Integrand size = 20, antiderivative size = 93 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^5} \, dx=-\frac {b c d^3}{12 x^3}-\frac {b c^2 d^3}{2 x^2}-\frac {7 b c^3 d^3}{4 x}-\frac {d^3 (1+c x)^4 (a+b \text {arctanh}(c x))}{4 x^4}+2 b c^4 d^3 \log (x)-2 b c^4 d^3 \log (1-c x) \] Output:
-1/12*b*c*d^3/x^3-1/2*b*c^2*d^3/x^2-7/4*b*c^3*d^3/x-1/4*d^3*(c*x+1)^4*(a+b *arctanh(c*x))/x^4+2*b*c^4*d^3*ln(x)-2*b*c^4*d^3*ln(-c*x+1)
Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.41 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^5} \, dx=-\frac {d^3 \left (6 a+24 a c x+2 b c x+36 a c^2 x^2+12 b c^2 x^2+24 a c^3 x^3+42 b c^3 x^3+6 b \left (1+4 c x+6 c^2 x^2+4 c^3 x^3\right ) \text {arctanh}(c x)-48 b c^4 x^4 \log (x)+45 b c^4 x^4 \log (1-c x)+3 b c^4 x^4 \log (1+c x)\right )}{24 x^4} \] Input:
Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^5,x]
Output:
-1/24*(d^3*(6*a + 24*a*c*x + 2*b*c*x + 36*a*c^2*x^2 + 12*b*c^2*x^2 + 24*a* c^3*x^3 + 42*b*c^3*x^3 + 6*b*(1 + 4*c*x + 6*c^2*x^2 + 4*c^3*x^3)*ArcTanh[c *x] - 48*b*c^4*x^4*Log[x] + 45*b*c^4*x^4*Log[1 - c*x] + 3*b*c^4*x^4*Log[1 + c*x]))/x^4
Time = 0.31 (sec) , antiderivative size = 76, 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, 99, 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.
| \(\displaystyle \int \frac {(c d x+d)^3 (a+b \text {arctanh}(c x))}{x^5} \, dx\) |
| \(\Big \downarrow \) 6498 |
| \(\displaystyle -b c \int -\frac {d^3 (c x+1)^3}{4 x^4 (1-c x)}dx-\frac {d^3 (c x+1)^4 (a+b \text {arctanh}(c x))}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} b c d^3 \int \frac {(c x+1)^3}{x^4 (1-c x)}dx-\frac {d^3 (c x+1)^4 (a+b \text {arctanh}(c x))}{4 x^4}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {1}{4} b c d^3 \int \left (-\frac {8 c^4}{c x-1}+\frac {8 c^3}{x}+\frac {7 c^2}{x^2}+\frac {4 c}{x^3}+\frac {1}{x^4}\right )dx-\frac {d^3 (c x+1)^4 (a+b \text {arctanh}(c x))}{4 x^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} b c d^3 \left (8 c^3 \log (x)-8 c^3 \log (1-c x)-\frac {7 c^2}{x}-\frac {2 c}{x^2}-\frac {1}{3 x^3}\right )-\frac {d^3 (c x+1)^4 (a+b \text {arctanh}(c x))}{4 x^4}\) |
Input:
Int[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^5,x]
Output:
-1/4*(d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x]))/x^4 + (b*c*d^3*(-1/3*1/x^3 - ( 2*c)/x^2 - (7*c^2)/x + 8*c^3*Log[x] - 8*c^3*Log[1 - c*x]))/4
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_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
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 = 0.39 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.48
| method | result | size |
