Integrand size = 22, antiderivative size = 479 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^7} \, dx=-\frac {b^2 c^2 d^3}{60 x^4}-\frac {b^2 c^3 d^3}{10 x^3}-\frac {61 b^2 c^4 d^3}{180 x^2}-\frac {37 b^2 c^5 d^3}{30 x}+\frac {37}{30} b^2 c^6 d^3 \text {arctanh}(c x)-\frac {b c d^3 (a+b \text {arctanh}(c x))}{15 x^5}-\frac {3 b c^2 d^3 (a+b \text {arctanh}(c x))}{10 x^4}-\frac {11 b c^3 d^3 (a+b \text {arctanh}(c x))}{18 x^3}-\frac {14 b c^4 d^3 (a+b \text {arctanh}(c x))}{15 x^2}-\frac {11 b c^5 d^3 (a+b \text {arctanh}(c x))}{6 x}-\frac {d^3 (a+b \text {arctanh}(c x))^2}{6 x^6}-\frac {3 c d^3 (a+b \text {arctanh}(c x))^2}{5 x^5}-\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))^2}{4 x^4}-\frac {c^3 d^3 (a+b \text {arctanh}(c x))^2}{3 x^3}+\frac {28}{15} a b c^6 d^3 \log (x)+\frac {113}{45} b^2 c^6 d^3 \log (x)+\frac {37}{20} b c^6 d^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )+\frac {1}{60} b c^6 d^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )-\frac {113}{90} b^2 c^6 d^3 \log \left (1-c^2 x^2\right )-\frac {14}{15} b^2 c^6 d^3 \operatorname {PolyLog}(2,-c x)+\frac {14}{15} b^2 c^6 d^3 \operatorname {PolyLog}(2,c x)+\frac {37}{40} b^2 c^6 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-\frac {1}{120} b^2 c^6 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right ) \] Output:
-1/60*b^2*c^2*d^3/x^4-1/10*b^2*c^3*d^3/x^3-61/180*b^2*c^4*d^3/x^2-37/30*b^ 2*c^5*d^3/x+37/30*b^2*c^6*d^3*arctanh(c*x)-1/15*b*c*d^3*(a+b*arctanh(c*x)) /x^5-3/10*b*c^2*d^3*(a+b*arctanh(c*x))/x^4-11/18*b*c^3*d^3*(a+b*arctanh(c* x))/x^3-14/15*b*c^4*d^3*(a+b*arctanh(c*x))/x^2-11/6*b*c^5*d^3*(a+b*arctanh (c*x))/x-1/6*d^3*(a+b*arctanh(c*x))^2/x^6-3/5*c*d^3*(a+b*arctanh(c*x))^2/x ^5-3/4*c^2*d^3*(a+b*arctanh(c*x))^2/x^4-1/3*c^3*d^3*(a+b*arctanh(c*x))^2/x ^3+28/15*a*b*c^6*d^3*ln(x)+113/45*b^2*c^6*d^3*ln(x)+37/20*b*c^6*d^3*(a+b*a rctanh(c*x))*ln(2/(-c*x+1))+1/60*b*c^6*d^3*(a+b*arctanh(c*x))*ln(2/(c*x+1) )-113/90*b^2*c^6*d^3*ln(-c^2*x^2+1)-14/15*b^2*c^6*d^3*polylog(2,-c*x)+14/1 5*b^2*c^6*d^3*polylog(2,c*x)+37/40*b^2*c^6*d^3*polylog(2,1-2/(-c*x+1))-1/1 20*b^2*c^6*d^3*polylog(2,1-2/(c*x+1))
Time = 0.91 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.84 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^7} \, dx=-\frac {d^3 \left (30 a^2+108 a^2 c x+12 a b c x+135 a^2 c^2 x^2+54 a b c^2 x^2+3 b^2 c^2 x^2+60 a^2 c^3 x^3+110 a b c^3 x^3+18 b^2 c^3 x^3+168 a b c^4 x^4+61 b^2 c^4 x^4+330 a b c^5 x^5+222 b^2 c^5 x^5-64 b^2 c^6 x^6+3 b^2 \left (10+36 c x+45 c^2 x^2+20 c^3 x^3-111 c^6 x^6\right ) \text {arctanh}(c x)^2+2 b \text {arctanh}(c x) \left (3 a \left (10+36 c x+45 c^2 x^2+20 c^3 x^3\right )+b c x \left (6+27 c x+55 c^2 x^2+84 c^3 x^3+165 c^4 x^4-111 c^5 x^5\right )-168 b c^6 x^6 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )-336 a b c^6 x^6 \log (c x)+165 a b c^6 x^6 \log (1-c x)-165 a b c^6 x^6 \log (1+c x)-452 b^2 c^6 x^6 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+168 a b c^6 x^6 \log \left (1-c^2 x^2\right )+168 b^2 c^6 x^6 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )}{180 x^6} \] Input:
Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^7,x]
Output:
-1/180*(d^3*(30*a^2 + 108*a^2*c*x + 12*a*b*c*x + 135*a^2*c^2*x^2 + 54*a*b* c^2*x^2 + 3*b^2*c^2*x^2 + 60*a^2*c^3*x^3 + 110*a*b*c^3*x^3 + 18*b^2*c^3*x^ 3 + 168*a*b*c^4*x^4 + 61*b^2*c^4*x^4 + 330*a*b*c^5*x^5 + 222*b^2*c^5*x^5 - 64*b^2*c^6*x^6 + 3*b^2*(10 + 36*c*x + 45*c^2*x^2 + 20*c^3*x^3 - 111*c^6*x ^6)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(3*a*(10 + 36*c*x + 45*c^2*x^2 + 20* c^3*x^3) + b*c*x*(6 + 27*c*x + 55*c^2*x^2 + 84*c^3*x^3 + 165*c^4*x^4 - 111 *c^5*x^5) - 168*b*c^6*x^6*Log[1 - E^(-2*ArcTanh[c*x])]) - 336*a*b*c^6*x^6* Log[c*x] + 165*a*b*c^6*x^6*Log[1 - c*x] - 165*a*b*c^6*x^6*Log[1 + c*x] - 4 52*b^2*c^6*x^6*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 168*a*b*c^6*x^6*Log[1 - c^2* x^2] + 168*b^2*c^6*x^6*PolyLog[2, E^(-2*ArcTanh[c*x])]))/x^6
Time = 0.97 (sec) , antiderivative size = 449, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6500, 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.
| \(\displaystyle \int \frac {(c d x+d)^3 (a+b \text {arctanh}(c x))^2}{x^7} \, dx\) |
| \(\Big \downarrow \) 6500 |
| \(\displaystyle -2 b c \int \left (-\frac {37 d^3 (a+b \text {arctanh}(c x)) c^6}{40 (1-c x)}+\frac {d^3 (a+b \text {arctanh}(c x)) c^6}{120 (c x+1)}-\frac {14 d^3 (a+b \text {arctanh}(c x)) c^5}{15 x}-\frac {11 d^3 (a+b \text {arctanh}(c x)) c^4}{12 x^2}-\frac {14 d^3 (a+b \text {arctanh}(c x)) c^3}{15 x^3}-\frac {11 d^3 (a+b \text {arctanh}(c x)) c^2}{12 x^4}-\frac {3 d^3 (a+b \text {arctanh}(c x)) c}{5 x^5}-\frac {d^3 (a+b \text {arctanh}(c x))}{6 x^6}\right )dx-\frac {c^3 d^3 (a+b \text {arctanh}(c x))^2}{3 x^3}-\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))^2}{4 x^4}-\frac {d^3 (a+b \text {arctanh}(c x))^2}{6 x^6}-\frac {3 c d^3 (a+b \text {arctanh}(c x))^2}{5 x^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {c^3 d^3 (a+b \text {arctanh}(c x))^2}{3 x^3}-\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))^2}{4 x^4}-2 b c \left (-\frac {37}{40} c^5 d^3 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))-\frac {1}{120} c^5 d^3 \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))+\frac {11 c^4 d^3 (a+b \text {arctanh}(c x))}{12 x}+\frac {7 c^3 d^3 (a+b \text {arctanh}(c x))}{15 x^2}+\frac {11 c^2 d^3 (a+b \text {arctanh}(c x))}{36 x^3}+\frac {d^3 (a+b \text {arctanh}(c x))}{30 x^5}+\frac {3 c d^3 (a+b \text {arctanh}(c x))}{20 x^4}-\frac {14}{15} a c^5 d^3 \log (x)-\frac {37}{60} b c^5 d^3 \text {arctanh}(c x)+\frac {7}{15} b c^5 d^3 \operatorname {PolyLog}(2,-c x)-\frac {7}{15} b c^5 d^3 \operatorname {PolyLog}(2,c x)-\frac {37}{80} b c^5 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+\frac {1}{240} b c^5 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )-\frac {113}{90} b c^5 d^3 \log (x)+\frac {37 b c^4 d^3}{60 x}+\frac {61 b c^3 d^3}{360 x^2}+\frac {b c^2 d^3}{20 x^3}+\frac {113}{180} b c^5 d^3 \log \left (1-c^2 x^2\right )+\frac {b c d^3}{120 x^4}\right )-\frac {d^3 (a+b \text {arctanh}(c x))^2}{6 x^6}-\frac {3 c d^3 (a+b \text {arctanh}(c x))^2}{5 x^5}\) |
