Integrand size = 22, antiderivative size = 355 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x} \, dx=3 a b c d^3 x+\frac {1}{3} b^2 c d^3 x-\frac {1}{3} b^2 d^3 \text {arctanh}(c x)+3 b^2 c d^3 x \text {arctanh}(c x)+\frac {1}{3} b c^2 d^3 x^2 (a+b \text {arctanh}(c x))+\frac {11}{6} d^3 (a+b \text {arctanh}(c x))^2+3 c d^3 x (a+b \text {arctanh}(c x))^2+\frac {3}{2} c^2 d^3 x^2 (a+b \text {arctanh}(c x))^2+\frac {1}{3} c^3 d^3 x^3 (a+b \text {arctanh}(c x))^2+2 d^3 (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )-\frac {20}{3} b d^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )+\frac {3}{2} b^2 d^3 \log \left (1-c^2 x^2\right )-\frac {10}{3} b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-b d^3 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+b d^3 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )+\frac {1}{2} b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 d^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right ) \]
3*a*b*c*d^3*x+1/3*b^2*c*d^3*x-1/3*b^2*d^3*arctanh(c*x)+3*b^2*c*d^3*x*arcta nh(c*x)+1/3*b*c^2*d^3*x^2*(a+b*arctanh(c*x))+11/6*d^3*(a+b*arctanh(c*x))^2 +3*c*d^3*x*(a+b*arctanh(c*x))^2+3/2*c^2*d^3*x^2*(a+b*arctanh(c*x))^2+1/3*c ^3*d^3*x^3*(a+b*arctanh(c*x))^2-2*d^3*(a+b*arctanh(c*x))^2*arctanh(-1+2/(- c*x+1))-20/3*b*d^3*(a+b*arctanh(c*x))*ln(2/(-c*x+1))+3/2*b^2*d^3*ln(-c^2*x ^2+1)-10/3*b^2*d^3*polylog(2,1-2/(-c*x+1))-b*d^3*(a+b*arctanh(c*x))*polylo g(2,1-2/(-c*x+1))+b*d^3*(a+b*arctanh(c*x))*polylog(2,-1+2/(-c*x+1))+1/2*b^ 2*d^3*polylog(3,1-2/(-c*x+1))-1/2*b^2*d^3*polylog(3,-1+2/(-c*x+1))
Result contains complex when optimal does not.
Time = 0.60 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.26 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x} \, dx=\frac {1}{24} d^3 \left (i b^2 \pi ^3+72 a^2 c x+72 a b c x+8 b^2 c x+36 a^2 c^2 x^2+8 a b c^2 x^2+8 a^2 c^3 x^3-8 b^2 \text {arctanh}(c x)+144 a b c x \text {arctanh}(c x)+72 b^2 c x \text {arctanh}(c x)+72 a b c^2 x^2 \text {arctanh}(c x)+8 b^2 c^2 x^2 \text {arctanh}(c x)+16 a b c^3 x^3 \text {arctanh}(c x)-116 b^2 \text {arctanh}(c x)^2+72 b^2 c x \text {arctanh}(c x)^2+36 b^2 c^2 x^2 \text {arctanh}(c x)^2+8 b^2 c^3 x^3 \text {arctanh}(c x)^2-16 b^2 \text {arctanh}(c x)^3-160 b^2 \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-24 b^2 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+24 b^2 \text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+24 a^2 \log (c x)+36 a b \log (1-c x)-36 a b \log (1+c x)+72 a b \log \left (1-c^2 x^2\right )+36 b^2 \log \left (1-c^2 x^2\right )+8 a b \log \left (-1+c^2 x^2\right )+8 b^2 (10+3 \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+24 b^2 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )-24 a b \operatorname {PolyLog}(2,-c x)+24 a b \operatorname {PolyLog}(2,c x)+12 b^2 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )-12 b^2 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right ) \]
(d^3*(I*b^2*Pi^3 + 72*a^2*c*x + 72*a*b*c*x + 8*b^2*c*x + 36*a^2*c^2*x^2 + 8*a*b*c^2*x^2 + 8*a^2*c^3*x^3 - 8*b^2*ArcTanh[c*x] + 144*a*b*c*x*ArcTanh[c *x] + 72*b^2*c*x*ArcTanh[c*x] + 72*a*b*c^2*x^2*ArcTanh[c*x] + 8*b^2*c^2*x^ 2*ArcTanh[c*x] + 16*a*b*c^3*x^3*ArcTanh[c*x] - 116*b^2*ArcTanh[c*x]^2 + 72 *b^2*c*x*ArcTanh[c*x]^2 + 36*b^2*c^2*x^2*ArcTanh[c*x]^2 + 8*b^2*c^3*x^3*Ar cTanh[c*x]^2 - 16*b^2*ArcTanh[c*x]^3 - 160*b^2*ArcTanh[c*x]*Log[1 + E^(-2* ArcTanh[c*x])] - 24*b^2*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 24*b ^2*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + 24*a^2*Log[c*x] + 36*a*b*L og[1 - c*x] - 36*a*b*Log[1 + c*x] + 72*a*b*Log[1 - c^2*x^2] + 36*b^2*Log[1 - c^2*x^2] + 8*a*b*Log[-1 + c^2*x^2] + 8*b^2*(10 + 3*ArcTanh[c*x])*PolyLo g[2, -E^(-2*ArcTanh[c*x])] + 24*b^2*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c *x])] - 24*a*b*PolyLog[2, -(c*x)] + 24*a*b*PolyLog[2, c*x] + 12*b^2*PolyLo g[3, -E^(-2*ArcTanh[c*x])] - 12*b^2*PolyLog[3, E^(2*ArcTanh[c*x])]))/24
