3.72 \(\int \frac{(d+c d x) (a+b \tanh ^{-1}(c x))^2}{x} \, dx\)
Optimal. Leaf size=191 \[ -b d \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b d \text{PolyLog}\left (2,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )+b^2 (-d) \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )+\frac{1}{2} b^2 d \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 d \text{PolyLog}\left (3,\frac{2}{1-c x}-1\right )+d \left (a+b \tanh ^{-1}(c x)\right )^2+c d x \left (a+b \tanh ^{-1}(c x)\right )^2+2 d \tanh ^{-1}\left (1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2-2 b d \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right ) \]
[Out]
d*(a + b*ArcTanh[c*x])^2 + c*d*x*(a + b*ArcTanh[c*x])^2 + 2*d*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)]
- 2*b*d*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)] - b^2*d*PolyLog[2, 1 - 2/(1 - c*x)] - b*d*(a + b*ArcTanh[c*x])*P
olyLog[2, 1 - 2/(1 - c*x)] + b*d*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] + (b^2*d*PolyLog[3, 1 - 2/(
1 - c*x)])/2 - (b^2*d*PolyLog[3, -1 + 2/(1 - c*x)])/2
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Rubi [A] time = 0.450618, antiderivative size = 191, normalized size of antiderivative =
1., number of steps used = 13, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) =
0.55, Rules used = {5940, 5910, 5984, 5918, 2402, 2315, 5914, 6052, 5948, 6058, 6610}
\[ -b d \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b d \text{PolyLog}\left (2,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )+b^2 (-d) \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )+\frac{1}{2} b^2 d \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 d \text{PolyLog}\left (3,\frac{2}{1-c x}-1\right )+d \left (a+b \tanh ^{-1}(c x)\right )^2+c d x \left (a+b \tanh ^{-1}(c x)\right )^2+2 d \tanh ^{-1}\left (1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2-2 b d \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right ) \]
Antiderivative was successfully verified.
[In]
Int[((d + c*d*x)*(a + b*ArcTanh[c*x])^2)/x,x]
[Out]
d*(a + b*ArcTanh[c*x])^2 + c*d*x*(a + b*ArcTanh[c*x])^2 + 2*d*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)]
- 2*b*d*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)] - b^2*d*PolyLog[2, 1 - 2/(1 - c*x)] - b*d*(a + b*ArcTanh[c*x])*P
olyLog[2, 1 - 2/(1 - c*x)] + b*d*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] + (b^2*d*PolyLog[3, 1 - 2/(
1 - c*x)])/2 - (b^2*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 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 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 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} \, dx &=\int \left (c d \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{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} \, dx+(c d) \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=c d x \left (a+b \tanh ^{-1}(c x)\right )^2+2 d \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-(4 b c d) \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-\left (2 b c^2 d\right ) \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=d \left (a+b \tanh ^{-1}(c x)\right )^2+c d x \left (a+b \tanh ^{-1}(c x)\right )^2+2 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)}{1-c x} \, dx+(2 b c d) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-(2 b c d) \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\\ &=d \left (a+b \tanh ^{-1}(c x)\right )^2+c d x \left (a+b \tanh ^{-1}(c x)\right )^2+2 d \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )-b d \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b d \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )+\left (b^2 c 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 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 d\right ) \int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=d \left (a+b \tanh ^{-1}(c x)\right )^2+c d x \left (a+b \tanh ^{-1}(c x)\right )^2+2 d \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )-b d \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b d \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )+\frac{1}{2} b^2 d \text{Li}_3\left (1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 d \text{Li}_3\left (-1+\frac{2}{1-c x}\right )-\left (2 b^2 d\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c x}\right )\\ &=d \left (a+b \tanh ^{-1}(c x)\right )^2+c d x \left (a+b \tanh ^{-1}(c x)\right )^2+2 d \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )-b^2 d \text{Li}_2\left (1-\frac{2}{1-c x}\right )-b d \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b d \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )+\frac{1}{2} b^2 d \text{Li}_3\left (1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 d \text{Li}_3\left (-1+\frac{2}{1-c x}\right )\\ \end{align*}
Mathematica [C] time = 0.530007, size = 228, normalized size = 1.19 \[ d \left (a b (\text{PolyLog}(2,c x)-\text{PolyLog}(2,-c x))+b^2 \left (\text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+\tanh ^{-1}(c x) \left ((c x-1) \tanh ^{-1}(c x)-2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )\right )+b^2 \left (\tanh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+\tanh ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \tanh ^{-1}(c x)}\right )+\frac{1}{2} \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(c x)}\right )-\frac{2}{3} \tanh ^{-1}(c x)^3-\tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+\tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )+\frac{i \pi ^3}{24}\right )+a^2 c x+a^2 \log (c x)+a b \left (\log \left (1-c^2 x^2\right )+2 c x \tanh ^{-1}(c x)\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
Integrate[((d + c*d*x)*(a + b*ArcTanh[c*x])^2)/x,x]
[Out]
d*(a^2*c*x + a^2*Log[c*x] + a*b*(2*c*x*ArcTanh[c*x] + Log[1 - c^2*x^2]) + b^2*(ArcTanh[c*x]*((-1 + c*x)*ArcTan
h[c*x] - 2*Log[1 + E^(-2*ArcTanh[c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + a*b*(-PolyLog[2, -(c*x)] + Poly