Input:
Int[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^7,x]
Output:
-1/6*(d^3*(a + b*ArcTanh[c*x])^2)/x^6 - (3*c*d^3*(a + b*ArcTanh[c*x])^2)/( 5*x^5) - (3*c^2*d^3*(a + b*ArcTanh[c*x])^2)/(4*x^4) - (c^3*d^3*(a + b*ArcT anh[c*x])^2)/(3*x^3) - 2*b*c*((b*c*d^3)/(120*x^4) + (b*c^2*d^3)/(20*x^3) + (61*b*c^3*d^3)/(360*x^2) + (37*b*c^4*d^3)/(60*x) - (37*b*c^5*d^3*ArcTanh[ c*x])/60 + (d^3*(a + b*ArcTanh[c*x]))/(30*x^5) + (3*c*d^3*(a + b*ArcTanh[c *x]))/(20*x^4) + (11*c^2*d^3*(a + b*ArcTanh[c*x]))/(36*x^3) + (7*c^3*d^3*( a + b*ArcTanh[c*x]))/(15*x^2) + (11*c^4*d^3*(a + b*ArcTanh[c*x]))/(12*x) - (14*a*c^5*d^3*Log[x])/15 - (113*b*c^5*d^3*Log[x])/90 - (37*c^5*d^3*(a + b *ArcTanh[c*x])*Log[2/(1 - c*x)])/40 - (c^5*d^3*(a + b*ArcTanh[c*x])*Log[2/ (1 + c*x)])/120 + (113*b*c^5*d^3*Log[1 - c^2*x^2])/180 + (7*b*c^5*d^3*Poly Log[2, -(c*x)])/15 - (7*b*c^5*d^3*PolyLog[2, c*x])/15 - (37*b*c^5*d^3*Poly Log[2, 1 - 2/(1 - c*x)])/80 + (b*c^5*d^3*PolyLog[2, 1 - 2/(1 + c*x)])/240)
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_.)*((d_.) + (e _.)*(x_))^(q_), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Si mp[(a + b*ArcTanh[c*x])^p u, x] - Simp[b*c*p Int[ExpandIntegrand[(a + b *ArcTanh[c*x])^(p - 1), u/(1 - c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 - e^2, 0] && IntegersQ[m, q] && N eQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]
Time = 1.16 (sec) , antiderivative size = 468, normalized size of antiderivative = 0.98
| method | result | size |
| parts | \(d^{3} a^{2} \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^{2} c^{6} \left (-\frac {11 \,\operatorname {arctanh}\left (c x \right )}{6 c x}-\frac {14 \,\operatorname {arctanh}\left (c x \right )}{15 c^{2} x^{2}}-\frac {11 \,\operatorname {arctanh}\left (c x \right )}{18 c^{3} x^{3}}-\frac {337 \ln \left (c x -1\right )}{180}-\frac {23 \ln \left (c x +1\right )}{36}-\frac {1}{60 c^{4} x^{4}}-\frac {37}{30 c x}+\frac {28 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )}{15}-\frac {14 \ln \left (c x \right ) \ln \left (c x +1\right )}{15}-\frac {1}{10 c^{3} x^{3}}-\frac {61}{180 c^{2} x^{2}}-\frac {3 \operatorname {arctanh}\left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{6 c^{6} x^{6}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {3 \operatorname {arctanh}\left (c x \right )^{2}}{5 c^{5} x^{5}}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{60}-\frac {37 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{20}-\frac {\operatorname {arctanh}\left (c x \right )}{15 c^{5} x^{5}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{10 c^{4} x^{4}}+\frac {14 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{15}-\frac {37 \ln \left (c x -1\right )^{2}}{80}+\frac {\ln \left (c x +1\right )^{2}}{240}+\frac {113 \ln \left (c x \right )}{45}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{120}+\frac {37 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{40}-\frac {14 \operatorname {dilog}\left (c x \right )}{15}-\frac {14 \operatorname {dilog}\left (c x +1\right )}{15}\right )+2 d^{3} b a \,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 )\) | \(468\) |
