Integrand size = 18, antiderivative size = 387 \[ \int (d+e x)^2 (a+b \text {arctanh}(c x))^3 \, dx=\frac {a b^2 e^2 x}{c^2}+\frac {b^3 e^2 x \text {arctanh}(c x)}{c^2}+\frac {3 b d e (a+b \text {arctanh}(c x))^2}{c^2}-\frac {b e^2 (a+b \text {arctanh}(c x))^2}{2 c^3}+\frac {3 b d e x (a+b \text {arctanh}(c x))^2}{c}+\frac {b e^2 x^2 (a+b \text {arctanh}(c x))^2}{2 c}+\frac {\left (3 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^3}{3 c^3}-\frac {d \left (d^2+\frac {3 e^2}{c^2}\right ) (a+b \text {arctanh}(c x))^3}{3 e}+\frac {(d+e x)^3 (a+b \text {arctanh}(c x))^3}{3 e}-\frac {6 b^2 d e (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {b \left (3 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 e^2 \log \left (1-c^2 x^2\right )}{2 c^3}-\frac {3 b^3 d e \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^2}-\frac {b^2 \left (3 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 \left (3 c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 c^3} \]
a*b^2*e^2*x/c^2+b^3*e^2*x*arctanh(c*x)/c^2+3*b*d*e*(a+b*arctanh(c*x))^2/c^ 2-1/2*b*e^2*(a+b*arctanh(c*x))^2/c^3+3*b*d*e*x*(a+b*arctanh(c*x))^2/c+1/2* b*e^2*x^2*(a+b*arctanh(c*x))^2/c+1/3*(3*c^2*d^2+e^2)*(a+b*arctanh(c*x))^3/ c^3-1/3*d*(d^2+3*e^2/c^2)*(a+b*arctanh(c*x))^3/e+1/3*(e*x+d)^3*(a+b*arctan h(c*x))^3/e-6*b^2*d*e*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^2-b*(3*c^2*d^2+e ^2)*(a+b*arctanh(c*x))^2*ln(2/(-c*x+1))/c^3+1/2*b^3*e^2*ln(-c^2*x^2+1)/c^3 -3*b^3*d*e*polylog(2,1-2/(-c*x+1))/c^2-b^2*(3*c^2*d^2+e^2)*(a+b*arctanh(c* x))*polylog(2,1-2/(-c*x+1))/c^3+1/2*b^3*(3*c^2*d^2+e^2)*polylog(3,1-2/(-c* x+1))/c^3
Time = 0.80 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.53 \[ \int (d+e x)^2 (a+b \text {arctanh}(c x))^3 \, dx=\frac {6 a^2 c^2 d (a c d+3 b e) x+3 a^2 c^2 e (2 a c d+b e) x^2+2 a^3 c^3 e^2 x^3+6 a^2 b c^3 x \left (3 d^2+3 d e x+e^2 x^2\right ) \text {arctanh}(c x)+3 a^2 b \left (3 c^2 d^2+3 c d e+e^2\right ) \log (1-c x)+3 a^2 b \left (3 c^2 d^2-3 c d e+e^2\right ) \log (1+c x)+18 a b^2 c d e \left (2 c x \text {arctanh}(c x)+\left (-1+c^2 x^2\right ) \text {arctanh}(c x)^2+\log \left (1-c^2 x^2\right )\right )-6 b^3 c d e \left (\text {arctanh}(c x) \left ((3-3 c x) \text {arctanh}(c x)+\left (1-c^2 x^2\right ) \text {arctanh}(c x)^2+6 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )-3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+6 a b^2 e^2 \left (c x+\left (-1+c^3 x^3\right ) \text {arctanh}(c x)^2+\text {arctanh}(c x) \left (-1+c^2 x^2-2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+18 a b^2 c^2 d^2 \left (\text {arctanh}(c x) \left ((-1+c x) \text {arctanh}(c x)-2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+6 b^3 c^2 d^2 \left (\text {arctanh}(c x)^2 \left ((-1+c x) \text {arctanh}(c x)-3 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+3 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )\right )+b^3 e^2 \left (6 c x \text {arctanh}(c x)-3 \text {arctanh}(c x)^2+3 c^2 x^2 \text {arctanh}(c x)^2-2 \text {arctanh}(c x)^3+2 c^3 x^3 \text {arctanh}(c x)^3-6 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+3 \log \left (1-c^2 x^2\right )+6 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )\right )}{6 c^3} \]
