Integrand size = 20, antiderivative size = 192 \[ \int x^3 (d+c d x)^3 (a+b \text {arctanh}(c x)) \, dx=\frac {3 b d^3 x}{4 c^3}+\frac {13 b d^3 x^2}{35 c^2}+\frac {b d^3 x^3}{4 c}+\frac {13}{70} b d^3 x^4+\frac {1}{10} b c d^3 x^5+\frac {1}{42} b c^2 d^3 x^6+\frac {1}{4} d^3 x^4 (a+b \text {arctanh}(c x))+\frac {3}{5} c d^3 x^5 (a+b \text {arctanh}(c x))+\frac {1}{2} c^2 d^3 x^6 (a+b \text {arctanh}(c x))+\frac {1}{7} c^3 d^3 x^7 (a+b \text {arctanh}(c x))+\frac {209 b d^3 \log (1-c x)}{280 c^4}-\frac {b d^3 \log (1+c x)}{280 c^4} \] Output:
3/4*b*d^3*x/c^3+13/35*b*d^3*x^2/c^2+1/4*b*d^3*x^3/c+13/70*b*d^3*x^4+1/10*b *c*d^3*x^5+1/42*b*c^2*d^3*x^6+1/4*d^3*x^4*(a+b*arctanh(c*x))+3/5*c*d^3*x^5 *(a+b*arctanh(c*x))+1/2*c^2*d^3*x^6*(a+b*arctanh(c*x))+1/7*c^3*d^3*x^7*(a+ b*arctanh(c*x))+209/280*b*d^3*ln(-c*x+1)/c^4-1/280*b*d^3*ln(c*x+1)/c^4
Time = 0.07 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.79 \[ \int x^3 (d+c d x)^3 (a+b \text {arctanh}(c x)) \, dx=\frac {d^3 \left (630 b c x+312 b c^2 x^2+210 b c^3 x^3+210 a c^4 x^4+156 b c^4 x^4+504 a c^5 x^5+84 b c^5 x^5+420 a c^6 x^6+20 b c^6 x^6+120 a c^7 x^7+6 b c^4 x^4 \left (35+84 c x+70 c^2 x^2+20 c^3 x^3\right ) \text {arctanh}(c x)+627 b \log (1-c x)-3 b \log (1+c x)\right )}{840 c^4} \] Input:
Integrate[x^3*(d + c*d*x)^3*(a + b*ArcTanh[c*x]),x]
Output:
(d^3*(630*b*c*x + 312*b*c^2*x^2 + 210*b*c^3*x^3 + 210*a*c^4*x^4 + 156*b*c^ 4*x^4 + 504*a*c^5*x^5 + 84*b*c^5*x^5 + 420*a*c^6*x^6 + 20*b*c^6*x^6 + 120* a*c^7*x^7 + 6*b*c^4*x^4*(35 + 84*c*x + 70*c^2*x^2 + 20*c^3*x^3)*ArcTanh[c* x] + 627*b*Log[1 - c*x] - 3*b*Log[1 + c*x]))/(840*c^4)
Time = 0.48 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6498, 27, 2333, 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.
