Integrand size = 20, antiderivative size = 196 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^7} \, dx=-\frac {b c d^3}{30 x^5}-\frac {3 b c^2 d^3}{20 x^4}-\frac {11 b c^3 d^3}{36 x^3}-\frac {7 b c^4 d^3}{15 x^2}-\frac {11 b c^5 d^3}{12 x}-\frac {d^3 (a+b \text {arctanh}(c x))}{6 x^6}-\frac {3 c d^3 (a+b \text {arctanh}(c x))}{5 x^5}-\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))}{4 x^4}-\frac {c^3 d^3 (a+b \text {arctanh}(c x))}{3 x^3}+\frac {14}{15} b c^6 d^3 \log (x)-\frac {37}{40} b c^6 d^3 \log (1-c x)-\frac {1}{120} b c^6 d^3 \log (1+c x) \] Output:
-1/30*b*c*d^3/x^5-3/20*b*c^2*d^3/x^4-11/36*b*c^3*d^3/x^3-7/15*b*c^4*d^3/x^ 2-11/12*b*c^5*d^3/x-1/6*d^3*(a+b*arctanh(c*x))/x^6-3/5*c*d^3*(a+b*arctanh( c*x))/x^5-3/4*c^2*d^3*(a+b*arctanh(c*x))/x^4-1/3*c^3*d^3*(a+b*arctanh(c*x) )/x^3+14/15*b*c^6*d^3*ln(x)-37/40*b*c^6*d^3*ln(-c*x+1)-1/120*b*c^6*d^3*ln( c*x+1)
Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.76 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^7} \, dx=-\frac {d^3 \left (60 a+216 a c x+12 b c x+270 a c^2 x^2+54 b c^2 x^2+120 a c^3 x^3+110 b c^3 x^3+168 b c^4 x^4+330 b c^5 x^5+6 b \left (10+36 c x+45 c^2 x^2+20 c^3 x^3\right ) \text {arctanh}(c x)-336 b c^6 x^6 \log (x)+333 b c^6 x^6 \log (1-c x)+3 b c^6 x^6 \log (1+c x)\right )}{360 x^6} \] Input:
Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^7,x]
Output:
-1/360*(d^3*(60*a + 216*a*c*x + 12*b*c*x + 270*a*c^2*x^2 + 54*b*c^2*x^2 + 120*a*c^3*x^3 + 110*b*c^3*x^3 + 168*b*c^4*x^4 + 330*b*c^5*x^5 + 6*b*(10 + 36*c*x + 45*c^2*x^2 + 20*c^3*x^3)*ArcTanh[c*x] - 336*b*c^6*x^6*Log[x] + 33 3*b*c^6*x^6*Log[1 - c*x] + 3*b*c^6*x^6*Log[1 + c*x]))/x^6
Time = 0.47 (sec) , antiderivative size = 161, 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.
| \(\displaystyle \int \frac {(c d x+d)^3 (a+b \text {arctanh}(c x))}{x^7} \, dx\) |
| \(\Big \downarrow \) 6498 |
| \(\displaystyle -b c \int -\frac {d^3 \left (20 c^3 x^3+45 c^2 x^2+36 c x+10\right )}{60 x^6 \left (1-c^2 x^2\right )}dx-\frac {c^3 d^3 (a+b \text {arctanh}(c x))}{3 x^3}-\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))}{4 x^4}-\frac {d^3 (a+b \text {arctanh}(c x))}{6 x^6}-\frac {3 c d^3 (a+b \text {arctanh}(c x))}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{60} b c d^3 \int \frac {20 c^3 x^3+45 c^2 x^2+36 c x+10}{x^6 \left (1-c^2 x^2\right )}dx-\frac {c^3 d^3 (a+b \text {arctanh}(c x))}{3 x^3}-\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))}{4 x^4}-\frac {d^3 (a+b \text {arctanh}(c x))}{6 x^6}-\frac {3 c d^3 (a+b \text {arctanh}(c x))}{5 x^5}\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle \frac {1}{60} b c d^3 \int \left (-\frac {111 c^6}{2 (c x-1)}-\frac {c^6}{2 (c x+1)}+\frac {56 c^5}{x}+\frac {55 c^4}{x^2}+\frac {56 c^3}{x^3}+\frac {55 c^2}{x^4}+\frac {36 c}{x^5}+\frac {10}{x^6}\right )dx-\frac {c^3 d^3 (a+b \text {arctanh}(c x))}{3 x^3}-\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))}{4 x^4}-\frac {d^3 (a+b \text {arctanh}(c x))}{6 x^6}-\frac {3 c d^3 (a+b \text {arctanh}(c x))}{5 x^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {c^3 d^3 (a+b \text {arctanh}(c x))}{3 x^3}-\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))}{4 x^4}-\frac {d^3 (a+b \text {arctanh}(c x))}{6 x^6}-\frac {3 c d^3 (a+b \text {arctanh}(c x))}{5 x^5}+\frac {1}{60} b c d^3 \left (56 c^5 \log (x)-\frac {111}{2} c^5 \log (1-c x)-\frac {1}{2} c^5 \log (c x+1)-\frac {55 c^4}{x}-\frac {28 c^3}{x^2}-\frac {55 c^2}{3 x^3}-\frac {9 c}{x^4}-\frac {2}{x^5}\right )\) |
Input:
Int[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^7,x]
Output:
-1/6*(d^3*(a + b*ArcTanh[c*x]))/x^6 - (3*c*d^3*(a + b*ArcTanh[c*x]))/(5*x^ 5) - (3*c^2*d^3*(a + b*ArcTanh[c*x]))/(4*x^4) - (c^3*d^3*(a + b*ArcTanh[c* x]))/(3*x^3) + (b*c*d^3*(-2/x^5 - (9*c)/x^4 - (55*c^2)/(3*x^3) - (28*c^3)/ x^2 - (55*c^4)/x + 56*c^5*Log[x] - (111*c^5*Log[1 - c*x])/2 - (c^5*Log[1 + c*x])/2))/60
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 = 0.41 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.79
| method | result | size |
| parts | \(d^{3} a \left (-\frac {1}{6 x^{6}}-\frac {3 c^{2}}{4 x^{4}}-\frac {3 c}{5 x^{5}}-\frac {c^{3}}{3 x^{3}}\right )+d^{3} b \,c^{6} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{4 c^{4} x^{4}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{5 c^{5} x^{5}}-\frac {\operatorname {arctanh}\left (c x \right )}{6 c^{6} x^{6}}-\frac {37 \ln \left (c x -1\right )}{40}-\frac {1}{30 c^{5} x^{5}}-\frac {3}{20 c^{4} x^{4}}-\frac {11}{36 c^{3} x^{3}}-\frac {7}{15 c^{2} x^{2}}-\frac {11}{12 c x}+\frac {14 \ln \left (c x \right )}{15}-\frac {\ln \left (c x +1\right )}{120}\right )\) | \(154\) |
| derivativedivides | \(c^{6} \left (d^{3} a \left (-\frac {1}{3 c^{3} x^{3}}-\frac {3}{4 c^{4} x^{4}}-\frac {3}{5 c^{5} x^{5}}-\frac {1}{6 c^{6} x^{6}}\right )+d^{3} b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{4 c^{4} x^{4}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{5 c^{5} x^{5}}-\frac {\operatorname {arctanh}\left (c x \right )}{6 c^{6} x^{6}}-\frac {37 \ln \left (c x -1\right )}{40}-\frac {1}{30 c^{5} x^{5}}-\frac {3}{20 c^{4} x^{4}}-\frac {11}{36 c^{3} x^{3}}-\frac {7}{15 c^{2} x^{2}}-\frac {11}{12 c x}+\frac {14 \ln \left (c x \right )}{15}-\frac {\ln \left (c x +1\right )}{120}\right )\right )\) | \(160\) |
