Integrand size = 20, antiderivative size = 137 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^6} \, dx=-\frac {b c d^3}{20 x^4}-\frac {b c^2 d^3}{4 x^3}-\frac {3 b c^3 d^3}{5 x^2}-\frac {5 b c^4 d^3}{4 x}-\frac {d^3 (1+c x)^4 (a+b \text {arctanh}(c x))}{5 x^5}+\frac {c d^3 (1+c x)^4 (a+b \text {arctanh}(c x))}{20 x^4}+\frac {6}{5} b c^5 d^3 \log (x)-\frac {6}{5} b c^5 d^3 \log (1-c x) \]
-1/20*b*c*d^3/x^4-1/4*b*c^2*d^3/x^3-3/5*b*c^3*d^3/x^2-5/4*b*c^4*d^3/x-1/5* d^3*(c*x+1)^4*(a+b*arctanh(c*x))/x^5+1/20*c*d^3*(c*x+1)^4*(a+b*arctanh(c*x ))/x^4+6/5*b*c^5*d^3*ln(x)-6/5*b*c^5*d^3*ln(-c*x+1)
Time = 0.09 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.02 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^6} \, dx=-\frac {d^3 \left (8 a+30 a c x+2 b c x+40 a c^2 x^2+10 b c^2 x^2+20 a c^3 x^3+24 b c^3 x^3+50 b c^4 x^4+2 b \left (4+15 c x+20 c^2 x^2+10 c^3 x^3\right ) \text {arctanh}(c x)-48 b c^5 x^5 \log (x)+49 b c^5 x^5 \log (1-c x)-b c^5 x^5 \log (1+c x)\right )}{40 x^5} \]
-1/40*(d^3*(8*a + 30*a*c*x + 2*b*c*x + 40*a*c^2*x^2 + 10*b*c^2*x^2 + 20*a* c^3*x^3 + 24*b*c^3*x^3 + 50*b*c^4*x^4 + 2*b*(4 + 15*c*x + 20*c^2*x^2 + 10* c^3*x^3)*ArcTanh[c*x] - 48*b*c^5*x^5*Log[x] + 49*b*c^5*x^5*Log[1 - c*x] - b*c^5*x^5*Log[1 + c*x]))/x^5
Time = 0.33 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.79, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6498, 27, 165, 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.
| \(\displaystyle \int \frac {(c d x+d)^3 (a+b \text {arctanh}(c x))}{x^6} \, dx\) |
| \(\Big \downarrow \) 6498 |
| \(\displaystyle -b c \int -\frac {d^3 (4-c x) (c x+1)^3}{20 x^5 (1-c x)}dx-\frac {d^3 (c x+1)^4 (a+b \text {arctanh}(c x))}{5 x^5}+\frac {c d^3 (c x+1)^4 (a+b \text {arctanh}(c x))}{20 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{20} b c d^3 \int \frac {(4-c x) (c x+1)^3}{x^5 (1-c x)}dx-\frac {d^3 (c x+1)^4 (a+b \text {arctanh}(c x))}{5 x^5}+\frac {c d^3 (c x+1)^4 (a+b \text {arctanh}(c x))}{20 x^4}\) |
| \(\Big \downarrow \) 165 |
| \(\displaystyle \frac {1}{20} b c d^3 \int \left (-\frac {24 c^5}{c x-1}+\frac {24 c^4}{x}+\frac {25 c^3}{x^2}+\frac {24 c^2}{x^3}+\frac {15 c}{x^4}+\frac {4}{x^5}\right )dx-\frac {d^3 (c x+1)^4 (a+b \text {arctanh}(c x))}{5 x^5}+\frac {c d^3 (c x+1)^4 (a+b \text {arctanh}(c x))}{20 x^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^3 (c x+1)^4 (a+b \text {arctanh}(c x))}{5 x^5}+\frac {c d^3 (c x+1)^4 (a+b \text {arctanh}(c x))}{20 x^4}+\frac {1}{20} b c d^3 \left (24 c^4 \log (x)-24 c^4 \log (1-c x)-\frac {25 c^3}{x}-\frac {12 c^2}{x^2}-\frac {5 c}{x^3}-\frac {1}{x^4}\right )\) |
-1/5*(d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x]))/x^5 + (c*d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x]))/(20*x^4) + (b*c*d^3*(-x^(-4) - (5*c)/x^3 - (12*c^2)/x^2 - (25*c^3)/x + 24*c^4*Log[x] - 24*c^4*Log[1 - c*x]))/20
3.1.29.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d* x)^n*(e + f*x)^p*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && (IntegersQ[m, n, p] || (IGtQ[n, 0] && IGtQ[p, 0]))
