Integrand size = 22, antiderivative size = 448 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^3} \, dx=-\frac {b^2 c}{16 d^3 (1+c x)^2}-\frac {19 b^2 c}{16 d^3 (1+c x)}+\frac {19 b^2 c \text {arctanh}(c x)}{16 d^3}-\frac {b c (a+b \text {arctanh}(c x))}{4 d^3 (1+c x)^2}-\frac {9 b c (a+b \text {arctanh}(c x))}{4 d^3 (1+c x)}+\frac {17 c (a+b \text {arctanh}(c x))^2}{8 d^3}-\frac {(a+b \text {arctanh}(c x))^2}{d^3 x}-\frac {c (a+b \text {arctanh}(c x))^2}{2 d^3 (1+c x)^2}-\frac {2 c (a+b \text {arctanh}(c x))^2}{d^3 (1+c x)}-\frac {6 c (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )}{d^3}-\frac {3 c (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{d^3}+\frac {2 b c (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )}{d^3}+\frac {3 b c (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{d^3}-\frac {3 b c (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )}{d^3}+\frac {3 b c (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{d^3}-\frac {b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )}{d^3}-\frac {3 b^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d^3}+\frac {3 b^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right )}{2 d^3}+\frac {3 b^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 d^3} \]
-1/16*b^2*c/d^3/(c*x+1)^2-19/16*b^2*c/d^3/(c*x+1)+19/16*b^2*c*arctanh(c*x) /d^3-1/4*b*c*(a+b*arctanh(c*x))/d^3/(c*x+1)^2-9/4*b*c*(a+b*arctanh(c*x))/d ^3/(c*x+1)+17/8*c*(a+b*arctanh(c*x))^2/d^3-(a+b*arctanh(c*x))^2/d^3/x-1/2* c*(a+b*arctanh(c*x))^2/d^3/(c*x+1)^2-2*c*(a+b*arctanh(c*x))^2/d^3/(c*x+1)+ 6*c*(a+b*arctanh(c*x))^2*arctanh(-1+2/(-c*x+1))/d^3-3*c*(a+b*arctanh(c*x)) ^2*ln(2/(c*x+1))/d^3+2*b*c*(a+b*arctanh(c*x))*ln(2-2/(c*x+1))/d^3+3*b*c*(a +b*arctanh(c*x))*polylog(2,1-2/(-c*x+1))/d^3-3*b*c*(a+b*arctanh(c*x))*poly log(2,-1+2/(-c*x+1))/d^3+3*b*c*(a+b*arctanh(c*x))*polylog(2,1-2/(c*x+1))/d ^3-b^2*c*polylog(2,-1+2/(c*x+1))/d^3-3/2*b^2*c*polylog(3,1-2/(-c*x+1))/d^3 +3/2*b^2*c*polylog(3,-1+2/(-c*x+1))/d^3+3/2*b^2*c*polylog(3,1-2/(c*x+1))/d ^3
Result contains complex when optimal does not.