| \(\displaystyle \int x^3 (c d x+d)^3 (a+b \text {arctanh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6498 |
| \(\displaystyle -b c \int \frac {d^3 x^4 \left (20 c^3 x^3+70 c^2 x^2+84 c x+35\right )}{140 \left (1-c^2 x^2\right )}dx+\frac {1}{7} c^3 d^3 x^7 (a+b \text {arctanh}(c x))+\frac {1}{2} c^2 d^3 x^6 (a+b \text {arctanh}(c x))+\frac {3}{5} c d^3 x^5 (a+b \text {arctanh}(c x))+\frac {1}{4} d^3 x^4 (a+b \text {arctanh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{140} b c d^3 \int \frac {x^4 \left (20 c^3 x^3+70 c^2 x^2+84 c x+35\right )}{1-c^2 x^2}dx+\frac {1}{7} c^3 d^3 x^7 (a+b \text {arctanh}(c x))+\frac {1}{2} c^2 d^3 x^6 (a+b \text {arctanh}(c x))+\frac {3}{5} c d^3 x^5 (a+b \text {arctanh}(c x))+\frac {1}{4} d^3 x^4 (a+b \text {arctanh}(c x))\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle -\frac {1}{140} b c d^3 \int \left (-20 c x^5-70 x^4-\frac {104 x^3}{c}-\frac {105 x^2}{c^2}-\frac {104 x}{c^3}+\frac {104 c x+105}{c^4 \left (1-c^2 x^2\right )}-\frac {105}{c^4}\right )dx+\frac {1}{7} c^3 d^3 x^7 (a+b \text {arctanh}(c x))+\frac {1}{2} c^2 d^3 x^6 (a+b \text {arctanh}(c x))+\frac {3}{5} c d^3 x^5 (a+b \text {arctanh}(c x))+\frac {1}{4} d^3 x^4 (a+b \text {arctanh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{7} c^3 d^3 x^7 (a+b \text {arctanh}(c x))+\frac {1}{2} c^2 d^3 x^6 (a+b \text {arctanh}(c x))+\frac {3}{5} c d^3 x^5 (a+b \text {arctanh}(c x))+\frac {1}{4} d^3 x^4 (a+b \text {arctanh}(c x))-\frac {1}{140} b c d^3 \left (\frac {105 \text {arctanh}(c x)}{c^5}-\frac {105 x}{c^4}-\frac {52 x^2}{c^3}-\frac {35 x^3}{c^2}-\frac {52 \log \left (1-c^2 x^2\right )}{c^5}-\frac {10 c x^6}{3}-\frac {26 x^4}{c}-14 x^5\right )\) |
Input:
Int[x^3*(d + c*d*x)^3*(a + b*ArcTanh[c*x]),x]
Output:
(d^3*x^4*(a + b*ArcTanh[c*x]))/4 + (3*c*d^3*x^5*(a + b*ArcTanh[c*x]))/5 + (c^2*d^3*x^6*(a + b*ArcTanh[c*x]))/2 + (c^3*d^3*x^7*(a + b*ArcTanh[c*x]))/ 7 - (b*c*d^3*((-105*x)/c^4 - (52*x^2)/c^3 - (35*x^3)/c^2 - (26*x^4)/c - 14 *x^5 - (10*c*x^6)/3 + (105*ArcTanh[c*x])/c^5 - (52*Log[1 - c^2*x^2])/c^5)) /140
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*( x_))^(q_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Simp[( a + b*ArcTanh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(1 - c^2*x ^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && Intege rQ[2*m] && ((IGtQ[m, 0] && IGtQ[q, 0]) || (ILtQ[m + q + 1, 0] && LtQ[m*q, 0 ]))
Time = 0.59 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.79
| method | result | size |
