Integrand size = 22, antiderivative size = 361 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^2} \, dx=a b c^2 d^3 x+b^2 c^2 d^3 x \text {arctanh}(c x)+\frac {7}{2} c d^3 (a+b \text {arctanh}(c x))^2-\frac {d^3 (a+b \text {arctanh}(c x))^2}{x}+3 c^2 d^3 x (a+b \text {arctanh}(c x))^2+\frac {1}{2} c^3 d^3 x^2 (a+b \text {arctanh}(c x))^2+6 c d^3 (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )-6 b c d^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )+\frac {1}{2} b^2 c d^3 \log \left (1-c^2 x^2\right )+2 b c d^3 (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )-3 b^2 c d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-3 b c d^3 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+3 b c d^3 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )-b^2 c d^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )+\frac {3}{2} b^2 c d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {3}{2} b^2 c d^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right ) \]
a*b*c^2*d^3*x+b^2*c^2*d^3*x*arctanh(c*x)+7/2*c*d^3*(a+b*arctanh(c*x))^2-d^ 3*(a+b*arctanh(c*x))^2/x+3*c^2*d^3*x*(a+b*arctanh(c*x))^2+1/2*c^3*d^3*x^2* (a+b*arctanh(c*x))^2-6*c*d^3*(a+b*arctanh(c*x))^2*arctanh(-1+2/(-c*x+1))-6 *b*c*d^3*(a+b*arctanh(c*x))*ln(2/(-c*x+1))+1/2*b^2*c*d^3*ln(-c^2*x^2+1)+2* b*c*d^3*(a+b*arctanh(c*x))*ln(2-2/(c*x+1))-3*b^2*c*d^3*polylog(2,1-2/(-c*x +1))-3*b*c*d^3*(a+b*arctanh(c*x))*polylog(2,1-2/(-c*x+1))+3*b*c*d^3*(a+b*a rctanh(c*x))*polylog(2,-1+2/(-c*x+1))-b^2*c*d^3*polylog(2,-1+2/(c*x+1))+3/ 2*b^2*c*d^3*polylog(3,1-2/(-c*x+1))-3/2*b^2*c*d^3*polylog(3,-1+2/(-c*x+1))
Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.33 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^2} \, dx=\frac {d^3 \left (-8 a^2+i b^2 c \pi ^3 x+24 a^2 c^2 x^2+8 a b c^2 x^2+4 a^2 c^3 x^3-16 a b \text {arctanh}(c x)+48 a b c^2 x^2 \text {arctanh}(c x)+8 b^2 c^2 x^2 \text {arctanh}(c x)+8 a b c^3 x^3 \text {arctanh}(c x)-8 b^2 \text {arctanh}(c x)^2-20 b^2 c x \text {arctanh}(c x)^2+24 b^2 c^2 x^2 \text {arctanh}(c x)^2+4 b^2 c^3 x^3 \text {arctanh}(c x)^2-16 b^2 c x \text {arctanh}(c x)^3+16 b^2 c x \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-48 b^2 c x \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-24 b^2 c x \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+24 b^2 c x \text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+24 a^2 c x \log (x)+16 a b c x \log (c x)+4 a b c x \log (1-c x)-4 a b c x \log (1+c x)+16 a b c x \log \left (1-c^2 x^2\right )+4 b^2 c x \log \left (1-c^2 x^2\right )+24 b^2 c x (1+\text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )-8 b^2 c x \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+24 b^2 c x \text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )-24 a b c x \operatorname {PolyLog}(2,-c x)+24 a b c x \operatorname {PolyLog}(2,c x)+12 b^2 c x \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )-12 b^2 c x \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right )}{8 x} \]