| default | \(c^{6} \left (d^{3} a \left (-\frac {1}{3 c^{3} x^{3}}-\frac {3}{4 c^{4} x^{4}}-\frac {3}{5 c^{5} x^{5}}-\frac {1}{6 c^{6} x^{6}}\right )+d^{3} b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{4 c^{4} x^{4}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{5 c^{5} x^{5}}-\frac {\operatorname {arctanh}\left (c x \right )}{6 c^{6} x^{6}}-\frac {37 \ln \left (c x -1\right )}{40}-\frac {1}{30 c^{5} x^{5}}-\frac {3}{20 c^{4} x^{4}}-\frac {11}{36 c^{3} x^{3}}-\frac {7}{15 c^{2} x^{2}}-\frac {11}{12 c x}+\frac {14 \ln \left (c x \right )}{15}-\frac {\ln \left (c x +1\right )}{120}\right )\right )\) | \(160\) |
| risch | \(-\frac {d^{3} b \left (20 x^{3} c^{3}+45 c^{2} x^{2}+36 c x +10\right ) \ln \left (c x +1\right )}{120 x^{6}}+\frac {d^{3} \left (336 b \,c^{6} \ln \left (-x \right ) x^{6}-333 b \,c^{6} x^{6} \ln \left (-c x +1\right )-3 b \,c^{6} \ln \left (c x +1\right ) x^{6}-330 b \,c^{5} x^{5}-168 b \,c^{4} x^{4}+60 b \,x^{3} \ln \left (-c x +1\right ) c^{3}-120 a \,c^{3} x^{3}-110 b \,c^{3} x^{3}+135 b \,c^{2} x^{2} \ln \left (-c x +1\right )-270 a \,c^{2} x^{2}-54 b \,c^{2} x^{2}+108 b c x \ln \left (-c x +1\right )-216 a c x -12 b c x +30 b \ln \left (-c x +1\right )-60 a \right )}{360 x^{6}}\) | \(213\) |
| parallelrisch | \(\frac {168 b \,c^{6} d^{3} \ln \left (x \right ) x^{6}-168 \ln \left (c x -1\right ) x^{6} b \,c^{6} d^{3}-3 b \,c^{6} d^{3} \operatorname {arctanh}\left (c x \right ) x^{6}-84 c^{6} d^{3} x^{6} b -165 b \,c^{5} d^{3} x^{5}-84 b \,c^{4} d^{3} x^{4}-60 d^{3} b \,\operatorname {arctanh}\left (c x \right ) x^{3} c^{3}-60 a \,c^{3} d^{3} x^{3}-55 b \,c^{3} d^{3} x^{3}-135 x^{2} \operatorname {arctanh}\left (c x \right ) b \,c^{2} d^{3}-135 a \,c^{2} d^{3} x^{2}-27 b \,c^{2} d^{3} x^{2}-108 b c \,d^{3} x \,\operatorname {arctanh}\left (c x \right )-108 d^{3} a c x -6 b c \,d^{3} x -30 b \,d^{3} \operatorname {arctanh}\left (c x \right )-30 d^{3} a}{180 x^{6}}\) | \(215\) |
Input:
int((c*d*x+d)^3*(a+b*arctanh(c*x))/x^7,x,method=_RETURNVERBOSE)
Output:
d^3*a*(-1/6/x^6-3/4*c^2/x^4-3/5*c/x^5-1/3*c^3/x^3)+d^3*b*c^6*(-1/3*arctanh (c*x)/c^3/x^3-3/4*arctanh(c*x)/c^4/x^4-3/5*arctanh(c*x)/c^5/x^5-1/6*arctan h(c*x)/c^6/x^6-37/40*ln(c*x-1)-1/30/c^5/x^5-3/20/c^4/x^4-11/36/c^3/x^3-7/1 5/c^2/x^2-11/12/c/x+14/15*ln(c*x)-1/120*ln(c*x+1))