| parts | \(d^{3} a \left (-\frac {1}{4 x^{4}}-\frac {c^{3}}{x}-\frac {3 c^{2}}{2 x^{2}}-\frac {c}{x^{3}}\right )+d^{3} b \,c^{4} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c^{3} x^{3}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\operatorname {arctanh}\left (c x \right )}{4 c^{4} x^{4}}-\frac {15 \ln \left (c x -1\right )}{8}-\frac {1}{12 c^{3} x^{3}}-\frac {1}{2 c^{2} x^{2}}-\frac {7}{4 c x}+2 \ln \left (c x \right )-\frac {\ln \left (c x +1\right )}{8}\right )\) | \(138\) |
| derivativedivides | \(c^{4} \left (d^{3} a \left (-\frac {1}{c^{3} x^{3}}-\frac {3}{2 c^{2} x^{2}}-\frac {1}{c x}-\frac {1}{4 c^{4} x^{4}}\right )+d^{3} b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c^{3} x^{3}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\operatorname {arctanh}\left (c x \right )}{4 c^{4} x^{4}}-\frac {15 \ln \left (c x -1\right )}{8}-\frac {1}{12 c^{3} x^{3}}-\frac {1}{2 c^{2} x^{2}}-\frac {7}{4 c x}+2 \ln \left (c x \right )-\frac {\ln \left (c x +1\right )}{8}\right )\right )\) | \(144\) |
| default | \(c^{4} \left (d^{3} a \left (-\frac {1}{c^{3} x^{3}}-\frac {3}{2 c^{2} x^{2}}-\frac {1}{c x}-\frac {1}{4 c^{4} x^{4}}\right )+d^{3} b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c^{3} x^{3}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\operatorname {arctanh}\left (c x \right )}{4 c^{4} x^{4}}-\frac {15 \ln \left (c x -1\right )}{8}-\frac {1}{12 c^{3} x^{3}}-\frac {1}{2 c^{2} x^{2}}-\frac {7}{4 c x}+2 \ln \left (c x \right )-\frac {\ln \left (c x +1\right )}{8}\right )\right )\) | \(144\) |
| risch | \(-\frac {d^{3} b \left (4 x^{3} c^{3}+6 c^{2} x^{2}+4 c x +1\right ) \ln \left (c x +1\right )}{8 x^{4}}+\frac {d^{3} \left (48 b \,c^{4} \ln \left (-x \right ) x^{4}-3 b \,c^{4} \ln \left (c x +1\right ) x^{4}-45 b \,x^{4} \ln \left (-c x +1\right ) c^{4}+12 b \,x^{3} \ln \left (-c x +1\right ) c^{3}-24 a \,c^{3} x^{3}-42 b \,c^{3} x^{3}+18 b \,c^{2} x^{2} \ln \left (-c x +1\right )-36 a \,c^{2} x^{2}-12 b \,c^{2} x^{2}+12 b c x \ln \left (-c x +1\right )-24 a c x -2 b c x +3 b \ln \left (-c x +1\right )-6 a \right )}{24 x^{4}}\) | \(195\) |
| parallelrisch | \(\frac {24 b \,c^{4} d^{3} \ln \left (x \right ) x^{4}-24 \ln \left (c x -1\right ) x^{4} b \,c^{4} d^{3}-3 d^{3} b \,\operatorname {arctanh}\left (c x \right ) x^{4} c^{4}-18 a \,c^{4} d^{3} x^{4}-6 b \,c^{4} d^{3} x^{4}-12 d^{3} b \,\operatorname {arctanh}\left (c x \right ) x^{3} c^{3}-12 a \,c^{3} d^{3} x^{3}-21 b \,c^{3} d^{3} x^{3}-18 x^{2} \operatorname {arctanh}\left (c x \right ) b \,c^{2} d^{3}-18 a \,c^{2} d^{3} x^{2}-6 b \,c^{2} d^{3} x^{2}-12 b c \,d^{3} x \,\operatorname {arctanh}\left (c x \right )-12 d^{3} a c x -b c \,d^{3} x -3 b \,d^{3} \operatorname {arctanh}\left (c x \right )-3 d^{3} a}{12 x^{4}}\) | \(203\) |
Input:
int((c*d*x+d)^3*(a+b*arctanh(c*x))/x^5,x,method=_RETURNVERBOSE)
