3.1.73 \(\int \frac {(d+c d x) (a+b \tanh ^{-1}(c x))^2}{x^2} \, dx\) [73]
Optimal. Leaf size=201 \[ c d \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c d \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+2 b c d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-b c d \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+b c d \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )-b^2 c d \text {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )+\frac {1}{2} b^2 c d \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c d \text {PolyLog}\left (3,-1+\frac {2}{1-c x}\right ) \]
[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/(-c*x+1))+2*b*c*d*(a
+b*arctanh(c*x))*ln(2-2/(c*x+1))-b*c*d*(a+b*arctanh(c*x))*polylog(2,1-2/(-c*x+1))+b*c*d*(a+b*arctanh(c*x))*pol
ylog(2,-1+2/(-c*x+1))-b^2*c*d*polylog(2,-1+2/(c*x+1))+1/2*b^2*c*d*polylog(3,1-2/(-c*x+1))-1/2*b^2*c*d*polylog(
3,-1+2/(-c*x+1))
________________________________________________________________________________________
Rubi [A]
time = 0.36, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6087, 6037,
6135, 6079, 2497, 6033, 6199, 6095, 6205, 6745} \begin {gather*} -b c d \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b c d \text {Li}_2\left (\frac {2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\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 )+b^2 (-c) d \text {Li}_2\left (\frac {2}{c x+1}-1\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 (\frac {2}{1-c x}-1\right ) \end {gather*}
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 2497
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 6033
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 6037
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTanh[c*
x^n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))
), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1
]
Rule 6079
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 6087
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 6095
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 6135
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 6199
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 6205
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 6745
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] Result contains complex when optimal does not.
time = 0.29, size = 249, normalized size = 1.24 \begin {gather*} -\frac {d \left (a^2-a^2 c x \log (x)+a b \left (2 \tanh ^{-1}(c x)+c x \left (-2 \log (c x)+\log \left (1-c^2 x^2\right )\right )\right )+b^2 \left (\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 )+c x \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )\right )+a b c x (\text {PolyLog}(2,-c x)-\text {PolyLog}(2,c x))-b^2 c x \left (\frac {i \pi ^3}{24}-\frac {2}{3} \tanh ^{-1}(c x)^3-\tanh ^{-1}(c x)^2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )+\tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )+\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 )\right )\right )}{x} \end {gather*}
Antiderivative was successfully verified.
[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] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 3.75, size = 3070, normalized size = 15.27
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\text {Expression too large to display}\) |
\(3070\) |
| default |
\(\text {Expression too large to display}\) |
\(3070\) |
| | |
|
|
|
|
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,method=_RETURNVERBOSE)
[Out]
c*(-1/8*I*d*b^2*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*dilog((c*x+1)^2/(-c^2*x^2+1
))-d*a*b*ln(c*x-1)-1/8*I*d*b^2*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*polylog(2,-(
c*x+1)^2/(-c^2*x^2+1))-1/8*I*d*b^2*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*polylog(
2,(c*x+1)^2/(-c^2*x^2+1))+1/8*I*d*b^2*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*dilog
(1+(c*x+1)^2/(-c^2*x^2+1))-2*d*a*b*arctanh(c*x)/c/x-1/8*I*d*b^2*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1))*csgn(I/(
1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))*polylog(2,(c*x+1)^2/(
-c^2*x^2+1))-1/8*I*d*b^2*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1))*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*((c*x
+1)^2/(-c^2*x^2+1)-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))*dilog((c*x+1)^2/(-c^2*x^2+1))-1/8*I*d*b^2*Pi*csgn(I*((c*x+1)
^2/(-c^2*x^2+1)-1))*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/(1+(c*x+1)^2/(-c^2*x^
2+1)))*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+1/8*I*d*b^2*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1))*csgn(I/(1+(c*x+1)^
2/(-c^2*x^2+1)))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))*dilog(1+(c*x+1)^2/(-c^2*x^2+1))
+1/4*I*d*b^2*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/(1+(c*x+1)^2/(-c^2*x^2+1)
))^2*arctanh(c*x)*ln(1-(c*x+1)^2/(-c^2*x^2+1))+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)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*arctanh(c*x)*ln(1-(c*x+1)^2/(-c^2*x^2+1))-d*a*b*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/(1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*((c*x+1)^2/(-c^2*x^
2+1)-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))*arctanh(c*x)^2-1/2*I*d*b^2*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*((
c*x+1)^2/(-c^2*x^2+1)-1)/(1+(c*x+1)^2/(-c^2*x^2+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)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*arctanh(c*x)^2+d*a^2*ln(c*x)-d*b^2*a
rctanh(c*x)^2+1/2*d*b^2*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))-1/4*d*b^2*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+1/8*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)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*po
lylog(2,(c*x+1)^2/(-c^2*x^2+1))+1/8*I*d*b^2*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*((c*x+1)^2/(-c^2*x^2+
1)-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*polylog(2,(c*x+1)^2/(-c^2*x^2+1))+1/8*I*d*b^2*Pi*csgn(I/(1+(c*x+1)^2/(-c^2
*x^2+1)))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*dilog((c*x+1)^2/(-c^2*x^2+1))+1/8*I*
d*b^2*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*po
lylog(2,-(c*x+1)^2/(-c^2*x^2+1))+1/8*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)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+1/8*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)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*dilog((c*x+1)^2/(-c^2*x^2+1))-1/8*
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)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*
dilog(1+(c*x+1)^2/(-c^2*x^2+1))-1/4*I*d*b^2*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^3
*arctanh(c*x)*ln(1-(c*x+1)^2/(-c^2*x^2+1))-1/8*I*d*b^2*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*((c*x+1)^2
/(-c^2*x^2+1)-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*dilog(1+(c*x+1)^2/(-c^2*x^2+1))+3/2*d*b^2*arctanh(c*x)*ln(1-(c*
x+1)^2/(-c^2*x^2+1))+d*b^2*arctanh(c*x)*polylog(2,(c*x+1)^2/(-c^2*x^2+1))+d*b^2*arctanh(c*x)^2*ln(1-(c*x+1)^2/
(-c^2*x^2+1))+2*d*a*b*ln(c*x)-1/4*I*d*b^2*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1))*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+
1)))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))*arctanh(c*x)*ln(1-(c*x+1)^2/(-c^2*x^2+1))-d
*b^2*arctanh(c*x)^2/c/x-d*a*b*dilog(c*x)-d*a*b*dilog(c*x+1)+d*b^2*arctanh(c*x)^2*ln(c*x)-d*b^2*arctanh(c*x)*po
lylog(2,-(c*x+1)^2/(-c^2*x^2+1))-d*b^2*arctanh(c*x)^2*ln((c*x+1)^2/(-c^2*x^2+1)-1)-1/4*d*b^2*dilog((c*x+1)^2/(
-c^2*x^2+1))+1/4*d*b^2*dilog(1+(c*x+1)^2/(-c^2*x^2+1))-1/2*d*b^2*polylog(3,(c*x+1)^2/(-c^2*x^2+1))+3/4*d*b^2*p
olylog(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)/(1+(c*x+1)^2/(-c^2*x^2+1)))^
3*arctanh(c*x)^2+2*d*a*b*arctanh(c*x)*ln(c*x)-d*a*b*ln(c*x)*ln(c*x+1)-d*a^2/c/x)
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
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)
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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)
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 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 {gather*}
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))
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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)
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((a + b*atanh(c*x))^2*(d + c*d*x))/x^2,x)
[Out]
int(((a + b*atanh(c*x))^2*(d + c*d*x))/x^2, x)
________________________________________________________________________________________