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 = 1.14 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.07
| method | result | size |
| parts | \(d^{3} a \left (-\frac {3 c}{4 x^{4}}-\frac {c^{2}}{x^{3}}-\frac {c^{3}}{2 x^{2}}-\frac {1}{5 x^{5}}\right )+d^{3} b \,c^{5} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{5 c^{5} x^{5}}-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{3} x^{3}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{4 c^{4} x^{4}}+\frac {\ln \left (c x +1\right )}{40}-\frac {49 \ln \left (c x -1\right )}{40}-\frac {1}{20 c^{4} x^{4}}-\frac {1}{4 c^{3} x^{3}}-\frac {3}{5 c^{2} x^{2}}-\frac {5}{4 c x}+\frac {6 \ln \left (c x \right )}{5}\right )\) | \(146\) |
| derivativedivides | \(c^{5} \left (d^{3} a \left (-\frac {1}{5 c^{5} x^{5}}-\frac {1}{2 c^{2} x^{2}}-\frac {1}{c^{3} x^{3}}-\frac {3}{4 c^{4} x^{4}}\right )+d^{3} b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{5 c^{5} x^{5}}-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{3} x^{3}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{4 c^{4} x^{4}}+\frac {\ln \left (c x +1\right )}{40}-\frac {49 \ln \left (c x -1\right )}{40}-\frac {1}{20 c^{4} x^{4}}-\frac {1}{4 c^{3} x^{3}}-\frac {3}{5 c^{2} x^{2}}-\frac {5}{4 c x}+\frac {6 \ln \left (c x \right )}{5}\right )\right )\) | \(152\) |
| default | \(c^{5} \left (d^{3} a \left (-\frac {1}{5 c^{5} x^{5}}-\frac {1}{2 c^{2} x^{2}}-\frac {1}{c^{3} x^{3}}-\frac {3}{4 c^{4} x^{4}}\right )+d^{3} b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{5 c^{5} x^{5}}-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{3} x^{3}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{4 c^{4} x^{4}}+\frac {\ln \left (c x +1\right )}{40}-\frac {49 \ln \left (c x -1\right )}{40}-\frac {1}{20 c^{4} x^{4}}-\frac {1}{4 c^{3} x^{3}}-\frac {3}{5 c^{2} x^{2}}-\frac {5}{4 c x}+\frac {6 \ln \left (c x \right )}{5}\right )\right )\) | \(152\) |
| risch | \(-\frac {d^{3} b \left (10 c^{3} x^{3}+20 c^{2} x^{2}+15 c x +4\right ) \ln \left (c x +1\right )}{40 x^{5}}+\frac {d^{3} \left (b \,c^{5} \ln \left (c x +1\right ) x^{5}-49 b \,x^{5} \ln \left (-c x +1\right ) c^{5}+48 b \,c^{5} \ln \left (-x \right ) x^{5}-50 b \,c^{4} x^{4}+10 b \,x^{3} \ln \left (-c x +1\right ) c^{3}-20 c^{3} x^{3} a -24 b \,c^{3} x^{3}+20 b \,x^{2} \ln \left (-c x +1\right ) c^{2}-40 a \,c^{2} x^{2}-10 b \,c^{2} x^{2}+15 b c x \ln \left (-c x +1\right )-30 c x a -2 b c x +4 b \ln \left (-c x +1\right )-8 a \right )}{40 x^{5}}\) | \(203\) |
| parallelrisch | \(-\frac {24 \ln \left (c x -1\right ) x^{5} b \,c^{5} d^{3}-24 b \,c^{5} d^{3} \ln \left (x \right ) x^{5}-b \,c^{5} d^{3} \operatorname {arctanh}\left (c x \right ) x^{5}+10 a \,c^{5} d^{3} x^{5}+12 b \,c^{5} d^{3} x^{5}+25 b \,c^{4} d^{3} x^{4}+10 x^{3} \operatorname {arctanh}\left (c x \right ) b \,d^{3} c^{3}+10 a \,c^{3} d^{3} x^{3}+12 b \,c^{3} d^{3} x^{3}+20 x^{2} \operatorname {arctanh}\left (c x \right ) b \,c^{2} d^{3}+20 a \,c^{2} d^{3} x^{2}+5 b \,c^{2} d^{3} x^{2}+15 b c \,d^{3} x \,\operatorname {arctanh}\left (c x \right )+15 d^{3} a c x +b c \,d^{3} x +4 b \,d^{3} \operatorname {arctanh}\left (c x \right )+4 d^{3} a}{20 x^{5}}\) | \(214\) |