Output:
d^3*a*(-1/4/x^4-c^3/x-3/2*c^2/x^2-c/x^3)+d^3*b*c^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)/c^4/x^4-15/8*ln (c*x-1)-1/12/c^3/x^3-1/2/c^2/x^2-7/4/c/x+2*ln(c*x)-1/8*ln(c*x+1))
Time = 0.10 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.75 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^5} \, dx=-\frac {3 \, b c^{4} d^{3} x^{4} \log \left (c x + 1\right ) + 45 \, b c^{4} d^{3} x^{4} \log \left (c x - 1\right ) - 48 \, b c^{4} d^{3} x^{4} \log \left (x\right ) + 6 \, {\left (4 \, a + 7 \, b\right )} c^{3} d^{3} x^{3} + 12 \, {\left (3 \, a + b\right )} c^{2} d^{3} x^{2} + 2 \, {\left (12 \, a + b\right )} c d^{3} x + 6 \, a d^{3} + 3 \, {\left (4 \, b c^{3} d^{3} x^{3} + 6 \, b c^{2} d^{3} x^{2} + 4 \, b c d^{3} x + b d^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{24 \, x^{4}} \] Input:
integrate((c*d*x+d)^3*(a+b*arctanh(c*x))/x^5,x, algorithm="fricas")
Output:
-1/24*(3*b*c^4*d^3*x^4*log(c*x + 1) + 45*b*c^4*d^3*x^4*log(c*x - 1) - 48*b *c^4*d^3*x^4*log(x) + 6*(4*a + 7*b)*c^3*d^3*x^3 + 12*(3*a + b)*c^2*d^3*x^2 + 2*(12*a + b)*c*d^3*x + 6*a*d^3 + 3*(4*b*c^3*d^3*x^3 + 6*b*c^2*d^3*x^2 + 4*b*c*d^3*x + b*d^3)*log(-(c*x + 1)/(c*x - 1)))/x^4
Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (92) = 184\).
Time = 0.54 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.23 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^5} \, dx=\begin {cases} - \frac {a c^{3} d^{3}}{x} - \frac {3 a c^{2} d^{3}}{2 x^{2}} - \frac {a c d^{3}}{x^{3}} - \frac {a d^{3}}{4 x^{4}} + 2 b c^{4} d^{3} \log {\left (x \right )} - 2 b c^{4} d^{3} \log {\left (x - \frac {1}{c} \right )} - \frac {b c^{4} d^{3} \operatorname {atanh}{\left (c x \right )}}{4} - \frac {b c^{3} d^{3} \operatorname {atanh}{\left (c x \right )}}{x} - \frac {7 b c^{3} d^{3}}{4 x} - \frac {3 b c^{2} d^{3} \operatorname {atanh}{\left (c x \right )}}{2 x^{2}} - \frac {b c^{2} d^{3}}{2 x^{2}} - \frac {b c d^{3} \operatorname {atanh}{\left (c x \right )}}{x^{3}} - \frac {b c d^{3}}{12 x^{3}} - \frac {b d^{3} \operatorname {atanh}{\left (c x \right )}}{4 x^{4}} & \text {for}\: c \neq 0 \\- \frac {a d^{3}}{4 x^{4}} & \text {otherwise} \end {cases} \] Input:
integrate((c*d*x+d)**3*(a+b*atanh(c*x))/x**5,x)
Output:
Piecewise((-a*c**3*d**3/x - 3*a*c**2*d**3/(2*x**2) - a*c*d**3/x**3 - a*d** 3/(4*x**4) + 2*b*c**4*d**3*log(x) - 2*b*c**4*d**3*log(x - 1/c) - b*c**4*d* *3*atanh(c*x)/4 - b*c**3*d**3*atanh(c*x)/x - 7*b*c**3*d**3/(4*x) - 3*b*c** 2*d**3*atanh(c*x)/(2*x**2) - b*c**2*d**3/(2*x**2) - b*c*d**3*atanh(c*x)/x* *3 - b*c*d**3/(12*x**3) - b*d**3*atanh(c*x)/(4*x**4), Ne(c, 0)), (-a*d**3/ (4*x**4), True))
Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (85) = 170\).
