Integrand size = 22, antiderivative size = 371 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^2} \, dx=-\frac {b^2 c}{2 d^2 (1+c x)}+\frac {b^2 c \text {arctanh}(c x)}{2 d^2}-\frac {b c (a+b \text {arctanh}(c x))}{d^2 (1+c x)}+\frac {3 c (a+b \text {arctanh}(c x))^2}{2 d^2}-\frac {(a+b \text {arctanh}(c x))^2}{d^2 x}-\frac {c (a+b \text {arctanh}(c x))^2}{d^2 (1+c x)}-\frac {4 c (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )}{d^2}-\frac {2 c (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{d^2}+\frac {2 b c (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )}{d^2}+\frac {2 b c (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{d^2}-\frac {2 b c (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )}{d^2}+\frac {2 b c (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{d^2}-\frac {b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )}{d^2}-\frac {b^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{d^2}+\frac {b^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right )}{d^2}+\frac {b^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{d^2} \]
-1/2*b^2*c/d^2/(c*x+1)+1/2*b^2*c*arctanh(c*x)/d^2-b*c*(a+b*arctanh(c*x))/d ^2/(c*x+1)+3/2*c*(a+b*arctanh(c*x))^2/d^2-(a+b*arctanh(c*x))^2/d^2/x-c*(a+ b*arctanh(c*x))^2/d^2/(c*x+1)+4*c*(a+b*arctanh(c*x))^2*arctanh(-1+2/(-c*x+ 1))/d^2-2*c*(a+b*arctanh(c*x))^2*ln(2/(c*x+1))/d^2+2*b*c*(a+b*arctanh(c*x) )*ln(2-2/(c*x+1))/d^2+2*b*c*(a+b*arctanh(c*x))*polylog(2,1-2/(-c*x+1))/d^2 -2*b*c*(a+b*arctanh(c*x))*polylog(2,-1+2/(-c*x+1))/d^2+2*b*c*(a+b*arctanh( c*x))*polylog(2,1-2/(c*x+1))/d^2-b^2*c*polylog(2,-1+2/(c*x+1))/d^2-b^2*c*p olylog(3,1-2/(-c*x+1))/d^2+b^2*c*polylog(3,-1+2/(-c*x+1))/d^2+b^2*c*polylo g(3,1-2/(c*x+1))/d^2
Result contains complex when optimal does not.
Time = 1.91 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^2} \, dx=\frac {-\frac {12 a^2}{x}-\frac {12 a^2 c}{1+c x}-24 a^2 c \log (x)+24 a^2 c \log (1+c x)+b^2 c \left (-i \pi ^3+12 \text {arctanh}(c x)^2-\frac {12 \text {arctanh}(c x)^2}{c x}+16 \text {arctanh}(c x)^3-3 \cosh (2 \text {arctanh}(c x))-6 \text {arctanh}(c x) \cosh (2 \text {arctanh}(c x))-6 \text {arctanh}(c x)^2 \cosh (2 \text {arctanh}(c x))+24 \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-24 \text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )-12 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )-24 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )+12 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )+3 \sinh (2 \text {arctanh}(c x))+6 \text {arctanh}(c x) \sinh (2 \text {arctanh}(c x))+6 \text {arctanh}(c x)^2 \sinh (2 \text {arctanh}(c x))\right )+24 a b c \left (\operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+\frac {1}{4} \left (-\cosh (2 \text {arctanh}(c x))+\text {arctanh}(c x) \left (-\frac {4}{c x}+\frac {4 c x}{1-c^2 x^2}-2 \cosh (2 \text {arctanh}(c x))-8 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )+4 \log (c x)-2 \log \left (1-c^2 x^2\right )+\sinh (2 \text {arctanh}(c x))\right )\right )}{12 d^2} \]
