Integrand size = 19, antiderivative size = 206 \[ \int (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\frac {7}{2} a b d^3 x+b^2 d^3 x+\frac {1}{12} b^2 c d^3 x^2-\frac {b^2 d^3 \text {arctanh}(c x)}{c}+\frac {7}{2} b^2 d^3 x \text {arctanh}(c x)+b c d^3 x^2 (a+b \text {arctanh}(c x))+\frac {1}{6} b c^2 d^3 x^3 (a+b \text {arctanh}(c x))+\frac {d^3 (1+c x)^4 (a+b \text {arctanh}(c x))^2}{4 c}-\frac {4 b d^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{c}+\frac {11 b^2 d^3 \log \left (1-c^2 x^2\right )}{6 c}-\frac {2 b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c} \] Output:
7/2*a*b*d^3*x+b^2*d^3*x+1/12*b^2*c*d^3*x^2-b^2*d^3*arctanh(c*x)/c+7/2*b^2* d^3*x*arctanh(c*x)+b*c*d^3*x^2*(a+b*arctanh(c*x))+1/6*b*c^2*d^3*x^3*(a+b*a rctanh(c*x))+1/4*d^3*(c*x+1)^4*(a+b*arctanh(c*x))^2/c-4*b*d^3*(a+b*arctanh (c*x))*ln(2/(-c*x+1))/c+11/6*b^2*d^3*ln(-c^2*x^2+1)/c-2*b^2*d^3*polylog(2, 1-2/(-c*x+1))/c
Time = 1.50 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.42 \[ \int (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\frac {d^3 \left (-b^2+12 a^2 c x+42 a b c x+12 b^2 c x+18 a^2 c^2 x^2+12 a b c^2 x^2+b^2 c^2 x^2+12 a^2 c^3 x^3+2 a b c^3 x^3+3 a^2 c^4 x^4+3 b^2 \left (-15+4 c x+6 c^2 x^2+4 c^3 x^3+c^4 x^4\right ) \text {arctanh}(c x)^2+2 b \text {arctanh}(c x) \left (3 a c x \left (4+6 c x+4 c^2 x^2+c^3 x^3\right )+b \left (-6+21 c x+6 c^2 x^2+c^3 x^3\right )-24 b \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+21 a b \log (1-c x)-21 a b \log (1+c x)+12 a b \log \left (1-c^2 x^2\right )+22 b^2 \log \left (1-c^2 x^2\right )+12 a b \log \left (-1+c^2 x^2\right )+24 b^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )}{12 c} \] Input:
Integrate[(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2,x]
Output:
(d^3*(-b^2 + 12*a^2*c*x + 42*a*b*c*x + 12*b^2*c*x + 18*a^2*c^2*x^2 + 12*a* b*c^2*x^2 + b^2*c^2*x^2 + 12*a^2*c^3*x^3 + 2*a*b*c^3*x^3 + 3*a^2*c^4*x^4 + 3*b^2*(-15 + 4*c*x + 6*c^2*x^2 + 4*c^3*x^3 + c^4*x^4)*ArcTanh[c*x]^2 + 2* b*ArcTanh[c*x]*(3*a*c*x*(4 + 6*c*x + 4*c^2*x^2 + c^3*x^3) + b*(-6 + 21*c*x + 6*c^2*x^2 + c^3*x^3) - 24*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 21*a*b*Log[ 1 - c*x] - 21*a*b*Log[1 + c*x] + 12*a*b*Log[1 - c^2*x^2] + 22*b^2*Log[1 - c^2*x^2] + 12*a*b*Log[-1 + c^2*x^2] + 24*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x ])]))/(12*c)
Time = 0.47 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.96, 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.