d^3*a*(-3/4*c/x^4-c^2/x^3-1/2*c^3/x^2-1/5/x^5)+d^3*b*c^5*(-1/5/c^5/x^5*arc tanh(c*x)-1/2/c^2/x^2*arctanh(c*x)-1/c^3/x^3*arctanh(c*x)-3/4/c^4/x^4*arct anh(c*x)+1/40*ln(c*x+1)-49/40*ln(c*x-1)-1/20/c^4/x^4-1/4/c^3/x^3-3/5/c^2/x ^2-5/4/c/x+6/5*ln(c*x))
Time = 0.28 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.28 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^6} \, dx=\frac {b c^{5} d^{3} x^{5} \log \left (c x + 1\right ) - 49 \, b c^{5} d^{3} x^{5} \log \left (c x - 1\right ) + 48 \, b c^{5} d^{3} x^{5} \log \left (x\right ) - 50 \, b c^{4} d^{3} x^{4} - 4 \, {\left (5 \, a + 6 \, b\right )} c^{3} d^{3} x^{3} - 10 \, {\left (4 \, a + b\right )} c^{2} d^{3} x^{2} - 2 \, {\left (15 \, a + b\right )} c d^{3} x - 8 \, a d^{3} - {\left (10 \, b c^{3} d^{3} x^{3} + 20 \, b c^{2} d^{3} x^{2} + 15 \, b c d^{3} x + 4 \, b d^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{40 \, x^{5}} \]
1/40*(b*c^5*d^3*x^5*log(c*x + 1) - 49*b*c^5*d^3*x^5*log(c*x - 1) + 48*b*c^ 5*d^3*x^5*log(x) - 50*b*c^4*d^3*x^4 - 4*(5*a + 6*b)*c^3*d^3*x^3 - 10*(4*a + b)*c^2*d^3*x^2 - 2*(15*a + b)*c*d^3*x - 8*a*d^3 - (10*b*c^3*d^3*x^3 + 20 *b*c^2*d^3*x^2 + 15*b*c*d^3*x + 4*b*d^3)*log(-(c*x + 1)/(c*x - 1)))/x^5
Time = 0.57 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.70 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^6} \, dx=\begin {cases} - \frac {a c^{3} d^{3}}{2 x^{2}} - \frac {a c^{2} d^{3}}{x^{3}} - \frac {3 a c d^{3}}{4 x^{4}} - \frac {a d^{3}}{5 x^{5}} + \frac {6 b c^{5} d^{3} \log {\left (x \right )}}{5} - \frac {6 b c^{5} d^{3} \log {\left (x - \frac {1}{c} \right )}}{5} + \frac {b c^{5} d^{3} \operatorname {atanh}{\left (c x \right )}}{20} - \frac {5 b c^{4} d^{3}}{4 x} - \frac {b c^{3} d^{3} \operatorname {atanh}{\left (c x \right )}}{2 x^{2}} - \frac {3 b c^{3} d^{3}}{5 x^{2}} - \frac {b c^{2} d^{3} \operatorname {atanh}{\left (c x \right )}}{x^{3}} - \frac {b c^{2} d^{3}}{4 x^{3}} - \frac {3 b c d^{3} \operatorname {atanh}{\left (c x \right )}}{4 x^{4}} - \frac {b c d^{3}}{20 x^{4}} - \frac {b d^{3} \operatorname {atanh}{\left (c x \right )}}{5 x^{5}} & \text {for}\: c \neq 0 \\- \frac {a d^{3}}{5 x^{5}} & \text {otherwise} \end {cases} \]
Piecewise((-a*c**3*d**3/(2*x**2) - a*c**2*d**3/x**3 - 3*a*c*d**3/(4*x**4) - a*d**3/(5*x**5) + 6*b*c**5*d**3*log(x)/5 - 6*b*c**5*d**3*log(x - 1/c)/5 + b*c**5*d**3*atanh(c*x)/20 - 5*b*c**4*d**3/(4*x) - b*c**3*d**3*atanh(c*x) /(2*x**2) - 3*b*c**3*d**3/(5*x**2) - b*c**2*d**3*atanh(c*x)/x**3 - b*c**2* d**3/(4*x**3) - 3*b*c*d**3*atanh(c*x)/(4*x**4) - b*c*d**3/(20*x**4) - b*d* *3*atanh(c*x)/(5*x**5), Ne(c, 0)), (-a*d**3/(5*x**5), True))
Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (121) = 242\).