Time = 0.04 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.45 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^5} \, dx=-\frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x}\right )} b c^{3} d^{3} + \frac {3}{4} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} b c^{2} d^{3} - \frac {1}{2} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{3}}\right )} b c d^{3} - \frac {a c^{3} d^{3}}{x} + \frac {1}{24} \, {\left ({\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac {2 \, {\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c - \frac {6 \, \operatorname {artanh}\left (c x\right )}{x^{4}}\right )} b d^{3} - \frac {3 \, a c^{2} d^{3}}{2 \, x^{2}} - \frac {a c d^{3}}{x^{3}} - \frac {a d^{3}}{4 \, x^{4}} \] Input:
integrate((c*d*x+d)^3*(a+b*arctanh(c*x))/x^5,x, algorithm="maxima")
Output:
-1/2*(c*(log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x)*b*c^3*d^3 + 3/4* ((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arctanh(c*x)/x^2)*b*c^2*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*d^3 - a*c^3*d^3/x + 1/24*((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*d^3 - 3/2*a*c^2*d^3/x^ 2 - a*c*d^3/x^3 - 1/4*a*d^3/x^4
Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (85) = 170\).
Time = 0.13 (sec) , antiderivative size = 431, normalized size of antiderivative = 4.63 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^5} \, dx=\frac {1}{3} \, {\left (6 \, b c^{3} d^{3} \log \left (-\frac {c x + 1}{c x - 1} - 1\right ) - 6 \, b c^{3} d^{3} \log \left (-\frac {c x + 1}{c x - 1}\right ) + \frac {6 \, {\left (\frac {4 \, {\left (c x + 1\right )}^{3} b c^{3} d^{3}}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )}^{2} b c^{3} d^{3}}{{\left (c x - 1\right )}^{2}} + \frac {4 \, {\left (c x + 1\right )} b c^{3} d^{3}}{c x - 1} + b c^{3} d^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {4 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {4 \, {\left (c x + 1\right )}}{c x - 1} + 1} + \frac {\frac {48 \, {\left (c x + 1\right )}^{3} a c^{3} d^{3}}{{\left (c x - 1\right )}^{3}} + \frac {72 \, {\left (c x + 1\right )}^{2} a c^{3} d^{3}}{{\left (c x - 1\right )}^{2}} + \frac {48 \, {\left (c x + 1\right )} a c^{3} d^{3}}{c x - 1} + 12 \, a c^{3} d^{3} + \frac {18 \, {\left (c x + 1\right )}^{3} b c^{3} d^{3}}{{\left (c x - 1\right )}^{3}} + \frac {45 \, {\left (c x + 1\right )}^{2} b c^{3} d^{3}}{{\left (c x - 1\right )}^{2}} + \frac {38 \, {\left (c x + 1\right )} b c^{3} d^{3}}{c x - 1} + 11 \, b c^{3} d^{3}}{\frac {{\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {4 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {4 \, {\left (c x + 1\right )}}{c x - 1} + 1}\right )} c \] Input:
integrate((c*d*x+d)^3*(a+b*arctanh(c*x))/x^5,x, algorithm="giac")