((-12*a^2)/x - (12*a^2*c)/(1 + c*x) - 24*a^2*c*Log[x] + 24*a^2*c*Log[1 + c *x] + b^2*c*((-I)*Pi^3 + 12*ArcTanh[c*x]^2 - (12*ArcTanh[c*x]^2)/(c*x) + 1 6*ArcTanh[c*x]^3 - 3*Cosh[2*ArcTanh[c*x]] - 6*ArcTanh[c*x]*Cosh[2*ArcTanh[ c*x]] - 6*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] + 24*ArcTanh[c*x]*Log[1 - E^ (-2*ArcTanh[c*x])] - 24*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] - 12*Po lyLog[2, E^(-2*ArcTanh[c*x])] - 24*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c* x])] + 12*PolyLog[3, E^(2*ArcTanh[c*x])] + 3*Sinh[2*ArcTanh[c*x]] + 6*ArcT anh[c*x]*Sinh[2*ArcTanh[c*x]] + 6*ArcTanh[c*x]^2*Sinh[2*ArcTanh[c*x]]) + 2 4*a*b*c*(PolyLog[2, E^(-2*ArcTanh[c*x])] + (-Cosh[2*ArcTanh[c*x]] + ArcTan h[c*x]*(-4/(c*x) + (4*c*x)/(1 - c^2*x^2) - 2*Cosh[2*ArcTanh[c*x]] - 8*Log[ 1 - E^(-2*ArcTanh[c*x])]) + 4*Log[c*x] - 2*Log[1 - c^2*x^2] + Sinh[2*ArcTa nh[c*x]])/4))/(12*d^2)
Time = 1.10 (sec) , antiderivative size = 371, 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.
\(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (c d x+d)^2} \, dx\) |
\(\Big \downarrow \) 6502 |
\(\displaystyle \int \left (\frac {2 c^2 (a+b \text {arctanh}(c x))^2}{d^2 (c x+1)}+\frac {c^2 (a+b \text {arctanh}(c x))^2}{d^2 (c x+1)^2}+\frac {(a+b \text {arctanh}(c x))^2}{d^2 x^2}-\frac {2 c (a+b \text {arctanh}(c x))^2}{d^2 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 b c \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{d^2}-\frac {2 b c \operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))}{d^2}+\frac {2 b c \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d^2}-\frac {b c (a+b \text {arctanh}(c x))}{d^2 (c x+1)}-\frac {(a+b \text {arctanh}(c x))^2}{d^2 x}-\frac {c (a+b \text {arctanh}(c x))^2}{d^2 (c x+1)}+\frac {3 c (a+b \text {arctanh}(c x))^2}{2 d^2}-\frac {4 c \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{d^2}+\frac {2 b c \log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d^2}-\frac {2 c \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{d^2}+\frac {b^2 c \text {arctanh}(c x)}{2 d^2}-\frac {b^2 c \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )}{d^2}-\frac {b^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{d^2}+\frac {b^2 c \operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right )}{d^2}+\frac {b^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{d^2}-\frac {b^2 c}{2 d^2 (c x+1)}\) |
-1/2*(b^2*c)/(d^2*(1 + c*x)) + (b^2*c*ArcTanh[c*x])/(2*d^2) - (b*c*(a + b* ArcTanh[c*x]))/(d^2*(1 + c*x)) + (3*c*(a + b*ArcTanh[c*x])^2)/(2*d^2) - (a + b*ArcTanh[c*x])^2/(d^2*x) - (c*(a + b*ArcTanh[c*x])^2)/(d^2*(1 + c*x)) - (4*c*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)])/d^2 - (2*c*(a + b* ArcTanh[c*x])^2*Log[2/(1 + c*x)])/d^2 + (2*b*c*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)])/d^2 + (2*b*c*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c* x)])/d^2 - (2*b*c*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)])/d^2 + (2*b*c*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/d^2 - (b^2*c*Pol yLog[2, -1 + 2/(1 + c*x)])/d^2 - (b^2*c*PolyLog[3, 1 - 2/(1 - c*x)])/d^2 + (b^2*c*PolyLog[3, -1 + 2/(1 - c*x)])/d^2 + (b^2*c*PolyLog[3, 1 - 2/(1 + c *x)])/d^2
3.2.9.3.1 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 = 3.82 (sec) , antiderivative size = 4256, normalized size of antiderivative = 11.47
method | result | size |
parts | \(\text {Expression too large to display}\) | \(4256\) |