Log[2, c*x]) + b^2*((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*ArcTanh[c*x])]/2))
________________________________________________________________________________________
Maple [C] time = 0.403, size = 3644, normalized size = 19.1 \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,x)
[Out]
a^2*d*ln(c*x)-1/2*d*b^2*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))-d*b^2*dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-2*d*b^2
*polylog(3,-(c*x+1)/(-c^2*x^2+1)^(1/2))+d*b^2*arctanh(c*x)^2-d*b^2*dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+1/2*d
*b^2*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))-2*d*b^2*polylog(3,(c*x+1)/(-c^2*x^2+1)^(1/2))+1/2*I*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+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-d*a*b*dilog(c*x)+2*d*b^2*arctanh(c*x)*polylog(2,(
c*x+1)/(-c^2*x^2+1)^(1/2))+d*a*b*ln(c*x-1)+d*a*b*ln(c*x+1)+d*b^2*arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2
))+a^2*d*c*x+2*d*b^2*arctanh(c*x)*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))+d*b^2*arctanh(c*x)^2*ln(1-(c*x+1)/(-c
^2*x^2+1)^(1/2))-d*b^2*arctanh(c*x)*ln((c*x+1)^2/(-c^2*x^2+1)+1)-d*b^2*arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1
)^(1/2))-d*b^2*arctanh(c*x)^2*ln((c*x+1)^2/(-c^2*x^2+1)-1)-d*b^2*arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2
))+d*b^2*arctanh(c*x)^2*ln(c*x)-d*b^2*arctanh(c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))-d*a*b*dilog(c*x+1)+2*d*a
*b*arctanh(c*x)*c*x+1/2*I*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(1-I*(
c*x+1)/(-c^2*x^2+1)^(1/2))+1/2*I*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*arct
anh(c*x)^2+1/2*I*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(1+I*(c*x+1)/(-
c^2*x^2+1)^(1/2))-1/4*I*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))+2*d*a*b*arctanh(c*x)*ln(c*x)-d*a*b*ln(c*x)*ln(c*x+1)-1/2*I*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(1-I*(c*x+1)/(-c^2*x^2+1)^
(1/2))+1/4*I*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))+1/2*I*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+I*(c*x+1)/(-c^2*x^2+1)^(1/2))+d*b^2*arctanh(c*x)^2*c*x+1/2*I*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-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+1/4*I*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))-1/2*I*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((c*x+1)^2/(-c^2*x^2+1)+1)-1/2*I*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((c*x+1)^2/(-c^2*x^2+1)+1)-1/2*I*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(1-I*(c*x+1)/(-c^2*
x^2+1)^(1/2))+1/2*I*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
-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-1/2*I*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)^2-1/2*I*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(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-1/2*I*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)^2
-1/2*I*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(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))+1/2*I*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)^2-1/2*I*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-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-1/2*I*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+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-1/2*I*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
-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+1/2*I*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(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))+1/2*
I*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(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-1/2*I*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+I*(c*x+1)/(-c^2*x
^2+1)^(1/2))+1/2*I*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((c*x+1)^2/(-c^2*x^2+1)+1)-1/4*I*d*b^2*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1))*c
sgn(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))+1/2*I*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((c*x+1)^2/(-c^2*x^2+1)+1)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \, b^{2} c d x \log \left (-c x + 1\right )^{2} + a^{2} c d x +{\left (2 \, c x \operatorname{artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} a b d + a^{2} d \log \left (x\right ) - \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 d x - a b d\right )} \log \left (c x + 1\right ) - 2 \,{\left (b^{2} c^{2} d x^{2} + 2 \, a b c d x - 2 \, a b d +{\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^{2} - x\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,x, algorithm="maxima")
[Out]
1/4*b^2*c*d*x*log(-c*x + 1)^2 + a^2*c*d*x + (2*c*x*arctanh(c*x) + log(-c^2*x^2 + 1))*a*b*d + a^2*d*log(x) - in
tegrate(-1/4*((b^2*c^2*d*x^2 - b^2*d)*log(c*x + 1)^2 + 4*(a*b*c*d*x - a*b*d)*log(c*x + 1) - 2*(b^2*c^2*d*x^2 +
2*a*b*c*d*x - 2*a*b*d + (b^2*c^2*d*x^2 - b^2*d)*log(c*x + 1))*log(-c*x + 1))/(c*x^2 - x), 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}, 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,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, x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} d \left (\int a^{2} c\, dx + \int \frac{a^{2}}{x}\, dx + \int b^{2} c \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int \frac{b^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{x}\, dx + \int 2 a b c \operatorname{atanh}{\left (c x \right )}\, dx + \int \frac{2 a b \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,x)
[Out]
d*(Integral(a**2*c, x) + Integral(a**2/x, x) + Integral(b**2*c*atanh(c*x)**2, x) + Integral(b**2*atanh(c*x)**2
/x, x) + Integral(2*a*b*c*atanh(c*x), x) + Integral(2*a*b*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}\,{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,x, algorithm="giac")
[Out]
integrate((c*d*x + d)*(b*arctanh(c*x) + a)^2/x, x)