Integrand size = 21, antiderivative size = 385 \[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{d+e x} \, dx=\frac {a b x}{c e}+\frac {b^2 x \text {arctanh}(c x)}{c e}-\frac {d (a+b \text {arctanh}(c x))^2}{c e^2}-\frac {(a+b \text {arctanh}(c x))^2}{2 c^2 e}-\frac {d x (a+b \text {arctanh}(c x))^2}{e^2}+\frac {x^2 (a+b \text {arctanh}(c x))^2}{2 e}+\frac {2 b d (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{c e^2}-\frac {d^2 (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{e^3}+\frac {d^2 (a+b \text {arctanh}(c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^3}+\frac {b^2 \log \left (1-c^2 x^2\right )}{2 c^2 e}+\frac {b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c e^2}+\frac {b d^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{e^3}-\frac {b d^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^3}+\frac {b^2 d^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 e^3}-\frac {b^2 d^2 \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^3} \]
a*b*x/c/e+b^2*x*arctanh(c*x)/c/e-d*(a+b*arctanh(c*x))^2/c/e^2-1/2*(a+b*arc tanh(c*x))^2/c^2/e-d*x*(a+b*arctanh(c*x))^2/e^2+1/2*x^2*(a+b*arctanh(c*x)) ^2/e+2*b*d*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c/e^2-d^2*(a+b*arctanh(c*x))^ 2*ln(2/(c*x+1))/e^3+d^2*(a+b*arctanh(c*x))^2*ln(2*c*(e*x+d)/(c*d+e)/(c*x+1 ))/e^3+1/2*b^2*ln(-c^2*x^2+1)/c^2/e+b^2*d*polylog(2,1-2/(-c*x+1))/c/e^2+b* d^2*(a+b*arctanh(c*x))*polylog(2,1-2/(c*x+1))/e^3-b*d^2*(a+b*arctanh(c*x)) *polylog(2,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/e^3+1/2*b^2*d^2*polylog(3,1-2/(c *x+1))/e^3-1/2*b^2*d^2*polylog(3,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/e^3
Result contains complex when optimal does not.
Time = 13.92 (sec) , antiderivative size = 1414, normalized size of antiderivative = 3.67 \[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{d+e x} \, dx =\text {Too large to display} \]
-((a^2*d*x)/e^2) + (a^2*x^2)/(2*e) + (a^2*d^2*Log[d + e*x])/e^3 + (a*b*(c* e^2*x + I*c^2*d^2*Pi*ArcTanh[c*x] - 2*c^2*d*e*x*ArcTanh[c*x] - e^2*(1 - c^ 2*x^2)*ArcTanh[c*x] + 2*c^2*d^2*ArcTanh[(c*d)/e]*ArcTanh[c*x] - c^2*d^2*Ar cTanh[c*x]^2 + c*d*e*ArcTanh[c*x]^2 - (c*d*Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTa nh[c*x]^2)/E^ArcTanh[(c*d)/e] - 2*c^2*d^2*ArcTanh[c*x]*Log[1 + E^(-2*ArcTa nh[c*x])] - I*c^2*d^2*Pi*Log[1 + E^(2*ArcTanh[c*x])] + 2*c^2*d^2*ArcTanh[( c*d)/e]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + 2*c^2*d^2*ArcT anh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + 2*c*d*e*Log[1 /Sqrt[1 - c^2*x^2]] + I*c^2*d^2*Pi*Log[1/Sqrt[1 - c^2*x^2]] - 2*c^2*d^2*Ar cTanh[(c*d)/e]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] + c^2*d^2*Poly Log[2, -E^(-2*ArcTanh[c*x])] - c^2*d^2*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/(c^2*e^3) + (b^2*((6*c*e^2*x*ArcTanh[c*x] + 6*c*d*e*Ar cTanh[c*x]^2 - 6*c^2*d*e*x*ArcTanh[c*x]^2 - 3*e^2*(1 - c^2*x^2)*ArcTanh[c* x]^2 - 2*c^2*d^2*ArcTanh[c*x]^3 + 2*c*d*e*ArcTanh[c*x]^3 + 12*c*d*e*ArcTan h[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 6*c^2*d^2*ArcTanh[c*x]^2*Log[1 + E^( -2*ArcTanh[c*x])] - 6*e^2*Log[1/Sqrt[1 - c^2*x^2]] + 6*c*d*(-e + c*d*ArcTa nh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 3*c^2*d^2*PolyLog[3, -E^(-2*Ar cTanh[c*x])])/(6*e^3) - (c*d*(-(c*d) + e)*(c*d + e)*(-6*c*d*ArcTanh[c*x]^3 + 2*e*ArcTanh[c*x]^3 - (4*Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^3)/E^Arc Tanh[(c*d)/e] - 6*c*d*ArcTanh[c*x]^2*Log[1 - (Sqrt[c*d + e]*E^ArcTanh[c...
