Integrand size = 22, antiderivative size = 385 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^3} \, dx=-\frac {b c d^3 (a+b \text {arctanh}(c x))}{x}+\frac {9}{2} c^2 d^3 (a+b \text {arctanh}(c x))^2-\frac {d^3 (a+b \text {arctanh}(c x))^2}{2 x^2}-\frac {3 c d^3 (a+b \text {arctanh}(c x))^2}{x}+c^3 d^3 x (a+b \text {arctanh}(c x))^2+6 c^2 d^3 (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )+b^2 c^2 d^3 \log (x)-2 b c^2 d^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^2 d^3 \log \left (1-c^2 x^2\right )+6 b c^2 d^3 (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )-b^2 c^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-3 b c^2 d^3 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+3 b c^2 d^3 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )-3 b^2 c^2 d^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )+\frac {3}{2} b^2 c^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {3}{2} b^2 c^2 d^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right ) \]
-b*c*d^3*(a+b*arctanh(c*x))/x+9/2*c^2*d^3*(a+b*arctanh(c*x))^2-1/2*d^3*(a+ b*arctanh(c*x))^2/x^2-3*c*d^3*(a+b*arctanh(c*x))^2/x+c^3*d^3*x*(a+b*arctan h(c*x))^2-6*c^2*d^3*(a+b*arctanh(c*x))^2*arctanh(-1+2/(-c*x+1))+b^2*c^2*d^ 3*ln(x)-2*b*c^2*d^3*(a+b*arctanh(c*x))*ln(2/(-c*x+1))-1/2*b^2*c^2*d^3*ln(- c^2*x^2+1)+6*b*c^2*d^3*(a+b*arctanh(c*x))*ln(2-2/(c*x+1))-b^2*c^2*d^3*poly log(2,1-2/(-c*x+1))-3*b*c^2*d^3*(a+b*arctanh(c*x))*polylog(2,1-2/(-c*x+1)) +3*b*c^2*d^3*(a+b*arctanh(c*x))*polylog(2,-1+2/(-c*x+1))-3*b^2*c^2*d^3*pol ylog(2,-1+2/(c*x+1))+3/2*b^2*c^2*d^3*polylog(3,1-2/(-c*x+1))-3/2*b^2*c^2*d ^3*polylog(3,-1+2/(-c*x+1))
Result contains complex when optimal does not.
Time = 0.79 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.20 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^3} \, dx=\frac {1}{2} d^3 \left (-\frac {a^2}{x^2}-\frac {6 a^2 c}{x}+2 a^2 c^3 x+6 a^2 c^2 \log (x)-\frac {a b (2 \text {arctanh}(c x)+c x (2+c x \log (1-c x)-c x \log (1+c x)))}{x^2}+\frac {b^2 \left (-2 c x \text {arctanh}(c x)+\left (-1+c^2 x^2\right ) \text {arctanh}(c x)^2+2 c^2 x^2 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )\right )}{x^2}+2 a b c^2 \left (2 c x \text {arctanh}(c x)+\log \left (1-c^2 x^2\right )\right )-\frac {6 a b c \left (2 \text {arctanh}(c x)+c x \left (-2 \log (c x)+\log \left (1-c^2 x^2\right )\right )\right )}{x}+2 b^2 c^2 \left (\text {arctanh}(c x) \left ((-1+c x) \text {arctanh}(c x)-2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+\frac {6 b^2 c \left (\text {arctanh}(c x) \left ((-1+c x) \text {arctanh}(c x)+2 c x \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )-c x \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )}{x}-6 a b c^2 (\operatorname {PolyLog}(2,-c x)-\operatorname {PolyLog}(2,c x))+6 b^2 c^2 \left (\frac {i \pi ^3}{24}-\frac {2}{3} \text {arctanh}(c x)^3-\text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right )\right ) \]
(d^3*(-(a^2/x^2) - (6*a^2*c)/x + 2*a^2*c^3*x + 6*a^2*c^2*Log[x] - (a*b*(2* ArcTanh[c*x] + c*x*(2 + c*x*Log[1 - c*x] - c*x*Log[1 + c*x])))/x^2 + (b^2* (-2*c*x*ArcTanh[c*x] + (-1 + c^2*x^2)*ArcTanh[c*x]^2 + 2*c^2*x^2*Log[(c*x) /Sqrt[1 - c^2*x^2]]))/x^2 + 2*a*b*c^2*(2*c*x*ArcTanh[c*x] + Log[1 - c^2*x^ 2]) - (6*a*b*c*(2*ArcTanh[c*x] + c*x*(-2*Log[c*x] + Log[1 - c^2*x^2])))/x + 2*b^2*c^2*(ArcTanh[c*x]*((-1 + c*x)*ArcTanh[c*x] - 2*Log[1 + E^(-2*ArcTa nh[c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + (6*b^2*c*(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])]))/x - 6*a*b*c^2*(PolyLog[2, -(c*x)] - PolyLog[2, c*x]) + 6*b^2*c^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])] + A rcTanh[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*Ar cTanh[c*x])]/2)))/2
Time = 1.06 (sec) , antiderivative size = 385, 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.
