Integrand size = 18, antiderivative size = 306 \[ \int (1+c x)^3 (a+b \text {arctanh}(c x))^3 \, dx=3 a b^2 x+\frac {b^3 x}{4}-\frac {b^3 \text {arctanh}(c x)}{4 c}+3 b^3 x \text {arctanh}(c x)+\frac {1}{4} b^2 c x^2 (a+b \text {arctanh}(c x))+\frac {4 b (a+b \text {arctanh}(c x))^2}{c}+\frac {21}{4} b x (a+b \text {arctanh}(c x))^2+\frac {3}{2} b c x^2 (a+b \text {arctanh}(c x))^2+\frac {1}{4} b c^2 x^3 (a+b \text {arctanh}(c x))^2+\frac {(1+c x)^4 (a+b \text {arctanh}(c x))^3}{4 c}-\frac {11 b^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {6 b (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \log \left (1-c^2 x^2\right )}{2 c}-\frac {11 b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c}-\frac {6 b^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{c} \] Output:
3*a*b^2*x+1/4*b^3*x-1/4*b^3*arctanh(c*x)/c+3*b^3*x*arctanh(c*x)+1/4*b^2*c* x^2*(a+b*arctanh(c*x))+4*b*(a+b*arctanh(c*x))^2/c+21/4*b*x*(a+b*arctanh(c* x))^2+3/2*b*c*x^2*(a+b*arctanh(c*x))^2+1/4*b*c^2*x^3*(a+b*arctanh(c*x))^2+ 1/4*(c*x+1)^4*(a+b*arctanh(c*x))^3/c-11*b^2*(a+b*arctanh(c*x))*ln(2/(-c*x+ 1))/c-6*b*(a+b*arctanh(c*x))^2*ln(2/(-c*x+1))/c+3/2*b^3*ln(-c^2*x^2+1)/c-1 1/2*b^3*polylog(2,1-2/(-c*x+1))/c-6*b^2*(a+b*arctanh(c*x))*polylog(2,1-2/( -c*x+1))/c+3*b^3*polylog(3,1-2/(-c*x+1))/c
Leaf count is larger than twice the leaf count of optimal. \(644\) vs. \(2(306)=612\).
Time = 1.93 (sec) , antiderivative size = 644, normalized size of antiderivative = 2.10 \[ \int (1+c x)^3 (a+b \text {arctanh}(c x))^3 \, dx=\frac {-2 a b^2+8 a^3 c x+42 a^2 b c x+24 a b^2 c x+2 b^3 c x+12 a^3 c^2 x^2+12 a^2 b c^2 x^2+2 a b^2 c^2 x^2+8 a^3 c^3 x^3+2 a^2 b c^3 x^3+2 a^3 c^4 x^4-24 a b^2 \text {arctanh}(c x)-2 b^3 \text {arctanh}(c x)+24 a^2 b c x \text {arctanh}(c x)+84 a b^2 c x \text {arctanh}(c x)+24 b^3 c x \text {arctanh}(c x)+36 a^2 b c^2 x^2 \text {arctanh}(c x)+24 a b^2 c^2 x^2 \text {arctanh}(c x)+2 b^3 c^2 x^2 \text {arctanh}(c x)+24 a^2 b c^3 x^3 \text {arctanh}(c x)+4 a b^2 c^3 x^3 \text {arctanh}(c x)+6 a^2 b c^4 x^4 \text {arctanh}(c x)-90 a b^2 \text {arctanh}(c x)^2-56 b^3 \text {arctanh}(c x)^2+24 a b^2 c x \text {arctanh}(c x)^2+42 b^3 c x \text {arctanh}(c x)^2+36 a b^2 c^2 x^2 \text {arctanh}(c x)^2+12 b^3 c^2 x^2 \text {arctanh}(c x)^2+24 a b^2 c^3 x^3 \text {arctanh}(c x)^2+2 b^3 c^3 x^3 \text {arctanh}(c x)^2+6 a b^2 c^4 x^4 \text {arctanh}(c x)^2-30 b^3 \text {arctanh}(c x)^3+8 b^3 c x \text {arctanh}(c x)^3+12 b^3 c^2 x^2 \text {arctanh}(c x)^3+8 b^3 c^3 x^3 \text {arctanh}(c x)^3+2 b^3 c^4 x^4 \text {arctanh}(c x)^3-96 a b^2 \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-88 b^3 \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-48 b^3 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+45 a^2 b \log (1-c x)+3 a^2 b \log (1+c x)+44 a b^2 \log \left (1-c^2 x^2\right )+12 b^3 \log \left (1-c^2 x^2\right )+4 b^2 (12 a+11 b+12 b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+24 b^3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )}{8 c} \] Input:
Integrate[(1 + c*x)^3*(a + b*ArcTanh[c*x])^3,x]
Output:
(-2*a*b^2 + 8*a^3*c*x + 42*a^2*b*c*x + 24*a*b^2*c*x + 2*b^3*c*x + 12*a^3*c ^2*x^2 + 12*a^2*b*c^2*x^2 + 2*a*b^2*c^2*x^2 + 8*a^3*c^3*x^3 + 2*a^2*b*c^3* x^3 + 2*a^3*c^4*x^4 - 24*a*b^2*ArcTanh[c*x] - 2*b^3*ArcTanh[c*x] + 24*a^2* b*c*x*ArcTanh[c*x] + 84*a*b^2*c*x*ArcTanh[c*x] + 24*b^3*c*x*ArcTanh[c*x] + 36*a^2*b*c^2*x^2*ArcTanh[c*x] + 24*a*b^2*c^2*x^2*ArcTanh[c*x] + 2*b^3*c^2 *x^2*ArcTanh[c*x] + 24*a^2*b*c^3*x^3*ArcTanh[c*x] + 4*a*b^2*c^3*x^3*ArcTan h[c*x] + 6*a^2*b*c^4*x^4*ArcTanh[c*x] - 90*a*b^2*ArcTanh[c*x]^2 - 56*b^3*A rcTanh[c*x]^2 + 24*a*b^2*c*x*ArcTanh[c*x]^2 + 42*b^3*c*x*ArcTanh[c*x]^2 + 36*a*b^2*c^2*x^2*ArcTanh[c*x]^2 + 12*b^3*c^2*x^2*ArcTanh[c*x]^2 + 24*a*b^2 *c^3*x^3*ArcTanh[c*x]^2 + 2*b^3*c^3*x^3*ArcTanh[c*x]^2 + 6*a*b^2*c^4*x^4*A rcTanh[c*x]^2 - 30*b^3*ArcTanh[c*x]^3 + 8*b^3*c*x*ArcTanh[c*x]^3 + 12*b^3* c^2*x^2*ArcTanh[c*x]^3 + 8*b^3*c^3*x^3*ArcTanh[c*x]^3 + 2*b^3*c^4*x^4*ArcT anh[c*x]^3 - 96*a*b^2*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 88*b^3*A rcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 48*b^3*ArcTanh[c*x]^2*Log[1 + E ^(-2*ArcTanh[c*x])] + 45*a^2*b*Log[1 - c*x] + 3*a^2*b*Log[1 + c*x] + 44*a* b^2*Log[1 - c^2*x^2] + 12*b^3*Log[1 - c^2*x^2] + 4*b^2*(12*a + 11*b + 12*b *ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 24*b^3*PolyLog[3, -E^(-2 *ArcTanh[c*x])])/(8*c)
Time = 0.98 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
| \(\displaystyle \int (c x+1)^3 (a+b \text {arctanh}(c x))^3 \, dx\) |
| \(\Big \downarrow \) 6480 |
| \(\displaystyle \frac {(c x+1)^4 (a+b \text {arctanh}(c x))^3}{4 c}-\frac {3}{4} b \int \left (-c^2 x^2 (a+b \text {arctanh}(c x))^2-4 c x (a+b \text {arctanh}(c x))^2+\frac {8 (c x+1) (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}-7 (a+b \text {arctanh}(c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(c x+1)^4 (a+b \text {arctanh}(c x))^3}{4 c}-\frac {3}{4} b \left (-\frac {1}{3} c^2 x^3 (a+b \text {arctanh}(c x))^2+\frac {8 b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}-2 c x^2 (a+b \text {arctanh}(c x))^2-\frac {1}{3} b c x^2 (a+b \text {arctanh}(c x))-7 x (a+b \text {arctanh}(c x))^2-\frac {16 (a+b \text {arctanh}(c x))^2}{3 c}+\frac {8 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{c}+\frac {44 b \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{3 c}-4 a b x-4 b^2 x \text {arctanh}(c x)+\frac {b^2 \text {arctanh}(c x)}{3 c}-\frac {2 b^2 \log \left (1-c^2 x^2\right )}{c}+\frac {22 b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c}-\frac {4 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{c}-\frac {b^2 x}{3}\right )\) |
Input:
Int[(1 + c*x)^3*(a + b*ArcTanh[c*x])^3,x]
Output:
((1 + c*x)^4*(a + b*ArcTanh[c*x])^3)/(4*c) - (3*b*(-4*a*b*x - (b^2*x)/3 + (b^2*ArcTanh[c*x])/(3*c) - 4*b^2*x*ArcTanh[c*x] - (b*c*x^2*(a + b*ArcTanh[ c*x]))/3 - (16*(a + b*ArcTanh[c*x])^2)/(3*c) - 7*x*(a + b*ArcTanh[c*x])^2 - 2*c*x^2*(a + b*ArcTanh[c*x])^2 - (c^2*x^3*(a + b*ArcTanh[c*x])^2)/3 + (4 4*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(3*c) + (8*(a + b*ArcTanh[c*x]) ^2*Log[2/(1 - c*x)])/c - (2*b^2*Log[1 - c^2*x^2])/c + (22*b^2*PolyLog[2, 1 - 2/(1 - c*x)])/(3*c) + (8*b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c *x)])/c - (4*b^2*PolyLog[3, 1 - 2/(1 - c*x)])/c))/4
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 = 2.19 (sec) , antiderivative size = 713, normalized size of antiderivative = 2.33