(d^3*(-8*a^2 + I*b^2*c*Pi^3*x + 24*a^2*c^2*x^2 + 8*a*b*c^2*x^2 + 4*a^2*c^3 *x^3 - 16*a*b*ArcTanh[c*x] + 48*a*b*c^2*x^2*ArcTanh[c*x] + 8*b^2*c^2*x^2*A rcTanh[c*x] + 8*a*b*c^3*x^3*ArcTanh[c*x] - 8*b^2*ArcTanh[c*x]^2 - 20*b^2*c *x*ArcTanh[c*x]^2 + 24*b^2*c^2*x^2*ArcTanh[c*x]^2 + 4*b^2*c^3*x^3*ArcTanh[ c*x]^2 - 16*b^2*c*x*ArcTanh[c*x]^3 + 16*b^2*c*x*ArcTanh[c*x]*Log[1 - E^(-2 *ArcTanh[c*x])] - 48*b^2*c*x*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 2 4*b^2*c*x*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 24*b^2*c*x*ArcTanh [c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + 24*a^2*c*x*Log[x] + 16*a*b*c*x*Log[c *x] + 4*a*b*c*x*Log[1 - c*x] - 4*a*b*c*x*Log[1 + c*x] + 16*a*b*c*x*Log[1 - c^2*x^2] + 4*b^2*c*x*Log[1 - c^2*x^2] + 24*b^2*c*x*(1 + ArcTanh[c*x])*Pol yLog[2, -E^(-2*ArcTanh[c*x])] - 8*b^2*c*x*PolyLog[2, E^(-2*ArcTanh[c*x])] + 24*b^2*c*x*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - 24*a*b*c*x*Poly Log[2, -(c*x)] + 24*a*b*c*x*PolyLog[2, c*x] + 12*b^2*c*x*PolyLog[3, -E^(-2 *ArcTanh[c*x])] - 12*b^2*c*x*PolyLog[3, E^(2*ArcTanh[c*x])]))/(8*x)
Time = 1.04 (sec) , antiderivative size = 361, 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.
| \(\displaystyle \int \frac {(c d x+d)^3 (a+b \text {arctanh}(c x))^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (c^3 d^3 x (a+b \text {arctanh}(c x))^2+3 c^2 d^3 (a+b \text {arctanh}(c x))^2+\frac {d^3 (a+b \text {arctanh}(c x))^2}{x^2}+\frac {3 c d^3 (a+b \text {arctanh}(c x))^2}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} c^3 d^3 x^2 (a+b \text {arctanh}(c x))^2+3 c^2 d^3 x (a+b \text {arctanh}(c x))^2-3 b c d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+3 b c d^3 \operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))+\frac {7}{2} c d^3 (a+b \text {arctanh}(c x))^2-\frac {d^3 (a+b \text {arctanh}(c x))^2}{x}+6 c d^3 \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2-6 b c d^3 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+2 b c d^3 \log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))+a b c^2 d^3 x+b^2 c^2 d^3 x \text {arctanh}(c x)+\frac {1}{2} b^2 c d^3 \log \left (1-c^2 x^2\right )-3 b^2 c d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-b^2 c d^3 \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )+\frac {3}{2} b^2 c d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {3}{2} b^2 c d^3 \operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right )\) |
a*b*c^2*d^3*x + b^2*c^2*d^3*x*ArcTanh[c*x] + (7*c*d^3*(a + b*ArcTanh[c*x]) ^2)/2 - (d^3*(a + b*ArcTanh[c*x])^2)/x + 3*c^2*d^3*x*(a + b*ArcTanh[c*x])^ 2 + (c^3*d^3*x^2*(a + b*ArcTanh[c*x])^2)/2 + 6*c*d^3*(a + b*ArcTanh[c*x])^ 2*ArcTanh[1 - 2/(1 - c*x)] - 6*b*c*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x )] + (b^2*c*d^3*Log[1 - c^2*x^2])/2 + 2*b*c*d^3*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - 3*b^2*c*d^3*PolyLog[2, 1 - 2/(1 - c*x)] - 3*b*c*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)] + 3*b*c*d^3*(a + b*ArcTanh[c* x])*PolyLog[2, -1 + 2/(1 - c*x)] - b^2*c*d^3*PolyLog[2, -1 + 2/(1 + c*x)] + (3*b^2*c*d^3*PolyLog[3, 1 - 2/(1 - c*x)])/2 - (3*b^2*c*d^3*PolyLog[3, -1 + 2/(1 - c*x)])/2
3.1.89.3.1 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 = 9.07 (sec) , antiderivative size = 1012, normalized size of antiderivative = 2.80
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1012\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1014\) |
| default | \(\text {Expression too large to display}\) | \(1014\) |
