Integrand size = 22, antiderivative size = 396 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d^3}{3 x}+\frac {1}{3} b^2 c^3 d^3 \text {arctanh}(c x)-\frac {b c d^3 (a+b \text {arctanh}(c x))}{3 x^2}-\frac {3 b c^2 d^3 (a+b \text {arctanh}(c x))}{x}+\frac {29}{6} c^3 d^3 (a+b \text {arctanh}(c x))^2-\frac {d^3 (a+b \text {arctanh}(c x))^2}{3 x^3}-\frac {3 c d^3 (a+b \text {arctanh}(c x))^2}{2 x^2}-\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))^2}{x}+2 c^3 d^3 (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )+3 b^2 c^3 d^3 \log (x)-\frac {3}{2} b^2 c^3 d^3 \log \left (1-c^2 x^2\right )+\frac {20}{3} b c^3 d^3 (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )-b c^3 d^3 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+b c^3 d^3 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )-\frac {10}{3} b^2 c^3 d^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )+\frac {1}{2} b^2 c^3 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^3 d^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right ) \]
-1/3*b^2*c^2*d^3/x+1/3*b^2*c^3*d^3*arctanh(c*x)-1/3*b*c*d^3*(a+b*arctanh(c *x))/x^2-3*b*c^2*d^3*(a+b*arctanh(c*x))/x+29/6*c^3*d^3*(a+b*arctanh(c*x))^ 2-1/3*d^3*(a+b*arctanh(c*x))^2/x^3-3/2*c*d^3*(a+b*arctanh(c*x))^2/x^2-3*c^ 2*d^3*(a+b*arctanh(c*x))^2/x-2*c^3*d^3*(a+b*arctanh(c*x))^2*arctanh(-1+2/( -c*x+1))+3*b^2*c^3*d^3*ln(x)-3/2*b^2*c^3*d^3*ln(-c^2*x^2+1)+20/3*b*c^3*d^3 *(a+b*arctanh(c*x))*ln(2-2/(c*x+1))-b*c^3*d^3*(a+b*arctanh(c*x))*polylog(2 ,1-2/(-c*x+1))+b*c^3*d^3*(a+b*arctanh(c*x))*polylog(2,-1+2/(-c*x+1))-10/3* b^2*c^3*d^3*polylog(2,-1+2/(c*x+1))+1/2*b^2*c^3*d^3*polylog(3,1-2/(-c*x+1) )-1/2*b^2*c^3*d^3*polylog(3,-1+2/(-c*x+1))
Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.44 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=\frac {d^3 \left (-8 a^2-36 a^2 c x-8 a b c x-72 a^2 c^2 x^2-72 a b c^2 x^2-8 b^2 c^2 x^2+i b^2 c^3 \pi ^3 x^3-16 a b \text {arctanh}(c x)-72 a b c x \text {arctanh}(c x)-8 b^2 c x \text {arctanh}(c x)-144 a b c^2 x^2 \text {arctanh}(c x)-72 b^2 c^2 x^2 \text {arctanh}(c x)+8 b^2 c^3 x^3 \text {arctanh}(c x)-8 b^2 \text {arctanh}(c x)^2-36 b^2 c x \text {arctanh}(c x)^2-72 b^2 c^2 x^2 \text {arctanh}(c x)^2+116 b^2 c^3 x^3 \text {arctanh}(c x)^2-16 b^2 c^3 x^3 \text {arctanh}(c x)^3+160 b^2 c^3 x^3 \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-24 b^2 c^3 x^3 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+24 b^2 c^3 x^3 \text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+24 a^2 c^3 x^3 \log (x)+160 a b c^3 x^3 \log (c x)-36 a b c^3 x^3 \log (1-c x)+36 a b c^3 x^3 \log (1+c x)+72 b^2 c^3 x^3 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )-80 a b c^3 x^3 \log \left (1-c^2 x^2\right )+24 b^2 c^3 x^3 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )-80 b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+24 b^2 c^3 x^3 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )-24 a b c^3 x^3 \operatorname {PolyLog}(2,-c x)+24 a b c^3 x^3 \operatorname {PolyLog}(2,c x)+12 b^2 c^3 x^3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )-12 b^2 c^3 x^3 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right )}{24 x^3} \]
