Integrand size = 22, antiderivative size = 260 \[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=-\frac {b^2}{2 c^3 d^2 (1+c x)}+\frac {b^2 \text {arctanh}(c x)}{2 c^3 d^2}-\frac {b (a+b \text {arctanh}(c x))}{c^3 d^2 (1+c x)}+\frac {3 (a+b \text {arctanh}(c x))^2}{2 c^3 d^2}+\frac {x (a+b \text {arctanh}(c x))^2}{c^2 d^2}-\frac {(a+b \text {arctanh}(c x))^2}{c^3 d^2 (1+c x)}-\frac {2 b (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{c^3 d^2}+\frac {2 (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{c^3 d^2}-\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^3 d^2}-\frac {2 b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c^3 d^2}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{c^3 d^2} \]
-1/2*b^2/c^3/d^2/(c*x+1)+1/2*b^2*arctanh(c*x)/c^3/d^2-b*(a+b*arctanh(c*x)) /c^3/d^2/(c*x+1)+3/2*(a+b*arctanh(c*x))^2/c^3/d^2+x*(a+b*arctanh(c*x))^2/c ^2/d^2-(a+b*arctanh(c*x))^2/c^3/d^2/(c*x+1)-2*b*(a+b*arctanh(c*x))*ln(2/(- c*x+1))/c^3/d^2+2*(a+b*arctanh(c*x))^2*ln(2/(c*x+1))/c^3/d^2-b^2*polylog(2 ,1-2/(-c*x+1))/c^3/d^2-2*b*(a+b*arctanh(c*x))*polylog(2,1-2/(c*x+1))/c^3/d ^2-b^2*polylog(3,1-2/(c*x+1))/c^3/d^2
Time = 0.71 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.13 \[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {4 a^2 c x-\frac {4 a^2}{1+c x}-8 a^2 \log (1+c x)+b^2 \left (-4 \text {arctanh}(c x)^2+4 c x \text {arctanh}(c x)^2-\cosh (2 \text {arctanh}(c x))-2 \text {arctanh}(c x) \cosh (2 \text {arctanh}(c x))-2 \text {arctanh}(c x)^2 \cosh (2 \text {arctanh}(c x))-8 \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+8 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+(4-8 \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )-4 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )+\sinh (2 \text {arctanh}(c x))+2 \text {arctanh}(c x) \sinh (2 \text {arctanh}(c x))+2 \text {arctanh}(c x)^2 \sinh (2 \text {arctanh}(c x))\right )+2 a b \left (-\cosh (2 \text {arctanh}(c x))+2 \log \left (1-c^2 x^2\right )-4 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+\sinh (2 \text {arctanh}(c x))+2 \text {arctanh}(c x) \left (2 c x-\cosh (2 \text {arctanh}(c x))+4 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+\sinh (2 \text {arctanh}(c x))\right )\right )}{4 c^3 d^2} \]
(4*a^2*c*x - (4*a^2)/(1 + c*x) - 8*a^2*Log[1 + c*x] + b^2*(-4*ArcTanh[c*x] ^2 + 4*c*x*ArcTanh[c*x]^2 - Cosh[2*ArcTanh[c*x]] - 2*ArcTanh[c*x]*Cosh[2*A rcTanh[c*x]] - 2*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] - 8*ArcTanh[c*x]*Log[ 1 + E^(-2*ArcTanh[c*x])] + 8*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + (4 - 8*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 4*PolyLog[3, -E^( -2*ArcTanh[c*x])] + Sinh[2*ArcTanh[c*x]] + 2*ArcTanh[c*x]*Sinh[2*ArcTanh[c *x]] + 2*ArcTanh[c*x]^2*Sinh[2*ArcTanh[c*x]]) + 2*a*b*(-Cosh[2*ArcTanh[c*x ]] + 2*Log[1 - c^2*x^2] - 4*PolyLog[2, -E^(-2*ArcTanh[c*x])] + Sinh[2*ArcT anh[c*x]] + 2*ArcTanh[c*x]*(2*c*x - Cosh[2*ArcTanh[c*x]] + 4*Log[1 + E^(-2 *ArcTanh[c*x])] + Sinh[2*ArcTanh[c*x]])))/(4*c^3*d^2)
Time = 0.76 (sec) , antiderivative size = 260, 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.