Time = 0.74 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.
| \(\displaystyle \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{d+e x} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (\frac {d^2 (a+b \text {arctanh}(c x))^2}{e^2 (d+e x)}-\frac {d (a+b \text {arctanh}(c x))^2}{e^2}+\frac {x (a+b \text {arctanh}(c x))^2}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(a+b \text {arctanh}(c x))^2}{2 c^2 e}+\frac {b d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{e^3}-\frac {b d^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{e^3}-\frac {d^2 \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{e^3}+\frac {d^2 (a+b \text {arctanh}(c x))^2 \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e^3}-\frac {d x (a+b \text {arctanh}(c x))^2}{e^2}-\frac {d (a+b \text {arctanh}(c x))^2}{c e^2}+\frac {2 b d \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c e^2}+\frac {x^2 (a+b \text {arctanh}(c x))^2}{2 e}+\frac {a b x}{c e}+\frac {b^2 x \text {arctanh}(c x)}{c e}+\frac {b^2 \log \left (1-c^2 x^2\right )}{2 c^2 e}+\frac {b^2 d^2 \operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{2 e^3}-\frac {b^2 d^2 \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 e^3}+\frac {b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c e^2}\) |
(a*b*x)/(c*e) + (b^2*x*ArcTanh[c*x])/(c*e) - (d*(a + b*ArcTanh[c*x])^2)/(c *e^2) - (a + b*ArcTanh[c*x])^2/(2*c^2*e) - (d*x*(a + b*ArcTanh[c*x])^2)/e^ 2 + (x^2*(a + b*ArcTanh[c*x])^2)/(2*e) + (2*b*d*(a + b*ArcTanh[c*x])*Log[2 /(1 - c*x)])/(c*e^2) - (d^2*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/e^3 + (d^2*(a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e ^3 + (b^2*Log[1 - c^2*x^2])/(2*c^2*e) + (b^2*d*PolyLog[2, 1 - 2/(1 - c*x)] )/(c*e^2) + (b*d^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/e^3 - (b*d^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e^3 + (b^2*d^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*e^3) - (b^2*d^2*P olyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e^3)
3.2.54.3.1 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 = 39.65 (sec) , antiderivative size = 1573, normalized size of antiderivative = 4.09
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1573\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1577\) |
| default | \(\text {Expression too large to display}\) | \(1577\) |