| derivativedivides | \(c^{6} \left (d^{3} a^{2} \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^{2} \left (-\frac {11 \,\operatorname {arctanh}\left (c x \right )}{6 c x}-\frac {14 \,\operatorname {arctanh}\left (c x \right )}{15 c^{2} x^{2}}-\frac {11 \,\operatorname {arctanh}\left (c x \right )}{18 c^{3} x^{3}}-\frac {337 \ln \left (c x -1\right )}{180}-\frac {23 \ln \left (c x +1\right )}{36}-\frac {1}{60 c^{4} x^{4}}-\frac {37}{30 c x}+\frac {28 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )}{15}-\frac {14 \ln \left (c x \right ) \ln \left (c x +1\right )}{15}-\frac {1}{10 c^{3} x^{3}}-\frac {61}{180 c^{2} x^{2}}-\frac {3 \operatorname {arctanh}\left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{6 c^{6} x^{6}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {3 \operatorname {arctanh}\left (c x \right )^{2}}{5 c^{5} x^{5}}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{60}-\frac {37 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{20}-\frac {\operatorname {arctanh}\left (c x \right )}{15 c^{5} x^{5}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{10 c^{4} x^{4}}+\frac {14 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{15}-\frac {37 \ln \left (c x -1\right )^{2}}{80}+\frac {\ln \left (c x +1\right )^{2}}{240}+\frac {113 \ln \left (c x \right )}{45}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{120}+\frac {37 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{40}-\frac {14 \operatorname {dilog}\left (c x \right )}{15}-\frac {14 \operatorname {dilog}\left (c x +1\right )}{15}\right )+2 d^{3} b a \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 )\) | \(471\) |
| default | \(c^{6} \left (d^{3} a^{2} \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^{2} \left (-\frac {11 \,\operatorname {arctanh}\left (c x \right )}{6 c x}-\frac {14 \,\operatorname {arctanh}\left (c x \right )}{15 c^{2} x^{2}}-\frac {11 \,\operatorname {arctanh}\left (c x \right )}{18 c^{3} x^{3}}-\frac {337 \ln \left (c x -1\right )}{180}-\frac {23 \ln \left (c x +1\right )}{36}-\frac {1}{60 c^{4} x^{4}}-\frac {37}{30 c x}+\frac {28 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )}{15}-\frac {14 \ln \left (c x \right ) \ln \left (c x +1\right )}{15}-\frac {1}{10 c^{3} x^{3}}-\frac {61}{180 c^{2} x^{2}}-\frac {3 \operatorname {arctanh}\left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{6 c^{6} x^{6}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {3 \operatorname {arctanh}\left (c x \right )^{2}}{5 c^{5} x^{5}}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{60}-\frac {37 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{20}-\frac {\operatorname {arctanh}\left (c x \right )}{15 c^{5} x^{5}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{10 c^{4} x^{4}}+\frac {14 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{15}-\frac {37 \ln \left (c x -1\right )^{2}}{80}+\frac {\ln \left (c x +1\right )^{2}}{240}+\frac {113 \ln \left (c x \right )}{45}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{120}+\frac {37 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{40}-\frac {14 \operatorname {dilog}\left (c x \right )}{15}-\frac {14 \operatorname {dilog}\left (c x +1\right )}{15}\right )+2 d^{3} b a \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 )\) | \(471\) |