derivativedivides | \(\text {Expression too large to display}\) | \(4258\) |
default | \(\text {Expression too large to display}\) | \(4258\) |
a^2/d^2*(-c/(c*x+1)+2*ln(c*x+1)*c-1/x-2*c*ln(x))+b^2/d^2*c*(3/2*dilog(1+(c *x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)^2*ln(c*x+1)-2*arctanh(c*x)^2*ln(2 )-4*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))-1/2*arctanh(c*x)^2-1/c/x *arctanh(c*x)^2-3/2*dilog((c*x+1)/(-c^2*x^2+1)^(1/2))+4/3*arctanh(c*x)^3+1 /4*(c*x-1)/(c*x+1)+1/2*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))+4*polylog(3, -(c*x+1)/(-c^2*x^2+1)^(1/2))+1/2*polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))+4*p olylog(3,(c*x+1)/(-c^2*x^2+1)^(1/2))-2*ln(c*x)*arctanh(c*x)^2+2*arctanh(c* x)^2*ln((c*x+1)^2/(-c^2*x^2+1)-1)-2*arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+ 1)^(1/2))-4*arctanh(c*x)*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))-2*arctanh( c*x)^2*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))-4*arctanh(c*x)*polylog(2,(c*x+1)/( -c^2*x^2+1)^(1/2))+1/2*arctanh(c*x)*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arc tanh(c*x)*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+2*ln(2)*dilog((c*x+1)/(-c^2*x^2 +1)^(1/2))-2*ln(2)*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+2*ln(2)*polylog(2,- (c*x+1)/(-c^2*x^2+1)^(1/2))+2*ln(2)*polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))+ 2*ln(2)*arctanh(c*x)*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))-1/(c*x+1)*arctanh(c* x)^2+1/2*arctanh(c*x)*(c*x-1)/(c*x+1)+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))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2 /(c^2*x^2-1)))*(arctanh(c*x)^2-arctanh(c*x)*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2 ))-arctanh(c*x)*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))-polylog(2,-(c*x+1)/(-c^2* x^2+1)^(1/2))-polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2)))+I*Pi*csgn(I*(-(c*x...
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{2} x^{2}} \,d x } \]
integral((b^2*arctanh(c*x)^2 + 2*a*b*arctanh(c*x) + a^2)/(c^2*d^2*x^4 + 2* c*d^2*x^3 + d^2*x^2), x)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^2} \, dx=\frac {\int \frac {a^{2}}{c^{2} x^{4} + 2 c x^{3} + x^{2}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c^{2} x^{4} + 2 c x^{3} + x^{2}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{4} + 2 c x^{3} + x^{2}}\, dx}{d^{2}} \]
(Integral(a**2/(c**2*x**4 + 2*c*x**3 + x**2), x) + Integral(b**2*atanh(c*x )**2/(c**2*x**4 + 2*c*x**3 + x**2), x) + Integral(2*a*b*atanh(c*x)/(c**2*x **4 + 2*c*x**3 + x**2), x))/d**2
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{2} x^{2}} \,d x } \]
-a^2*((2*c*x + 1)/(c*d^2*x^2 + d^2*x) - 2*c*log(c*x + 1)/d^2 + 2*c*log(x)/ d^2) - 1/4*(2*b^2*c*x + b^2 - 2*(b^2*c^2*x^2 + b^2*c*x)*log(c*x + 1))*log( -c*x + 1)^2/(c*d^2*x^2 + d^2*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) + 2*(2*b^2*c^3*x^3 + 3*b^2*c^2*x^2 + 2*a*b - (2*a*b*c - b^2*c)*x - (2*b^2*c^4*x^4 + 4*b^2*c^3*x^3 + 2*b^2*c^ 2*x^2 + b^2*c*x - b^2)*log(c*x + 1))*log(-c*x + 1))/(c^3*d^2*x^5 + c^2*d^2 *x^4 - c*d^2*x^3 - d^2*x^2), x)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{2} x^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{x^2\,{\left (d+c\,d\,x\right )}^2} \,d x \]