| \(\displaystyle \int (c d x+d)^3 (a+b \text {arctanh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6480 |
| \(\displaystyle \frac {d^3 (c x+1)^4 (a+b \text {arctanh}(c x))^2}{4 c}-\frac {b \int \left (-c^2 x^2 (a+b \text {arctanh}(c x)) d^4-4 c x (a+b \text {arctanh}(c x)) d^4+\frac {8 (c x+1) (a+b \text {arctanh}(c x)) d^4}{1-c^2 x^2}-7 (a+b \text {arctanh}(c x)) d^4\right )dx}{2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^3 (c x+1)^4 (a+b \text {arctanh}(c x))^2}{4 c}-\frac {b \left (-\frac {1}{3} c^2 d^4 x^3 (a+b \text {arctanh}(c x))-2 c d^4 x^2 (a+b \text {arctanh}(c x))+\frac {8 d^4 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}-7 a d^4 x-7 b d^4 x \text {arctanh}(c x)+\frac {2 b d^4 \text {arctanh}(c x)}{c}-\frac {11 b d^4 \log \left (1-c^2 x^2\right )}{3 c}+\frac {4 b d^4 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c}-\frac {1}{6} b c d^4 x^2-2 b d^4 x\right )}{2 d}\) |
Input:
Int[(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2,x]
Output:
(d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x])^2)/(4*c) - (b*(-7*a*d^4*x - 2*b*d^4* x - (b*c*d^4*x^2)/6 + (2*b*d^4*ArcTanh[c*x])/c - 7*b*d^4*x*ArcTanh[c*x] - 2*c*d^4*x^2*(a + b*ArcTanh[c*x]) - (c^2*d^4*x^3*(a + b*ArcTanh[c*x]))/3 + (8*d^4*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c - (11*b*d^4*Log[1 - c^2*x^ 2])/(3*c) + (4*b*d^4*PolyLog[2, 1 - 2/(1 - c*x)])/c))/(2*d)
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 = 0.65 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.34
| method | result | size |
| derivativedivides | \(\frac {\frac {d^{3} a^{2} \left (c x +1\right )^{4}}{4}+d^{3} b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{4} x^{4}}{4}+\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}+\frac {3 \operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\operatorname {arctanh}\left (c x \right )^{2} c x +\frac {\operatorname {arctanh}\left (c x \right )^{2}}{4}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{6}+\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}+\frac {7 \,\operatorname {arctanh}\left (c x \right ) c x}{2}+4 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )+\frac {\left (c x -1\right )^{2}}{12}+\frac {7 c x}{6}-\frac {7}{6}+\frac {7 \ln \left (c x -1\right )}{3}+\frac {4 \ln \left (c x +1\right )}{3}+\ln \left (c x -1\right )^{2}-2 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )-2 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right )+2 d^{3} a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}+\operatorname {arctanh}\left (c x \right ) c x +\frac {\operatorname {arctanh}\left (c x \right )}{4}+\frac {x^{3} c^{3}}{12}+\frac {c^{2} x^{2}}{2}+\frac {7 c x}{4}+2 \ln \left (c x -1\right )\right )}{c}\) | \(276\) |
