Integrand size = 22, antiderivative size = 394 \[ \int \frac {x^4 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=-\frac {2 a b x}{c^4 d^2}+\frac {b^2 x}{3 c^4 d^2}-\frac {b^2}{2 c^5 d^2 (1+c x)}+\frac {b^2 \text {arctanh}(c x)}{6 c^5 d^2}-\frac {2 b^2 x \text {arctanh}(c x)}{c^4 d^2}+\frac {b x^2 (a+b \text {arctanh}(c x))}{3 c^3 d^2}-\frac {b (a+b \text {arctanh}(c x))}{c^5 d^2 (1+c x)}+\frac {29 (a+b \text {arctanh}(c x))^2}{6 c^5 d^2}+\frac {3 x (a+b \text {arctanh}(c x))^2}{c^4 d^2}-\frac {x^2 (a+b \text {arctanh}(c x))^2}{c^3 d^2}+\frac {x^3 (a+b \text {arctanh}(c x))^2}{3 c^2 d^2}-\frac {(a+b \text {arctanh}(c x))^2}{c^5 d^2 (1+c x)}-\frac {20 b (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{3 c^5 d^2}+\frac {4 (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {b^2 \log \left (1-c^2 x^2\right )}{c^5 d^2}-\frac {10 b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^5 d^2}-\frac {4 b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {2 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{c^5 d^2} \]
-2*a*b*x/c^4/d^2+1/3*b^2*x/c^4/d^2-1/2*b^2/c^5/d^2/(c*x+1)+1/6*b^2*arctanh (c*x)/c^5/d^2-2*b^2*x*arctanh(c*x)/c^4/d^2+1/3*b*x^2*(a+b*arctanh(c*x))/c^ 3/d^2-b*(a+b*arctanh(c*x))/c^5/d^2/(c*x+1)+29/6*(a+b*arctanh(c*x))^2/c^5/d ^2+3*x*(a+b*arctanh(c*x))^2/c^4/d^2-x^2*(a+b*arctanh(c*x))^2/c^3/d^2+1/3*x ^3*(a+b*arctanh(c*x))^2/c^2/d^2-(a+b*arctanh(c*x))^2/c^5/d^2/(c*x+1)-20/3* b*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^5/d^2+4*(a+b*arctanh(c*x))^2*ln(2/(c *x+1))/c^5/d^2-b^2*ln(-c^2*x^2+1)/c^5/d^2-10/3*b^2*polylog(2,1-2/(-c*x+1)) /c^5/d^2-4*b*(a+b*arctanh(c*x))*polylog(2,1-2/(c*x+1))/c^5/d^2-2*b^2*polyl og(3,1-2/(c*x+1))/c^5/d^2
Time = 1.30 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.08 \[ \int \frac {x^4 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {36 a^2 c x-12 a^2 c^2 x^2+4 a^2 c^3 x^3-\frac {12 a^2}{1+c x}-48 a^2 \log (1+c x)+b^2 \left (4 c x-4 \text {arctanh}(c x)-24 c x \text {arctanh}(c x)+4 c^2 x^2 \text {arctanh}(c x)-28 \text {arctanh}(c x)^2+36 c x \text {arctanh}(c x)^2-12 c^2 x^2 \text {arctanh}(c x)^2+4 c^3 x^3 \text {arctanh}(c x)^2-3 \cosh (2 \text {arctanh}(c x))-6 \text {arctanh}(c x) \cosh (2 \text {arctanh}(c x))-6 \text {arctanh}(c x)^2 \cosh (2 \text {arctanh}(c x))-80 \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+48 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-12 \log \left (1-c^2 x^2\right )-8 (-5+6 \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )-24 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )+3 \sinh (2 \text {arctanh}(c x))+6 \text {arctanh}(c x) \sinh (2 \text {arctanh}(c x))+6 \text {arctanh}(c x)^2 \sinh (2 \text {arctanh}(c x))\right )+2 a b \left (-2-12 c x+2 c^2 x^2-3 \cosh (2 \text {arctanh}(c x))+20 \log \left (1-c^2 x^2\right )-24 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+3 \sinh (2 \text {arctanh}(c x))+2 \text {arctanh}(c x) \left (6+18 c x-6 c^2 x^2+2 c^3 x^3-3 \cosh (2 \text {arctanh}(c x))+24 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+3 \sinh (2 \text {arctanh}(c x))\right )\right )}{12 c^5 d^2} \]
