Integrand size = 22, antiderivative size = 480 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3 (d+c d x)^2} \, dx=\frac {b^2 c^2}{2 d^2 (1+c x)}-\frac {b^2 c^2 \text {arctanh}(c x)}{2 d^2}-\frac {b c (a+b \text {arctanh}(c x))}{d^2 x}+\frac {b c^2 (a+b \text {arctanh}(c x))}{d^2 (1+c x)}-\frac {2 c^2 (a+b \text {arctanh}(c x))^2}{d^2}-\frac {(a+b \text {arctanh}(c x))^2}{2 d^2 x^2}+\frac {2 c (a+b \text {arctanh}(c x))^2}{d^2 x}+\frac {c^2 (a+b \text {arctanh}(c x))^2}{d^2 (1+c x)}+\frac {6 c^2 (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}+\frac {3 c^2 (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^2}-\frac {4 b c^2 (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )}{d^2}-\frac {3 b c^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{d^2}+\frac {3 b c^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )}{d^2}-\frac {3 b c^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{d^2}+\frac {2 b^2 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )}{d^2}+\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d^2}-\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right )}{2 d^2}-\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 d^2} \]
1/2*b^2*c^2/d^2/(c*x+1)-1/2*b^2*c^2*arctanh(c*x)/d^2-b*c*(a+b*arctanh(c*x) )/d^2/x+b*c^2*(a+b*arctanh(c*x))/d^2/(c*x+1)-2*c^2*(a+b*arctanh(c*x))^2/d^ 2-1/2*(a+b*arctanh(c*x))^2/d^2/x^2+2*c*(a+b*arctanh(c*x))^2/d^2/x+c^2*(a+b *arctanh(c*x))^2/d^2/(c*x+1)-6*c^2*(a+b*arctanh(c*x))^2*arctanh(-1+2/(-c*x +1))/d^2+b^2*c^2*ln(x)/d^2+3*c^2*(a+b*arctanh(c*x))^2*ln(2/(c*x+1))/d^2-1/ 2*b^2*c^2*ln(-c^2*x^2+1)/d^2-4*b*c^2*(a+b*arctanh(c*x))*ln(2-2/(c*x+1))/d^ 2-3*b*c^2*(a+b*arctanh(c*x))*polylog(2,1-2/(-c*x+1))/d^2+3*b*c^2*(a+b*arct anh(c*x))*polylog(2,-1+2/(-c*x+1))/d^2-3*b*c^2*(a+b*arctanh(c*x))*polylog( 2,1-2/(c*x+1))/d^2+2*b^2*c^2*polylog(2,-1+2/(c*x+1))/d^2+3/2*b^2*c^2*polyl og(3,1-2/(-c*x+1))/d^2-3/2*b^2*c^2*polylog(3,-1+2/(-c*x+1))/d^2-3/2*b^2*c^ 2*polylog(3,1-2/(c*x+1))/d^2
Result contains complex when optimal does not.
Time = 1.96 (sec) , antiderivative size = 452, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3 (d+c d x)^2} \, dx=\frac {-\frac {4 a^2}{x^2}+\frac {16 a^2 c}{x}+\frac {8 a^2 c^2}{1+c x}+24 a^2 c^2 \log (x)-24 a^2 c^2 \log (1+c x)+b^2 c^2 \left (i \pi ^3-\frac {8 \text {arctanh}(c x)}{c x}-12 \text {arctanh}(c x)^2-\frac {4 \text {arctanh}(c x)^2}{c^2 x^2}+\frac {16 \text {arctanh}(c x)^2}{c x}-16 \text {arctanh}(c x)^3+2 \cosh (2 \text {arctanh}(c x))+4 \text {arctanh}(c x) \cosh (2 \text {arctanh}(c x))+4 \text {arctanh}(c x)^2 \cosh (2 \text {arctanh}(c x))-32 \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )+24 \text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+8 \log (c x)-4 \log \left (1-c^2 x^2\right )+16 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+24 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )-12 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )-2 \sinh (2 \text {arctanh}(c x))-4 \text {arctanh}(c x) \sinh (2 \text {arctanh}(c x))-4 \text {arctanh}(c x)^2 \sinh (2 \text {arctanh}(c x))\right )+\frac {4 a b \left (-6 c^2 x^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+c x \left (-2+c x \cosh (2 \text {arctanh}(c x))-8 c x \log (c x)+4 c x \log \left (1-c^2 x^2\right )-c x \sinh (2 \text {arctanh}(c x))\right )+2 \text {arctanh}(c x) \left (-1+4 c x+c^2 x^2+c^2 x^2 \cosh (2 \text {arctanh}(c x))+6 c^2 x^2 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-c^2 x^2 \sinh (2 \text {arctanh}(c x))\right )\right )}{x^2}}{8 d^2} \]