Time = 0.09 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.96 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^7} \, dx=-\frac {3 \, b c^{6} d^{3} x^{6} \log \left (c x + 1\right ) + 333 \, b c^{6} d^{3} x^{6} \log \left (c x - 1\right ) - 336 \, b c^{6} d^{3} x^{6} \log \left (x\right ) + 330 \, b c^{5} d^{3} x^{5} + 168 \, b c^{4} d^{3} x^{4} + 10 \, {\left (12 \, a + 11 \, b\right )} c^{3} d^{3} x^{3} + 54 \, {\left (5 \, a + b\right )} c^{2} d^{3} x^{2} + 12 \, {\left (18 \, a + b\right )} c d^{3} x + 60 \, a d^{3} + 3 \, {\left (20 \, b c^{3} d^{3} x^{3} + 45 \, b c^{2} d^{3} x^{2} + 36 \, b c d^{3} x + 10 \, b d^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{360 \, x^{6}} \] Input:
integrate((c*d*x+d)^3*(a+b*arctanh(c*x))/x^7,x, algorithm="fricas")
Output:
-1/360*(3*b*c^6*d^3*x^6*log(c*x + 1) + 333*b*c^6*d^3*x^6*log(c*x - 1) - 33 6*b*c^6*d^3*x^6*log(x) + 330*b*c^5*d^3*x^5 + 168*b*c^4*d^3*x^4 + 10*(12*a + 11*b)*c^3*d^3*x^3 + 54*(5*a + b)*c^2*d^3*x^2 + 12*(18*a + b)*c*d^3*x + 6 0*a*d^3 + 3*(20*b*c^3*d^3*x^3 + 45*b*c^2*d^3*x^2 + 36*b*c*d^3*x + 10*b*d^3 )*log(-(c*x + 1)/(c*x - 1)))/x^6
Time = 0.99 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.31 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^7} \, dx=\begin {cases} - \frac {a c^{3} d^{3}}{3 x^{3}} - \frac {3 a c^{2} d^{3}}{4 x^{4}} - \frac {3 a c d^{3}}{5 x^{5}} - \frac {a d^{3}}{6 x^{6}} + \frac {14 b c^{6} d^{3} \log {\left (x \right )}}{15} - \frac {14 b c^{6} d^{3} \log {\left (x - \frac {1}{c} \right )}}{15} - \frac {b c^{6} d^{3} \operatorname {atanh}{\left (c x \right )}}{60} - \frac {11 b c^{5} d^{3}}{12 x} - \frac {7 b c^{4} d^{3}}{15 x^{2}} - \frac {b c^{3} d^{3} \operatorname {atanh}{\left (c x \right )}}{3 x^{3}} - \frac {11 b c^{3} d^{3}}{36 x^{3}} - \frac {3 b c^{2} d^{3} \operatorname {atanh}{\left (c x \right )}}{4 x^{4}} - \frac {3 b c^{2} d^{3}}{20 x^{4}} - \frac {3 b c d^{3} \operatorname {atanh}{\left (c x \right )}}{5 x^{5}} - \frac {b c d^{3}}{30 x^{5}} - \frac {b d^{3} \operatorname {atanh}{\left (c x \right )}}{6 x^{6}} & \text {for}\: c \neq 0 \\- \frac {a d^{3}}{6 x^{6}} & \text {otherwise} \end {cases} \] Input:
integrate((c*d*x+d)**3*(a+b*atanh(c*x))/x**7,x)
Output:
Piecewise((-a*c**3*d**3/(3*x**3) - 3*a*c**2*d**3/(4*x**4) - 3*a*c*d**3/(5* x**5) - a*d**3/(6*x**6) + 14*b*c**6*d**3*log(x)/15 - 14*b*c**6*d**3*log(x - 1/c)/15 - b*c**6*d**3*atanh(c*x)/60 - 11*b*c**5*d**3/(12*x) - 7*b*c**4*d **3/(15*x**2) - b*c**3*d**3*atanh(c*x)/(3*x**3) - 11*b*c**3*d**3/(36*x**3) - 3*b*c**2*d**3*atanh(c*x)/(4*x**4) - 3*b*c**2*d**3/(20*x**4) - 3*b*c*d** 3*atanh(c*x)/(5*x**5) - b*c*d**3/(30*x**5) - b*d**3*atanh(c*x)/(6*x**6), N e(c, 0)), (-a*d**3/(6*x**6), True))