(6*a^2*c^2*d*(a*c*d + 3*b*e)*x + 3*a^2*c^2*e*(2*a*c*d + b*e)*x^2 + 2*a^3*c ^3*e^2*x^3 + 6*a^2*b*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcTanh[c*x] + 3*a^ 2*b*(3*c^2*d^2 + 3*c*d*e + e^2)*Log[1 - c*x] + 3*a^2*b*(3*c^2*d^2 - 3*c*d* e + e^2)*Log[1 + c*x] + 18*a*b^2*c*d*e*(2*c*x*ArcTanh[c*x] + (-1 + c^2*x^2 )*ArcTanh[c*x]^2 + Log[1 - c^2*x^2]) - 6*b^3*c*d*e*(ArcTanh[c*x]*((3 - 3*c *x)*ArcTanh[c*x] + (1 - c^2*x^2)*ArcTanh[c*x]^2 + 6*Log[1 + E^(-2*ArcTanh[ c*x])]) - 3*PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 6*a*b^2*e^2*(c*x + (-1 + c ^3*x^3)*ArcTanh[c*x]^2 + ArcTanh[c*x]*(-1 + c^2*x^2 - 2*Log[1 + E^(-2*ArcT anh[c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 18*a*b^2*c^2*d^2*(ArcTan h[c*x]*((-1 + c*x)*ArcTanh[c*x] - 2*Log[1 + E^(-2*ArcTanh[c*x])]) + PolyLo g[2, -E^(-2*ArcTanh[c*x])]) + 6*b^3*c^2*d^2*(ArcTanh[c*x]^2*((-1 + c*x)*Ar cTanh[c*x] - 3*Log[1 + E^(-2*ArcTanh[c*x])]) + 3*ArcTanh[c*x]*PolyLog[2, - E^(-2*ArcTanh[c*x])] + (3*PolyLog[3, -E^(-2*ArcTanh[c*x])])/2) + b^3*e^2*( 6*c*x*ArcTanh[c*x] - 3*ArcTanh[c*x]^2 + 3*c^2*x^2*ArcTanh[c*x]^2 - 2*ArcTa nh[c*x]^3 + 2*c^3*x^3*ArcTanh[c*x]^3 - 6*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcT anh[c*x])] + 3*Log[1 - c^2*x^2] + 6*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh [c*x])] + 3*PolyLog[3, -E^(-2*ArcTanh[c*x])]))/(6*c^3)
Time = 1.04 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6480, 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.
\(\displaystyle \int (d+e x)^2 (a+b \text {arctanh}(c x))^3 \, dx\) |
\(\Big \downarrow \) 6480 |
\(\displaystyle \frac {(d+e x)^3 (a+b \text {arctanh}(c x))^3}{3 e}-\frac {b c \int \left (-\frac {x (a+b \text {arctanh}(c x))^2 e^3}{c^2}-\frac {3 d (a+b \text {arctanh}(c x))^2 e^2}{c^2}+\frac {\left (d \left (c^2 d^2+3 e^2\right )+e \left (3 c^2 d^2+e^2\right ) x\right ) (a+b \text {arctanh}(c x))^2}{c^2 \left (1-c^2 x^2\right )}\right )dx}{e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(d+e x)^3 (a+b \text {arctanh}(c x))^3}{3 e}-\frac {b c \left (\frac {e^3 (a+b \text {arctanh}(c x))^2}{2 c^4}-\frac {3 d e^2 (a+b \text {arctanh}(c x))^2}{c^3}+\frac {6 b d e^2 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c^3}-\frac {3 d e^2 x (a+b \text {arctanh}(c x))^2}{c^2}-\frac {e^3 x^2 (a+b \text {arctanh}(c x))^2}{2 c^2}+\frac {b e \left (3 c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c^4}-\frac {e \left (3 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^3}{3 b c^4}+\frac {e \left (3 c^2 d^2+e^2\right ) \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{c^4}+\frac {d \left (c^2 d^2+3 e^2\right ) (a+b \text {arctanh}(c x))^3}{3 b c^3}-\frac {a b e^3 x}{c^3}-\frac {b^2 e^3 x \text {arctanh}(c x)}{c^3}+\frac {3 b^2 d e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^3}-\frac {b^2 e \left (3 c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 c^4}-\frac {b^2 e^3 \log \left (1-c^2 x^2\right )}{2 c^4}\right )}{e}\) |
