3.1.32 \(\int \frac {(a+b \tanh ^{-1}(c x))^3}{x^3} \, dx\) [32]
Optimal. Leaf size=123 \[ \frac {3}{2} b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x}+\frac {1}{2} c^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-\frac {3}{2} b^3 c^2 \text {PolyLog}\left (2,-1+\frac {2}{1+c x}\right ) \]
[Out]
3/2*b*c^2*(a+b*arctanh(c*x))^2-3/2*b*c*(a+b*arctanh(c*x))^2/x+1/2*c^2*(a+b*arctanh(c*x))^3-1/2*(a+b*arctanh(c*
x))^3/x^2+3*b^2*c^2*(a+b*arctanh(c*x))*ln(2-2/(c*x+1))-3/2*b^3*c^2*polylog(2,-1+2/(c*x+1))
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Rubi [A]
time = 0.21, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6037, 6129,
6135, 6079, 2497, 6095} \begin {gather*} 3 b^2 c^2 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{2} b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x}-\frac {3}{2} b^3 c^2 \text {Li}_2\left (\frac {2}{c x+1}-1\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*ArcTanh[c*x])^3/x^3,x]
[Out]
(3*b*c^2*(a + b*ArcTanh[c*x])^2)/2 - (3*b*c*(a + b*ArcTanh[c*x])^2)/(2*x) + (c^2*(a + b*ArcTanh[c*x])^3)/2 - (
a + b*ArcTanh[c*x])^3/(2*x^2) + 3*b^2*c^2*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - (3*b^3*c^2*PolyLog[2, -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 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 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 6129
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d
, Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x], x] - Dist[e/(d*f^2), 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] && LtQ[m, -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]
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{x^3} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+\frac {1}{2} (3 b c) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+\frac {1}{2} (3 b c) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx+\frac {1}{2} \left (3 b c^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx\\ &=-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x}+\frac {1}{2} c^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+\left (3 b^2 c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx\\ &=\frac {3}{2} b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x}+\frac {1}{2} c^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+\left (3 b^2 c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx\\ &=\frac {3}{2} b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x}+\frac {1}{2} c^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-\left (3 b^3 c^3\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=\frac {3}{2} b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x}+\frac {1}{2} c^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-\frac {3}{2} b^3 c^2 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 192, normalized size = 1.56 \begin {gather*} \frac {6 b^2 (-1+c x) (a+a c x+b c x) \tanh ^{-1}(c x)^2+2 b^3 \left (-1+c^2 x^2\right ) \tanh ^{-1}(c x)^3-6 b \tanh ^{-1}(c x) \left (a^2+2 a b c x-2 b^2 c^2 x^2 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )\right )+a \left (-2 a^2-6 a b c x-3 a b c^2 x^2 \log (1-c x)+3 a b c^2 x^2 \log (1+c x)+12 b^2 c^2 x^2 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )\right )-6 b^3 c^2 x^2 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*ArcTanh[c*x])^3/x^3,x]
[Out]
(6*b^2*(-1 + c*x)*(a + a*c*x + b*c*x)*ArcTanh[c*x]^2 + 2*b^3*(-1 + c^2*x^2)*ArcTanh[c*x]^3 - 6*b*ArcTanh[c*x]*
(a^2 + 2*a*b*c*x - 2*b^2*c^2*x^2*Log[1 - E^(-2*ArcTanh[c*x])]) + a*(-2*a^2 - 6*a*b*c*x - 3*a*b*c^2*x^2*Log[1 -