Time = 0.04 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.39 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^7} \, dx=-\frac {1}{6} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{3}}\right )} b c^{3} d^{3} + \frac {1}{8} \, {\left ({\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac {2 \, {\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c - \frac {6 \, \operatorname {artanh}\left (c x\right )}{x^{4}}\right )} b c^{2} d^{3} - \frac {3}{20} \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} - 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) + \frac {2 \, c^{2} x^{2} + 1}{x^{4}}\right )} c + \frac {4 \, \operatorname {artanh}\left (c x\right )}{x^{5}}\right )} b c d^{3} + \frac {1}{180} \, {\left ({\left (15 \, c^{5} \log \left (c x + 1\right ) - 15 \, c^{5} \log \left (c x - 1\right ) - \frac {2 \, {\left (15 \, c^{4} x^{4} + 5 \, c^{2} x^{2} + 3\right )}}{x^{5}}\right )} c - \frac {30 \, \operatorname {artanh}\left (c x\right )}{x^{6}}\right )} b d^{3} - \frac {a c^{3} d^{3}}{3 \, x^{3}} - \frac {3 \, a c^{2} d^{3}}{4 \, x^{4}} - \frac {3 \, a c d^{3}}{5 \, x^{5}} - \frac {a d^{3}}{6 \, x^{6}} \] Input:
integrate((c*d*x+d)^3*(a+b*arctanh(c*x))/x^7,x, algorithm="maxima")
Output:
-1/6*((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^3*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^2*d^3 - 3/20*((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*c*d^3 + 1/180*((15*c^5*log(c*x + 1) - 15*c^5*log(c*x - 1) - 2*(15*c^4*x^4 + 5*c ^2*x^2 + 3)/x^5)*c - 30*arctanh(c*x)/x^6)*b*d^3 - 1/3*a*c^3*d^3/x^3 - 3/4* a*c^2*d^3/x^4 - 3/5*a*c*d^3/x^5 - 1/6*a*d^3/x^6
Leaf count of result is larger than twice the leaf count of optimal. 634 vs. \(2 (172) = 344\).
Time = 0.13 (sec) , antiderivative size = 634, normalized size of antiderivative = 3.23 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^7} \, dx =\text {Too large to display} \] Input:
integrate((c*d*x+d)^3*(a+b*arctanh(c*x))/x^7,x, algorithm="giac")
Output:
1/45*(42*b*c^5*d^3*log(-(c*x + 1)/(c*x - 1) - 1) - 42*b*c^5*d^3*log(-(c*x + 1)/(c*x - 1)) + 6*(60*(c*x + 1)^5*b*c^5*d^3/(c*x - 1)^5 + 90*(c*x + 1)^4 *b*c^5*d^3/(c*x - 1)^4 + 140*(c*x + 1)^3*b*c^5*d^3/(c*x - 1)^3 + 105*(c*x + 1)^2*b*c^5*d^3/(c*x - 1)^2 + 42*(c*x + 1)*b*c^5*d^3/(c*x - 1) + 7*b*c^5* d^3)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^6/(c*x - 1)^6 + 6*(c*x + 1)^5/(c *x - 1)^5 + 15*(c*x + 1)^4/(c*x - 1)^4 + 20*(c*x + 1)^3/(c*x - 1)^3 + 15*( c*x + 1)^2/(c*x - 1)^2 + 6*(c*x + 1)/(c*x - 1) + 1) + (720*(c*x + 1)^5*a*c ^5*d^3/(c*x - 1)^5 + 1080*(c*x + 1)^4*a*c^5*d^3/(c*x - 1)^4 + 1680*(c*x + 1)^3*a*c^5*d^3/(c*x - 1)^3 + 1260*(c*x + 1)^2*a*c^5*d^3/(c*x - 1)^2 + 504* (c*x + 1)*a*c^5*d^3/(c*x - 1) + 84*a*c^5*d^3 + 318*(c*x + 1)^5*b*c^5*d^3/( c*x - 1)^5 + 1119*(c*x + 1)^4*b*c^5*d^3/(c*x - 1)^4 + 1742*(c*x + 1)^3*b*c ^5*d^3/(c*x - 1)^3 + 1464*(c*x + 1)^2*b*c^5*d^3/(c*x - 1)^2 + 636*(c*x + 1 )*b*c^5*d^3/(c*x - 1) + 113*b*c^5*d^3)/((c*x + 1)^6/(c*x - 1)^6 + 6*(c*x + 1)^5/(c*x - 1)^5 + 15*(c*x + 1)^4/(c*x - 1)^4 + 20*(c*x + 1)^3/(c*x - 1)^ 3 + 15*(c*x + 1)^2/(c*x - 1)^2 + 6*(c*x + 1)/(c*x - 1) + 1))*c