Time = 1.09 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6502, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(c d x+d)^3 (a+b \text {arctanh}(c x))^2}{x} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (c^3 d^3 x^2 (a+b \text {arctanh}(c x))^2+3 c^2 d^3 x (a+b \text {arctanh}(c x))^2+3 c d^3 (a+b \text {arctanh}(c x))^2+\frac {d^3 (a+b \text {arctanh}(c x))^2}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} c^3 d^3 x^3 (a+b \text {arctanh}(c x))^2+\frac {3}{2} c^2 d^3 x^2 (a+b \text {arctanh}(c x))^2+\frac {1}{3} b c^2 d^3 x^2 (a+b \text {arctanh}(c x))-b d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+b d^3 \operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))+3 c d^3 x (a+b \text {arctanh}(c x))^2+\frac {11}{6} d^3 (a+b \text {arctanh}(c x))^2+2 d^3 \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2-\frac {20}{3} b d^3 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+3 a b c d^3 x-\frac {1}{3} b^2 d^3 \text {arctanh}(c x)+3 b^2 c d^3 x \text {arctanh}(c x)+\frac {3}{2} b^2 d^3 \log \left (1-c^2 x^2\right )-\frac {10}{3} b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+\frac {1}{2} b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 d^3 \operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right )+\frac {1}{3} b^2 c d^3 x\) |
3*a*b*c*d^3*x + (b^2*c*d^3*x)/3 - (b^2*d^3*ArcTanh[c*x])/3 + 3*b^2*c*d^3*x *ArcTanh[c*x] + (b*c^2*d^3*x^2*(a + b*ArcTanh[c*x]))/3 + (11*d^3*(a + b*Ar cTanh[c*x])^2)/6 + 3*c*d^3*x*(a + b*ArcTanh[c*x])^2 + (3*c^2*d^3*x^2*(a + b*ArcTanh[c*x])^2)/2 + (c^3*d^3*x^3*(a + b*ArcTanh[c*x])^2)/3 + 2*d^3*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] - (20*b*d^3*(a + b*ArcTanh[c*x ])*Log[2/(1 - c*x)])/3 + (3*b^2*d^3*Log[1 - c^2*x^2])/2 - (10*b^2*d^3*Poly Log[2, 1 - 2/(1 - c*x)])/3 - b*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/( 1 - c*x)] + b*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] + (b^2 *d^3*PolyLog[3, 1 - 2/(1 - c*x)])/2 - (b^2*d^3*PolyLog[3, -1 + 2/(1 - c*x) ])/2
3.1.88.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(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])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 7.53 (sec) , antiderivative size = 959, normalized size of antiderivative = 2.70
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(959\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(961\) |
| default | \(\text {Expression too large to display}\) | \(961\) |
d^3*a^2*(1/3*c^3*x^3+3/2*c^2*x^2+3*c*x+ln(x))+d^3*b^2*(3*c*x*arctanh(c*x)^ 2-1/3-20/3*dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-20/3*dilog(1-I*(c*x+1)/(- c^2*x^2+1)^(1/2))+1/3*c*x+3/2*c^2*x^2*arctanh(c*x)^2+11/6*arctanh(c*x)^2-3 *ln(1+(c*x+1)^2/(-c^2*x^2+1))-20/3*arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1 )^(1/2))-20/3*arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+1/2*polylog( 3,-(c*x+1)^2/(-c^2*x^2+1))-2*polylog(3,-(c*x+1)/(-c^2*x^2+1)^(1/2))-2*poly log(3,(c*x+1)/(-c^2*x^2+1)^(1/2))+ln(c*x)*arctanh(c*x)^2-arctanh(c*x)*poly log(2,-(c*x+1)^2/(-c^2*x^2+1))-arctanh(c*x)^2*ln((c*x+1)^2/(-c^2*x^2+1)-1) +arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)*polylog(2, -(c*x+1)/(-c^2*x^2+1)^(1/2))+arctanh(c*x)^2*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2 ))+2*arctanh(c*x)*polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))+1/3*arctanh(c*x)^2 *c^3*x^3+1/2*I*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))*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+11/3*(c*x+1)*arctanh(c*x)-1/2*I*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*a rctanh(c*x)^2-1/2*I*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+1/2*I*Pi*csg n(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 +1/3*(c*x-3)*(c*x+1)*arctanh(c*x))+2*d^3*a*b*(1/3*c^3*x^3*arctanh(c*x)+3/2 *c^2*x^2*arctanh(c*x)+3*c*x*arctanh(c*x)+ln(c*x)*arctanh(c*x)-1/2*dilog...