Input:
int((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^7,x,method=_RETURNVERBOSE)
Output:
d^3*a^2*(-1/6/x^6-3/4*c^2/x^4-3/5*c/x^5-1/3*c^3/x^3)+d^3*b^2*c^6*(-11/6*ar ctanh(c*x)/c/x-14/15*arctanh(c*x)/c^2/x^2-11/18*arctanh(c*x)/c^3/x^3-337/1 80*ln(c*x-1)-23/36*ln(c*x+1)-1/60/c^4/x^4-37/30/c/x+28/15*arctanh(c*x)*ln( c*x)-14/15*ln(c*x)*ln(c*x+1)-1/10/c^3/x^3-61/180/c^2/x^2-3/4*arctanh(c*x)^ 2/c^4/x^4-1/6*arctanh(c*x)^2/c^6/x^6-1/3*arctanh(c*x)^2/c^3/x^3-3/5*arctan h(c*x)^2/c^5/x^5-1/60*arctanh(c*x)*ln(c*x+1)-37/20*arctanh(c*x)*ln(c*x-1)- 1/15*arctanh(c*x)/c^5/x^5-3/10*arctanh(c*x)/c^4/x^4+14/15*dilog(1/2*c*x+1/ 2)-37/80*ln(c*x-1)^2+1/240*ln(c*x+1)^2+113/45*ln(c*x)-1/120*(ln(c*x+1)-ln( 1/2*c*x+1/2))*ln(-1/2*c*x+1/2)+37/40*ln(c*x-1)*ln(1/2*c*x+1/2)-14/15*dilog (c*x)-14/15*dilog(c*x+1))+2*d^3*b*a*c^6*(-1/3*arctanh(c*x)/c^3/x^3-3/4*arc tanh(c*x)/c^4/x^4-3/5*arctanh(c*x)/c^5/x^5-1/6*arctanh(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^3-7/15/c^2/x^2-11/12/c/x+1 4/15*ln(c*x)-1/120*ln(c*x+1))
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^7} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{7}} \,d x } \] Input:
integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^7,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)*a rctanh(c*x))/x^7, x)
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^7} \, dx=d^{3} \left (\int \frac {a^{2}}{x^{7}}\, dx + \int \frac {3 a^{2} c}{x^{6}}\, dx + \int \frac {3 a^{2} c^{2}}{x^{5}}\, dx + \int \frac {a^{2} c^{3}}{x^{4}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{7}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x^{7}}\, dx + \int \frac {3 b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}}{x^{6}}\, dx + \int \frac {3 b^{2} c^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{5}}\, dx + \int \frac {b^{2} c^{3} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {6 a b c \operatorname {atanh}{\left (c x \right )}}{x^{6}}\, dx + \int \frac {6 a b c^{2} \operatorname {atanh}{\left (c x \right )}}{x^{5}}\, dx + \int \frac {2 a b c^{3} \operatorname {atanh}{\left (c x \right )}}{x^{4}}\, dx\right ) \] Input:
integrate((c*d*x+d)**3*(a+b*atanh(c*x))**2/x**7,x)
Output:
d**3*(Integral(a**2/x**7, x) + Integral(3*a**2*c/x**6, x) + Integral(3*a** 2*c**2/x**5, x) + Integral(a**2*c**3/x**4, x) + Integral(b**2*atanh(c*x)** 2/x**7, x) + Integral(2*a*b*atanh(c*x)/x**7, x) + Integral(3*b**2*c*atanh( c*x)**2/x**6, x) + Integral(3*b**2*c**2*atanh(c*x)**2/x**5, x) + Integral( b**2*c**3*atanh(c*x)**2/x**4, x) + Integral(6*a*b*c*atanh(c*x)/x**6, x) + Integral(6*a*b*c**2*atanh(c*x)/x**5, x) + Integral(2*a*b*c**3*atanh(c*x)/x **4, x))
Leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (427) = 854\).