(d^3*(-8*a^2 - 36*a^2*c*x - 8*a*b*c*x - 72*a^2*c^2*x^2 - 72*a*b*c^2*x^2 - 8*b^2*c^2*x^2 + I*b^2*c^3*Pi^3*x^3 - 16*a*b*ArcTanh[c*x] - 72*a*b*c*x*ArcT anh[c*x] - 8*b^2*c*x*ArcTanh[c*x] - 144*a*b*c^2*x^2*ArcTanh[c*x] - 72*b^2* c^2*x^2*ArcTanh[c*x] + 8*b^2*c^3*x^3*ArcTanh[c*x] - 8*b^2*ArcTanh[c*x]^2 - 36*b^2*c*x*ArcTanh[c*x]^2 - 72*b^2*c^2*x^2*ArcTanh[c*x]^2 + 116*b^2*c^3*x ^3*ArcTanh[c*x]^2 - 16*b^2*c^3*x^3*ArcTanh[c*x]^3 + 160*b^2*c^3*x^3*ArcTan h[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - 24*b^2*c^3*x^3*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 24*b^2*c^3*x^3*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTan h[c*x])] + 24*a^2*c^3*x^3*Log[x] + 160*a*b*c^3*x^3*Log[c*x] - 36*a*b*c^3*x ^3*Log[1 - c*x] + 36*a*b*c^3*x^3*Log[1 + c*x] + 72*b^2*c^3*x^3*Log[(c*x)/S qrt[1 - c^2*x^2]] - 80*a*b*c^3*x^3*Log[1 - c^2*x^2] + 24*b^2*c^3*x^3*ArcTa nh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 80*b^2*c^3*x^3*PolyLog[2, E^(-2 *ArcTanh[c*x])] + 24*b^2*c^3*x^3*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x] )] - 24*a*b*c^3*x^3*PolyLog[2, -(c*x)] + 24*a*b*c^3*x^3*PolyLog[2, c*x] + 12*b^2*c^3*x^3*PolyLog[3, -E^(-2*ArcTanh[c*x])] - 12*b^2*c^3*x^3*PolyLog[3 , E^(2*ArcTanh[c*x])]))/(24*x^3)
Time = 1.21 (sec) , antiderivative size = 396, 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.
| \(\displaystyle \int \frac {(c d x+d)^3 (a+b \text {arctanh}(c x))^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (\frac {c^3 d^3 (a+b \text {arctanh}(c x))^2}{x}+\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))^2}{x^2}+\frac {d^3 (a+b \text {arctanh}(c x))^2}{x^4}+\frac {3 c d^3 (a+b \text {arctanh}(c x))^2}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -b c^3 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+b c^3 d^3 \operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))+\frac {29}{6} c^3 d^3 (a+b \text {arctanh}(c x))^2+2 c^3 d^3 \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2+\frac {20}{3} b c^3 d^3 \log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))-\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))^2}{x}-\frac {3 b c^2 d^3 (a+b \text {arctanh}(c x))}{x}-\frac {d^3 (a+b \text {arctanh}(c x))^2}{3 x^3}-\frac {3 c d^3 (a+b \text {arctanh}(c x))^2}{2 x^2}-\frac {b c d^3 (a+b \text {arctanh}(c x))}{3 x^2}+\frac {1}{3} b^2 c^3 d^3 \text {arctanh}(c x)-\frac {10}{3} b^2 c^3 d^3 \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )+\frac {1}{2} b^2 c^3 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^3 d^3 \operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right )+3 b^2 c^3 d^3 \log (x)-\frac {b^2 c^2 d^3}{3 x}-\frac {3}{2} b^2 c^3 d^3 \log \left (1-c^2 x^2\right )\) |
-1/3*(b^2*c^2*d^3)/x + (b^2*c^3*d^3*ArcTanh[c*x])/3 - (b*c*d^3*(a + b*ArcT anh[c*x]))/(3*x^2) - (3*b*c^2*d^3*(a + b*ArcTanh[c*x]))/x + (29*c^3*d^3*(a + b*ArcTanh[c*x])^2)/6 - (d^3*(a + b*ArcTanh[c*x])^2)/(3*x^3) - (3*c*d^3* (a + b*ArcTanh[c*x])^2)/(2*x^2) - (3*c^2*d^3*(a + b*ArcTanh[c*x])^2)/x + 2 *c^3*d^3*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] + 3*b^2*c^3*d^3*L og[x] - (3*b^2*c^3*d^3*Log[1 - c^2*x^2])/2 + (20*b*c^3*d^3*(a + b*ArcTanh[ c*x])*Log[2 - 2/(1 + c*x)])/3 - b*c^3*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)] + b*c^3*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c *x)] - (10*b^2*c^3*d^3*PolyLog[2, -1 + 2/(1 + c*x)])/3 + (b^2*c^3*d^3*Poly Log[3, 1 - 2/(1 - c*x)])/2 - (b^2*c^3*d^3*PolyLog[3, -1 + 2/(1 - c*x)])/2
3.1.91.3.1 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 = 9.35 (sec) , antiderivative size = 1197, normalized size of antiderivative = 3.02
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1197\) |
| default | \(\text {Expression too large to display}\) | \(1197\) |
| parts | \(\text {Expression too large to display}\) | \(1255\) |