| parts | \(d^{3} a \left (\frac {1}{7} c^{3} x^{7}+\frac {1}{2} c^{2} x^{6}+\frac {3}{5} c \,x^{5}+\frac {1}{4} x^{4}\right )+\frac {d^{3} b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arctanh}\left (c x \right ) c^{6} x^{6}}{2}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {c^{6} x^{6}}{42}+\frac {c^{5} x^{5}}{10}+\frac {13 c^{4} x^{4}}{70}+\frac {x^{3} c^{3}}{4}+\frac {13 c^{2} x^{2}}{35}+\frac {3 c x}{4}+\frac {209 \ln \left (c x -1\right )}{280}-\frac {\ln \left (c x +1\right )}{280}\right )}{c^{4}}\) | \(152\) |
| derivativedivides | \(\frac {d^{3} a \left (\frac {1}{7} c^{7} x^{7}+\frac {1}{2} c^{6} x^{6}+\frac {3}{5} c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+d^{3} b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arctanh}\left (c x \right ) c^{6} x^{6}}{2}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {c^{6} x^{6}}{42}+\frac {c^{5} x^{5}}{10}+\frac {13 c^{4} x^{4}}{70}+\frac {x^{3} c^{3}}{4}+\frac {13 c^{2} x^{2}}{35}+\frac {3 c x}{4}+\frac {209 \ln \left (c x -1\right )}{280}-\frac {\ln \left (c x +1\right )}{280}\right )}{c^{4}}\) | \(158\) |
| default | \(\frac {d^{3} a \left (\frac {1}{7} c^{7} x^{7}+\frac {1}{2} c^{6} x^{6}+\frac {3}{5} c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+d^{3} b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arctanh}\left (c x \right ) c^{6} x^{6}}{2}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {c^{6} x^{6}}{42}+\frac {c^{5} x^{5}}{10}+\frac {13 c^{4} x^{4}}{70}+\frac {x^{3} c^{3}}{4}+\frac {13 c^{2} x^{2}}{35}+\frac {3 c x}{4}+\frac {209 \ln \left (c x -1\right )}{280}-\frac {\ln \left (c x +1\right )}{280}\right )}{c^{4}}\) | \(158\) |
| parallelrisch | \(\frac {60 b \,c^{7} d^{3} \operatorname {arctanh}\left (c x \right ) x^{7}+60 c^{7} d^{3} x^{7} a +210 b \,c^{6} d^{3} \operatorname {arctanh}\left (c x \right ) x^{6}+210 a \,c^{6} d^{3} x^{6}+10 c^{6} d^{3} x^{6} b +252 b \,c^{5} d^{3} \operatorname {arctanh}\left (c x \right ) x^{5}+252 a \,c^{5} d^{3} x^{5}+42 b \,c^{5} d^{3} x^{5}+105 d^{3} b \,\operatorname {arctanh}\left (c x \right ) x^{4} c^{4}+105 a \,c^{4} d^{3} x^{4}+78 b \,c^{4} d^{3} x^{4}+105 b \,c^{3} d^{3} x^{3}+156 b \,c^{2} d^{3} x^{2}+315 b c \,d^{3} x +312 \ln \left (c x -1\right ) b \,d^{3}-3 b \,d^{3} \operatorname {arctanh}\left (c x \right )+156 d^{3} b}{420 c^{4}}\) | \(215\) |
| risch | \(\frac {d^{3} b \,x^{4} \left (20 x^{3} c^{3}+70 c^{2} x^{2}+84 c x +35\right ) \ln \left (c x +1\right )}{280}-\frac {d^{3} c^{3} b \,x^{7} \ln \left (-c x +1\right )}{14}+\frac {d^{3} c^{3} a \,x^{7}}{7}-\frac {d^{3} c^{2} b \,x^{6} \ln \left (-c x +1\right )}{4}+\frac {d^{3} c^{2} a \,x^{6}}{2}+\frac {b \,c^{2} d^{3} x^{6}}{42}-\frac {3 d^{3} c b \,x^{5} \ln \left (-c x +1\right )}{10}+\frac {3 d^{3} c a \,x^{5}}{5}+\frac {b c \,d^{3} x^{5}}{10}-\frac {d^{3} b \,x^{4} \ln \left (-c x +1\right )}{8}+\frac {d^{3} a \,x^{4}}{4}+\frac {13 b \,d^{3} x^{4}}{70}+\frac {b \,d^{3} x^{3}}{4 c}+\frac {13 b \,d^{3} x^{2}}{35 c^{2}}+\frac {3 b \,d^{3} x}{4 c^{3}}+\frac {209 b \,d^{3} \ln \left (-c x +1\right )}{280 c^{4}}-\frac {b \,d^{3} \ln \left (c x +1\right )}{280 c^{4}}\) | \(249\) |