Time = 0.51 (sec) , antiderivative size = 961, normalized size of antiderivative = 2.01 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^7} \, dx=\text {Too large to display} \] Input:
integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^7,x, algorithm="maxima")
Output:
-14/15*(log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2))*b^2*c^6*d ^3 - 14/15*(log(c*x)*log(-c*x + 1) + dilog(-c*x + 1))*b^2*c^6*d^3 + 14/15* (log(c*x + 1)*log(-c*x) + dilog(c*x + 1))*b^2*c^6*d^3 - 23/60*b^2*c^6*d^3* log(c*x + 1) - 97/60*b^2*c^6*d^3*log(c*x - 1) + 2*b^2*c^6*d^3*log(x) - 1/3 *((c^2*log(c^2*x^2 - 1) - c^2*log(x^2) + 1/x^2)*c + 2*arctanh(c*x)/x^3)*a* b*c^3*d^3 + 1/4*((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)*a*b*c^2*d^3 - 3/10*((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)*a*b*c*d ^3 + 1/90*((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)*a*b*d^3 + 1/360*((184*c^4*log(x ) - (15*c^4*x^4*log(c*x + 1)^2 + 15*c^4*x^4*log(c*x - 1)^2 + 92*c^4*x^4*lo g(c*x - 1) + 32*c^2*x^2 - 2*(15*c^4*x^4*log(c*x - 1) - 46*c^4*x^4)*log(c*x + 1) + 6)/x^4)*c^2 + 4*(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*arctanh(c*x))*b^2*d^3 - 1/3*a^2*c^3*d^3/x ^3 - 3/4*a^2*c^2*d^3/x^4 - 3/5*a^2*c*d^3/x^5 - 1/6*b^2*d^3*arctanh(c*x)^2/ x^6 - 1/6*a^2*d^3/x^6 - 1/240*(296*b^2*c^5*d^3*x^4 + 60*b^2*c^4*d^3*x^3 + 24*b^2*c^3*d^3*x^2 + (11*b^2*c^6*d^3*x^5 + 20*b^2*c^3*d^3*x^2 + 45*b^2*c^2 *d^3*x + 36*b^2*c*d^3)*log(c*x + 1)^2 - (101*b^2*c^6*d^3*x^5 - 20*b^2*c^3* d^3*x^2 - 45*b^2*c^2*d^3*x - 36*b^2*c*d^3)*log(-c*x + 1)^2 + 4*(45*b^2*c^5 *d^3*x^4 + 28*b^2*c^4*d^3*x^3 + 15*b^2*c^3*d^3*x^2 + 9*b^2*c^2*d^3*x)*l...