(36*a^2*c*x - 12*a^2*c^2*x^2 + 4*a^2*c^3*x^3 - (12*a^2)/(1 + c*x) - 48*a^2 *Log[1 + c*x] + b^2*(4*c*x - 4*ArcTanh[c*x] - 24*c*x*ArcTanh[c*x] + 4*c^2* x^2*ArcTanh[c*x] - 28*ArcTanh[c*x]^2 + 36*c*x*ArcTanh[c*x]^2 - 12*c^2*x^2* ArcTanh[c*x]^2 + 4*c^3*x^3*ArcTanh[c*x]^2 - 3*Cosh[2*ArcTanh[c*x]] - 6*Arc Tanh[c*x]*Cosh[2*ArcTanh[c*x]] - 6*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] - 8 0*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] + 48*ArcTanh[c*x]^2*Log[1 + E^ (-2*ArcTanh[c*x])] - 12*Log[1 - c^2*x^2] - 8*(-5 + 6*ArcTanh[c*x])*PolyLog [2, -E^(-2*ArcTanh[c*x])] - 24*PolyLog[3, -E^(-2*ArcTanh[c*x])] + 3*Sinh[2 *ArcTanh[c*x]] + 6*ArcTanh[c*x]*Sinh[2*ArcTanh[c*x]] + 6*ArcTanh[c*x]^2*Si nh[2*ArcTanh[c*x]]) + 2*a*b*(-2 - 12*c*x + 2*c^2*x^2 - 3*Cosh[2*ArcTanh[c* x]] + 20*Log[1 - c^2*x^2] - 24*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 3*Sinh[2 *ArcTanh[c*x]] + 2*ArcTanh[c*x]*(6 + 18*c*x - 6*c^2*x^2 + 2*c^3*x^3 - 3*Co sh[2*ArcTanh[c*x]] + 24*Log[1 + E^(-2*ArcTanh[c*x])] + 3*Sinh[2*ArcTanh[c* x]])))/(12*c^5*d^2)
Time = 1.11 (sec) , antiderivative size = 394, 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.
| \(\displaystyle \int \frac {x^4 (a+b \text {arctanh}(c x))^2}{(c d x+d)^2} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (-\frac {4 (a+b \text {arctanh}(c x))^2}{c^4 d^2 (c x+1)}+\frac {3 (a+b \text {arctanh}(c x))^2}{c^4 d^2}+\frac {(a+b \text {arctanh}(c x))^2}{c^4 d^2 (c x+1)^2}-\frac {2 x (a+b \text {arctanh}(c x))^2}{c^3 d^2}+\frac {x^2 (a+b \text {arctanh}(c x))^2}{c^2 d^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 b \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{c^5 d^2}-\frac {(a+b \text {arctanh}(c x))^2}{c^5 d^2 (c x+1)}+\frac {29 (a+b \text {arctanh}(c x))^2}{6 c^5 d^2}-\frac {b (a+b \text {arctanh}(c x))}{c^5 d^2 (c x+1)}-\frac {20 b \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{3 c^5 d^2}+\frac {4 \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{c^5 d^2}+\frac {3 x (a+b \text {arctanh}(c x))^2}{c^4 d^2}-\frac {x^2 (a+b \text {arctanh}(c x))^2}{c^3 d^2}+\frac {b x^2 (a+b \text {arctanh}(c x))}{3 c^3 d^2}+\frac {x^3 (a+b \text {arctanh}(c x))^2}{3 c^2 d^2}-\frac {2 a b x}{c^4 d^2}+\frac {b^2 \text {arctanh}(c x)}{6 c^5 d^2}-\frac {2 b^2 x \text {arctanh}(c x)}{c^4 d^2}-\frac {10 b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^5 d^2}-\frac {2 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{c^5 d^2}-\frac {b^2}{2 c^5 d^2 (c x+1)}+\frac {b^2 x}{3 c^4 d^2}-\frac {b^2 \log \left (1-c^2 x^2\right )}{c^5 d^2}\) |
(-2*a*b*x)/(c^4*d^2) + (b^2*x)/(3*c^4*d^2) - b^2/(2*c^5*d^2*(1 + c*x)) + ( b^2*ArcTanh[c*x])/(6*c^5*d^2) - (2*b^2*x*ArcTanh[c*x])/(c^4*d^2) + (b*x^2* (a + b*ArcTanh[c*x]))/(3*c^3*d^2) - (b*(a + b*ArcTanh[c*x]))/(c^5*d^2*(1 + c*x)) + (29*(a + b*ArcTanh[c*x])^2)/(6*c^5*d^2) + (3*x*(a + b*ArcTanh[c*x ])^2)/(c^4*d^2) - (x^2*(a + b*ArcTanh[c*x])^2)/(c^3*d^2) + (x^3*(a + b*Arc Tanh[c*x])^2)/(3*c^2*d^2) - (a + b*ArcTanh[c*x])^2/(c^5*d^2*(1 + c*x)) - ( 20*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(3*c^5*d^2) + (4*(a + b*ArcTan h[c*x])^2*Log[2/(1 + c*x)])/(c^5*d^2) - (b^2*Log[1 - c^2*x^2])/(c^5*d^2) - (10*b^2*PolyLog[2, 1 - 2/(1 - c*x)])/(3*c^5*d^2) - (4*b*(a + b*ArcTanh[c* x])*PolyLog[2, 1 - 2/(1 + c*x)])/(c^5*d^2) - (2*b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(c^5*d^2)
3.2.3.3.1 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 = 10.43 (sec) , antiderivative size = 1050, normalized size of antiderivative = 2.66
