Integrand size = 19, antiderivative size = 175 \[ \int (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=2 a b d^2 x+\frac {1}{3} b^2 d^2 x-\frac {b^2 d^2 \text {arctanh}(c x)}{3 c}+2 b^2 d^2 x \text {arctanh}(c x)+\frac {1}{3} b c d^2 x^2 (a+b \text {arctanh}(c x))+\frac {d^2 (1+c x)^3 (a+b \text {arctanh}(c x))^2}{3 c}-\frac {8 b d^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{3 c}+\frac {b^2 d^2 \log \left (1-c^2 x^2\right )}{c}-\frac {4 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c} \]
2*a*b*d^2*x+1/3*b^2*d^2*x-1/3*b^2*d^2*arctanh(c*x)/c+2*b^2*d^2*x*arctanh(c *x)+1/3*b*c*d^2*x^2*(a+b*arctanh(c*x))+1/3*d^2*(c*x+1)^3*(a+b*arctanh(c*x) )^2/c-8/3*b*d^2*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c+b^2*d^2*ln(-c^2*x^2+1) /c-4/3*b^2*d^2*polylog(2,1-2/(-c*x+1))/c
Time = 0.64 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.30 \[ \int (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\frac {d^2 \left (3 a^2 c x+6 a b c x+b^2 c x+3 a^2 c^2 x^2+a b c^2 x^2+a^2 c^3 x^3+b^2 \left (-7+3 c x+3 c^2 x^2+c^3 x^3\right ) \text {arctanh}(c x)^2+b \text {arctanh}(c x) \left (2 a c x \left (3+3 c x+c^2 x^2\right )+b \left (-1+6 c x+c^2 x^2\right )-8 b \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+3 a b \log (1-c x)-3 a b \log (1+c x)+3 a b \log \left (1-c^2 x^2\right )+3 b^2 \log \left (1-c^2 x^2\right )+a b \log \left (-1+c^2 x^2\right )+4 b^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )}{3 c} \]
(d^2*(3*a^2*c*x + 6*a*b*c*x + b^2*c*x + 3*a^2*c^2*x^2 + a*b*c^2*x^2 + a^2* c^3*x^3 + b^2*(-7 + 3*c*x + 3*c^2*x^2 + c^3*x^3)*ArcTanh[c*x]^2 + b*ArcTan h[c*x]*(2*a*c*x*(3 + 3*c*x + c^2*x^2) + b*(-1 + 6*c*x + c^2*x^2) - 8*b*Log [1 + E^(-2*ArcTanh[c*x])]) + 3*a*b*Log[1 - c*x] - 3*a*b*Log[1 + c*x] + 3*a *b*Log[1 - c^2*x^2] + 3*b^2*Log[1 - c^2*x^2] + a*b*Log[-1 + c^2*x^2] + 4*b ^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(3*c)
Time = 0.40 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, 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.