Input:
int(x^3*(c*d*x+d)^3*(a+b*arctanh(c*x)),x,method=_RETURNVERBOSE)
Output:
d^3*a*(1/7*c^3*x^7+1/2*c^2*x^6+3/5*c*x^5+1/4*x^4)+d^3*b/c^4*(1/7*arctanh(c *x)*c^7*x^7+1/2*arctanh(c*x)*c^6*x^6+3/5*arctanh(c*x)*c^5*x^5+1/4*arctanh( c*x)*c^4*x^4+1/42*c^6*x^6+1/10*c^5*x^5+13/70*c^4*x^4+1/4*x^3*c^3+13/35*c^2 *x^2+3/4*c*x+209/280*ln(c*x-1)-1/280*ln(c*x+1))
Time = 0.10 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.99 \[ \int x^3 (d+c d x)^3 (a+b \text {arctanh}(c x)) \, dx=\frac {120 \, a c^{7} d^{3} x^{7} + 20 \, {\left (21 \, a + b\right )} c^{6} d^{3} x^{6} + 84 \, {\left (6 \, a + b\right )} c^{5} d^{3} x^{5} + 6 \, {\left (35 \, a + 26 \, b\right )} c^{4} d^{3} x^{4} + 210 \, b c^{3} d^{3} x^{3} + 312 \, b c^{2} d^{3} x^{2} + 630 \, b c d^{3} x - 3 \, b d^{3} \log \left (c x + 1\right ) + 627 \, b d^{3} \log \left (c x - 1\right ) + 3 \, {\left (20 \, b c^{7} d^{3} x^{7} + 70 \, b c^{6} d^{3} x^{6} + 84 \, b c^{5} d^{3} x^{5} + 35 \, b c^{4} d^{3} x^{4}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{840 \, c^{4}} \] Input:
integrate(x^3*(c*d*x+d)^3*(a+b*arctanh(c*x)),x, algorithm="fricas")
Output:
1/840*(120*a*c^7*d^3*x^7 + 20*(21*a + b)*c^6*d^3*x^6 + 84*(6*a + b)*c^5*d^ 3*x^5 + 6*(35*a + 26*b)*c^4*d^3*x^4 + 210*b*c^3*d^3*x^3 + 312*b*c^2*d^3*x^ 2 + 630*b*c*d^3*x - 3*b*d^3*log(c*x + 1) + 627*b*d^3*log(c*x - 1) + 3*(20* b*c^7*d^3*x^7 + 70*b*c^6*d^3*x^6 + 84*b*c^5*d^3*x^5 + 35*b*c^4*d^3*x^4)*lo g(-(c*x + 1)/(c*x - 1)))/c^4
Time = 0.57 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.27 \[ \int x^3 (d+c d x)^3 (a+b \text {arctanh}(c x)) \, dx=\begin {cases} \frac {a c^{3} d^{3} x^{7}}{7} + \frac {a c^{2} d^{3} x^{6}}{2} + \frac {3 a c d^{3} x^{5}}{5} + \frac {a d^{3} x^{4}}{4} + \frac {b c^{3} d^{3} x^{7} \operatorname {atanh}{\left (c x \right )}}{7} + \frac {b c^{2} d^{3} x^{6} \operatorname {atanh}{\left (c x \right )}}{2} + \frac {b c^{2} d^{3} x^{6}}{42} + \frac {3 b c d^{3} x^{5} \operatorname {atanh}{\left (c x \right )}}{5} + \frac {b c d^{3} x^{5}}{10} + \frac {b d^{3} x^{4} \operatorname {atanh}{\left (c x \right )}}{4} + \frac {13 b d^{3} x^{4}}{70} + \frac {b d^{3} x^{3}}{4 c} + \frac {13 b d^{3} x^{2}}{35 c^{2}} + \frac {3 b d^{3} x}{4 c^{3}} + \frac {26 b d^{3} \log {\left (x - \frac {1}{c} \right )}}{35 c^{4}} - \frac {b d^{3} \operatorname {atanh}{\left (c x \right )}}{140 c^{4}} & \text {for}\: c \neq 0 \\\frac {a d^{3} x^{4}}{4} & \text {otherwise} \end {cases} \] Input:
integrate(x**3*(c*d*x+d)**3*(a+b*atanh(c*x)),x)
Output:
Piecewise((a*c**3*d**3*x**7/7 + a*c**2*d**3*x**6/2 + 3*a*c*d**3*x**5/5 + a *d**3*x**4/4 + b*c**3*d**3*x**7*atanh(c*x)/7 + b*c**2*d**3*x**6*atanh(c*x) /2 + b*c**2*d**3*x**6/42 + 3*b*c*d**3*x**5*atanh(c*x)/5 + b*c*d**3*x**5/10 + b*d**3*x**4*atanh(c*x)/4 + 13*b*d**3*x**4/70 + b*d**3*x**3/(4*c) + 13*b *d**3*x**2/(35*c**2) + 3*b*d**3*x/(4*c**3) + 26*b*d**3*log(x - 1/c)/(35*c* *4) - b*d**3*atanh(c*x)/(140*c**4), Ne(c, 0)), (a*d**3*x**4/4, True))
Time = 0.03 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.48 \[ \int x^3 (d+c d x)^3 (a+b \text {arctanh}(c x)) \, dx=\frac {1}{7} \, a c^{3} d^{3} x^{7} + \frac {1}{2} \, a c^{2} d^{3} x^{6} + \frac {3}{5} \, a c d^{3} x^{5} + \frac {1}{84} \, {\left (12 \, x^{7} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, c^{4} x^{6} + 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} + \frac {6 \, \log \left (c^{2} x^{2} - 1\right )}{c^{8}}\right )}\right )} b c^{3} d^{3} + \frac {1}{4} \, a d^{3} x^{4} + \frac {1}{60} \, {\left (30 \, x^{6} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b c^{2} d^{3} + \frac {3}{20} \, {\left (4 \, x^{5} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b c d^{3} + \frac {1}{24} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b d^{3} \] Input:
integrate(x^3*(c*d*x+d)^3*(a+b*arctanh(c*x)),x, algorithm="maxima")
Output:
1/7*a*c^3*d^3*x^7 + 1/2*a*c^2*d^3*x^6 + 3/5*a*c*d^3*x^5 + 1/84*(12*x^7*arc tanh(c*x) + c*((2*c^4*x^6 + 3*c^2*x^4 + 6*x^2)/c^6 + 6*log(c^2*x^2 - 1)/c^ 8))*b*c^3*d^3 + 1/4*a*d^3*x^4 + 1/60*(30*x^6*arctanh(c*x) + c*(2*(3*c^4*x^ 5 + 5*c^2*x^3 + 15*x)/c^6 - 15*log(c*x + 1)/c^7 + 15*log(c*x - 1)/c^7))*b* c^2*d^3 + 3/20*(4*x^5*arctanh(c*x) + c*((c^2*x^4 + 2*x^2)/c^4 + 2*log(c^2* x^2 - 1)/c^6))*b*c*d^3 + 1/24*(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))*b*d^3
Leaf count of result is larger than twice the leaf count of optimal. 722 vs. \(2 (168) = 336\).
Time = 0.13 (sec) , antiderivative size = 722, normalized size of antiderivative = 3.76 \[ \int x^3 (d+c d x)^3 (a+b \text {arctanh}(c x)) \, dx =\text {Too large to display} \] Input:
integrate(x^3*(c*d*x+d)^3*(a+b*arctanh(c*x)),x, algorithm="giac")
Output:
1/105*c*(6*(140*(c*x + 1)^6*b*d^3/(c*x - 1)^6 - 210*(c*x + 1)^5*b*d^3/(c*x - 1)^5 + 490*(c*x + 1)^4*b*d^3/(c*x - 1)^4 - 455*(c*x + 1)^3*b*d^3/(c*x - 1)^3 + 273*(c*x + 1)^2*b*d^3/(c*x - 1)^2 - 91*(c*x + 1)*b*d^3/(c*x - 1) + 13*b*d^3)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^7*c^5/(c*x - 1)^7 - 7*(c*x + 1)^6*c^5/(c*x - 1)^6 + 21*(c*x + 1)^5*c^5/(c*x - 1)^5 - 35*(c*x + 1)^4* c^5/(c*x - 1)^4 + 35*(c*x + 1)^3*c^5/(c*x - 1)^3 - 21*(c*x + 1)^2*c^5/(c*x - 1)^2 + 7*(c*x + 1)*c^5/(c*x - 1) - c^5) + (1680*(c*x + 1)^6*a*d^3/(c*x - 1)^6 - 2520*(c*x + 1)^5*a*d^3/(c*x - 1)^5 + 5880*(c*x + 1)^4*a*d^3/(c*x - 1)^4 - 5460*(c*x + 1)^3*a*d^3/(c*x - 1)^3 + 3276*(c*x + 1)^2*a*d^3/(c*x - 1)^2 - 1092*(c*x + 1)*a*d^3/(c*x - 1) + 156*a*d^3 + 762*(c*x + 1)^6*b*d^ 3/(c*x - 1)^6 - 3063*(c*x + 1)^5*b*d^3/(c*x - 1)^5 + 5959*(c*x + 1)^4*b*d^ 3/(c*x - 1)^4 - 6694*(c*x + 1)^3*b*d^3/(c*x - 1)^3 + 4344*(c*x + 1)^2*b*d^ 3/(c*x - 1)^2 - 1539*(c*x + 1)*b*d^3/(c*x - 1) + 231*b*d^3)/((c*x + 1)^7*c ^5/(c*x - 1)^7 - 7*(c*x + 1)^6*c^5/(c*x - 1)^6 + 21*(c*x + 1)^5*c^5/(c*x - 1)^5 - 35*(c*x + 1)^4*c^5/(c*x - 1)^4 + 35*(c*x + 1)^3*c^5/(c*x - 1)^3 - 21*(c*x + 1)^2*c^5/(c*x - 1)^2 + 7*(c*x + 1)*c^5/(c*x - 1) - c^5) - 78*b*d ^3*log(-(c*x + 1)/(c*x - 1) + 1)/c^5 + 78*b*d^3*log(-(c*x + 1)/(c*x - 1))/ c^5)
Time = 3.80 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.92 \[ \int x^3 (d+c d x)^3 (a+b \text {arctanh}(c x)) \, dx=\frac {\frac {13\,b\,c^2\,d^3\,x^2}{35}-\frac {d^3\,\left (315\,b\,\mathrm {atanh}\left (c\,x\right )-156\,b\,\ln \left (c^2\,x^2-1\right )\right )}{420}+\frac {b\,c^3\,d^3\,x^3}{4}+\frac {3\,b\,c\,d^3\,x}{4}}{c^4}+\frac {d^3\,\left (105\,a\,x^4+78\,b\,x^4+105\,b\,x^4\,\mathrm {atanh}\left (c\,x\right )\right )}{420}+\frac {c^3\,d^3\,\left (60\,a\,x^7+60\,b\,x^7\,\mathrm {atanh}\left (c\,x\right )\right )}{420}+\frac {c\,d^3\,\left (252\,a\,x^5+42\,b\,x^5+252\,b\,x^5\,\mathrm {atanh}\left (c\,x\right )\right )}{420}+\frac {c^2\,d^3\,\left (210\,a\,x^6+10\,b\,x^6+210\,b\,x^6\,\mathrm {atanh}\left (c\,x\right )\right )}{420} \] Input:
int(x^3*(a + b*atanh(c*x))*(d + c*d*x)^3,x)
Output:
((13*b*c^2*d^3*x^2)/35 - (d^3*(315*b*atanh(c*x) - 156*b*log(c^2*x^2 - 1))) /420 + (b*c^3*d^3*x^3)/4 + (3*b*c*d^3*x)/4)/c^4 + (d^3*(105*a*x^4 + 78*b*x ^4 + 105*b*x^4*atanh(c*x)))/420 + (c^3*d^3*(60*a*x^7 + 60*b*x^7*atanh(c*x) ))/420 + (c*d^3*(252*a*x^5 + 42*b*x^5 + 252*b*x^5*atanh(c*x)))/420 + (c^2* d^3*(210*a*x^6 + 10*b*x^6 + 210*b*x^6*atanh(c*x)))/420
Time = 0.17 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.87 \[ \int x^3 (d+c d x)^3 (a+b \text {arctanh}(c x)) \, dx=\frac {d^{3} \left (60 \mathit {atanh} \left (c x \right ) b \,c^{7} x^{7}+210 \mathit {atanh} \left (c x \right ) b \,c^{6} x^{6}+252 \mathit {atanh} \left (c x \right ) b \,c^{5} x^{5}+105 \mathit {atanh} \left (c x \right ) b \,c^{4} x^{4}-3 \mathit {atanh} \left (c x \right ) b +312 \,\mathrm {log}\left (c^{2} x -c \right ) b +60 a \,c^{7} x^{7}+210 a \,c^{6} x^{6}+252 a \,c^{5} x^{5}+105 a \,c^{4} x^{4}+10 b \,c^{6} x^{6}+42 b \,c^{5} x^{5}+78 b \,c^{4} x^{4}+105 b \,c^{3} x^{3}+156 b \,c^{2} x^{2}+315 b c x \right )}{420 c^{4}} \] Input:
int(x^3*(c*d*x+d)^3*(a+b*atanh(c*x)),x)
Output:
(d**3*(60*atanh(c*x)*b*c**7*x**7 + 210*atanh(c*x)*b*c**6*x**6 + 252*atanh( c*x)*b*c**5*x**5 + 105*atanh(c*x)*b*c**4*x**4 - 3*atanh(c*x)*b + 312*log(c **2*x - c)*b + 60*a*c**7*x**7 + 210*a*c**6*x**6 + 252*a*c**5*x**5 + 105*a* c**4*x**4 + 10*b*c**6*x**6 + 42*b*c**5*x**5 + 78*b*c**4*x**4 + 105*b*c**3* x**3 + 156*b*c**2*x**2 + 315*b*c*x))/(420*c**4)