Integrand size = 18, antiderivative size = 240 \[ \int (1+c x)^2 (a+b \text {arctanh}(c x))^3 \, dx=a b^2 x+b^3 x \text {arctanh}(c x)+\frac {5 b (a+b \text {arctanh}(c x))^2}{2 c}+3 b x (a+b \text {arctanh}(c x))^2+\frac {1}{2} b c x^2 (a+b \text {arctanh}(c x))^2+\frac {(1+c x)^3 (a+b \text {arctanh}(c x))^3}{3 c}-\frac {6 b^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {4 b (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1-c x}\right )}{c}+\frac {b^3 \log \left (1-c^2 x^2\right )}{2 c}-\frac {3 b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c}-\frac {4 b^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c}+\frac {2 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{c} \] Output:
a*b^2*x+b^3*x*arctanh(c*x)+5/2*b*(a+b*arctanh(c*x))^2/c+3*b*x*(a+b*arctanh (c*x))^2+1/2*b*c*x^2*(a+b*arctanh(c*x))^2+1/3*(c*x+1)^3*(a+b*arctanh(c*x)) ^3/c-6*b^2*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c-4*b*(a+b*arctanh(c*x))^2*ln (2/(-c*x+1))/c+1/2*b^3*ln(-c^2*x^2+1)/c-3*b^3*polylog(2,1-2/(-c*x+1))/c-4* b^2*(a+b*arctanh(c*x))*polylog(2,1-2/(-c*x+1))/c+2*b^3*polylog(3,1-2/(-c*x +1))/c
Leaf count is larger than twice the leaf count of optimal. \(488\) vs. \(2(240)=480\).
Time = 1.60 (sec) , antiderivative size = 488, normalized size of antiderivative = 2.03 \[ \int (1+c x)^2 (a+b \text {arctanh}(c x))^3 \, dx=\frac {6 a^3 c x+18 a^2 b c x+6 a b^2 c x+6 a^3 c^2 x^2+3 a^2 b c^2 x^2+2 a^3 c^3 x^3-6 a b^2 \text {arctanh}(c x)+18 a^2 b c x \text {arctanh}(c x)+36 a b^2 c x \text {arctanh}(c x)+6 b^3 c x \text {arctanh}(c x)+18 a^2 b c^2 x^2 \text {arctanh}(c x)+6 a b^2 c^2 x^2 \text {arctanh}(c x)+6 a^2 b c^3 x^3 \text {arctanh}(c x)-42 a b^2 \text {arctanh}(c x)^2-21 b^3 \text {arctanh}(c x)^2+18 a b^2 c x \text {arctanh}(c x)^2+18 b^3 c x \text {arctanh}(c x)^2+18 a b^2 c^2 x^2 \text {arctanh}(c x)^2+3 b^3 c^2 x^2 \text {arctanh}(c x)^2+6 a b^2 c^3 x^3 \text {arctanh}(c x)^2-14 b^3 \text {arctanh}(c x)^3+6 b^3 c x \text {arctanh}(c x)^3+6 b^3 c^2 x^2 \text {arctanh}(c x)^3+2 b^3 c^3 x^3 \text {arctanh}(c x)^3-48 a b^2 \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-36 b^3 \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-24 b^3 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+21 a^2 b \log (1-c x)+3 a^2 b \log (1+c x)+18 a b^2 \log \left (1-c^2 x^2\right )+3 b^3 \log \left (1-c^2 x^2\right )+6 b^2 (4 a+3 b+4 b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+12 b^3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )}{6 c} \] Input:
Integrate[(1 + c*x)^2*(a + b*ArcTanh[c*x])^3,x]
Output:
(6*a^3*c*x + 18*a^2*b*c*x + 6*a*b^2*c*x + 6*a^3*c^2*x^2 + 3*a^2*b*c^2*x^2 + 2*a^3*c^3*x^3 - 6*a*b^2*ArcTanh[c*x] + 18*a^2*b*c*x*ArcTanh[c*x] + 36*a* b^2*c*x*ArcTanh[c*x] + 6*b^3*c*x*ArcTanh[c*x] + 18*a^2*b*c^2*x^2*ArcTanh[c *x] + 6*a*b^2*c^2*x^2*ArcTanh[c*x] + 6*a^2*b*c^3*x^3*ArcTanh[c*x] - 42*a*b ^2*ArcTanh[c*x]^2 - 21*b^3*ArcTanh[c*x]^2 + 18*a*b^2*c*x*ArcTanh[c*x]^2 + 18*b^3*c*x*ArcTanh[c*x]^2 + 18*a*b^2*c^2*x^2*ArcTanh[c*x]^2 + 3*b^3*c^2*x^ 2*ArcTanh[c*x]^2 + 