| \(\displaystyle \int (c d x+d)^2 (a+b \text {arctanh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6480 |
| \(\displaystyle \frac {d^2 (c x+1)^3 (a+b \text {arctanh}(c x))^2}{3 c}-\frac {2 b \int \left (-c x (a+b \text {arctanh}(c x)) d^3+\frac {4 (c x+1) (a+b \text {arctanh}(c x)) d^3}{1-c^2 x^2}-3 (a+b \text {arctanh}(c x)) d^3\right )dx}{3 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 (c x+1)^3 (a+b \text {arctanh}(c x))^2}{3 c}-\frac {2 b \left (-\frac {1}{2} c d^3 x^2 (a+b \text {arctanh}(c x))+\frac {4 d^3 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}-3 a d^3 x-3 b d^3 x \text {arctanh}(c x)+\frac {b d^3 \text {arctanh}(c x)}{2 c}-\frac {3 b d^3 \log \left (1-c^2 x^2\right )}{2 c}+\frac {2 b d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c}-\frac {1}{2} b d^3 x\right )}{3 d}\) |
(d^2*(1 + c*x)^3*(a + b*ArcTanh[c*x])^2)/(3*c) - (2*b*(-3*a*d^3*x - (b*d^3 *x)/2 + (b*d^3*ArcTanh[c*x])/(2*c) - 3*b*d^3*x*ArcTanh[c*x] - (c*d^3*x^2*( a + b*ArcTanh[c*x]))/2 + (4*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c - (3*b*d^3*Log[1 - c^2*x^2])/(2*c) + (2*b*d^3*PolyLog[2, 1 - 2/(1 - c*x)])/ c))/(3*d)
3.1.79.3.1 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]
Time = 1.28 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.27
| method | result | size |
| derivativedivides | \(\frac {\frac {d^{2} a^{2} \left (c x +1\right )^{3}}{3}+d^{2} b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+c^{2} x^{2} \operatorname {arctanh}\left (c x \right )^{2}+c x \operatorname {arctanh}\left (c x \right )^{2}+\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3}+\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{3}+2 c x \,\operatorname {arctanh}\left (c x \right )+\frac {8 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{3}+\frac {c x}{3}-\frac {1}{3}+\frac {5 \ln \left (c x +1\right )}{6}+\frac {7 \ln \left (c x -1\right )}{6}-\frac {4 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {4 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {2 \ln \left (c x -1\right )^{2}}{3}\right )+2 a b \,d^{2} \left (\frac {c^{3} x^{3} \operatorname {arctanh}\left (c x \right )}{3}+c^{2} x^{2} \operatorname {arctanh}\left (c x \right )+c x \,\operatorname {arctanh}\left (c x \right )+\frac {\operatorname {arctanh}\left (c x \right )}{3}+\frac {c^{2} x^{2}}{6}+c x +\frac {4 \ln \left (c x -1\right )}{3}\right )}{c}\) | \(223\) |
| default | \(\frac {\frac {d^{2} a^{2} \left (c x +1\right )^{3}}{3}+d^{2} b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+c^{2} x^{2} \operatorname {arctanh}\left (c x \right )^{2}+c x \operatorname {arctanh}\left (c x \right )^{2}+\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3}+\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{3}+2 c x \,\operatorname {arctanh}\left (c x \right )+\frac {8 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{3}+\frac {c x}{3}-\frac {1}{3}+\frac {5 \ln \left (c x +1\right )}{6}+\frac {7 \ln \left (c x -1\right )}{6}-\frac {4 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {4 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {2 \ln \left (c x -1\right )^{2}}{3}\right )+2 a b \,d^{2} \left (\frac {c^{3} x^{3} \operatorname {arctanh}\left (c x \right )}{3}+c^{2} x^{2} \operatorname {arctanh}\left (c x \right )+c x \,\operatorname {arctanh}\left (c x \right )+\frac {\operatorname {arctanh}\left (c x \right )}{3}+\frac {c^{2} x^{2}}{6}+c x +\frac {4 \ln \left (c x -1\right )}{3}\right )}{c}\) | \(223\) |