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{3} \left (c x +1\right )^{4}}{4}+b^{3} \left (-6 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}+6 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}+\frac {3 \operatorname {arctanh}\left (c x \right )^{3} c^{2} x^{2}}{2}-\frac {1}{4}+\frac {c x}{4}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{4}+\frac {\operatorname {arctanh}\left (c x \right )^{3}}{4}+\frac {3 \operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {21 \operatorname {arctanh}\left (c x \right )^{2} c x}{4}-11 \,\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-11 \,\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 )^{2} \ln \left (c x -1\right )+\operatorname {arctanh}\left (c x \right )^{3} c x +\frac {\operatorname {arctanh}\left (c x \right )^{3} c^{4} x^{4}}{4}+\operatorname {arctanh}\left (c x \right )^{3} c^{3} x^{3}+4 \operatorname {arctanh}\left (c x \right )^{2}+3 \operatorname {polylog}\left (3, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+\frac {7 \left (c x +1\right ) \operatorname {arctanh}\left (c x \right )}{2}-3 \ln \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )-6 \,\operatorname {arctanh}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+\frac {\left (c x -3\right ) \left (c x +1\right ) \operatorname {arctanh}\left (c x \right )}{4}-6 i \pi \operatorname {arctanh}\left (c x \right )^{2}-6 \ln \left (2\right ) \operatorname {arctanh}\left (c x \right )^{2}-11 \operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-11 \operatorname {dilog}\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^{4} x^{4}}{4}+\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}+\frac {3 \operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\operatorname {arctanh}\left (c x \right )^{2} c x +\frac {\operatorname {arctanh}\left (c x \right )^{2}}{4}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{6}+\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}+\frac {7 \,\operatorname {arctanh}\left (c x \right ) c x}{2}+4 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )+\frac {\left (c x -1\right )^{2}}{12}+\frac {7 c x}{6}-\frac {7}{6}+\frac {7 \ln \left (c x -1\right )}{3}+\frac {4 \ln \left (c x +1\right )}{3}+\ln \left (c x -1\right )^{2}-2 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )-2 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right )+3 a^{2} b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}+\operatorname {arctanh}\left (c x \right ) c x +\frac {\operatorname {arctanh}\left (c x \right )}{4}+\frac {x^{3} c^{3}}{12}+\frac {c^{2} x^{2}}{2}+\frac {7 c x}{4}+2 \ln \left (c x -1\right )\right )}{c}\) | \(713\) |
| default | \(\frac {\frac {a^{3} \left (c x +1\right )^{4}}{4}+b^{3} \left (-6 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}+6 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}+\frac {3 \operatorname {arctanh}\left (c x \right )^{3} c^{2} x^{2}}{2}-\frac {1}{4}+\frac {c x}{4}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{4}+\frac {\operatorname {arctanh}\left (c x \right )^{3}}{4}+\frac {3 \operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {21 \operatorname {arctanh}\left (c x \right )^{2} c x}{4}-11 \,\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-11 \,\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 )^{2} \ln \left (c x -1\right )+\operatorname {arctanh}\left (c x \right )^{3} c x +\frac {\operatorname {arctanh}\left (c x \right )^{3} c^{4} x^{4}}{4}+\operatorname {arctanh}\left (c x \right )^{3} c^{3} x^{3}+4 \operatorname {arctanh}\left (c x \right )^{2}+3 \operatorname {polylog}\left (3, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+\frac {7 \left (c x +1\right ) \operatorname {arctanh}\left (c x \right )}{2}-3 \ln \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )-6 \,\operatorname {arctanh}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+\frac {\left (c x -3\right ) \left (c x +1\right ) \operatorname {arctanh}\left (c x \right )}{4}-6 i \pi \operatorname {arctanh}\left (c x \right )^{2}-6 \ln \left (2\right ) \operatorname {arctanh}\left (c x \right )^{2}-11 \operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-11 \operatorname {dilog}\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^{4} x^{4}}{4}+\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}+\frac {3 \operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\operatorname {arctanh}\left (c x \right )^{2} c x +\frac {\operatorname {arctanh}\left (c x \right )^{2}}{4}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{6}+\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}+\frac {7 \,\operatorname {arctanh}\left (c x \right ) c x}{2}+4 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )+\frac {\left (c x -1\right )^{2}}{12}+\frac {7 c x}{6}-\frac {7}{6}+\frac {7 \ln \left (c x -1\right )}{3}+\frac {4 \ln \left (c x +1\right )}{3}+\ln \left (c x -1\right )^{2}-2 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )-2 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right )+3 a^{2} b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}+\operatorname {arctanh}\left (c x \right ) c x +\frac {\operatorname {arctanh}\left (c x \right )}{4}+\frac {x^{3} c^{3}}{12}+\frac {c^{2} x^{2}}{2}+\frac {7 c x}{4}+2 \ln \left (c x -1\right )\right )}{c}\) | \(713\) |
| parts | \(\frac {a^{3} \left (c x +1\right )^{4}}{4 c}+\frac {b^{3} \left (-6 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}+6 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}+\frac {3 \operatorname {arctanh}\left (c x \right )^{3} c^{2} x^{2}}{2}-\frac {1}{4}+\frac {c x}{4}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{4}+\frac {\operatorname {arctanh}\left (c x \right )^{3}}{4}+\frac {3 \operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {21 \operatorname {arctanh}\left (c x \right )^{2} c x}{4}-11 \,\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-11 \,\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 )^{2} \ln \left (c x -1\right )+\operatorname {arctanh}\left (c x \right )^{3} c x +\frac {\operatorname {arctanh}\left (c x \right )^{3} c^{4} x^{4}}{4}+\operatorname {arctanh}\left (c x \right )^{3} c^{3} x^{3}+4 \operatorname {arctanh}\left (c x \right )^{2}+3 \operatorname {polylog}\left (3, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+\frac {7 \left (c x +1\right ) \operatorname {arctanh}\left (c x \right )}{2}-3 \ln \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )-6 \,\operatorname {arctanh}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+\frac {\left (c x -3\right ) \left (c x +1\right ) \operatorname {arctanh}\left (c x \right )}{4}-6 i \pi \operatorname {arctanh}\left (c x \right )^{2}-6 \ln \left (2\right ) \operatorname {arctanh}\left (c x \right )^{2}-11 \operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-11 \operatorname {dilog}\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^{4} x^{4}}{4}+\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}+\frac {3 \operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\operatorname {arctanh}\left (c x \right )^{2} c x +\frac {\operatorname {arctanh}\left (c x \right )^{2}}{4}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{6}+\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}+\frac {7 \,\operatorname {arctanh}\left (c x \right ) c x}{2}+4 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )+\frac {\left (c x -1\right )^{2}}{12}+\frac {7 c x}{6}-\frac {7}{6}+\frac {7 \ln \left (c x -1\right )}{3}+\frac {4 \ln \left (c x +1\right )}{3}+\ln \left (c x -1\right )^{2}-2 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )-2 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right )}{c}+\frac {3 a^{2} b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}+\operatorname {arctanh}\left (c x \right ) c x +\frac {\operatorname {arctanh}\left (c x \right )}{4}+\frac {x^{3} c^{3}}{12}+\frac {c^{2} x^{2}}{2}+\frac {7 c x}{4}+2 \ln \left (c x -1\right )\right )}{c}\) | \(721\) |
Input:
int((c*x+1)^3*(a+b*arctanh(c*x))^3,x,method=_RETURNVERBOSE)
Output:
1/c*(1/4*a^3*(c*x+1)^4+b^3*(3/2*arctanh(c*x)^3*c^2*x^2-1/4-11*arctanh(c*x) *ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-11*arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x ^2+1)^(1/2))-11*dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-11*dilog(1-I*(c*x+1) /(-c^2*x^2+1)^(1/2))+1/4*c*x+1/4*arctanh(c*x)^2*c^3*x^3+1/4*arctanh(c*x)^3 +3/2*arctanh(c*x)^2*c^2*x^2+21/4*arctanh(c*x)^2*c*x+6*arctanh(c*x)^2*ln(c* x-1)+arctanh(c*x)^3*c*x+1/4*arctanh(c*x)^3*c^4*x^4+arctanh(c*x)^3*c^3*x^3- 6*I*Pi*arctanh(c*x)^2+4*arctanh(c*x)^2+3*polylog(3,-(c*x+1)^2/(-c^2*x^2+1) )+7/2*(c*x+1)*arctanh(c*x)-3*ln(1+(c*x+1)^2/(-c^2*x^2+1))-6*arctanh(c*x)*p olylog(2,-(c*x+1)^2/(-c^2*x^2+1))-6*I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1))) ^3*arctanh(c*x)^2+6*I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))^2*arctanh(c*x)^ 2+1/4*(c*x-3)*(c*x+1)*arctanh(c*x)-6*ln(2)*arctanh(c*x)^2)+3*a*b^2*(1/4*ar ctanh(c*x)^2*c^4*x^4+arctanh(c*x)^2*c^3*x^3+3/2*arctanh(c*x)^2*c^2*x^2+arc tanh(c*x)^2*c*x+1/4*arctanh(c*x)^2+1/6*arctanh(c*x)*c^3*x^3+arctanh(c*x)*c ^2*x^2+7/2*arctanh(c*x)*c*x+4*arctanh(c*x)*ln(c*x-1)+1/12*(c*x-1)^2+7/6*c* x-7/6+7/3*ln(c*x-1)+4/3*ln(c*x+1)+ln(c*x-1)^2-2*dilog(1/2*c*x+1/2)-2*ln(c* x-1)*ln(1/2*c*x+1/2))+3*a^2*b*(1/4*arctanh(c*x)*c^4*x^4+arctanh(c*x)*c^3*x ^3+3/2*arctanh(c*x)*c^2*x^2+arctanh(c*x)*c*x+1/4*arctanh(c*x)+1/12*x^3*c^3 +1/2*c^2*x^2+7/4*c*x+2*ln(c*x-1)))
\[ \int (1+c x)^3 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (c x + 1\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} \,d x } \] Input:
integrate((c*x+1)^3*(a+b*arctanh(c*x))^3,x, algorithm="fricas")
Output:
integral(a^3*c^3*x^3 + 3*a^3*c^2*x^2 + 3*a^3*c*x + (b^3*c^3*x^3 + 3*b^3*c^ 2*x^2 + 3*b^3*c*x + b^3)*arctanh(c*x)^3 + a^3 + 3*(a*b^2*c^3*x^3 + 3*a*b^2 *c^2*x^2 + 3*a*b^2*c*x + a*b^2)*arctanh(c*x)^2 + 3*(a^2*b*c^3*x^3 + 3*a^2* b*c^2*x^2 + 3*a^2*b*c*x + a^2*b)*arctanh(c*x), x)
\[ \int (1+c x)^3 (a+b \text {arctanh}(c x))^3 \, dx=\int \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3} \left (c x + 1\right )^{3}\, dx \] Input:
integrate((c*x+1)**3*(a+b*atanh(c*x))**3,x)
Output:
Integral((a + b*atanh(c*x))**3*(c*x + 1)**3, x)
\[ \int (1+c x)^3 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (c x + 1\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} \,d x } \] Input:
integrate((c*x+1)^3*(a+b*arctanh(c*x))^3,x, algorithm="maxima")
Output:
1/4*a^3*c^3*x^4 + a^3*c^2*x^3 + 1/8*(6*x^4*arctanh(c*x) + c*(2*(c^2*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c*x - 1)/c^5))*a^2*b*c^3 + 3/2*(2*x^ 3*arctanh(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*a^2*b*c^2 + 3/2*a^3*c *x^2 + 9/4*(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 - 1/32*((b^3*c^4*x^4 + 4*b^3*c^3*x^3 + 6*b^3*c^2*x^2 + 4*b^3*c*x - 15*b^3)*log(-c*x + 1)^3 - (6*a*b^2*c^4*x^4 + 2*(12*a*b^2*c^3 + b^3*c^3)*x^ 3 + 12*(3*a*b^2*c^2 + b^3*c^2)*x^2 + 6*(4*a*b^2*c + 7*b^3*c)*x + 3*(b^3*c^ 4*x^4 + 4*b^3*c^3*x^3 + 6*b^3*c^2*x^2 + 4*b^3*c*x + b^3)*log(c*x + 1))*log (-c*x + 1)^2)/c - integrate(-1/16*(2*(b^3*c^4*x^4 + 2*b^3*c^3*x^3 - 2*b^3* c*x - b^3)*log(c*x + 1)^3 + 12*(a*b^2*c^4*x^4 + 2*a*b^2*c^3*x^3 - 2*a*b^2* c*x - a*b^2)*log(c*x + 1)^2 - (6*a*b^2*c^4*x^4 + 2*(12*a*b^2*c^3 + b^3*c^3 )*x^3 + 12*(3*a*b^2*c^2 + b^3*c^2)*x^2 + 6*(b^3*c^4*x^4 + 2*b^3*c^3*x^3 - 2*b^3*c*x - b^3)*log(c*x + 1)^2 + 6*(4*a*b^2*c + 7*b^3*c)*x + 3*(6*b^3*c^2 *x^2 + (8*a*b^2*c^4 + b^3*c^4)*x^4 + 4*(4*a*b^2*c^3 + b^3*c^3)*x^3 - 8*a*b ^2 + b^3 - 4*(4*a*b^2*c - b^3*c)*x)*log(c*x + 1))*log(-c*x + 1))/(c*x - 1) , x)
\[ \int (1+c x)^3 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (c x + 1\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} \,d x } \] Input:
integrate((c*x+1)^3*(a+b*arctanh(c*x))^3,x, algorithm="giac")
Output:
integrate((c*x + 1)^3*(b*arctanh(c*x) + a)^3, x)
Timed out. \[ \int (1+c x)^3 (a+b \text {arctanh}(c x))^3 \, dx=\int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3\,{\left (c\,x+1\right )}^3 \,d x \] Input:
int((a + b*atanh(c*x))^3*(c*x + 1)^3,x)
Output:
int((a + b*atanh(c*x))^3*(c*x + 1)^3, x)
\[ \int (1+c x)^3 (a+b \text {arctanh}(c x))^3 \, dx =\text {Too large to display} \] Input:
int((c*x+1)^3*(a+b*atanh(c*x))^3,x)
Output:
(atanh(c*x)**3*b**3*c**4*x**4 + 4*atanh(c*x)**3*b**3*c**3*x**3 + 6*atanh(c *x)**3*b**3*c**2*x**2 + 4*atanh(c*x)**3*b**3*c*x - 7*atanh(c*x)**3*b**3 + 3*atanh(c*x)**2*a*b**2*c**4*x**4 + 12*atanh(c*x)**2*a*b**2*c**3*x**3 + 18* atanh(c*x)**2*a*b**2*c**2*x**2 + 12*atanh(c*x)**2*a*b**2*c*x - 21*atanh(c* x)**2*a*b**2 + atanh(c*x)**2*b**3*c**3*x**3 + 6*atanh(c*x)**2*b**3*c**2*x* *2 + 21*atanh(c*x)**2*b**3*c*x - 6*atanh(c*x)**2*b**3 + 3*atanh(c*x)*a**2* b*c**4*x**4 + 12*atanh(c*x)*a**2*b*c**3*x**3 + 18*atanh(c*x)*a**2*b*c**2*x **2 + 12*atanh(c*x)*a**2*b*c*x + 3*atanh(c*x)*a**2*b + 2*atanh(c*x)*a*b**2 *c**3*x**3 + 12*atanh(c*x)*a*b**2*c**2*x**2 + 42*atanh(c*x)*a*b**2*c*x + 3 2*atanh(c*x)*a*b**2 + atanh(c*x)*b**3*c**2*x**2 + 12*atanh(c*x)*b**3*c*x + 11*atanh(c*x)*b**3 + 48*int((atanh(c*x)*x)/(c**2*x**2 - 1),x)*a*b**2*c**2 + 44*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 + 44*log(c **2*x - c)*a*b**2 + 12*log(c**2*x - c)*b**3 + a**3*c**4*x**4 + 4*a**3*c**3 *x**3 + 6*a**3*c**2*x**2 + 4*a**3*c*x + a**2*b*c**3*x**3 + 6*a**2*b*c**2*x **2 + 21*a**2*b*c*x + a*b**2*c**2*x**2 + 12*a*b**2*c*x + b**3*c*x)/(4*c)