Time = 0.19 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.82 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^6} \, dx=\frac {1}{4} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} b c^{3} d^{3} - \frac {1}{2} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{3}}\right )} b c^{2} d^{3} + \frac {1}{8} \, {\left ({\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac {2 \, {\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c - \frac {6 \, \operatorname {artanh}\left (c x\right )}{x^{4}}\right )} b c d^{3} - \frac {1}{20} \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} - 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) + \frac {2 \, c^{2} x^{2} + 1}{x^{4}}\right )} c + \frac {4 \, \operatorname {artanh}\left (c x\right )}{x^{5}}\right )} b d^{3} - \frac {a c^{3} d^{3}}{2 \, x^{2}} - \frac {a c^{2} d^{3}}{x^{3}} - \frac {3 \, a c d^{3}}{4 \, x^{4}} - \frac {a d^{3}}{5 \, x^{5}} \]
1/4*((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arctanh(c*x)/x^2)*b*c^3 *d^3 - 1/2*((c^2*log(c^2*x^2 - 1) - c^2*log(x^2) + 1/x^2)*c + 2*arctanh(c* x)/x^3)*b*c^2*d^3 + 1/8*((3*c^3*log(c*x + 1) - 3*c^3*log(c*x - 1) - 2*(3*c ^2*x^2 + 1)/x^3)*c - 6*arctanh(c*x)/x^4)*b*c*d^3 - 1/20*((2*c^4*log(c^2*x^ 2 - 1) - 2*c^4*log(x^2) + (2*c^2*x^2 + 1)/x^4)*c + 4*arctanh(c*x)/x^5)*b*d ^3 - 1/2*a*c^3*d^3/x^2 - a*c^2*d^3/x^3 - 3/4*a*c*d^3/x^4 - 1/5*a*d^3/x^5
Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (121) = 242\).
Time = 0.29 (sec) , antiderivative size = 533, normalized size of antiderivative = 3.89 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^6} \, dx=\frac {1}{5} \, {\left (6 \, b c^{4} d^{3} \log \left (-\frac {c x + 1}{c x - 1} - 1\right ) - 6 \, b c^{4} d^{3} \log \left (-\frac {c x + 1}{c x - 1}\right ) + \frac {2 \, {\left (\frac {20 \, {\left (c x + 1\right )}^{4} b c^{4} d^{3}}{{\left (c x - 1\right )}^{4}} + \frac {30 \, {\left (c x + 1\right )}^{3} b c^{4} d^{3}}{{\left (c x - 1\right )}^{3}} + \frac {30 \, {\left (c x + 1\right )}^{2} b c^{4} d^{3}}{{\left (c x - 1\right )}^{2}} + \frac {15 \, {\left (c x + 1\right )} b c^{4} d^{3}}{c x - 1} + 3 \, b c^{4} d^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{5}}{{\left (c x - 1\right )}^{5}} + \frac {5 \, {\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {10 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {10 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {5 \, {\left (c x + 1\right )}}{c x - 1} + 1} + \frac {\frac {80 \, {\left (c x + 1\right )}^{4} a c^{4} d^{3}}{{\left (c x - 1\right )}^{4}} + \frac {120 \, {\left (c x + 1\right )}^{3} a c^{4} d^{3}}{{\left (c x - 1\right )}^{3}} + \frac {120 \, {\left (c x + 1\right )}^{2} a c^{4} d^{3}}{{\left (c x - 1\right )}^{2}} + \frac {60 \, {\left (c x + 1\right )} a c^{4} d^{3}}{c