Time = 2.36 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^3} \, dx=-\frac {\frac {64 a^2}{x}+\frac {32 a^2 c}{(1+c x)^2}+\frac {128 a^2 c}{1+c x}+192 a^2 c \log (x)-192 a^2 c \log (1+c x)-b^2 c \left (-8 i \pi ^3+64 \text {arctanh}(c x)^2-\frac {64 \text {arctanh}(c x)^2}{c x}+128 \text {arctanh}(c x)^3-40 \cosh (2 \text {arctanh}(c x))-80 \text {arctanh}(c x) \cosh (2 \text {arctanh}(c x))-80 \text {arctanh}(c x)^2 \cosh (2 \text {arctanh}(c x))-\cosh (4 \text {arctanh}(c x))-4 \text {arctanh}(c x) \cosh (4 \text {arctanh}(c x))-8 \text {arctanh}(c x)^2 \cosh (4 \text {arctanh}(c x))+128 \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-192 \text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )-64 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )-192 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )+96 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )+40 \sinh (2 \text {arctanh}(c x))+80 \text {arctanh}(c x) \sinh (2 \text {arctanh}(c x))+80 \text {arctanh}(c x)^2 \sinh (2 \text {arctanh}(c x))+\sinh (4 \text {arctanh}(c x))+4 \text {arctanh}(c x) \sinh (4 \text {arctanh}(c x))+8 \text {arctanh}(c x)^2 \sinh (4 \text {arctanh}(c x))\right )+\frac {4 a b \left (-48 c x \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+c x \left (20 \cosh (2 \text {arctanh}(c x))+\cosh (4 \text {arctanh}(c x))-32 \log (c x)+16 \log \left (1-c^2 x^2\right )-20 \sinh (2 \text {arctanh}(c x))-\sinh (4 \text {arctanh}(c x))\right )+4 \text {arctanh}(c x) \left (8+10 c x \cosh (2 \text {arctanh}(c x))+c x \cosh (4 \text {arctanh}(c x))+24 c x \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-10 c x \sinh (2 \text {arctanh}(c x))-c x \sinh (4 \text {arctanh}(c x))\right )\right )}{x}}{64 d^3} \]
-1/64*((64*a^2)/x + (32*a^2*c)/(1 + c*x)^2 + (128*a^2*c)/(1 + c*x) + 192*a ^2*c*Log[x] - 192*a^2*c*Log[1 + c*x] - b^2*c*((-8*I)*Pi^3 + 64*ArcTanh[c*x ]^2 - (64*ArcTanh[c*x]^2)/(c*x) + 128*ArcTanh[c*x]^3 - 40*Cosh[2*ArcTanh[c *x]] - 80*ArcTanh[c*x]*Cosh[2*ArcTanh[c*x]] - 80*ArcTanh[c*x]^2*Cosh[2*Arc Tanh[c*x]] - Cosh[4*ArcTanh[c*x]] - 4*ArcTanh[c*x]*Cosh[4*ArcTanh[c*x]] - 8*ArcTanh[c*x]^2*Cosh[4*ArcTanh[c*x]] + 128*ArcTanh[c*x]*Log[1 - E^(-2*Arc Tanh[c*x])] - 192*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] - 64*PolyLog[ 2, E^(-2*ArcTanh[c*x])] - 192*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] + 96*PolyLog[3, E^(2*ArcTanh[c*x])] + 40*Sinh[2*ArcTanh[c*x]] + 80*ArcTanh [c*x]*Sinh[2*ArcTanh[c*x]] + 80*ArcTanh[c*x]^2*Sinh[2*ArcTanh[c*x]] + Sinh [4*ArcTanh[c*x]] + 4*ArcTanh[c*x]*Sinh[4*ArcTanh[c*x]] + 8*ArcTanh[c*x]^2* Sinh[4*ArcTanh[c*x]]) + (4*a*b*(-48*c*x*PolyLog[2, E^(-2*ArcTanh[c*x])] + c*x*(20*Cosh[2*ArcTanh[c*x]] + Cosh[4*ArcTanh[c*x]] - 32*Log[c*x] + 16*Log [1 - c^2*x^2] - 20*Sinh[2*ArcTanh[c*x]] - Sinh[4*ArcTanh[c*x]]) + 4*ArcTan h[c*x]*(8 + 10*c*x*Cosh[2*ArcTanh[c*x]] + c*x*Cosh[4*ArcTanh[c*x]] + 24*c* x*Log[1 - E^(-2*ArcTanh[c*x])] - 10*c*x*Sinh[2*ArcTanh[c*x]] - c*x*Sinh[4* ArcTanh[c*x]])))/x)/d^3
Time = 1.25 (sec) , antiderivative size = 448, 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.