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
integral((a^2*c^3*d^3*x^3 + 3*a^2*c^2*d^3*x^2 + 3*a^2*c*d^3*x + a^2*d^3 + (b^2*c^3*d^3*x^3 + 3*b^2*c^2*d^3*x^2 + 3*b^2*c*d^3*x + b^2*d^3)*arctanh(c* x)^2 + 2*(a*b*c^3*d^3*x^3 + 3*a*b*c^2*d^3*x^2 + 3*a*b*c*d^3*x + a*b*d^3)*a rctanh(c*x))/x, x)
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x} \, dx=d^{3} \left (\int 3 a^{2} c\, dx + \int \frac {a^{2}}{x}\, dx + \int 3 a^{2} c^{2} x\, dx + \int a^{2} c^{3} x^{2}\, dx + \int 3 b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x}\, dx + \int 6 a b c \operatorname {atanh}{\left (c x \right )}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x}\, dx + \int 3 b^{2} c^{2} x \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{3} x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 6 a b c^{2} x \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a b c^{3} x^{2} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]
d**3*(Integral(3*a**2*c, x) + Integral(a**2/x, x) + Integral(3*a**2*c**2*x , x) + Integral(a**2*c**3*x**2, x) + Integral(3*b**2*c*atanh(c*x)**2, x) + Integral(b**2*atanh(c*x)**2/x, x) + Integral(6*a*b*c*atanh(c*x), x) + Int egral(2*a*b*atanh(c*x)/x, x) + Integral(3*b**2*c**2*x*atanh(c*x)**2, x) + Integral(b**2*c**3*x**2*atanh(c*x)**2, x) + Integral(6*a*b*c**2*x*atanh(c* x), x) + Integral(2*a*b*c**3*x**2*atanh(c*x), x))
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
1/3*a^2*c^3*d^3*x^3 + 3/2*a^2*c^2*d^3*x^2 + 3*a^2*c*d^3*x + 3*(2*c*x*arcta nh(c*x) + log(-c^2*x^2 + 1))*a*b*d^3 + a^2*d^3*log(x) + 1/24*(2*b^2*c^3*d^ 3*x^3 + 9*b^2*c^2*d^3*x^2 + 18*b^2*c*d^3*x)*log(-c*x + 1)^2 - integrate(-1 /12*(3*(b^2*c^4*d^3*x^4 + 2*b^2*c^3*d^3*x^3 - 2*b^2*c*d^3*x - b^2*d^3)*log (c*x + 1)^2 + 12*(a*b*c^4*d^3*x^4 + 2*a*b*c^3*d^3*x^3 - 3*a*b*c^2*d^3*x^2 + a*b*c*d^3*x - a*b*d^3)*log(c*x + 1) - (12*a*b*c*d^3*x - 12*a*b*d^3 + 2*( 6*a*b*c^4*d^3 + b^2*c^4*d^3)*x^4 + 3*(8*a*b*c^3*d^3 + 3*b^2*c^3*d^3)*x^3 - 18*(2*a*b*c^2*d^3 - b^2*c^2*d^3)*x^2 + 6*(b^2*c^4*d^3*x^4 + 2*b^2*c^3*d^3 *x^3 - 2*b^2*c*d^3*x - b^2*d^3)*log(c*x + 1))*log(-c*x + 1))/(c*x^2 - x), x)
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
Timed out. \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^3}{x} \,d x \]