((d + e*x)^3*(a + b*ArcTanh[c*x])^3)/(3*e) - (b*c*(-((a*b*e^3*x)/c^3) - (b ^2*e^3*x*ArcTanh[c*x])/c^3 - (3*d*e^2*(a + b*ArcTanh[c*x])^2)/c^3 + (e^3*( a + b*ArcTanh[c*x])^2)/(2*c^4) - (3*d*e^2*x*(a + b*ArcTanh[c*x])^2)/c^2 - (e^3*x^2*(a + b*ArcTanh[c*x])^2)/(2*c^2) - (e*(3*c^2*d^2 + e^2)*(a + b*Arc Tanh[c*x])^3)/(3*b*c^4) + (d*(c^2*d^2 + 3*e^2)*(a + b*ArcTanh[c*x])^3)/(3* b*c^3) + (6*b*d*e^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c^3 + (e*(3*c^2 *d^2 + e^2)*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c^4 - (b^2*e^3*Log[1 - c^2*x^2])/(2*c^4) + (3*b^2*d*e^2*PolyLog[2, 1 - 2/(1 - c*x)])/c^3 + (b*e *(3*c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c^4 - (b^2*e*(3*c^2*d^2 + e^2)*PolyLog[3, 1 - 2/(1 - c*x)])/(2*c^4)))/e
3.1.16.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_S ymbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTanh[c*x])^p/(e*(q + 1))), x] - Simp[b*c*(p/(e*(q + 1))) Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^(p - 1 ), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 15.45 (sec) , antiderivative size = 3848, normalized size of antiderivative = 9.94
method | result | size |
parts | \(\text {Expression too large to display}\) | \(3848\) |
derivativedivides | \(\text {Expression too large to display}\) | \(3873\) |
default | \(\text {Expression too large to display}\) | \(3873\) |
1/3*a^3*(e*x+d)^3/e+b^3/c*(1/3*c*e^2*arctanh(c*x)^3*x^3+c*e*arctanh(c*x)^3 *x^2*d+arctanh(c*x)^3*c*x*d^2+1/3*c/e*arctanh(c*x)^3*d^3-1/c^2/e*(-1/3*e^3 *arctanh(c*x)^3+1/2*e^3*arctanh(c*x)^2-3/4*I*e^2*c*csgn(I*(c*x+1)^2/(c^2*x ^2-1))^3*Pi*d*arctanh(c*x)^2-3/4*I*e^2*c*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-( c*x+1)^2/(c^2*x^2-1)))^3*Pi*d*arctanh(c*x)^2-3/2*I*e^2*c*csgn(I/(1-(c*x+1) ^2/(c^2*x^2-1)))^2*Pi*d*arctanh(c*x)^2+3/2*I*e^2*c*csgn(I/(1-(c*x+1)^2/(c^ 2*x^2-1)))^3*Pi*d*arctanh(c*x)^2-1/4*I*Pi*e^3*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+1/4*I*c^3*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)))^2*Pi*d^3*arctanh(c* x)^2+3/2*I*c^2*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))^3*e*Pi*d^2*arctanh(c*x)^2 -3/2*I*c^2*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))^2*e*Pi*d^2*arctanh(c*x)^2-1/4 *I*Pi*c^3*d^3*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+3/4*I*c^2*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/ (c^2*x^2-1)))^3*e*Pi*d^2*arctanh(c*x)^2+3/4*I*c^2*csgn(I*(c*x+1)^2/(c^2*x^ 2-1))^3*e*Pi*d^2*arctanh(c*x)^2-1/4*I*c^3*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*Pi*d^3*arctanh (c*x)^2-1/2*I*c^3*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1)^2/(c^2 *x^2-1))^2*Pi*d^3*arctanh(c*x)^2+3*c^2*ln(2)*e*d^2*arctanh(c*x)^2+3*polylo g(2,-(c*x+1)^2/(-c^2*x^2+1))*c^2*d^2*e*arctanh(c*x)+6*c*d*e^2*arctanh(c...