1/2*a^2/e*x^2-a^2/e^2*x*d+a^2*d^2/e^3*ln(e*x+d)+b^2/c^3*(1/2*c^3*arctanh(c *x)^2/e*x^2-c^3*arctanh(c*x)^2/e^2*x*d+c^3*arctanh(c*x)^2/e^3*d^2*ln(c*e*x +c*d)-2*c*(1/4/e*arctanh(c*x)^2+1/2/e^2*c*d*arctanh(c*x)^2-1/e^2*ln(1+I*(c *x+1)/(-c^2*x^2+1)^(1/2))*c*d*arctanh(c*x)-1/e^2*ln(1-I*(c*x+1)/(-c^2*x^2+ 1)^(1/2))*c*d*arctanh(c*x)+1/2/e*ln(1+(c*x+1)^2/(-c^2*x^2+1))-1/2*(c*x+1)* arctanh(c*x)/e-1/e^2*dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))*c*d-1/e^2*dilog (1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))*c*d-1/4*I/e^3*Pi*arctanh(c*x)^2*c^2*d^2*c sgn(I*(d*c*(1-(c*x+1)^2/(c^2*x^2-1))+e*(-(c*x+1)^2/(c^2*x^2-1)-1))/(1-(c*x +1)^2/(c^2*x^2-1)))^3+1/4*I/e^3*Pi*arctanh(c*x)^2*c^2*d^2*csgn(I*(d*c*(1-( c*x+1)^2/(c^2*x^2-1))+e*(-(c*x+1)^2/(c^2*x^2-1)-1)))*csgn(I*(d*c*(1-(c*x+1 )^2/(c^2*x^2-1))+e*(-(c*x+1)^2/(c^2*x^2-1)-1))/(1-(c*x+1)^2/(c^2*x^2-1)))^ 2+1/4*I/e^3*Pi*arctanh(c*x)^2*c^2*d^2*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*cs gn(I*(d*c*(1-(c*x+1)^2/(c^2*x^2-1))+e*(-(c*x+1)^2/(c^2*x^2-1)-1))/(1-(c*x+ 1)^2/(c^2*x^2-1)))^2-1/4*I/e^3*Pi*arctanh(c*x)^2*c^2*d^2*csgn(I/(1-(c*x+1) ^2/(c^2*x^2-1)))*csgn(I*(d*c*(1-(c*x+1)^2/(c^2*x^2-1))+e*(-(c*x+1)^2/(c^2* x^2-1)-1)))*csgn(I*(d*c*(1-(c*x+1)^2/(c^2*x^2-1))+e*(-(c*x+1)^2/(c^2*x^2-1 )-1))/(1-(c*x+1)^2/(c^2*x^2-1)))+1/2*d^2*c^2/e^3*arctanh(c*x)*polylog(2,-( c*x+1)^2/(-c^2*x^2+1))-1/4*d^2*c^2/e^3*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))+ 1/2*d^2*c^2/e^3*arctanh(c*x)^2*ln(d*c*(1+(c*x+1)^2/(-c^2*x^2+1))+e*((c*x+1 )^2/(-c^2*x^2+1)-1))-1/2*d^2*c^2/e^2/(c*d+e)*arctanh(c*x)^2*ln(1-(c*d+e...
\[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{2}}{e x + d} \,d x } \]
\[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{d+e x} \, dx=\int \frac {x^{2} \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{2}}{d + e x}\, dx \]
\[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{2}}{e x + d} \,d x } \]
1/2*a^2*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2) + 1/8*(b^2*e*x^2 - 2*b^2*d*x)*log(-c*x + 1)^2/e^2 - integrate(-1/4*((b^2*c*e^2*x^3 - b^2*e^2* x^2)*log(c*x + 1)^2 + 4*(a*b*c*e^2*x^3 - a*b*e^2*x^2)*log(c*x + 1) + (2*b^ 2*c*d^2*x - (4*a*b*c*e^2 + b^2*c*e^2)*x^3 + (b^2*c*d*e + 4*a*b*e^2)*x^2 - 2*(b^2*c*e^2*x^3 - b^2*e^2*x^2)*log(c*x + 1))*log(-c*x + 1))/(c*e^3*x^2 - d*e^2 + (c*d*e^2 - e^3)*x), x)
\[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{2}}{e x + d} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{d+e x} \, dx=\int \frac {x^2\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{d+e\,x} \,d x \]