\right )+a^2 (-c) x \log (x)+a^2+a b \left (c x \left (\log \left (1-c^2 x^2\right )-2 \log (c x)\right )+2 \tanh ^{-1}(c x)\right )\right )}{x} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[((d + c*d*x)*(a + b*ArcTanh[c*x])^2)/x^2,x]
[Out]
-((d*(a^2 - a^2*c*x*Log[x] + a*b*(2*ArcTanh[c*x] + c*x*(-2*Log[c*x] + Log[1 - c^2*x^2])) + b^2*(ArcTanh[c*x]*(
(1 - c*x)*ArcTanh[c*x] - 2*c*x*Log[1 - E^(-2*ArcTanh[c*x])]) + c*x*PolyLog[2, E^(-2*ArcTanh[c*x])]) + a*b*c*x*
(PolyLog[2, -(c*x)] - PolyLog[2, c*x]) - b^2*c*x*((I/24)*Pi^3 - (2*ArcTanh[c*x]^3)/3 - ArcTanh[c*x]^2*Log[1 +
E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x
])] + ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] + PolyLog[3, -E^(-2*ArcTanh[c*x])]/2 - PolyLog[3, E^(2*ArcTa
nh[c*x])]/2)))/x)
________________________________________________________________________________________
Maple [C] time = 0.82, size = 3104, normalized size = 15.4 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*d*x+d)*(a+b*arctanh(c*x))^2/x^2,x)
[Out]
-1/8*I*c*d*b^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)-1)/((c*x+1)^2/(-c^2*x^2+1)
+1))^2*dilog((c*x+1)^2/(-c^2*x^2+1)+1)+1/8*I*c*d*b^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)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+1/8*I*c*d*b^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)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*polylog(2,-(c*x+1)^2/(-c
^2*x^2+1))+1/8*I*c*d*b^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)-1)/((c*x+1)^2/(-
c^2*x^2+1)+1))^2*polylog(2,(c*x+1)^2/(-c^2*x^2+1))-1/8*I*c*d*b^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)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*dilog((c*x+1)^2/(-c^2*x^2+1)+1)+1/8*I*c*d*b^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)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*polylog(2,(c*x+
1)^2/(-c^2*x^2+1))+1/8*I*c*d*b^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)-1)/((c*x
+1)^2/(-c^2*x^2+1)+1))^2*dilog((c*x+1)^2/(-c^2*x^2+1))-1/2*I*c*d*b^2*Pi*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))*csg
n(I*((c*x+1)^2/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctanh(c*x)^2-1/2*I*c*d*b^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)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctanh(c*x)^2-1/4*I*c*d*b^2
*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^3*arctanh(c*x)*ln(1-(c*x+1)^2/(-c^2*x^2+1))+
1/8*I*c*d*b^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)-1)/((c*x+1)^2/(-c^2*x^2+1)+
1))^2*dilog((c*x+1)^2/(-c^2*x^2+1))+3/4*c*d*b^2*polylog(2,(c*x+1)^2/(-c^2*x^2+1))+1/4*c*d*b^2*dilog((c*x+1)^2/
(-c^2*x^2+1)+1)-1/2*c*d*b^2*polylog(3,(c*x+1)^2/(-c^2*x^2+1))-1/4*c*d*b^2*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))-c
*d*b^2*arctanh(c*x)^2+1/2*c*d*b^2*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))-1/4*c*d*b^2*dilog((c*x+1)^2/(-c^2*x^2+1))
+c*a^2*d*ln(c*x)-d*b^2*arctanh(c*x)^2/x-a^2*d/x+2*c*d*a*b*arctanh(c*x)*ln(c*x)-c*d*a*b*ln(c*x)*ln(c*x+1)-1/4*I
*c*d*b^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)+1))*csgn(I*((c*x+1)^2/(-c^2*x^2+
1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))*arctanh(c*x)*ln(1-(c*x+1)^2/(-c^2*x^2+1))-1/8*I*c*d*b^2*Pi*csgn(I*((c*x+1)^2
/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^3*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+1/8*I*c*d*b^2*Pi*csgn(I*((c*
x+1)^2/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^3*dilog((c*x+1)^2/(-c^2*x^2+1)+1)-1/8*I*c*d*b^2*Pi*csgn(I*(
(c*x+1)^2/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^3*polylog(2,(c*x+1)^2/(-c^2*x^2+1))-1/8*I*c*d*b^2*Pi*csg
n(I*((c*x+1)^2/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^3*dilog((c*x+1)^2/(-c^2*x^2+1))+1/2*I*c*d*b^2*Pi*cs
gn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^3*arctanh(c*x)^2+1/2*I*c*d*b^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)+1))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1