| \(\displaystyle \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{(c d x+d)^2} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (-\frac {2 (a+b \text {arctanh}(c x))^2}{c^2 d^2 (c x+1)}+\frac {(a+b \text {arctanh}(c x))^2}{c^2 d^2}+\frac {(a+b \text {arctanh}(c x))^2}{c^2 d^2 (c x+1)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{c^3 d^2}-\frac {b (a+b \text {arctanh}(c x))}{c^3 d^2 (c x+1)}-\frac {(a+b \text {arctanh}(c x))^2}{c^3 d^2 (c x+1)}+\frac {3 (a+b \text {arctanh}(c x))^2}{2 c^3 d^2}-\frac {2 b \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c^3 d^2}+\frac {2 \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{c^3 d^2}+\frac {x (a+b \text {arctanh}(c x))^2}{c^2 d^2}+\frac {b^2 \text {arctanh}(c x)}{2 c^3 d^2}-\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^3 d^2}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{c^3 d^2}-\frac {b^2}{2 c^3 d^2 (c x+1)}\) |
-1/2*b^2/(c^3*d^2*(1 + c*x)) + (b^2*ArcTanh[c*x])/(2*c^3*d^2) - (b*(a + b* ArcTanh[c*x]))/(c^3*d^2*(1 + c*x)) + (3*(a + b*ArcTanh[c*x])^2)/(2*c^3*d^2 ) + (x*(a + b*ArcTanh[c*x])^2)/(c^2*d^2) - (a + b*ArcTanh[c*x])^2/(c^3*d^2 *(1 + c*x)) - (2*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(c^3*d^2) + (2*( a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^3*d^2) - (b^2*PolyLog[2, 1 - 2/ (1 - c*x)])/(c^3*d^2) - (2*b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c* x)])/(c^3*d^2) - (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(c^3*d^2)
3.2.5.3.1 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 = 5.02 (sec) , antiderivative size = 2688, normalized size of antiderivative = 10.34
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(2688\) |
| default | \(\text {Expression too large to display}\) | \(2688\) |
| parts | \(\text {Expression too large to display}\) | \(2698\) |
1/c^3*(a^2/d^2*(c*x-2*ln(c*x+1)-1/(c*x+1))+b^2/d^2*(c*x*arctanh(c*x)^2-3/2 *arctanh(c*x)*ln(1+(c*x+1)^2/(-c^2*x^2+1))-1/2*dilog(1+I*(c*x+1)/(-c^2*x^2 +1)^(1/2))-1/2*dilog(1-I*(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 ))+3/2*arctanh(c*x)^2-3/4*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))-4/3*arctanh(c *x)^3+1/4*(c*x-1)/(c*x+1)-1/2*arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/ 2))-1/2*arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-polylog(3,-(c*x+1) ^2/(-c^2*x^2+1))+2*arctanh(c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+2*ln(2) *arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))+2*ln(2)*arctanh(c*x)*ln(1 -I*(c*x+1)/(-c^2*x^2+1)^(1/2))-2*ln(2)*arctanh(c*x)*ln(1+(c*x+1)^2/(-c^2*x ^2+1))+2*ln(2)*dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))+2*ln(2)*dilog(1-I*(c* x+1)/(-c^2*x^2+1)^(1/2))-ln(2)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))-1/(c*x+1 )*arctanh(c*x)^2+1/2*arctanh(c*x)*(c*x-1)/(c*x+1)-1/2*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))*csgn(I*(c*x+1)^2/(c^2*x^2 -1)/(1-(c*x+1)^2/(c^2*x^2-1)))*(2*arctanh(c*x)^2-2*arctanh(c*x)*ln(1+(c*x+ 1)^2/(-c^2*x^2+1))-polylog(2,-(c*x+1)^2/(-c^2*x^2+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)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^ (1/2))+arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+dilog(1+I*(c*x+1)/( -c^2*x^2+1)^(1/2))+dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2)))+2*I*Pi*csgn(I...
\[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{2}}{{\left (c d x + d\right )}^{2}} \,d x } \]
integral((b^2*x^2*arctanh(c*x)^2 + 2*a*b*x^2*arctanh(c*x) + a^2*x^2)/(c^2* d^2*x^2 + 2*c*d^2*x + d^2), x)
\[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {\int \frac {a^{2} x^{2}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac {b^{2} x^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac {2 a b x^{2} \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx}{d^{2}} \]
(Integral(a**2*x**2/(c**2*x**2 + 2*c*x + 1), x) + Integral(b**2*x**2*atanh (c*x)**2/(c**2*x**2 + 2*c*x + 1), x) + Integral(2*a*b*x**2*atanh(c*x)/(c** 2*x**2 + 2*c*x + 1), x))/d**2
\[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{2}}{{\left (c d x + d\right )}^{2}} \,d x } \]
-a^2*(1/(c^4*d^2*x + c^3*d^2) - x/(c^2*d^2) + 2*log(c*x + 1)/(c^3*d^2)) + 1/4*(b^2*c^2*x^2 + b^2*c*x - b^2 - 2*(b^2*c*x + b^2)*log(c*x + 1))*log(-c* x + 1)^2/(c^4*d^2*x + c^3*d^2) - integrate(-1/4*((b^2*c^3*x^3 - b^2*c^2*x^ 2)*log(c*x + 1)^2 + 4*(a*b*c^3*x^3 - a*b*c^2*x^2)*log(c*x + 1) - 2*((2*a*b *c^3 + b^2*c^3)*x^3 - 2*(a*b*c^2 - b^2*c^2)*x^2 - b^2 + (b^2*c^3*x^3 - 3*b ^2*c^2*x^2 - 4*b^2*c*x - 2*b^2)*log(c*x + 1))*log(-c*x + 1))/(c^5*d^2*x^3 + c^4*d^2*x^2 - c^3*d^2*x - c^2*d^2), x)
\[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{2}}{{\left (c d x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\int \frac {x^2\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\right )}^2} \,d x \]