c^3*(-8/3*b^2*d^3*arctanh(c*x)+1/2*d^3*b^2*polylog(3,-(c*x+1)^2/(-c^2*x^2+ 1))-11/6*d^3*b^2*arctanh(c*x)^2-d^3*b^2*arctanh(c*x)*polylog(2,-(c*x+1)^2/ (-c^2*x^2+1))+d^3*b^2*ln(c*x)*arctanh(c*x)^2+1/2*I*d^3*b^2*Pi*arctanh(c*x) ^2*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^3-20/3*d^3 *b^2*dilog((c*x+1)/(-c^2*x^2+1)^(1/2))+3*d^3*b^2*ln((c*x+1)/(-c^2*x^2+1)^( 1/2)-1)-3*d^3*b^2/c/x*arctanh(c*x)-3*d^3*b^2/c/x*arctanh(c*x)^2-3/2*d^3*b^ 2/c^2/x^2*arctanh(c*x)^2+20/3*d^3*b^2*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+ 3*d^3*b^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+d^3*a^2*(ln(c*x)-3/c/x-3/2/c^2/ x^2-1/3/c^3/x^3)-2*d^3*b^2*polylog(3,-(c*x+1)/(-c^2*x^2+1)^(1/2))-2*d^3*b^ 2*polylog(3,(c*x+1)/(-c^2*x^2+1)^(1/2))-1/3*d^3*b^2*arctanh(c*x)/c^2/x^2-1 /3*d^3*b^2*arctanh(c*x)^2/c^3/x^3-1/2*I*d^3*b^2*Pi*arctanh(c*x)^2*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)))^2-1/2*I*d^3*b^2*Pi*arctanh(c*x)^2*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+d^3 *b^2*arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+2*d^3*b^2*arctanh(c*x )*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))+d^3*b^2*arctanh(c*x)^2*ln(1-(c*x+ 1)/(-c^2*x^2+1)^(1/2))+2*d^3*b^2*arctanh(c*x)*polylog(2,(c*x+1)/(-c^2*x^2+ 1)^(1/2))+20/3*d^3*b^2*arctanh(c*x)*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+2*d^3 *a*b*(ln(c*x)*arctanh(c*x)-3/c/x*arctanh(c*x)-3/2/c^2/x^2*arctanh(c*x)-1/3 /c^3/x^3*arctanh(c*x)-11/12*ln(c*x+1)-29/12*ln(c*x-1)-1/6/c^2/x^2-3/2/c...
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
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^4, x)
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=d^{3} \left (\int \frac {a^{2}}{x^{4}}\, dx + \int \frac {3 a^{2} c}{x^{3}}\, dx + \int \frac {3 a^{2} c^{2}}{x^{2}}\, dx + \int \frac {a^{2} c^{3}}{x}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {3 b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {3 b^{2} c^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {b^{2} c^{3} \operatorname {atanh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {6 a b c \operatorname {atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {6 a b c^{2} \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {2 a b c^{3} \operatorname {atanh}{\left (c x \right )}}{x}\, dx\right ) \]
d**3*(Integral(a**2/x**4, x) + Integral(3*a**2*c/x**3, x) + Integral(3*a** 2*c**2/x**2, x) + Integral(a**2*c**3/x, x) + Integral(b**2*atanh(c*x)**2/x **4, x) + Integral(2*a*b*atanh(c*x)/x**4, x) + Integral(3*b**2*c*atanh(c*x )**2/x**3, x) + Integral(3*b**2*c**2*atanh(c*x)**2/x**2, x) + Integral(b** 2*c**3*atanh(c*x)**2/x, x) + Integral(6*a*b*c*atanh(c*x)/x**3, x) + Integr al(6*a*b*c**2*atanh(c*x)/x**2, x) + Integral(2*a*b*c**3*atanh(c*x)/x, x))
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
a^2*c^3*d^3*log(x) - 3*(c*(log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x )*a*b*c^2*d^3 + 3/2*((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arctanh (c*x)/x^2)*a*b*c*d^3 - 1/3*((c^2*log(c^2*x^2 - 1) - c^2*log(x^2) + 1/x^2)* c + 2*arctanh(c*x)/x^3)*a*b*d^3 - 3*a^2*c^2*d^3/x - 3/2*a^2*c*d^3/x^2 - 1/ 3*a^2*d^3/x^3 - 1/24*(18*b^2*c^2*d^3*x^2 + 9*b^2*c*d^3*x + 2*b^2*d^3)*log( -c*x + 1)^2/x^3 - 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 - a*b*c^3* d^3*x^3)*log(c*x + 1) - (12*a*b*c^4*d^3*x^4 - 9*b^2*c^2*d^3*x^2 - 2*b^2*c* d^3*x - 6*(2*a*b*c^3*d^3 + 3*b^2*c^3*d^3)*x^3 + 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^ 5 - x^4), x)
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^3}{x^4} \,d x \]