Output:
1/3*(6*b*c^3*d^3*log(-(c*x + 1)/(c*x - 1) - 1) - 6*b*c^3*d^3*log(-(c*x + 1 )/(c*x - 1)) + 6*(4*(c*x + 1)^3*b*c^3*d^3/(c*x - 1)^3 + 6*(c*x + 1)^2*b*c^ 3*d^3/(c*x - 1)^2 + 4*(c*x + 1)*b*c^3*d^3/(c*x - 1) + b*c^3*d^3)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^4/(c*x - 1)^4 + 4*(c*x + 1)^3/(c*x - 1)^3 + 6* (c*x + 1)^2/(c*x - 1)^2 + 4*(c*x + 1)/(c*x - 1) + 1) + (48*(c*x + 1)^3*a*c ^3*d^3/(c*x - 1)^3 + 72*(c*x + 1)^2*a*c^3*d^3/(c*x - 1)^2 + 48*(c*x + 1)*a *c^3*d^3/(c*x - 1) + 12*a*c^3*d^3 + 18*(c*x + 1)^3*b*c^3*d^3/(c*x - 1)^3 + 45*(c*x + 1)^2*b*c^3*d^3/(c*x - 1)^2 + 38*(c*x + 1)*b*c^3*d^3/(c*x - 1) + 11*b*c^3*d^3)/((c*x + 1)^4/(c*x - 1)^4 + 4*(c*x + 1)^3/(c*x - 1)^3 + 6*(c *x + 1)^2/(c*x - 1)^2 + 4*(c*x + 1)/(c*x - 1) + 1))*c
Time = 3.67 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.58 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^5} \, dx=\frac {d^3\,\left (21\,b\,c^4\,\mathrm {atanh}\left (c\,x\right )-12\,b\,c^4\,\ln \left (c^2\,x^2-1\right )+24\,b\,c^4\,\ln \left (x\right )\right )}{12}-\frac {\frac {d^3\,\left (3\,a+3\,b\,\mathrm {atanh}\left (c\,x\right )\right )}{12}+\frac {d^3\,x\,\left (12\,a\,c+b\,c+12\,b\,c\,\mathrm {atanh}\left (c\,x\right )\right )}{12}+\frac {d^3\,x^2\,\left (18\,a\,c^2+6\,b\,c^2+18\,b\,c^2\,\mathrm {atanh}\left (c\,x\right )\right )}{12}+\frac {d^3\,x^3\,\left (12\,a\,c^3+21\,b\,c^3+12\,b\,c^3\,\mathrm {atanh}\left (c\,x\right )\right )}{12}}{x^4} \] Input:
int(((a + b*atanh(c*x))*(d + c*d*x)^3)/x^5,x)
Output:
(d^3*(21*b*c^4*atanh(c*x) - 12*b*c^4*log(c^2*x^2 - 1) + 24*b*c^4*log(x)))/ 12 - ((d^3*(3*a + 3*b*atanh(c*x)))/12 + (d^3*x*(12*a*c + b*c + 12*b*c*atan h(c*x)))/12 + (d^3*x^2*(18*a*c^2 + 6*b*c^2 + 18*b*c^2*atanh(c*x)))/12 + (d ^3*x^3*(12*a*c^3 + 21*b*c^3 + 12*b*c^3*atanh(c*x)))/12)/x^4
Time = 0.17 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.54 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^5} \, dx=\frac {d^{3} \left (-3 \mathit {atanh} \left (c x \right ) b \,c^{4} x^{4}-12 \mathit {atanh} \left (c x \right ) b \,c^{3} x^{3}-18 \mathit {atanh} \left (c x \right ) b \,c^{2} x^{2}-12 \mathit {atanh} \left (c x \right ) b c x -3 \mathit {atanh} \left (c x \right ) b -24 \,\mathrm {log}\left (c^{2} x -c \right ) b \,c^{4} x^{4}+24 \,\mathrm {log}\left (x \right ) b \,c^{4} x^{4}-12 a \,c^{3} x^{3}-18 a \,c^{2} x^{2}-12 a c x -3 a -21 b \,c^{3} x^{3}-6 b \,c^{2} x^{2}-b c x \right )}{12 x^{4}} \] Input:
int((c*d*x+d)^3*(a+b*atanh(c*x))/x^5,x)
Output:
(d**3*( - 3*atanh(c*x)*b*c**4*x**4 - 12*atanh(c*x)*b*c**3*x**3 - 18*atanh( c*x)*b*c**2*x**2 - 12*atanh(c*x)*b*c*x - 3*atanh(c*x)*b - 24*log(c**2*x - c)*b*c**4*x**4 + 24*log(x)*b*c**4*x**4 - 12*a*c**3*x**3 - 18*a*c**2*x**2 - 12*a*c*x - 3*a - 21*b*c**3*x**3 - 6*b*c**2*x**2 - b*c*x))/(12*x**4)