\[ \int (d+e x)^2 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (e x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} \,d x } \]
integral(a^3*e^2*x^2 + 2*a^3*d*e*x + a^3*d^2 + (b^3*e^2*x^2 + 2*b^3*d*e*x + b^3*d^2)*arctanh(c*x)^3 + 3*(a*b^2*e^2*x^2 + 2*a*b^2*d*e*x + a*b^2*d^2)* arctanh(c*x)^2 + 3*(a^2*b*e^2*x^2 + 2*a^2*b*d*e*x + a^2*b*d^2)*arctanh(c*x ), x)
\[ \int (d+e x)^2 (a+b \text {arctanh}(c x))^3 \, dx=\int \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3} \left (d + e x\right )^{2}\, dx \]
\[ \int (d+e x)^2 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (e x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} \,d x } \]
1/3*a^3*e^2*x^3 + a^3*d*e*x^2 + 3/2*(2*x^2*arctanh(c*x) + c*(2*x/c^2 - log (c*x + 1)/c^3 + log(c*x - 1)/c^3))*a^2*b*d*e + 1/2*(2*x^3*arctanh(c*x) + c *(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*a^2*b*e^2 + a^3*d^2*x + 3/2*(2*c*x*arct anh(c*x) + log(-c^2*x^2 + 1))*a^2*b*d^2/c - 1/24*((b^3*c^3*e^2*x^3 + 3*b^3 *c^3*d*e*x^2 + 3*b^3*c^3*d^2*x - (3*c^2*d^2 + 3*c*d*e + e^2)*b^3)*log(-c*x + 1)^3 - 3*(2*a*b^2*c^3*e^2*x^3 + (6*a*b^2*c^3*d*e + b^3*c^2*e^2)*x^2 + 6 *(a*b^2*c^3*d^2 + b^3*c^2*d*e)*x + (b^3*c^3*e^2*x^3 + 3*b^3*c^3*d*e*x^2 + 3*b^3*c^3*d^2*x + (3*c^2*d^2 - 3*c*d*e + e^2)*b^3)*log(c*x + 1))*log(-c*x + 1)^2)/c^3 - integrate(-1/8*((b^3*c^3*e^2*x^3 - b^3*c^2*d^2 + (2*c^3*d*e - c^2*e^2)*b^3*x^2 + (c^3*d^2 - 2*c^2*d*e)*b^3*x)*log(c*x + 1)^3 + 6*(a*b^ 2*c^3*e^2*x^3 - a*b^2*c^2*d^2 + (2*c^3*d*e - c^2*e^2)*a*b^2*x^2 + (c^3*d^2 - 2*c^2*d*e)*a*b^2*x)*log(c*x + 1)^2 - (4*a*b^2*c^3*e^2*x^3 + 2*(6*a*b^2* c^3*d*e + b^3*c^2*e^2)*x^2 + 3*(b^3*c^3*e^2*x^3 - b^3*c^2*d^2 + (2*c^3*d*e - c^2*e^2)*b^3*x^2 + (c^3*d^2 - 2*c^2*d*e)*b^3*x)*log(c*x + 1)^2 + 12*(a* b^2*c^3*d^2 + b^3*c^2*d*e)*x - 2*(6*a*b^2*c^2*d^2 - (3*c^2*d^2 - 3*c*d*e + e^2)*b^3 - (6*a*b^2*c^3*e^2 + b^3*c^3*e^2)*x^3 - 3*(b^3*c^3*d*e + 2*(2*c^ 3*d*e - c^2*e^2)*a*b^2)*x^2 - 3*(b^3*c^3*d^2 + 2*(c^3*d^2 - 2*c^2*d*e)*a*b ^2)*x)*log(c*x + 1))*log(-c*x + 1))/(c^3*x - c^2), x)
\[ \int (d+e x)^2 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (e x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (d+e x)^2 (a+b \text {arctanh}(c x))^3 \, dx=\int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3\,{\left (d+e\,x\right )}^2 \,d x \]