((-4*a^2)/x^2 + (16*a^2*c)/x + (8*a^2*c^2)/(1 + c*x) + 24*a^2*c^2*Log[x] - 24*a^2*c^2*Log[1 + c*x] + b^2*c^2*(I*Pi^3 - (8*ArcTanh[c*x])/(c*x) - 12*A rcTanh[c*x]^2 - (4*ArcTanh[c*x]^2)/(c^2*x^2) + (16*ArcTanh[c*x]^2)/(c*x) - 16*ArcTanh[c*x]^3 + 2*Cosh[2*ArcTanh[c*x]] + 4*ArcTanh[c*x]*Cosh[2*ArcTan h[c*x]] + 4*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] - 32*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] + 24*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + 8*L og[c*x] - 4*Log[1 - c^2*x^2] + 16*PolyLog[2, E^(-2*ArcTanh[c*x])] + 24*Arc Tanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - 12*PolyLog[3, E^(2*ArcTanh[c*x] )] - 2*Sinh[2*ArcTanh[c*x]] - 4*ArcTanh[c*x]*Sinh[2*ArcTanh[c*x]] - 4*ArcT anh[c*x]^2*Sinh[2*ArcTanh[c*x]]) + (4*a*b*(-6*c^2*x^2*PolyLog[2, E^(-2*Arc Tanh[c*x])] + c*x*(-2 + c*x*Cosh[2*ArcTanh[c*x]] - 8*c*x*Log[c*x] + 4*c*x* Log[1 - c^2*x^2] - c*x*Sinh[2*ArcTanh[c*x]]) + 2*ArcTanh[c*x]*(-1 + 4*c*x + c^2*x^2 + c^2*x^2*Cosh[2*ArcTanh[c*x]] + 6*c^2*x^2*Log[1 - E^(-2*ArcTanh [c*x])] - c^2*x^2*Sinh[2*ArcTanh[c*x]])))/x^2)/(8*d^2)
Time = 1.27 (sec) , antiderivative size = 480, 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.
| \(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^2}{x^3 (c d x+d)^2} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (-\frac {3 c^3 (a+b \text {arctanh}(c x))^2}{d^2 (c x+1)}-\frac {c^3 (a+b \text {arctanh}(c x))^2}{d^2 (c x+1)^2}+\frac {3 c^2 (a+b \text {arctanh}(c x))^2}{d^2 x}+\frac {(a+b \text {arctanh}(c x))^2}{d^2 x^3}-\frac {2 c (a+b \text {arctanh}(c x))^2}{d^2 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 b c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{d^2}+\frac {3 b c^2 \operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))}{d^2}-\frac {3 b c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d^2}+\frac {c^2 (a+b \text {arctanh}(c x))^2}{d^2 (c x+1)}-\frac {2 c^2 (a+b \text {arctanh}(c x))^2}{d^2}+\frac {b c^2 (a+b \text {arctanh}(c x))}{d^2 (c x+1)}+\frac {6 c^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{d^2}+\frac {3 c^2 \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{d^2}-\frac {4 b c^2 \log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d^2}-\frac {(a+b \text {arctanh}(c x))^2}{2 d^2 x^2}+\frac {2 c (a+b \text {arctanh}(c x))^2}{d^2 x}-\frac {b c (a+b \text {arctanh}(c x))}{d^2 x}-\frac {b^2 c^2 \text {arctanh}(c x)}{2 d^2}+\frac {2 b^2 c^2 \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )}{d^2}+\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d^2}-\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right )}{2 d^2}-\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{2 d^2}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^2}+\frac {b^2 c^2}{2 d^2 (c x+1)}+\frac {b^2 c^2 \log (x)}{d^2}\) |