| \(\displaystyle \int \frac {(c d x+d)^3 (a+b \text {arctanh}(c x))^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (c^3 d^3 (a+b \text {arctanh}(c x))^2+\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))^2}{x}+\frac {d^3 (a+b \text {arctanh}(c x))^2}{x^3}+\frac {3 c d^3 (a+b \text {arctanh}(c x))^2}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle c^3 d^3 x (a+b \text {arctanh}(c x))^2-3 b c^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+3 b c^2 d^3 \operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))+\frac {9}{2} c^2 d^3 (a+b \text {arctanh}(c x))^2+6 c^2 d^3 \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2-2 b c^2 d^3 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+6 b c^2 d^3 \log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))-\frac {d^3 (a+b \text {arctanh}(c x))^2}{2 x^2}-\frac {3 c d^3 (a+b \text {arctanh}(c x))^2}{x}-\frac {b c d^3 (a+b \text {arctanh}(c x))}{x}-b^2 c^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-3 b^2 c^2 d^3 \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )+\frac {3}{2} b^2 c^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {3}{2} b^2 c^2 d^3 \operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right )-\frac {1}{2} b^2 c^2 d^3 \log \left (1-c^2 x^2\right )+b^2 c^2 d^3 \log (x)\) |
-((b*c*d^3*(a + b*ArcTanh[c*x]))/x) + (9*c^2*d^3*(a + b*ArcTanh[c*x])^2)/2 - (d^3*(a + b*ArcTanh[c*x])^2)/(2*x^2) - (3*c*d^3*(a + b*ArcTanh[c*x])^2) /x + c^3*d^3*x*(a + b*ArcTanh[c*x])^2 + 6*c^2*d^3*(a + b*ArcTanh[c*x])^2*A rcTanh[1 - 2/(1 - c*x)] + b^2*c^2*d^3*Log[x] - 2*b*c^2*d^3*(a + b*ArcTanh[ c*x])*Log[2/(1 - c*x)] - (b^2*c^2*d^3*Log[1 - c^2*x^2])/2 + 6*b*c^2*d^3*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - b^2*c^2*d^3*PolyLog[2, 1 - 2/(1 - c*x)] - 3*b*c^2*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)] + 3 *b*c^2*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] - 3*b^2*c^2*d ^3*PolyLog[2, -1 + 2/(1 + c*x)] + (3*b^2*c^2*d^3*PolyLog[3, 1 - 2/(1 - c*x )])/2 - (3*b^2*c^2*d^3*PolyLog[3, -1 + 2/(1 - c*x)])/2
3.1.90.3.1 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 = 7.99 (sec) , antiderivative size = 1086, normalized size of antiderivative = 2.82
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1086\) |
| default | \(\text {Expression too large to display}\) | \(1086\) |
| parts | \(\text {Expression too large to display}\) | \(1086\) |
c^2*(d^3*a^2*(c*x+3*ln(c*x)-3/c/x-1/2/c^2/x^2)+d^3*b^2*(c*x*arctanh(c*x)^2 -2*dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-2*dilog(1-I*(c*x+1)/(-c^2*x^2+1)^ (1/2))+6*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))-3/2*arctanh(c*x)^2-3/c/x*arct anh(c*x)^2-1/2/c^2/x^2*arctanh(c*x)^2-6*dilog((c*x+1)/(-c^2*x^2+1)^(1/2))- 2*arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-2*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))+ln(1+(c*x +1)/(-c^2*x^2+1)^(1/2))-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)*pol ylog(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)*polyl og(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))+6*arctanh(c *x)*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+ln((c*x+1)/(-c^2*x^2+1)^(1/2)-1)-1/2* (c*x-(-c^2*x^2+1)^(1/2)+1)/c/x*arctanh(c*x)-1/2*arctanh(c*x)*(c*x+(-c^2*x^ 2+1)^(1/2)+1)/c/x+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)^...
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
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^3, x)
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^3} \, dx=d^{3} \left (\int a^{2} c^{3}\, dx + \int \frac {a^{2}}{x^{3}}\, dx + \int \frac {3 a^{2} c}{x^{2}}\, dx + \int \frac {3 a^{2} c^{2}}{x}\, dx + \int b^{2} c^{3} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int 2 a b c^{3} \operatorname {atanh}{\left (c x \right )}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {3 b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {3 b^{2} c^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {6 a b c \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {6 a b c^{2} \operatorname {atanh}{\left (c x \right )}}{x}\, dx\right ) \]
d**3*(Integral(a**2*c**3, x) + Integral(a**2/x**3, x) + Integral(3*a**2*c/ x**2, x) + Integral(3*a**2*c**2/x, x) + Integral(b**2*c**3*atanh(c*x)**2, x) + Integral(b**2*atanh(c*x)**2/x**3, x) + Integral(2*a*b*c**3*atanh(c*x) , x) + Integral(2*a*b*atanh(c*x)/x**3, x) + Integral(3*b**2*c*atanh(c*x)** 2/x**2, x) + Integral(3*b**2*c**2*atanh(c*x)**2/x, x) + Integral(6*a*b*c*a tanh(c*x)/x**2, x) + Integral(6*a*b*c**2*atanh(c*x)/x, x))
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
a^2*c^3*d^3*x + (2*c*x*arctanh(c*x) + log(-c^2*x^2 + 1))*a*b*c^2*d^3 + 3*a ^2*c^2*d^3*log(x) - 3*(c*(log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x) *a*b*c*d^3 + 1/2*((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arctanh(c* x)/x^2)*a*b*d^3 - 3*a^2*c*d^3/x - 1/2*a^2*d^3/x^2 + 1/8*(2*b^2*c^3*d^3*x^3 - 6*b^2*c*d^3*x - b^2*d^3)*log(-c*x + 1)^2/x^2 - 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 + 1 2*(a*b*c^3*d^3*x^3 - a*b*c^2*d^3*x^2)*log(c*x + 1) - (2*b^2*c^4*d^3*x^4 + 12*a*b*c^3*d^3*x^3 - b^2*c*d^3*x - 6*(2*a*b*c^2*d^3 + b^2*c^2*d^3)*x^2 + 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^4 - x^3), x)
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^3}{x^3} \,d x \]