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^7} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{7}} \,d x } \] Input:
integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^7,x, algorithm="giac")
Output:
integrate((c*d*x + d)^3*(b*arctanh(c*x) + a)^2/x^7, x)
Timed out. \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^7} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^3}{x^7} \,d x \] Input:
int(((a + b*atanh(c*x))^2*(d + c*d*x)^3)/x^7,x)
Output:
int(((a + b*atanh(c*x))^2*(d + c*d*x)^3)/x^7, x)
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^7} \, dx=\frac {d^{3} \left (-230 \mathit {atanh} \left (c x \right ) b^{2} c^{6} x^{6}-452 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2} c^{6} x^{6}+452 \,\mathrm {log}\left (x \right ) b^{2} c^{6} x^{6}-60 \mathit {atanh} \left (c x \right ) a b -3 b^{2} c^{2} x^{2}-222 b^{2} c^{5} x^{5}+165 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{6} x^{6}-330 \mathit {atanh} \left (c x \right ) b^{2} c^{5} x^{5}-330 a b \,c^{5} x^{5}-110 \mathit {atanh} \left (c x \right ) b^{2} c^{3} x^{3}-12 \mathit {atanh} \left (c x \right ) b^{2} c x -110 a b \,c^{3} x^{3}-12 a b c x -18 b^{2} c^{3} x^{3}-108 \mathit {atanh} \left (c x \right )^{2} b^{2} c x -168 \mathit {atanh} \left (c x \right ) b^{2} c^{4} x^{4}-54 \mathit {atanh} \left (c x \right ) b^{2} c^{2} x^{2}-168 a b \,c^{4} x^{4}-54 a b \,c^{2} x^{2}-30 a^{2}-60 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{3} x^{3}-336 \,\mathrm {log}\left (c^{2} x -c \right ) a b \,c^{6} x^{6}+336 \,\mathrm {log}\left (x \right ) a b \,c^{6} x^{6}-61 b^{2} c^{4} x^{4}-336 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{2} x^{3}-x}d x \right ) b^{2} c^{6} x^{6}-60 a^{2} c^{3} x^{3}-216 \mathit {atanh} \left (c x \right ) a b c x -135 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{2} x^{2}-120 \mathit {atanh} \left (c x \right ) a b \,c^{3} x^{3}-108 a^{2} c x -30 \mathit {atanh} \left (c x \right )^{2} b^{2}-270 \mathit {atanh} \left (c x \right ) a b \,c^{2} x^{2}-135 a^{2} c^{2} x^{2}-6 \mathit {atanh} \left (c x \right ) a b \,c^{6} x^{6}\right )}{180 x^{6}} \] Input:
int((c*d*x+d)^3*(a+b*atanh(c*x))^2/x^7,x)
Output:
(d**3*(165*atanh(c*x)**2*b**2*c**6*x**6 - 60*atanh(c*x)**2*b**2*c**3*x**3 - 135*atanh(c*x)**2*b**2*c**2*x**2 - 108*atanh(c*x)**2*b**2*c*x - 30*atanh (c*x)**2*b**2 - 6*atanh(c*x)*a*b*c**6*x**6 - 120*atanh(c*x)*a*b*c**3*x**3 - 270*atanh(c*x)*a*b*c**2*x**2 - 216*atanh(c*x)*a*b*c*x - 60*atanh(c*x)*a* b - 230*atanh(c*x)*b**2*c**6*x**6 - 330*atanh(c*x)*b**2*c**5*x**5 - 168*at anh(c*x)*b**2*c**4*x**4 - 110*atanh(c*x)*b**2*c**3*x**3 - 54*atanh(c*x)*b* *2*c**2*x**2 - 12*atanh(c*x)*b**2*c*x - 336*int(atanh(c*x)/(c**2*x**3 - x) ,x)*b**2*c**6*x**6 - 336*log(c**2*x - c)*a*b*c**6*x**6 - 452*log(c**2*x - c)*b**2*c**6*x**6 + 336*log(x)*a*b*c**6*x**6 + 452*log(x)*b**2*c**6*x**6 - 60*a**2*c**3*x**3 - 135*a**2*c**2*x**2 - 108*a**2*c*x - 30*a**2 - 330*a*b *c**5*x**5 - 168*a*b*c**4*x**4 - 110*a*b*c**3*x**3 - 54*a*b*c**2*x**2 - 12 *a*b*c*x - 222*b**2*c**5*x**5 - 61*b**2*c**4*x**4 - 18*b**2*c**3*x**3 - 3* b**2*c**2*x**2))/(180*x**6)