| parts | \(\frac {d^{2} a^{2} \left (c x +1\right )^{3}}{3 c}+\frac {d^{2} b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+c^{2} x^{2} \operatorname {arctanh}\left (c x \right )^{2}+c x \operatorname {arctanh}\left (c x \right )^{2}+\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3}+\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{3}+2 c x \,\operatorname {arctanh}\left (c x \right )+\frac {8 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{3}+\frac {c x}{3}-\frac {1}{3}+\frac {5 \ln \left (c x +1\right )}{6}+\frac {7 \ln \left (c x -1\right )}{6}-\frac {4 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {4 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {2 \ln \left (c x -1\right )^{2}}{3}\right )}{c}+\frac {2 a b \,d^{2} \left (\frac {c^{3} x^{3} \operatorname {arctanh}\left (c x \right )}{3}+c^{2} x^{2} \operatorname {arctanh}\left (c x \right )+c x \,\operatorname {arctanh}\left (c x \right )+\frac {\operatorname {arctanh}\left (c x \right )}{3}+\frac {c^{2} x^{2}}{6}+c x +\frac {4 \ln \left (c x -1\right )}{3}\right )}{c}\) | \(228\) |
| risch | \(\frac {b^{2} d^{2} x}{3}+a^{2} d^{2} x +\frac {4 b^{2} \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right ) d^{2}}{3 c}+\frac {b \ln \left (-c x -1\right ) a \,d^{2}}{3 c}-\frac {4 b^{2} \ln \left (-c x +1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) d^{2}}{3 c}+\frac {4 b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) d^{2}}{3 c}-\frac {d^{2} b^{2}}{3 c}-\frac {7 d^{2} a^{2}}{3 c}-\frac {7 \ln \left (-c x +1\right )^{2} b^{2} d^{2}}{12 c}+\frac {7 \ln \left (-c x +1\right ) b^{2} d^{2}}{6 c}+\frac {\ln \left (-c x +1\right )^{2} x \,b^{2} d^{2}}{4}-\ln \left (-c x +1\right ) x \,b^{2} d^{2}+2 x a b \,d^{2}-\frac {7 a b \,d^{2}}{3 c}+\frac {7 \ln \left (-c x +1\right ) a b \,d^{2}}{3 c}-\ln \left (-c x +1\right ) x a b \,d^{2}+\frac {a b c \,d^{2} x^{2}}{3}-d^{2} c \ln \left (-c x +1\right ) a b \,x^{2}-\frac {d^{2} c^{2} \ln \left (-c x +1\right ) a b \,x^{3}}{3}+\left (-\frac {d^{2} \left (c x +1\right )^{3} b^{2} \ln \left (-c x +1\right )}{6 c}+\frac {d^{2} b \left (2 c^{3} x^{3} a +6 a \,c^{2} x^{2}+b \,c^{2} x^{2}+6 c x a +6 b c x +8 b \ln \left (-c x +1\right )\right )}{6 c}\right ) \ln \left (c x +1\right )+\frac {5 d^{2} b^{2} \ln \left (-c x -1\right )}{6 c}+d^{2} c \,x^{2} a^{2}+\frac {d^{2} c^{2} a^{2} x^{3}}{3}+\frac {d^{2} c \ln \left (-c x +1\right )^{2} b^{2} x^{2}}{4}-\frac {d^{2} c \ln \left (-c x +1\right ) b^{2} x^{2}}{6}+\frac {d^{2} c^{2} \ln \left (-c x +1\right )^{2} b^{2} x^{3}}{12}+\frac {d^{2} \left (c x +1\right )^{3} b^{2} \ln \left (c x +1\right )^{2}}{12 c}\) | \(518\) |
1/c*(1/3*d^2*a^2*(c*x+1)^3+d^2*b^2*(1/3*arctanh(c*x)^2*c^3*x^3+c^2*x^2*arc tanh(c*x)^2+c*x*arctanh(c*x)^2+1/3*arctanh(c*x)^2+1/3*c^2*x^2*arctanh(c*x) +2*c*x*arctanh(c*x)+8/3*arctanh(c*x)*ln(c*x-1)+1/3*c*x-1/3+5/6*ln(c*x+1)+7 /6*ln(c*x-1)-4/3*dilog(1/2*c*x+1/2)-4/3*ln(c*x-1)*ln(1/2*c*x+1/2)+2/3*ln(c *x-1)^2)+2*a*b*d^2*(1/3*c^3*x^3*arctanh(c*x)+c^2*x^2*arctanh(c*x)+c*x*arct anh(c*x)+1/3*arctanh(c*x)+1/6*c^2*x^2+c*x+4/3*ln(c*x-1)))
\[ \int (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\int { {\left (c d x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} \,d x } \]
integral(a^2*c^2*d^2*x^2 + 2*a^2*c*d^2*x + a^2*d^2 + (b^2*c^2*d^2*x^2 + 2* b^2*c*d^2*x + b^2*d^2)*arctanh(c*x)^2 + 2*(a*b*c^2*d^2*x^2 + 2*a*b*c*d^2*x + a*b*d^2)*arctanh(c*x), x)
\[ \int (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=d^{2} \left (\int a^{2}\, dx + \int b^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a^{2} c x\, dx + \int a^{2} c^{2} x^{2}\, dx + \int 2 b^{2} c x \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{2} x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 4 a b c x \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a b c^{2} x^{2} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]
d**2*(Integral(a**2, x) + Integral(b**2*atanh(c*x)**2, x) + Integral(2*a*b *atanh(c*x), x) + Integral(2*a**2*c*x, x) + Integral(a**2*c**2*x**2, x) + Integral(2*b**2*c*x*atanh(c*x)**2, x) + Integral(b**2*c**2*x**2*atanh(c*x) **2, x) + Integral(4*a*b*c*x*atanh(c*x), x) + Integral(2*a*b*c**2*x**2*ata nh(c*x), x))
Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (160) = 320\).