d^3*a^2*(1/2*c^3*x^2+3*c^2*x-1/x+3*c*ln(x))+d^3*b^2*c*(3*c*x*arctanh(c*x)^ 2-6*dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-6*dilog(1-I*(c*x+1)/(-c^2*x^2+1) ^(1/2))+1/2*c^2*x^2*arctanh(c*x)^2+2*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+3 /2*arctanh(c*x)^2-1/c/x*arctanh(c*x)^2-ln(1+(c*x+1)^2/(-c^2*x^2+1))-2*dilo g((c*x+1)/(-c^2*x^2+1)^(1/2))-6*arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^( 1/2))-6*arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+3/2*polylog(3,-(c* x+1)^2/(-c^2*x^2+1))-6*polylog(3,-(c*x+1)/(-c^2*x^2+1)^(1/2))-6*polylog(3, (c*x+1)/(-c^2*x^2+1)^(1/2))+3*ln(c*x)*arctanh(c*x)^2-3*arctanh(c*x)*polylo g(2,-(c*x+1)^2/(-c^2*x^2+1))-3*arctanh(c*x)^2*ln((c*x+1)^2/(-c^2*x^2+1)-1) +3*arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+6*arctanh(c*x)*polylog( 2,-(c*x+1)/(-c^2*x^2+1)^(1/2))+3*arctanh(c*x)^2*ln(1-(c*x+1)/(-c^2*x^2+1)^ (1/2))+6*arctanh(c*x)*polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))+(c*x+1)*arctan h(c*x)+2*arctanh(c*x)*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+3/2*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)^2-3/2*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*arctanh(c*x)^2-3/2*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+3/2*I*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^3*a*b*c*(1/2*c^2*x^2*arctanh(c*x)+3*c*x*arcta...
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^2} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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^2, x)
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^2} \, dx=d^{3} \left (\int 3 a^{2} c^{2}\, dx + \int \frac {a^{2}}{x^{2}}\, dx + \int \frac {3 a^{2} c}{x}\, dx + \int a^{2} c^{3} x\, dx + \int 3 b^{2} c^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int 6 a b c^{2} \operatorname {atanh}{\left (c x \right )}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {3 b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}}{x}\, dx + \int b^{2} c^{3} x \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int \frac {6 a b c \operatorname {atanh}{\left (c x \right )}}{x}\, dx + \int 2 a b c^{3} x \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]
d**3*(Integral(3*a**2*c**2, x) + Integral(a**2/x**2, x) + Integral(3*a**2* c/x, x) + Integral(a**2*c**3*x, x) + Integral(3*b**2*c**2*atanh(c*x)**2, x ) + Integral(b**2*atanh(c*x)**2/x**2, x) + Integral(6*a*b*c**2*atanh(c*x), x) + Integral(2*a*b*atanh(c*x)/x**2, x) + Integral(3*b**2*c*atanh(c*x)**2 /x, x) + Integral(b**2*c**3*x*atanh(c*x)**2, x) + Integral(6*a*b*c*atanh(c *x)/x, x) + Integral(2*a*b*c**3*x*atanh(c*x), x))
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^2} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
1/2*a^2*c^3*d^3*x^2 + 3*a^2*c^2*d^3*x + 3*(2*c*x*arctanh(c*x) + log(-c^2*x ^2 + 1))*a*b*c*d^3 + 3*a^2*c*d^3*log(x) - (c*(log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x)*a*b*d^3 - a^2*d^3/x + 1/8*(b^2*c^3*d^3*x^3 + 6*b^2*c^ 2*d^3*x^2 - 2*b^2*d^3)*log(-c*x + 1)^2/x - integrate(-1/4*((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 + 4*(a*b*c ^4*d^3*x^4 - a*b*c^3*d^3*x^3 + 3*a*b*c^2*d^3*x^2 - 3*a*b*c*d^3*x)*log(c*x + 1) - (12*a*b*c^2*d^3*x^2 + (4*a*b*c^4*d^3 + b^2*c^4*d^3)*x^4 - 2*(2*a*b* c^3*d^3 - 3*b^2*c^3*d^3)*x^3 - 2*(6*a*b*c*d^3 + b^2*c*d^3)*x + 2*(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^3 - x^2), x)
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^2} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^3}{x^2} \,d x \]