Time = 3.66 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.12 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^7} \, dx=\frac {14\,b\,c^6\,d^3\,\ln \left (x\right )}{15}-\frac {7\,b\,c^6\,d^3\,\ln \left (c^2\,x^2-1\right )}{15}-\frac {3\,a\,c^2\,d^3}{4\,x^4}-\frac {a\,c^3\,d^3}{3\,x^3}-\frac {3\,b\,c^2\,d^3}{20\,x^4}-\frac {11\,b\,c^3\,d^3}{36\,x^3}-\frac {7\,b\,c^4\,d^3}{15\,x^2}-\frac {11\,b\,c^5\,d^3}{12\,x}-\frac {a\,d^3}{6\,x^6}-\frac {3\,a\,c\,d^3}{5\,x^5}-\frac {b\,c\,d^3}{30\,x^5}-\frac {b\,d^3\,\mathrm {atanh}\left (c\,x\right )}{6\,x^6}-\frac {11\,b\,c^7\,d^3\,\mathrm {atan}\left (\frac {c^2\,x}{\sqrt {-c^2}}\right )}{12\,\sqrt {-c^2}}-\frac {3\,b\,c\,d^3\,\mathrm {atanh}\left (c\,x\right )}{5\,x^5}-\frac {3\,b\,c^2\,d^3\,\mathrm {atanh}\left (c\,x\right )}{4\,x^4}-\frac {b\,c^3\,d^3\,\mathrm {atanh}\left (c\,x\right )}{3\,x^3} \] Input:
int(((a + b*atanh(c*x))*(d + c*d*x)^3)/x^7,x)
Output:
(14*b*c^6*d^3*log(x))/15 - (7*b*c^6*d^3*log(c^2*x^2 - 1))/15 - (3*a*c^2*d^ 3)/(4*x^4) - (a*c^3*d^3)/(3*x^3) - (3*b*c^2*d^3)/(20*x^4) - (11*b*c^3*d^3) /(36*x^3) - (7*b*c^4*d^3)/(15*x^2) - (11*b*c^5*d^3)/(12*x) - (a*d^3)/(6*x^ 6) - (3*a*c*d^3)/(5*x^5) - (b*c*d^3)/(30*x^5) - (b*d^3*atanh(c*x))/(6*x^6) - (11*b*c^7*d^3*atan((c^2*x)/(-c^2)^(1/2)))/(12*(-c^2)^(1/2)) - (3*b*c*d^ 3*atanh(c*x))/(5*x^5) - (3*b*c^2*d^3*atanh(c*x))/(4*x^4) - (b*c^3*d^3*atan h(c*x))/(3*x^3)
Time = 0.17 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.82 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^7} \, dx=\frac {d^{3} \left (-3 \mathit {atanh} \left (c x \right ) b \,c^{6} x^{6}-60 \mathit {atanh} \left (c x \right ) b \,c^{3} x^{3}-135 \mathit {atanh} \left (c x \right ) b \,c^{2} x^{2}-108 \mathit {atanh} \left (c x \right ) b c x -30 \mathit {atanh} \left (c x \right ) b -168 \,\mathrm {log}\left (c^{2} x -c \right ) b \,c^{6} x^{6}+168 \,\mathrm {log}\left (x \right ) b \,c^{6} x^{6}-60 a \,c^{3} x^{3}-135 a \,c^{2} x^{2}-108 a c x -30 a -165 b \,c^{5} x^{5}-84 b \,c^{4} x^{4}-55 b \,c^{3} x^{3}-27 b \,c^{2} x^{2}-6 b c x \right )}{180 x^{6}} \] Input:
int((c*d*x+d)^3*(a+b*atanh(c*x))/x^7,x)
Output:
(d**3*( - 3*atanh(c*x)*b*c**6*x**6 - 60*atanh(c*x)*b*c**3*x**3 - 135*atanh (c*x)*b*c**2*x**2 - 108*atanh(c*x)*b*c*x - 30*atanh(c*x)*b - 168*log(c**2* x - c)*b*c**6*x**6 + 168*log(x)*b*c**6*x**6 - 60*a*c**3*x**3 - 135*a*c**2* x**2 - 108*a*c*x - 30*a - 165*b*c**5*x**5 - 84*b*c**4*x**4 - 55*b*c**3*x** 3 - 27*b*c**2*x**2 - 6*b*c*x))/(180*x**6)