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1050\) |
| default | \(\text {Expression too large to display}\) | \(1050\) |
| parts | \(\text {Expression too large to display}\) | \(1060\) |
1/c^5*(a^2/d^2*(1/3*c^3*x^3-c^2*x^2+3*c*x-4*ln(c*x+1)-1/(c*x+1))+b^2/d^2*( 3*c*x*arctanh(c*x)^2-1/3-20/3*dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-20/3*d ilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+1/3*c*x-c^2*x^2*arctanh(c*x)^2-4*arct anh(c*x)^2*ln(c*x+1)+4*arctanh(c*x)^2*ln(2)+8*arctanh(c*x)^2*ln((c*x+1)/(- c^2*x^2+1)^(1/2))+29/6*arctanh(c*x)^2+2*ln(1+(c*x+1)^2/(-c^2*x^2+1))-8/3*a rctanh(c*x)^3+1/4*(c*x-1)/(c*x+1)-20/3*arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x ^2+1)^(1/2))-20/3*arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-2*polylo g(3,-(c*x+1)^2/(-c^2*x^2+1))+4*arctanh(c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2 +1))+1/3*arctanh(c*x)^2*c^3*x^3-4/3*(c*x+1)*arctanh(c*x)+1/3*(c*x-3)*(c*x+ 1)*arctanh(c*x)-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)))*arcta nh(c*x)^2+2*I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^3 *arctanh(c*x)^2+2*I*Pi*csgn(I*(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)+2*I*Pi*csgn(I*(c*x+1 )/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*arctanh(c*x)^2+4*I*P i*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2*arcta nh(c*x)^2+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-(c*x+1)^2/(c^2*x^2-1)))^2*arctanh(c*x)^2-2*I*Pi*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*arc tanh(c*x)^2)+2*a*b/d^2*(1/3*c^3*x^3*arctanh(c*x)-c^2*x^2*arctanh(c*x)+3...
\[ \int \frac {x^4 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c d x + d\right )}^{2}} \,d x } \]
integral((b^2*x^4*arctanh(c*x)^2 + 2*a*b*x^4*arctanh(c*x) + a^2*x^4)/(c^2* d^2*x^2 + 2*c*d^2*x + d^2), x)
\[ \int \frac {x^4 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {\int \frac {a^{2} x^{4}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac {b^{2} x^{4} \operatorname {atanh}^{2}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac {2 a b x^{4} \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx}{d^{2}} \]
(Integral(a**2*x**4/(c**2*x**2 + 2*c*x + 1), x) + Integral(b**2*x**4*atanh (c*x)**2/(c**2*x**2 + 2*c*x + 1), x) + Integral(2*a*b*x**4*atanh(c*x)/(c** 2*x**2 + 2*c*x + 1), x))/d**2
\[ \int \frac {x^4 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c d x + d\right )}^{2}} \,d x } \]
-1/3*a^2*(3/(c^6*d^2*x + c^5*d^2) - (c^2*x^3 - 3*c*x^2 + 9*x)/(c^4*d^2) + 12*log(c*x + 1)/(c^5*d^2)) + 1/12*(b^2*c^4*x^4 - 2*b^2*c^3*x^3 + 6*b^2*c^2 *x^2 + 9*b^2*c*x - 3*b^2 - 12*(b^2*c*x + b^2)*log(c*x + 1))*log(-c*x + 1)^ 2/(c^6*d^2*x + c^5*d^2) - integrate(-1/12*(3*(b^2*c^5*x^5 - b^2*c^4*x^4)*l og(c*x + 1)^2 + 12*(a*b*c^5*x^5 - a*b*c^4*x^4)*log(c*x + 1) - 2*(4*b^2*c^3 *x^3 + 15*b^2*c^2*x^2 + (6*a*b*c^5 + b^2*c^5)*x^5 - (6*a*b*c^4 + b^2*c^4)* x^4 + 6*b^2*c*x - 3*b^2 + 3*(b^2*c^5*x^5 - b^2*c^4*x^4 - 4*b^2*c^2*x^2 - 8 *b^2*c*x - 4*b^2)*log(c*x + 1))*log(-c*x + 1))/(c^7*d^2*x^3 + c^6*d^2*x^2 - c^5*d^2*x - c^4*d^2), x)
\[ \int \frac {x^4 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c d x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^4 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\right )}^2} \,d x \]