| \(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (c d x+d)^3} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (\frac {3 c^2 (a+b \text {arctanh}(c x))^2}{d^3 (c x+1)}+\frac {2 c^2 (a+b \text {arctanh}(c x))^2}{d^3 (c x+1)^2}+\frac {c^2 (a+b \text {arctanh}(c x))^2}{d^3 (c x+1)^3}+\frac {(a+b \text {arctanh}(c x))^2}{d^3 x^2}-\frac {3 c (a+b \text {arctanh}(c x))^2}{d^3 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 b c \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{d^3}-\frac {3 b c \operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))}{d^3}+\frac {3 b c \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d^3}-\frac {9 b c (a+b \text {arctanh}(c x))}{4 d^3 (c x+1)}-\frac {b c (a+b \text {arctanh}(c x))}{4 d^3 (c x+1)^2}-\frac {(a+b \text {arctanh}(c x))^2}{d^3 x}-\frac {2 c (a+b \text {arctanh}(c x))^2}{d^3 (c x+1)}-\frac {c (a+b \text {arctanh}(c x))^2}{2 d^3 (c x+1)^2}+\frac {17 c (a+b \text {arctanh}(c x))^2}{8 d^3}-\frac {6 c \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{d^3}+\frac {2 b c \log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d^3}-\frac {3 c \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{d^3}+\frac {19 b^2 c \text {arctanh}(c x)}{16 d^3}-\frac {b^2 c \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )}{d^3}-\frac {3 b^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d^3}+\frac {3 b^2 c \operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right )}{2 d^3}+\frac {3 b^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{2 d^3}-\frac {19 b^2 c}{16 d^3 (c x+1)}-\frac {b^2 c}{16 d^3 (c x+1)^2}\) |
-1/16*(b^2*c)/(d^3*(1 + c*x)^2) - (19*b^2*c)/(16*d^3*(1 + c*x)) + (19*b^2* c*ArcTanh[c*x])/(16*d^3) - (b*c*(a + b*ArcTanh[c*x]))/(4*d^3*(1 + c*x)^2) - (9*b*c*(a + b*ArcTanh[c*x]))/(4*d^3*(1 + c*x)) + (17*c*(a + b*ArcTanh[c* x])^2)/(8*d^3) - (a + b*ArcTanh[c*x])^2/(d^3*x) - (c*(a + b*ArcTanh[c*x])^ 2)/(2*d^3*(1 + c*x)^2) - (2*c*(a + b*ArcTanh[c*x])^2)/(d^3*(1 + c*x)) - (6 *c*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)])/d^3 - (3*c*(a + b*ArcT anh[c*x])^2*Log[2/(1 + c*x)])/d^3 + (2*b*c*(a + b*ArcTanh[c*x])*Log[2 - 2/ (1 + c*x)])/d^3 + (3*b*c*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)]) /d^3 - (3*b*c*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)])/d^3 + (3* b*c*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/d^3 - (b^2*c*PolyLog [2, -1 + 2/(1 + c*x)])/d^3 - (3*b^2*c*PolyLog[3, 1 - 2/(1 - c*x)])/(2*d^3) + (3*b^2*c*PolyLog[3, -1 + 2/(1 - c*x)])/(2*d^3) + (3*b^2*c*PolyLog[3, 1 - 2/(1 + c*x)])/(2*d^3)
3.2.17.3.1 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 = 3.21 (sec) , antiderivative size = 4345, normalized size of antiderivative = 9.70
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(4345\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(4346\) |
| default | \(\text {Expression too large to display}\) | \(4346\) |