)+1))*arctanh(c*x)^2-1/8*I*c*d*b^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)+1))*cs
gn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))*dilog((c*x+1)^2/(-c^2*x^2+1))+1/4*I*c*d*b^2*Pi*csg
n(I*((c*x+1)^2/(-c^2*x^2+1)-1))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctanh(c*x)*l
n(1-(c*x+1)^2/(-c^2*x^2+1))+1/8*I*c*d*b^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)
+1))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))*dilog((c*x+1)^2/(-c^2*x^2+1)+1)-1/8*I*c*d*b
^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)+1))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/
((c*x+1)^2/(-c^2*x^2+1)+1))*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+1/4*I*c*d*b^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)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctanh(c*x)*ln(1-(c*x+1)^2/(-c^2*x^2+1))
-1/8*I*c*d*b^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)+1))*csgn(I*((c*x+1)^2/(-c^
2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))*polylog(2,(c*x+1)^2/(-c^2*x^2+1))-2*d*a*b*arctanh(c*x)/x-c*d*a*b*dilog
(c*x+1)+c*d*b^2*arctanh(c*x)*polylog(2,(c*x+1)^2/(-c^2*x^2+1))+c*d*b^2*arctanh(c*x)^2*ln(1-(c*x+1)^2/(-c^2*x^2
+1))+3/2*c*d*b^2*arctanh(c*x)*ln(1-(c*x+1)^2/(-c^2*x^2+1))-c*d*b^2*arctanh(c*x)^2*ln((c*x+1)^2/(-c^2*x^2+1)-1)
+c*d*b^2*arctanh(c*x)^2*ln(c*x)-c*d*b^2*arctanh(c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))-c*d*a*b*dilog(c*x)-c*d
*a*b*ln(c*x-1)+2*c*d*a*b*ln(c*x)-c*d*a*b*ln(c*x+1)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} c d \log \left (x\right ) -{\left (c{\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac{2 \, \operatorname{artanh}\left (c x\right )}{x}\right )} a b d - \frac{b^{2} d \log \left (-c x + 1\right )^{2}}{4 \, x} - \frac{a^{2} d}{x} - \int -\frac{{\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \log \left (c x + 1\right )^{2} + 4 \,{\left (a b c^{2} d x^{2} - a b c d x\right )} \log \left (c x + 1\right ) - 2 \,{\left (2 \, a b c^{2} d x^{2} -{\left (2 \, a b c d + b^{2} c d\right )} x +{\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{4 \,{\left (c x^{3} - x^{2}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*d*x+d)*(a+b*arctanh(c*x))^2/x^2,x, algorithm="maxima")
[Out]
a^2*c*d*log(x) - (c*(log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x)*a*b*d - 1/4*b^2*d*log(-c*x + 1)^2/x - a^
2*d/x - integrate(-1/4*((b^2*c^2*d*x^2 - b^2*d)*log(c*x + 1)^2 + 4*(a*b*c^2*d*x^2 - a*b*c*d*x)*log(c*x + 1) -
2*(2*a*b*c^2*d*x^2 - (2*a*b*c*d + b^2*c*d)*x + (b^2*c^2*d*x^2 - b^2*d)*log(c*x + 1))*log(-c*x + 1))/(c*x^3 - x
^2), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{2} c d x + a^{2} d +{\left (b^{2} c d x + b^{2} d\right )} \operatorname{artanh}\left (c x\right )^{2} + 2 \,{\left (a b c d x + a b d\right )} \operatorname{artanh}\left (c x\right )}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*d*x+d)*(a+b*arctanh(c*x))^2/x^2,x, algorithm="fricas")
[Out]
integral((a^2*c*d*x + a^2*d + (b^2*c*d*x + b^2*d)*arctanh(c*x)^2 + 2*(a*b*c*d*x + a*b*d)*arctanh(c*x))/x^2, x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} d \left (\int \frac{a^{2}}{x^{2}}\, dx + \int \frac{a^{2} c}{x}\, dx + \int \frac{b^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{2 a b \operatorname{atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{b^{2} c \operatorname{atanh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac{2 a b c \operatorname{atanh}{\left (c x \right )}}{x}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*d*x+d)*(a+b*atanh(c*x))**2/x**2,x)
[Out]
d*(Integral(a**2/x**2, x) + Integral(a**2*c/x, x) + Integral(b**2*atanh(c*x)**2/x**2, x) + Integral(2*a*b*atan
h(c*x)/x**2, x) + Integral(b**2*c*atanh(c*x)**2/x, x) + Integral(2*a*b*c*atanh(c*x)/x, x))
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c d x + d\right )}{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*d*x+d)*(a+b*arctanh(c*x))^2/x^2,x, algorithm="giac")
[Out]
integrate((c*d*x + d)*(b*arctanh(c*x) + a)^2/x^2, x)