x - 1} + 12 \, a c^{4} d^{3} + \frac {34 \, {\left (c x + 1\right )}^{4} b c^{4} d^{3}}{{\left (c x - 1\right )}^{4}} + \frac {103 \, {\left (c x + 1\right )}^{3} b c^{4} d^{3}}{{\left (c x - 1\right )}^{3}} + \frac {123 \, {\left (c x + 1\right )}^{2} b c^{4} d^{3}}{{\left (c x - 1\right )}^{2}} + \frac {69 \, {\left (c x + 1\right )} b c^{4} d^{3}}{c x - 1} + 15 \, b c^{4} d^{3}}{\frac {{\left (c x + 1\right )}^{5}}{{\left (c x - 1\right )}^{5}} + \frac {5 \, {\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {10 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {10 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {5 \, {\left (c x + 1\right )}}{c x - 1} + 1}\right )} c \]
1/5*(6*b*c^4*d^3*log(-(c*x + 1)/(c*x - 1) - 1) - 6*b*c^4*d^3*log(-(c*x + 1 )/(c*x - 1)) + 2*(20*(c*x + 1)^4*b*c^4*d^3/(c*x - 1)^4 + 30*(c*x + 1)^3*b* c^4*d^3/(c*x - 1)^3 + 30*(c*x + 1)^2*b*c^4*d^3/(c*x - 1)^2 + 15*(c*x + 1)* b*c^4*d^3/(c*x - 1) + 3*b*c^4*d^3)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^5/ (c*x - 1)^5 + 5*(c*x + 1)^4/(c*x - 1)^4 + 10*(c*x + 1)^3/(c*x - 1)^3 + 10* (c*x + 1)^2/(c*x - 1)^2 + 5*(c*x + 1)/(c*x - 1) + 1) + (80*(c*x + 1)^4*a*c ^4*d^3/(c*x - 1)^4 + 120*(c*x + 1)^3*a*c^4*d^3/(c*x - 1)^3 + 120*(c*x + 1) ^2*a*c^4*d^3/(c*x - 1)^2 + 60*(c*x + 1)*a*c^4*d^3/(c*x - 1) + 12*a*c^4*d^3 + 34*(c*x + 1)^4*b*c^4*d^3/(c*x - 1)^4 + 103*(c*x + 1)^3*b*c^4*d^3/(c*x - 1)^3 + 123*(c*x + 1)^2*b*c^4*d^3/(c*x - 1)^2 + 69*(c*x + 1)*b*c^4*d^3/(c* x - 1) + 15*b*c^4*d^3)/((c*x + 1)^5/(c*x - 1)^5 + 5*(c*x + 1)^4/(c*x - 1)^ 4 + 10*(c*x + 1)^3/(c*x - 1)^3 + 10*(c*x + 1)^2/(c*x - 1)^2 + 5*(c*x + 1)/ (c*x - 1) + 1))*c
Time = 3.45 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.70 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^6} \, dx=-\frac {4\,a\,d^3+4\,b\,d^3\,\mathrm {atanh}\left (c\,x\right )+20\,a\,c^2\,d^3\,x^2+10\,a\,c^3\,d^3\,x^3+10\,a\,c^5\,d^3\,x^5+5\,b\,c^2\,d^3\,x^2+12\,b\,c^3\,d^3\,x^3+25\,b\,c^4\,d^3\,x^4+12\,b\,c^5\,d^3\,x^5+15\,a\,c\,d^3\,x+b\,c\,d^3\,x-24\,b\,c^5\,d^3\,x^5\,\ln \left (x\right )+20\,b\,c^2\,d^3\,x^2\,\mathrm {atanh}\left (c\,x\right )+10\,b\,c^3\,d^3\,x^3\,\mathrm {atanh}\left (c\,x\right )+12\,b\,c^5\,d^3\,x^5\,\ln \left (c^2\,x^2-1\right )+15\,b\,c\,d^3\,x\,\mathrm {atanh}\left (c\,x\right )-25\,b\,c^4\,d^3\,x^5\,\mathrm {atan}\left (\frac {c^2\,x}{\sqrt {-c^2}}\right )\,\sqrt {-c^2}}{20\,x^5} \]
-(4*a*d^3 + 4*b*d^3*atanh(c*x) + 20*a*c^2*d^3*x^2 + 10*a*c^3*d^3*x^3 + 10* a*c^5*d^3*x^5 + 5*b*c^2*d^3*x^2 + 12*b*c^3*d^3*x^3 + 25*b*c^4*d^3*x^4 + 12 *b*c^5*d^3*x^5 + 15*a*c*d^3*x + b*c*d^3*x - 24*b*c^5*d^3*x^5*log(x) + 20*b *c^2*d^3*x^2*atanh(c*x) + 10*b*c^3*d^3*x^3*atanh(c*x) + 12*b*c^5*d^3*x^5*l og(c^2*x^2 - 1) + 15*b*c*d^3*x*atanh(c*x) - 25*b*c^4*d^3*x^5*atan((c^2*x)/ (-c^2)^(1/2))*(-c^2)^(1/2))/(20*x^5)