(b^2*c^2)/(2*d^2*(1 + c*x)) - (b^2*c^2*ArcTanh[c*x])/(2*d^2) - (b*c*(a + b *ArcTanh[c*x]))/(d^2*x) + (b*c^2*(a + b*ArcTanh[c*x]))/(d^2*(1 + c*x)) - ( 2*c^2*(a + b*ArcTanh[c*x])^2)/d^2 - (a + b*ArcTanh[c*x])^2/(2*d^2*x^2) + ( 2*c*(a + b*ArcTanh[c*x])^2)/(d^2*x) + (c^2*(a + b*ArcTanh[c*x])^2)/(d^2*(1 + c*x)) + (6*c^2*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)])/d^2 + ( b^2*c^2*Log[x])/d^2 + (3*c^2*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/d^2 - (b^2*c^2*Log[1 - c^2*x^2])/(2*d^2) - (4*b*c^2*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)])/d^2 - (3*b*c^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/d^2 + (3*b*c^2*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)])/ d^2 - (3*b*c^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/d^2 + (2* b^2*c^2*PolyLog[2, -1 + 2/(1 + c*x)])/d^2 + (3*b^2*c^2*PolyLog[3, 1 - 2/(1 - c*x)])/(2*d^2) - (3*b^2*c^2*PolyLog[3, -1 + 2/(1 - c*x)])/(2*d^2) - (3* b^2*c^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*d^2)
3.2.10.3.1 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 = 13.00 (sec) , antiderivative size = 1572, normalized size of antiderivative = 3.28
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1572\) |
| default | \(\text {Expression too large to display}\) | \(1572\) |
| parts | \(\text {Expression too large to display}\) | \(1577\) |
c^2*(a^2/d^2*(1/(c*x+1)-3*ln(c*x+1)-1/2/c^2/x^2+2/c/x+3*ln(c*x))+b^2/d^2*( -4*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))-3*arctanh(c*x)^2*ln(c*x+1)+3*arctan h(c*x)^2*ln(2)+6*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c *x)^2+2/c/x*arctanh(c*x)^2-1/2/c^2/x^2*arctanh(c*x)^2+4*dilog((c*x+1)/(-c^ 2*x^2+1)^(1/2))-2*arctanh(c*x)^3-1/4*(c*x-1)/(c*x+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)^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*arctan h(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)) -4*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)))*c sgn(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*arctan h(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+1/(c *x+1)*arctanh(c*x)^2-1/2*arctanh(c*x)*(c*x-1)/(c*x+1)+3/2*I*Pi*csgn(I*(...
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3 (d+c d x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{2} x^{3}} \,d x } \]
integral((b^2*arctanh(c*x)^2 + 2*a*b*arctanh(c*x) + a^2)/(c^2*d^2*x^5 + 2* c*d^2*x^4 + d^2*x^3), x)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3 (d+c d x)^2} \, dx=\frac {\int \frac {a^{2}}{c^{2} x^{5} + 2 c x^{4} + x^{3}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c^{2} x^{5} + 2 c x^{4} + x^{3}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{5} + 2 c x^{4} + x^{3}}\, dx}{d^{2}} \]
(Integral(a**2/(c**2*x**5 + 2*c*x**4 + x**3), x) + Integral(b**2*atanh(c*x )**2/(c**2*x**5 + 2*c*x**4 + x**3), x) + Integral(2*a*b*atanh(c*x)/(c**2*x **5 + 2*c*x**4 + x**3), x))/d**2
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3 (d+c d x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{2} x^{3}} \,d x } \]
-1/2*a^2*(6*c^2*log(c*x + 1)/d^2 - 6*c^2*log(x)/d^2 - (6*c^2*x^2 + 3*c*x - 1)/(c*d^2*x^3 + d^2*x^2)) + 1/8*(6*b^2*c^2*x^2 + 3*b^2*c*x - b^2 - 6*(b^2 *c^3*x^3 + b^2*c^2*x^2)*log(c*x + 1))*log(-c*x + 1)^2/(c*d^2*x^3 + d^2*x^2 ) + integrate(1/4*((b^2*c*x - b^2)*log(c*x + 1)^2 + 4*(a*b*c*x - a*b)*log( c*x + 1) - (6*b^2*c^4*x^4 + 9*b^2*c^3*x^3 + 2*b^2*c^2*x^2 - 4*a*b + (4*a*b *c - b^2*c)*x - 2*(3*b^2*c^5*x^5 + 6*b^2*c^4*x^4 + 3*b^2*c^3*x^3 - b^2*c*x + b^2)*log(c*x + 1))*log(-c*x + 1))/(c^3*d^2*x^6 + c^2*d^2*x^5 - c*d^2*x^ 4 - d^2*x^3), x)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3 (d+c d x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{2} x^{3}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3 (d+c d x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{x^3\,{\left (d+c\,d\,x\right )}^2} \,d x \]