6*a*b^2*c^3*x^3*ArcTanh[c*x]^2 - 14*b^3*ArcTanh[c*x]^3 + 6*b^3*c*x*ArcTanh[c*x]^3 + 6*b^3*c^2*x^2*ArcTanh[c*x]^3 + 2*b^3*c^3*x^3* ArcTanh[c*x]^3 - 48*a*b^2*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 36*b ^3*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 24*b^3*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 21*a^2*b*Log[1 - c*x] + 3*a^2*b*Log[1 + c*x] + 1 8*a*b^2*Log[1 - c^2*x^2] + 3*b^3*Log[1 - c^2*x^2] + 6*b^2*(4*a + 3*b + 4*b *ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 12*b^3*PolyLog[3, -E^(-2 *ArcTanh[c*x])])/(6*c)
Time = 0.74 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.98, 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.
| \(\displaystyle \int (c x+1)^2 (a+b \text {arctanh}(c x))^3 \, dx\) |
| \(\Big \downarrow \) 6480 |
| \(\displaystyle \frac {(c x+1)^3 (a+b \text {arctanh}(c x))^3}{3 c}-b \int \left (-c x (a+b \text {arctanh}(c x))^2+\frac {4 (c x+1) (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}-3 (a+b \text {arctanh}(c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(c x+1)^3 (a+b \text {arctanh}(c x))^3}{3 c}-b \left (\frac {4 b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}-\frac {1}{2} c x^2 (a+b \text {arctanh}(c x))^2-3 x (a+b \text {arctanh}(c x))^2-\frac {5 (a+b \text {arctanh}(c x))^2}{2 c}+\frac {6 b \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}+\frac {4 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{c}-a b x+b^2 (-x) \text {arctanh}(c x)-\frac {b^2 \log \left (1-c^2 x^2\right )}{2 c}+\frac {3 b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c}-\frac {2 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{c}\right )\) |
Input:
Int[(1 + c*x)^2*(a + b*ArcTanh[c*x])^3,x]
Output:
((1 + c*x)^3*(a + b*ArcTanh[c*x])^3)/(3*c) - b*(-(a*b*x) - b^2*x*ArcTanh[c *x] - (5*(a + b*ArcTanh[c*x])^2)/(2*c) - 3*x*(a + b*ArcTanh[c*x])^2 - (c*x ^2*(a + b*ArcTanh[c*x])^2)/2 + (6*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)]) /c + (4*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c - (b^2*Log[1 - c^2*x^2] )/(2*c) + (3*b^2*PolyLog[2, 1 - 2/(1 - c*x)])/c + (4*b*(a + b*ArcTanh[c*x] )*PolyLog[2, 1 - 2/(1 - c*x)])/c - (2*b^2*PolyLog[3, 1 - 2/(1 - c*x)])/c)
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 = 1.18 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.54
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (c x +1\right )^{3} a^{3}}{3}+b^{3} \left (\frac {\operatorname {arctanh}\left (c x \right )^{3} c^{3} x^{3}}{3}+\operatorname {arctanh}\left (c x \right )^{3} c^{2} x^{2}+\operatorname {arctanh}\left (c x \right )^{3} c x +\frac {\operatorname {arctanh}\left (c x \right )^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}}{2}+3 \operatorname {arctanh}\left (c x \right )^{2} c x +4 \operatorname {arctanh}\left (c x \right )^{2} \ln \left (c x -1\right )-4 \,\operatorname {arctanh}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )-4 i \pi \operatorname {arctanh}\left (c x \right )^{2}-6 \operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-6 \operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )+\frac {5 \operatorname {arctanh}\left (c x \right )^{2}}{2}+4 i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}}\right )}^{2} \operatorname {arctanh}\left (c x \right )^{2}+\left (c x +1\right ) \operatorname {arctanh}\left (c x \right )-4 i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}}\right )}^{3} \operatorname {arctanh}\left (c x \right )^{2}-\ln \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )-4 \ln \left (2\right ) \operatorname {arctanh}\left (c x \right )^{2}-6 \,\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-6 \,\operatorname {arctanh}\left (c x \right ) \ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right )+3 a \,b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+\operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}+\operatorname {arctanh}\left (c x \right )^{2} c x +\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{3}+2 \,\operatorname {arctanh}\left (c x \right ) c x +\frac {8 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{3}+\frac {c x}{3}-\frac {1}{3}+\frac {7 \ln \left (c x -1\right )}{6}+\frac {5 \ln \left (c x +1\right )}{6}+\frac {2 \ln \left (c x -1\right )^{2}}{3}-\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}\right )+3 a^{2} b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}+\operatorname {arctanh}\left (c x \right ) c x +\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}\) | \(610\) |
| default | \(\frac {\frac {\left (c x +1\right )^{3} a^{3}}{3}+b^{3} \left (\frac {\operatorname {arctanh}\left (c x \right )^{3} c^{3} x^{3}}{3}+\operatorname {arctanh}\left (c x \right )^{3} c^{2} x^{2}+\operatorname {arctanh}\left (c x \right )^{3} c x +\frac {\operatorname {arctanh}\left (c x \right )^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}}{2}+3 \operatorname {arctanh}\left (c x \right )^{2} c x +4 \operatorname {arctanh}\left (c x \right )^{2} \ln \left (c x -1\right )-4 \,\operatorname {arctanh}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )-4 i \pi \operatorname {arctanh}\left (c x \right )^{2}-6 \operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-6 \operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )+\frac {5 \operatorname {arctanh}\left (c x \right )^{2}}{2}+4 i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}}\right )}^{2} \operatorname {arctanh}\left (c x \right )^{2}+\left (c x +1\right ) \operatorname {arctanh}\left (c x \right )-4 i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}}\right )}^{3} \operatorname {arctanh}\left (c x \right )^{2}-\ln \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )-4 \ln \left (2\right ) \operatorname {arctanh}\left (c x \right )^{2}-6 \,\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-6 \,\operatorname {arctanh}\left (c x \right ) \ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right )+3 a \,b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+\operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}+\operatorname {arctanh}\left (c x \right )^{2} c x +\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{3}+2 \,\operatorname {arctanh}\left (c x \right ) c x +\frac {8 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{3}+\frac {c x}{3}-\frac {1}{3}+\frac {7 \ln \left (c x -1\right )}{6}+\frac {5 \ln \left (c x +1\right )}{6}+\frac {2 \ln \left (c x -1\right )^{2}}{3}-\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}\right )+3 a^{2} b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}+\operatorname {arctanh}\left (c x \right ) c x +\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}\) | \(610\) |
| parts | \(\frac {a^{3} \left (c x +1\right )^{3}}{3 c}+\frac {b^{3} \left (\frac {\operatorname {arctanh}\left (c x \right )^{3} c^{3} x^{3}}{3}+\operatorname {arctanh}\left (c x \right )^{3} c^{2} x^{2}+\operatorname {arctanh}\left (c x \right )^{3} c x +\frac {\operatorname {arctanh}\left (c x \right )^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}}{2}+3 \operatorname {arctanh}\left (c x \right )^{2} c x +4 \operatorname {arctanh}\left (c x \right )^{2} \ln \left (c x -1\right )-4 \,\operatorname {arctanh}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )-4 i \pi \operatorname {arctanh}\left (c x \right )^{2}-6 \operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-6 \operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )+\frac {5 \operatorname {arctanh}\left (c x \right )^{2}}{2}+4 i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}}\right )}^{2} \operatorname {arctanh}\left (c x \right )^{2}+\left (c x +1\right ) \operatorname {arctanh}\left (c x \right )-4 i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}}\right )}^{3} \operatorname {arctanh}\left (c x \right )^{2}-\ln \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )-4 \ln \left (2\right ) \operatorname {arctanh}\left (c x \right )^{2}-6 \,\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-6 \,\operatorname {arctanh}\left (c x \right ) \ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{c}+\frac {3 a \,b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+\operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}+\operatorname {arctanh}\left (c x \right )^{2} c x +\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{3}+2 \,\operatorname {arctanh}\left (c x \right ) c x +\frac {8 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{3}+\frac {c x}{3}-\frac {1}{3}+\frac {7 \ln \left (c x -1\right )}{6}+\frac {5 \ln \left (c x +1\right )}{6}+\frac {2 \ln \left (c x -1\right )^{2}}{3}-\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}\right )}{c}+\frac {3 a^{2} b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}+\operatorname {arctanh}\left (c x \right ) c x +\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}\) | \(618\) |
Input:
int((c*x+1)^2*(a+b*arctanh(c*x))^3,x,method=_RETURNVERBOSE)
Output:
1/c*(1/3*(c*x+1)^3*a^3+b^3*(1/3*arctanh(c*x)^3*c^3*x^3+arctanh(c*x)^3*c^2* x^2+arctanh(c*x)^3*c*x+1/3*arctanh(c*x)^3+1/2*arctanh(c*x)^2*c^2*x^2+3*arc tanh(c*x)^2*c*x+4*arctanh(c*x)^2*ln(c*x-1)-4*arctanh(c*x)*polylog(2,-(c*x+ 1)^2/(-c^2*x^2+1))+2*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))-4*I*Pi*arctanh(c*x )^2-6*dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-6*dilog(1-I*(c*x+1)/(-c^2*x^2+ 1)^(1/2))+5/2*arctanh(c*x)^2+4*I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))^2*ar ctanh(c*x)^2+(c*x+1)*arctanh(c*x)-4*I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1))) ^3*arctanh(c*x)^2-ln(1+(c*x+1)^2/(-c^2*x^2+1))-4*ln(2)*arctanh(c*x)^2-6*ar ctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-6*arctanh(c*x)*ln(1-I*(c*x+1 )/(-c^2*x^2+1)^(1/2)))+3*a*b^2*(1/3*arctanh(c*x)^2*c^3*x^3+arctanh(c*x)^2* c^2*x^2+arctanh(c*x)^2*c*x+1/3*arctanh(c*x)^2+1/3*arctanh(c*x)*c^2*x^2+2*a rctanh(c*x)*c*x+8/3*arctanh(c*x)*ln(c*x-1)+1/3*c*x-1/3+7/6*ln(c*x-1)+5/6*l n(c*x+1)+2/3*ln(c*x-1)^2-4/3*dilog(1/2*c*x+1/2)-4/3*ln(c*x-1)*ln(1/2*c*x+1 /2))+3*a^2*b*(1/3*arctanh(c*x)*c^3*x^3+arctanh(c*x)*c^2*x^2+arctanh(c*x)*c *x+1/3*arctanh(c*x)+1/6*c^2*x^2+c*x+4/3*ln(c*x-1)))
\[ \int (1+c x)^2 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (c x + 1\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} \,d x } \] Input:
integrate((c*x+1)^2*(a+b*arctanh(c*x))^3,x, algorithm="fricas")
Output:
integral(a^3*c^2*x^2 + 2*a^3*c*x + (b^3*c^2*x^2 + 2*b^3*c*x + b^3)*arctanh (c*x)^3 + a^3 + 3*(a*b^2*c^2*x^2 + 2*a*b^2*c*x + a*b^2)*arctanh(c*x)^2 + 3 *(a^2*b*c^2*x^2 + 2*a^2*b*c*x + a^2*b)*arctanh(c*x), x)
\[ \int (1+c x)^2 (a+b \text {arctanh}(c x))^3 \, dx=\int \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3} \left (c x + 1\right )^{2}\, dx \] Input:
integrate((c*x+1)**2*(a+b*atanh(c*x))**3,x)
Output:
Integral((a + b*atanh(c*x))**3*(c*x + 1)**2, x)
\[ \int (1+c x)^2 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (c x + 1\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} \,d x } \] Input:
integrate((c*x+1)^2*(a+b*arctanh(c*x))^3,x, algorithm="maxima")
Output:
1/3*a^3*c^2*x^3 + 1/2*(2*x^3*arctanh(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/ c^4))*a^2*b*c^2 + a^3*c*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*c + a^3*x + 3/2*(2*c*x*arctanh(c*x) + log(-c^2*x^2 + 1))*a^2*b/c - 1/24*((b^3*c^3*x^3 + 3*b^3*c^2*x^2 + 3*b^3 *c*x - 7*b^3)*log(-c*x + 1)^3 - 3*(2*a*b^2*c^3*x^3 + (6*a*b^2*c^2 + b^3*c^ 2)*x^2 + 6*(a*b^2*c + b^3*c)*x + (b^3*c^3*x^3 + 3*b^3*c^2*x^2 + 3*b^3*c*x + b^3)*log(c*x + 1))*log(-c*x + 1)^2)/c - integrate(-1/8*((b^3*c^3*x^3 + b ^3*c^2*x^2 - b^3*c*x - b^3)*log(c*x + 1)^3 + 6*(a*b^2*c^3*x^3 + a*b^2*c^2* x^2 - a*b^2*c*x - a*b^2)*log(c*x + 1)^2 - (4*a*b^2*c^3*x^3 + 2*(6*a*b^2*c^ 2 + b^3*c^2)*x^2 + 3*(b^3*c^3*x^3 + b^3*c^2*x^2 - b^3*c*x - b^3)*log(c*x + 1)^2 + 12*(a*b^2*c + b^3*c)*x + 2*((6*a*b^2*c^3 + b^3*c^3)*x^3 - 6*a*b^2 + b^3 + 3*(2*a*b^2*c^2 + b^3*c^2)*x^2 - 3*(2*a*b^2*c - b^3*c)*x)*log(c*x + 1))*log(-c*x + 1))/(c*x - 1), x)
\[ \int (1+c x)^2 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (c x + 1\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} \,d x } \] Input:
integrate((c*x+1)^2*(a+b*arctanh(c*x))^3,x, algorithm="giac")
Output:
integrate((c*x + 1)^2*(b*arctanh(c*x) + a)^3, x)
Timed out. \[ \int (1+c x)^2 (a+b \text {arctanh}(c x))^3 \, dx=\int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3\,{\left (c\,x+1\right )}^2 \,d x \] Input:
int((a + b*atanh(c*x))^3*(c*x + 1)^2,x)
Output:
int((a + b*atanh(c*x))^3*(c*x + 1)^2, x)
\[ \int (1+c x)^2 (a+b \text {arctanh}(c x))^3 \, dx=\frac {-6 \mathit {atanh} \left (c x \right )^{3} b^{3}+6 \mathit {atanh} \left (c x \right ) b^{3}+48 \left (\int \frac {\mathit {atanh} \left (c x \right ) x}{c^{2} x^{2}-1}d x \right ) a \,b^{2} c^{2}+2 \mathit {atanh} \left (c x \right )^{3} b^{3} c^{3} x^{3}+6 \mathit {atanh} \left (c x \right )^{3} b^{3} c x +3 \mathit {atanh} \left (c x \right )^{2} b^{3} c^{2} x^{2}+6 \mathit {atanh} \left (c x \right ) b^{3} c x +3 a^{2} b \,c^{2} x^{2}+6 a \,b^{2} c x +36 \mathit {atanh} \left (c x \right ) a \,b^{2} c x +18 \mathit {atanh} \left (c x \right )^{2} b^{3} c x +18 a^{2} b c x +6 a^{3} c x +2 a^{3} c^{3} x^{3}+6 a^{3} c^{2} x^{2}+36 \left (\int \frac {\mathit {atanh} \left (c x \right ) x}{c^{2} x^{2}-1}d x \right ) b^{3} c^{2}+24 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2} x}{c^{2} x^{2}-1}d x \right ) b^{3} c^{2}+6 \mathit {atanh} \left (c x \right )^{3} b^{3} c^{2} x^{2}+6 \mathit {atanh} \left (c x \right )^{2} a \,b^{2} c^{3} x^{3}+18 \mathit {atanh} \left (c x \right )^{2} a \,b^{2} c x +6 \mathit {atanh} \left (c x \right ) a^{2} b \,c^{3} x^{3}+18 \mathit {atanh} \left (c x \right )^{2} a \,b^{2} c^{2} x^{2}+18 \mathit {atanh} \left (c x \right ) a^{2} b \,c^{2} x^{2}-3 \mathit {atanh} \left (c x \right )^{2} b^{3}+6 \,\mathrm {log}\left (c^{2} x -c \right ) b^{3}-18 \mathit {atanh} \left (c x \right )^{2} a \,b^{2}+6 \mathit {atanh} \left (c x \right ) a^{2} b +30 \mathit {atanh} \left (c x \right ) a \,b^{2}+36 \,\mathrm {log}\left (c^{2} x -c \right ) a \,b^{2}+6 \mathit {atanh} \left (c x \right ) a \,b^{2} c^{2} x^{2}+18 \mathit {atanh} \left (c x \right ) a^{2} b c x +24 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} b}{6 c} \] Input:
int((c*x+1)^2*(a+b*atanh(c*x))^3,x)
Output:
(2*atanh(c*x)**3*b**3*c**3*x**3 + 6*atanh(c*x)**3*b**3*c**2*x**2 + 6*atanh (c*x)**3*b**3*c*x - 6*atanh(c*x)**3*b**3 + 6*atanh(c*x)**2*a*b**2*c**3*x** 3 + 18*atanh(c*x)**2*a*b**2*c**2*x**2 + 18*atanh(c*x)**2*a*b**2*c*x - 18*a tanh(c*x)**2*a*b**2 + 3*atanh(c*x)**2*b**3*c**2*x**2 + 18*atanh(c*x)**2*b* *3*c*x - 3*atanh(c*x)**2*b**3 + 6*atanh(c*x)*a**2*b*c**3*x**3 + 18*atanh(c *x)*a**2*b*c**2*x**2 + 18*atanh(c*x)*a**2*b*c*x + 6*atanh(c*x)*a**2*b + 6* atanh(c*x)*a*b**2*c**2*x**2 + 36*atanh(c*x)*a*b**2*c*x + 30*atanh(c*x)*a*b **2 + 6*atanh(c*x)*b**3*c*x + 6*atanh(c*x)*b**3 + 48*int((atanh(c*x)*x)/(c **2*x**2 - 1),x)*a*b**2*c**2 + 36*int((atanh(c*x)*x)/(c**2*x**2 - 1),x)*b* *3*c**2 + 24*int((atanh(c*x)**2*x)/(c**2*x**2 - 1),x)*b**3*c**2 + 24*log(c **2*x - c)*a**2*b + 36*log(c**2*x - c)*a*b**2 + 6*log(c**2*x - c)*b**3 + 2 *a**3*c**3*x**3 + 6*a**3*c**2*x**2 + 6*a**3*c*x + 3*a**2*b*c**2*x**2 + 18* a**2*b*c*x + 6*a*b**2*c*x)/(6*c)