c*x] + 3*a*b*c^2*x^2*Log[1 + c*x] + 12*b^2*c^2*x^2*Log[(c*x)/Sqrt[1 - c^2*x^2]]) - 6*b^3*c^2*x^2*PolyLog[2, E
^(-2*ArcTanh[c*x])])/(4*x^2)
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.58, size = 4871, normalized size = 39.60
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| method |
result |
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| derivativedivides |
\(\text {Expression too large to display}\) |
\(4871\) |
| default |
\(\text {Expression too large to display}\) |
\(4871\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*arctanh(c*x))^3/x^3,x,method=_RETURNVERBOSE)
[Out]
c^2*(-1/2*a^3/c^2/x^2-3/4*I*b^3*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*dilog(1+
(c*x+1)/(-c^2*x^2+1)^(1/2))-3/8*I*b^3*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)
))^3+3/4*I*b^3*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))-3/8*I*b^3*Pi*csgn(I
/(1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2
+1)))*arctanh(c*x)*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))-3/4*I*b^3*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*dilog((c
*x+1)/(-c^2*x^2+1)^(1/2))-3/4*I*b^3*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/
2))-3/8*I*b^3*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2)
)+3/8*I*b^3*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*dilog((c*x+1)/(-c^2*x^2+1)^(1/2))-3/8*I*b^3*Pi*csgn(I*(c*x+1)^2
/(c^2*x^2-1))^3*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+3/8*I*b^3*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*polylog(2,(c*
x+1)/(-c^2*x^2+1)^(1/2))+3/8*I*b^3*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*polylog(2,-(c
*x+1)/(-c^2*x^2+1)^(1/2))+3/8*I*b^3*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*polylog(2,(c
*x+1)/(-c^2*x^2+1)^(1/2))+3/8*I*b^3*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*dilog((c*x+1
)/(-c^2*x^2+1)^(1/2))+3/8*I*b^3*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))-3/4*
I*b^3*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))+3/4*I*b^3*Pi*csgn(I/(1+(c
*x+1)^2/(-c^2*x^2+1)))^2*polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))+3/4*I*b^3*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))
^2*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))+3/4*I*b^3*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*dilog((c*x+1)/(-c^
2*x^2+1)^(1/2))-3/4*I*b^3*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))-3/4*I*b^
3*Pi*arctanh(c*x)*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))-3/4*I*b^3*arctanh(c*x)^2*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1
)))^2+3/4*I*b^3*arctanh(c*x)^2*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^3-3/8*I*b^3*arctanh(c*x)^2*Pi*csgn(I*(c*x
+1)^2/(c^2*x^2-1))^3-3/2*a^2*b/c^2/x^2*arctanh(c*x)-3/2*a*b^2/c^2/x^2*arctanh(c*x)^2-3*a*b^2*arctanh(c*x)/c/x-
1/2*b^3/c^2/x^2*arctanh(c*x)^3-3/4*I*b^3*Pi*dilog((c*x+1)/(-c^2*x^2+1)^(1/2))+3/4*I*b^3*arctanh(c*x)^2*Pi+3/4*
I*b^3*Pi*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))-3/4*I*b^3*Pi*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))-3/4*I*b^3*Pi*
polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))-3/4*a^2*b*ln(c*x-1)+3/4*a^2*b*ln(c*x+1)-3/2*b^3*arctanh(c*x)^2*ln((c*x+1
)/(-c^2*x^2+1)^(1/2))-3/4*b^3*arctanh(c*x)^2*ln(c*x-1)+3/4*b^3*arctanh(c*x)^2*ln(c*x+1)-3/8*a*b^2*ln(c*x-1)^2-
3/8*a*b^2*ln(c*x+1)^2-3/2*a*b^2*ln(c*x-1)-3/2*a*b^2*ln(c*x+1)+3/2*a*b^2*arctanh(c*x)*ln(c*x+1)+3/4*a*b^2*ln(c*
x-1)*ln(1/2*c*x+1/2)+3/4*a*b^2*ln(-1/2*c*x+1/2)*ln(c*x+1)-3/4*a*b^2*ln(-1/2*c*x+1/2)*ln(1/2*c*x+1/2)-3/2*a*b^2