| default | \(\frac {\frac {d^{3} a^{2} \left (c x +1\right )^{4}}{4}+d^{3} b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{4} x^{4}}{4}+\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}+\frac {3 \operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\operatorname {arctanh}\left (c x \right )^{2} c x +\frac {\operatorname {arctanh}\left (c x \right )^{2}}{4}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{6}+\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}+\frac {7 \,\operatorname {arctanh}\left (c x \right ) c x}{2}+4 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )+\frac {\left (c x -1\right )^{2}}{12}+\frac {7 c x}{6}-\frac {7}{6}+\frac {7 \ln \left (c x -1\right )}{3}+\frac {4 \ln \left (c x +1\right )}{3}+\ln \left (c x -1\right )^{2}-2 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )-2 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right )+2 d^{3} a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}+\operatorname {arctanh}\left (c x \right ) c x +\frac {\operatorname {arctanh}\left (c x \right )}{4}+\frac {x^{3} c^{3}}{12}+\frac {c^{2} x^{2}}{2}+\frac {7 c x}{4}+2 \ln \left (c x -1\right )\right )}{c}\) | \(276\) |
| parts | \(\frac {d^{3} a^{2} \left (c x +1\right )^{4}}{4 c}+\frac {d^{3} b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{4} x^{4}}{4}+\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}+\frac {3 \operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\operatorname {arctanh}\left (c x \right )^{2} c x +\frac {\operatorname {arctanh}\left (c x \right )^{2}}{4}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{6}+\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}+\frac {7 \,\operatorname {arctanh}\left (c x \right ) c x}{2}+4 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )+\frac {\left (c x -1\right )^{2}}{12}+\frac {7 c x}{6}-\frac {7}{6}+\frac {7 \ln \left (c x -1\right )}{3}+\frac {4 \ln \left (c x +1\right )}{3}+\ln \left (c x -1\right )^{2}-2 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )-2 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right )}{c}+\frac {2 d^{3} a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}+\operatorname {arctanh}\left (c x \right ) c x +\frac {\operatorname {arctanh}\left (c x \right )}{4}+\frac {x^{3} c^{3}}{12}+\frac {c^{2} x^{2}}{2}+\frac {7 c x}{4}+2 \ln \left (c x -1\right )\right )}{c}\) | \(281\) |
| risch | \(\frac {2 b^{2} d^{3} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{c}-\frac {2 b^{2} d^{3} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{c}+\frac {b^{2} c \,d^{3} x^{2}}{12}+\frac {d^{3} c^{3} a^{2} x^{4}}{4}+d^{3} c^{2} a^{2} x^{3}+\frac {3 d^{3} c \,a^{2} x^{2}}{2}+\frac {4 d^{3} b^{2} \ln \left (-c x -1\right )}{3 c}-\frac {13 b^{2} d^{3}}{12 c}-\frac {15 d^{3} a^{2}}{4 c}+d^{3} a^{2} x +x \,b^{2} d^{3}+\frac {3 d^{3} c \ln \left (-c x +1\right )^{2} b^{2} x^{2}}{8}+\frac {d^{3} b \ln \left (-c x -1\right ) a}{4 c}+\frac {d^{3} \left (c x +1\right )^{4} b^{2} \ln \left (c x +1\right )^{2}}{16 c}+\frac {\ln \left (-c x +1\right )^{2} x \,b^{2} d^{3}}{4}-\frac {7 \ln \left (-c x +1\right ) x \,b^{2} d^{3}}{4}-\frac {15 \ln \left (-c x +1\right )^{2} b^{2} d^{3}}{16 c}+\frac {7 \ln \left (-c x +1\right ) b^{2} d^{3}}{3 c}-\ln \left (-c x +1\right ) x a b \,d^{3}+\frac {15 \ln \left (-c x +1\right ) a b \,d^{3}}{4 c}-\frac {14 a b \,d^{3}}{3 c}+\frac {7 x a b \,d^{3}}{2}+d^{3} c a b \,x^{2}+\frac {d^{3} c^{2} a b \,x^{3}}{6}-\frac {d^{3} c^{2} \ln \left (-c x +1\right ) b^{2} x^{3}}{12}-\frac {d^{3} c \ln \left (-c x +1\right ) b^{2} x^{2}}{2}+\frac {d^{3} c^{3} \ln \left (-c x +1\right )^{2} b^{2} x^{4}}{16}+\frac {d^{3} c^{2} \ln \left (-c x +1\right )^{2} b^{2} x^{3}}{4}+\left (-\frac {d^{3} \left (c x +1\right )^{4} b^{2} \ln \left (-c x +1\right )}{8 c}+\frac {d^{3} b \left (3 a \,c^{4} x^{4}+12 a \,c^{3} x^{3}+b \,c^{3} x^{3}+18 a \,c^{2} x^{2}+6 b \,c^{2} x^{2}+12 a c x +21 b c x +24 b \ln \left (-c x +1\right )\right )}{12 c}\right ) \ln \left (c x +1\right )-\frac {d^{3} c^{3} \ln \left (-c x +1\right ) a b \,x^{4}}{4}-d^{3} c^{2} \ln \left (-c x +1\right ) a b \,x^{3}-\frac {3 d^{3} c \ln \left (-c x +1\right ) a b \,x^{2}}{2}+\frac {2 b^{2} d^{3} \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{c}\) | \(637\) |
Input:
int((c*d*x+d)^3*(a+b*arctanh(c*x))^2,x,method=_RETURNVERBOSE)
Output:
1/c*(1/4*d^3*a^2*(c*x+1)^4+d^3*b^2*(1/4*arctanh(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+arctanh(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*ar ctanh(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))+2*d^3*a*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+ar ctanh(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)))
\[ \int (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\int { {\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2,x, algorithm="fricas")
Output:
integral(a^2*c^3*d^3*x^3 + 3*a^2*c^2*d^3*x^2 + 3*a^2*c*d^3*x + a^2*d^3 + ( b^2*c^3*d^3*x^3 + 3*b^2*c^2*d^3*x^2 + 3*b^2*c*d^3*x + b^2*d^3)*arctanh(c*x )^2 + 2*(a*b*c^3*d^3*x^3 + 3*a*b*c^2*d^3*x^2 + 3*a*b*c*d^3*x + a*b*d^3)*ar ctanh(c*x), x)
\[ \int (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=d^{3} \left (\int a^{2}\, dx + \int b^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b \operatorname {atanh}{\left (c x \right )}\, dx + \int 3 a^{2} c x\, dx + \int 3 a^{2} c^{2} x^{2}\, dx + \int a^{2} c^{3} x^{3}\, dx + \int 3 b^{2} c x \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 3 b^{2} c^{2} x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{3} x^{3} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 6 a b c x \operatorname {atanh}{\left (c x \right )}\, dx + \int 6 a b c^{2} x^{2} \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a b c^{3} x^{3} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \] Input:
integrate((c*d*x+d)**3*(a+b*atanh(c*x))**2,x)
Output:
d**3*(Integral(a**2, x) + Integral(b**2*atanh(c*x)**2, x) + Integral(2*a*b *atanh(c*x), x) + Integral(3*a**2*c*x, x) + Integral(3*a**2*c**2*x**2, x) + Integral(a**2*c**3*x**3, x) + Integral(3*b**2*c*x*atanh(c*x)**2, x) + In tegral(3*b**2*c**2*x**2*atanh(c*x)**2, x) + Integral(b**2*c**3*x**3*atanh( c*x)**2, x) + Integral(6*a*b*c*x*atanh(c*x), x) + Integral(6*a*b*c**2*x**2 *atanh(c*x), x) + Integral(2*a*b*c**3*x**3*atanh(c*x), x))
Leaf count of result is larger than twice the leaf count of optimal. 627 vs. \(2 (191) = 382\).
Time = 0.21 (sec) , antiderivative size = 627, normalized size of antiderivative = 3.04 \[ \int (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx =\text {Too large to display} \] Input:
integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2,x, algorithm="maxima")
Output:
1/4*a^2*c^3*d^3*x^4 + a^2*c^2*d^3*x^3 + 1/12*(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*b*c^3*d^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^3 + 3/2*a^2*c*d^3*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*b*c*d^3 + a^2*d^3*x + (2*c*x*arctanh(c*x) + log(- c^2*x^2 + 1))*a*b*d^3/c + 2*(log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2* c*x + 1/2))*b^2*d^3/c + 4/3*b^2*d^3*log(c*x + 1)/c + 7/3*b^2*d^3*log(c*x - 1)/c + 1/48*(4*b^2*c^2*d^3*x^2 + 48*b^2*c*d^3*x + 3*(b^2*c^4*d^3*x^4 + 4* b^2*c^3*d^3*x^3 + 6*b^2*c^2*d^3*x^2 + 4*b^2*c*d^3*x + b^2*d^3)*log(c*x + 1 )^2 + 3*(b^2*c^4*d^3*x^4 + 4*b^2*c^3*d^3*x^3 + 6*b^2*c^2*d^3*x^2 + 4*b^2*c *d^3*x - 15*b^2*d^3)*log(-c*x + 1)^2 + 4*(b^2*c^3*d^3*x^3 + 6*b^2*c^2*d^3* x^2 + 21*b^2*c*d^3*x)*log(c*x + 1) - 2*(2*b^2*c^3*d^3*x^3 + 12*b^2*c^2*d^3 *x^2 + 42*b^2*c*d^3*x + 3*(b^2*c^4*d^3*x^4 + 4*b^2*c^3*d^3*x^3 + 6*b^2*c^2 *d^3*x^2 + 4*b^2*c*d^3*x + b^2*d^3)*log(c*x + 1))*log(-c*x + 1))/c
\[ \int (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\int { {\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2,x, algorithm="giac")
Output:
integrate((c*d*x + d)^3*(b*arctanh(c*x) + a)^2, x)
Timed out. \[ \int (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^3 \,d x \] Input:
int((a + b*atanh(c*x))^2*(d + c*d*x)^3,x)
Output:
int((a + b*atanh(c*x))^2*(d + c*d*x)^3, x)
\[ \int (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\frac {d^{3} \left (48 \,\mathrm {log}\left (c^{2} x -c \right ) a b +12 b^{2} c x +6 \mathit {atanh} \left (c x \right ) a b +b^{2} c^{2} x^{2}+2 \mathit {atanh} \left (c x \right ) b^{2} c^{3} x^{3}+42 \mathit {atanh} \left (c x \right ) b^{2} c x +2 a b \,c^{3} x^{3}+42 a b c x +12 \mathit {atanh} \left (c x \right )^{2} b^{2} c x +12 \mathit {atanh} \left (c x \right ) b^{2} c^{2} x^{2}+12 a b \,c^{2} x^{2}+3 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{4} x^{4}+12 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{3} x^{3}+48 \left (\int \frac {\mathit {atanh} \left (c x \right ) x}{c^{2} x^{2}-1}d x \right ) b^{2} c^{2}+12 a^{2} c^{3} x^{3}+24 \mathit {atanh} \left (c x \right ) a b c x +6 \mathit {atanh} \left (c x \right ) a b \,c^{4} x^{4}+18 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{2} x^{2}+24 \mathit {atanh} \left (c x \right ) a b \,c^{3} x^{3}+12 a^{2} c x -21 \mathit {atanh} \left (c x \right )^{2} b^{2}+32 \mathit {atanh} \left (c x \right ) b^{2}+44 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2}+36 \mathit {atanh} \left (c x \right ) a b \,c^{2} x^{2}+18 a^{2} c^{2} x^{2}+3 a^{2} c^{4} x^{4}\right )}{12 c} \] Input:
int((c*d*x+d)^3*(a+b*atanh(c*x))^2,x)
Output:
(d**3*(3*atanh(c*x)**2*b**2*c**4*x**4 + 12*atanh(c*x)**2*b**2*c**3*x**3 + 18*atanh(c*x)**2*b**2*c**2*x**2 + 12*atanh(c*x)**2*b**2*c*x - 21*atanh(c*x )**2*b**2 + 6*atanh(c*x)*a*b*c**4*x**4 + 24*atanh(c*x)*a*b*c**3*x**3 + 36* atanh(c*x)*a*b*c**2*x**2 + 24*atanh(c*x)*a*b*c*x + 6*atanh(c*x)*a*b + 2*at anh(c*x)*b**2*c**3*x**3 + 12*atanh(c*x)*b**2*c**2*x**2 + 42*atanh(c*x)*b** 2*c*x + 32*atanh(c*x)*b**2 + 48*int((atanh(c*x)*x)/(c**2*x**2 - 1),x)*b**2 *c**2 + 48*log(c**2*x - c)*a*b + 44*log(c**2*x - c)*b**2 + 3*a**2*c**4*x** 4 + 12*a**2*c**3*x**3 + 18*a**2*c**2*x**2 + 12*a**2*c*x + 2*a*b*c**3*x**3 + 12*a*b*c**2*x**2 + 42*a*b*c*x + b**2*c**2*x**2 + 12*b**2*c*x))/(12*c)