a^2/d^3*(-1/2/(c*x+1)^2*c-2*c/(c*x+1)+3*ln(c*x+1)*c-1/x-3*c*ln(x))+b^2/d^3 *c*(17/8*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+3*arctanh(c*x)^2*ln(c*x+1)-3* arctanh(c*x)^2*ln(2)-6*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))+1/8*a rctanh(c*x)^2-1/c/x*arctanh(c*x)^2-17/8*dilog((c*x+1)/(-c^2*x^2+1)^(1/2))+ 2*arctanh(c*x)^3+5/8*(c*x-1)/(c*x+1)-1/8*polylog(2,-(c*x+1)/(-c^2*x^2+1)^( 1/2))+6*polylog(3,-(c*x+1)/(-c^2*x^2+1)^(1/2))-1/8*polylog(2,(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*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)*p olylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))-1/64/(c*x+1)^2*(c*x-1)^2-1/8*arctanh( c*x)*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)*ln(1+(c*x+1)/(-c^2*x^ 2+1)^(1/2))+3*ln(2)*dilog((c*x+1)/(-c^2*x^2+1)^(1/2))-3*ln(2)*dilog(1+(c*x +1)/(-c^2*x^2+1)^(1/2))+3*ln(2)*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))+3*l n(2)*polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))+3*ln(2)*arctanh(c*x)*ln(1-(c*x+ 1)/(-c^2*x^2+1)^(1/2))-2/(c*x+1)*arctanh(c*x)^2+5/4*arctanh(c*x)*(c*x-1)/( c*x+1)-1/2/(c*x+1)^2*arctanh(c*x)^2-1/16*arctanh(c*x)*(c*x-1)^2/(c*x+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))*c sgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))*(arctanh(c*x)*ln(1+ (c*x+1)/(-c^2*x^2+1)^(1/2))-dilog((c*x+1)/(-c^2*x^2+1)^(1/2))+dilog(1+(...
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{3} x^{2}} \,d x } \]
integral((b^2*arctanh(c*x)^2 + 2*a*b*arctanh(c*x) + a^2)/(c^3*d^3*x^5 + 3* c^2*d^3*x^4 + 3*c*d^3*x^3 + d^3*x^2), x)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^3} \, dx=\frac {\int \frac {a^{2}}{c^{3} x^{5} + 3 c^{2} x^{4} + 3 c x^{3} + x^{2}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c^{3} x^{5} + 3 c^{2} x^{4} + 3 c x^{3} + x^{2}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{c^{3} x^{5} + 3 c^{2} x^{4} + 3 c x^{3} + x^{2}}\, dx}{d^{3}} \]
(Integral(a**2/(c**3*x**5 + 3*c**2*x**4 + 3*c*x**3 + x**2), x) + Integral( b**2*atanh(c*x)**2/(c**3*x**5 + 3*c**2*x**4 + 3*c*x**3 + x**2), x) + Integ ral(2*a*b*atanh(c*x)/(c**3*x**5 + 3*c**2*x**4 + 3*c*x**3 + x**2), x))/d**3
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{3} x^{2}} \,d x } \]
-1/2*a^2*((6*c^2*x^2 + 9*c*x + 2)/(c^2*d^3*x^3 + 2*c*d^3*x^2 + d^3*x) - 6* c*log(c*x + 1)/d^3 + 6*c*log(x)/d^3) - 1/8*(6*b^2*c^2*x^2 + 9*b^2*c*x + 2* b^2 - 6*(b^2*c^3*x^3 + 2*b^2*c^2*x^2 + b^2*c*x)*log(c*x + 1))*log(-c*x + 1 )^2/(c^2*d^3*x^3 + 2*c*d^3*x^2 + d^3*x) - 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 + 15*b^2* c^3*x^3 + 11*b^2*c^2*x^2 + 4*a*b - 2*(2*a*b*c - b^2*c)*x - 2*(3*b^2*c^5*x^ 5 + 9*b^2*c^4*x^4 + 9*b^2*c^3*x^3 + 3*b^2*c^2*x^2 + b^2*c*x - b^2)*log(c*x + 1))*log(-c*x + 1))/(c^4*d^3*x^6 + 2*c^3*d^3*x^5 - 2*c*d^3*x^3 - d^3*x^2 ), x)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{3} x^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{x^2\,{\left (d+c\,d\,x\right )}^3} \,d x \]