*arctanh(c*x)*ln(c*x-1)-3/2*b^3*dilog((c*x+1)/(-c^2*x^2+1)^(1/2))+3/2*b^3*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+
3/2*b^3*polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))+3/2*b^3*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))-3/2*a^2*b/c/x-3/2
*b^3*arctanh(c*x)^2/c/x-3/2*b^3*arctanh(c*x)^2+1/2*b^3*arctanh(c*x)^3+3*a*b^2*ln(c*x)+3*b^3*arctanh(c*x)*ln(1+
(c*x+1)/(-c^2*x^2+1)^(1/2))+3/2*b^3*arctanh(c*x)*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))+3/8*I*b^3*Pi*csgn(I*(c*x+1)^
2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*arctanh(c*x)*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))+3/4*I*b^3*Pi*csgn(I/
(1+(c*x+1)^2/(-c^2*x^2+1)))^2*arctanh(c*x)*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))-3/4*I*b^3*Pi*csgn(I/(1+(c*x+1)^2/(
-c^2*x^2+1)))^3*arctanh(c*x)*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))-3/8*I*b^3*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+(c*x+1)^2/(-c^2*x^2+1)))^2*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))-3/8*I*b^3*Pi*cs
gn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*polylog(2,(c*x+1)/(-c^2
*x^2+1)^(1/2))-3/8*I*b^3*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+
1)))^2*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))+3/8*I*b^3*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*arctanh(c*x)*ln(1-(
c*x+1)/(-c^2*x^2+1)^(1/2))+3/4*I*b^3*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*dil
og((c*x+1)/(-c^2*x^2+1)^(1/2))+3/8*I*b^3*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
+(c*x+1)^2/(-c^2*x^2+1)))^2*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))+3/4*I*b^3*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*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))+3/8*I*b^3*Pi*csgn(I*(c*x+1)^2/(c^2
*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))-3/8*I*
b^3*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*dilog((c*x+1)/
(-c^2*x^2+1)^(1/2))-3/8*I*b^3*arctanh(c*x)^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))-3/4*I*b^3*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2-3/8*I*b^3*
Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*...
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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((a+b*arctanh(c*x))^3/x^3,x, algorithm="maxima")
[Out]
3/4*((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arctanh(c*x)/x^2)*a^2*b + 3/8*((2*(log(c*x - 1) - 2)*log(c*
x + 1) - log(c*x + 1)^2 - log(c*x - 1)^2 - 4*log(c*x - 1) + 8*log(x))*c^2 + 4*(c*log(c*x + 1) - c*log(c*x - 1)
- 2/x)*c*arctanh(c*x))*a*b^2 - 1/16*b^3*(((c^2*x^2 - 1)*log(-c*x + 1)^3 + 3*(2*c*x - (c^2*x^2 - 1)*log(c*x +
1))*log(-c*x + 1)^2)/x^2 + 2*integrate(-((c*x - 1)*log(c*x + 1)^3 + 3*(2*c^2*x^2 - (c*x - 1)*log(c*x + 1)^2 -
(c^3*x^3 - c*x)*log(c*x + 1))*log(-c*x + 1))/(c*x^4 - x^3), x)) - 3/2*a*b^2*arctanh(c*x)^2/x^2 - 1/2*a^3/x^2
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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((a+b*arctanh(c*x))^3/x^3,x, algorithm="fricas")
[Out]
integral((b^3*arctanh(c*x)^3 + 3*a*b^2*arctanh(c*x)^2 + 3*a^2*b*arctanh(c*x) + a^3)/x^3, x)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*atanh(c*x))**3/x**3,x)
[Out]
Integral((a + b*atanh(c*x))**3/x**3, x)
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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((a+b*arctanh(c*x))^3/x^3,x, algorithm="giac")
[Out]
integrate((b*arctanh(c*x) + a)^3/x^3, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*atanh(c*x))^3/x^3,x)
[Out]
int((a + b*atanh(c*x))^3/x^3, x)
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