Time = 0.36 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.65 \[ \int (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\frac {1}{3} \, a^{2} c^{2} d^{2} x^{3} + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} a b c^{2} d^{2} + a^{2} c d^{2} x^{2} + {\left (2 \, x^{2} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} a b c d^{2} + a^{2} d^{2} x + \frac {{\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} a b d^{2}}{c} + \frac {4 \, {\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )} b^{2} d^{2}}{3 \, c} + \frac {5 \, b^{2} d^{2} \log \left (c x + 1\right )}{6 \, c} + \frac {7 \, b^{2} d^{2} \log \left (c x - 1\right )}{6 \, c} + \frac {4 \, b^{2} c d^{2} x + {\left (b^{2} c^{3} d^{2} x^{3} + 3 \, b^{2} c^{2} d^{2} x^{2} + 3 \, b^{2} c d^{2} x + b^{2} d^{2}\right )} \log \left (c x + 1\right )^{2} + {\left (b^{2} c^{3} d^{2} x^{3} + 3 \, b^{2} c^{2} d^{2} x^{2} + 3 \, b^{2} c d^{2} x - 7 \, b^{2} d^{2}\right )} \log \left (-c x + 1\right )^{2} + 2 \, {\left (b^{2} c^{2} d^{2} x^{2} + 6 \, b^{2} c d^{2} x\right )} \log \left (c x + 1\right ) - 2 \, {\left (b^{2} c^{2} d^{2} x^{2} + 6 \, b^{2} c d^{2} x + {\left (b^{2} c^{3} d^{2} x^{3} + 3 \, b^{2} c^{2} d^{2} x^{2} + 3 \, b^{2} c d^{2} x + b^{2} d^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{12 \, c} \]
1/3*a^2*c^2*d^2*x^3 + 1/3*(2*x^3*arctanh(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*a*b*c^2*d^2 + a^2*c*d^2*x^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*b*c*d^2 + a^2*d^2*x + (2*c*x*arct anh(c*x) + log(-c^2*x^2 + 1))*a*b*d^2/c + 4/3*(log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2))*b^2*d^2/c + 5/6*b^2*d^2*log(c*x + 1)/c + 7/6 *b^2*d^2*log(c*x - 1)/c + 1/12*(4*b^2*c*d^2*x + (b^2*c^3*d^2*x^3 + 3*b^2*c ^2*d^2*x^2 + 3*b^2*c*d^2*x + b^2*d^2)*log(c*x + 1)^2 + (b^2*c^3*d^2*x^3 + 3*b^2*c^2*d^2*x^2 + 3*b^2*c*d^2*x - 7*b^2*d^2)*log(-c*x + 1)^2 + 2*(b^2*c^ 2*d^2*x^2 + 6*b^2*c*d^2*x)*log(c*x + 1) - 2*(b^2*c^2*d^2*x^2 + 6*b^2*c*d^2 *x + (b^2*c^3*d^2*x^3 + 3*b^2*c^2*d^2*x^2 + 3*b^2*c*d^2*x + b^2*d^2)*log(c *x + 1))*log(-c*x + 1))/c
\[ \int (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\int { {\left (c d x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^2 \,d x \]