Integrand size = 14, antiderivative size = 245 \[ \int \left (c+d x^2\right )^4 \coth ^{-1}(a x) \, dx=\frac {d \left (420 a^6 c^3+378 a^4 c^2 d+180 a^2 c d^2+35 d^3\right ) x^2}{630 a^7}+\frac {d^2 \left (378 a^4 c^2+180 a^2 c d+35 d^2\right ) x^4}{1260 a^5}+\frac {d^3 \left (36 a^2 c+7 d\right ) x^6}{378 a^3}+\frac {d^4 x^8}{72 a}+c^4 x \coth ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \coth ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \coth ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \coth ^{-1}(a x)+\frac {1}{9} d^4 x^9 \coth ^{-1}(a x)+\frac {\left (315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right ) \log \left (1-a^2 x^2\right )}{630 a^9} \] Output:
1/630*d*(420*a^6*c^3+378*a^4*c^2*d+180*a^2*c*d^2+35*d^3)*x^2/a^7+1/1260*d^ 2*(378*a^4*c^2+180*a^2*c*d+35*d^2)*x^4/a^5+1/378*d^3*(36*a^2*c+7*d)*x^6/a^ 3+1/72*d^4*x^8/a+c^4*x*arccoth(a*x)+4/3*c^3*d*x^3*arccoth(a*x)+6/5*c^2*d^2 *x^5*arccoth(a*x)+4/7*c*d^3*x^7*arccoth(a*x)+1/9*d^4*x^9*arccoth(a*x)+1/63 0*(315*a^8*c^4+420*a^6*c^3*d+378*a^4*c^2*d^2+180*a^2*c*d^3+35*d^4)*ln(-a^2 *x^2+1)/a^9
Time = 0.06 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.87 \[ \int \left (c+d x^2\right )^4 \coth ^{-1}(a x) \, dx=\frac {a^2 d x^2 \left (420 d^3+30 a^2 d^2 \left (72 c+7 d x^2\right )+4 a^4 d \left (1134 c^2+270 c d x^2+35 d^2 x^4\right )+3 a^6 \left (1680 c^3+756 c^2 d x^2+240 c d^2 x^4+35 d^3 x^6\right )\right )+24 a^9 x \left (315 c^4+420 c^3 d x^2+378 c^2 d^2 x^4+180 c d^3 x^6+35 d^4 x^8\right ) \coth ^{-1}(a x)+12 \left (315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right ) \log \left (1-a^2 x^2\right )}{7560 a^9} \] Input:
Integrate[(c + d*x^2)^4*ArcCoth[a*x],x]
Output:
(a^2*d*x^2*(420*d^3 + 30*a^2*d^2*(72*c + 7*d*x^2) + 4*a^4*d*(1134*c^2 + 27 0*c*d*x^2 + 35*d^2*x^4) + 3*a^6*(1680*c^3 + 756*c^2*d*x^2 + 240*c*d^2*x^4 + 35*d^3*x^6)) + 24*a^9*x*(315*c^4 + 420*c^3*d*x^2 + 378*c^2*d^2*x^4 + 180 *c*d^3*x^6 + 35*d^4*x^8)*ArcCoth[a*x] + 12*(315*a^8*c^4 + 420*a^6*c^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*Log[1 - a^2*x^2])/(7560*a^9)
Time = 0.68 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6539, 27, 2331, 2389, 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.
| \(\displaystyle \int \coth ^{-1}(a x) \left (c+d x^2\right )^4 \, dx\) |
| \(\Big \downarrow \) 6539 |
| \(\displaystyle -a \int \frac {x \left (35 d^4 x^8+180 c d^3 x^6+378 c^2 d^2 x^4+420 c^3 d x^2+315 c^4\right )}{315 \left (1-a^2 x^2\right )}dx+c^4 x \coth ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \coth ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \coth ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \coth ^{-1}(a x)+\frac {1}{9} d^4 x^9 \coth ^{-1}(a x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{315} a \int \frac {x \left (35 d^4 x^8+180 c d^3 x^6+378 c^2 d^2 x^4+420 c^3 d x^2+315 c^4\right )}{1-a^2 x^2}dx+c^4 x \coth ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \coth ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \coth ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \coth ^{-1}(a x)+\frac {1}{9} d^4 x^9 \coth ^{-1}(a x)\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle -\frac {1}{630} a \int \frac {35 d^4 x^8+180 c d^3 x^6+378 c^2 d^2 x^4+420 c^3 d x^2+315 c^4}{1-a^2 x^2}dx^2+c^4 x \coth ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \coth ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \coth ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \coth ^{-1}(a x)+\frac {1}{9} d^4 x^9 \coth ^{-1}(a x)\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle -\frac {1}{630} a \int \left (-\frac {35 d^4 x^6}{a^2}-\frac {5 d^3 \left (36 c a^2+7 d\right ) x^4}{a^4}-\frac {d^2 \left (378 c^2 a^4+180 c d a^2+35 d^2\right ) x^2}{a^6}-\frac {d \left (420 c^3 a^6+378 c^2 d a^4+180 c d^2 a^2+35 d^3\right )}{a^8}+\frac {-315 c^4 a^8-420 c^3 d a^6-378 c^2 d^2 a^4-180 c d^3 a^2-35 d^4}{a^8 \left (a^2 x^2-1\right )}\right )dx^2+c^4 x \coth ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \coth ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \coth ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \coth ^{-1}(a x)+\frac {1}{9} d^4 x^9 \coth ^{-1}(a x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{630} a \left (-\frac {35 d^4 x^8}{4 a^2}-\frac {5 d^3 x^6 \left (36 a^2 c+7 d\right )}{3 a^4}-\frac {d^2 x^4 \left (378 a^4 c^2+180 a^2 c d+35 d^2\right )}{2 a^6}-\frac {d x^2 \left (420 a^6 c^3+378 a^4 c^2 d+180 a^2 c d^2+35 d^3\right )}{a^8}-\frac {\left (315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right ) \log \left (1-a^2 x^2\right )}{a^{10}}\right )+c^4 x \coth ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \coth ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \coth ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \coth ^{-1}(a x)+\frac {1}{9} d^4 x^9 \coth ^{-1}(a x)\) |
Input:
Int[(c + d*x^2)^4*ArcCoth[a*x],x]
Output:
c^4*x*ArcCoth[a*x] + (4*c^3*d*x^3*ArcCoth[a*x])/3 + (6*c^2*d^2*x^5*ArcCoth [a*x])/5 + (4*c*d^3*x^7*ArcCoth[a*x])/7 + (d^4*x^9*ArcCoth[a*x])/9 - (a*(- ((d*(420*a^6*c^3 + 378*a^4*c^2*d + 180*a^2*c*d^2 + 35*d^3)*x^2)/a^8) - (d^ 2*(378*a^4*c^2 + 180*a^2*c*d + 35*d^2)*x^4)/(2*a^6) - (5*d^3*(36*a^2*c + 7 *d)*x^6)/(3*a^4) - (35*d^4*x^8)/(4*a^2) - ((315*a^8*c^4 + 420*a^6*c^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*Log[1 - a^2*x^2])/a^10))/630
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Sym bol] :> With[{u = IntHide[(d + e*x^2)^q, x]}, Simp[(a + b*ArcCoth[c*x]) u , x] - Simp[b*c Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x], x]] /; Fre eQ[{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])
Time = 0.37 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00
| method | result | size |
| parts | \(\frac {d^{4} x^{9} \operatorname {arccoth}\left (x a \right )}{9}+\frac {4 c \,d^{3} x^{7} \operatorname {arccoth}\left (x a \right )}{7}+\frac {6 c^{2} d^{2} x^{5} \operatorname {arccoth}\left (x a \right )}{5}+\frac {4 c^{3} d \,x^{3} \operatorname {arccoth}\left (x a \right )}{3}+c^{4} x \,\operatorname {arccoth}\left (x a \right )+\frac {a \left (\frac {d \left (\frac {35}{4} a^{6} d^{3} x^{8}+60 a^{6} c \,d^{2} x^{6}+189 a^{6} c^{2} d \,x^{4}+420 a^{6} c^{3} x^{2}+\frac {35}{3} a^{4} d^{3} x^{6}+90 a^{4} c \,d^{2} x^{4}+378 a^{4} c^{2} d \,x^{2}+\frac {35}{2} a^{2} d^{3} x^{4}+180 a^{2} c \,d^{2} x^{2}+35 d^{3} x^{2}\right )}{2 a^{8}}+\frac {\left (315 a^{8} c^{4}+420 a^{6} c^{3} d +378 a^{4} c^{2} d^{2}+180 a^{2} c \,d^{3}+35 d^{4}\right ) \ln \left (a^{2} x^{2}-1\right )}{2 a^{10}}\right )}{315}\) | \(245\) |
| derivativedivides | \(\frac {\operatorname {arccoth}\left (x a \right ) c^{4} x a +\frac {4 a \,\operatorname {arccoth}\left (x a \right ) c^{3} d \,x^{3}}{3}+\frac {6 a \,\operatorname {arccoth}\left (x a \right ) c^{2} d^{2} x^{5}}{5}+\frac {4 a \,\operatorname {arccoth}\left (x a \right ) c \,d^{3} x^{7}}{7}+\frac {a \,\operatorname {arccoth}\left (x a \right ) d^{4} x^{9}}{9}+\frac {\frac {35 d^{4} x^{8} a^{8}}{8}+30 c \,a^{8} d^{3} x^{6}+\frac {189 c^{2} a^{8} d^{2} x^{4}}{2}+210 c^{3} a^{8} d \,x^{2}-\frac {\left (-315 a^{8} c^{4}-420 a^{6} c^{3} d -378 a^{4} c^{2} d^{2}-180 a^{2} c \,d^{3}-35 d^{4}\right ) \ln \left (x a +1\right )}{2}+\frac {\left (315 a^{8} c^{4}+420 a^{6} c^{3} d +378 a^{4} c^{2} d^{2}+180 a^{2} c \,d^{3}+35 d^{4}\right ) \ln \left (x a -1\right )}{2}+\frac {35 d^{4} x^{6} a^{6}}{6}+\frac {35 d^{4} x^{2} a^{2}}{2}+45 a^{6} c \,d^{3} x^{4}+90 a^{4} c \,d^{3} x^{2}+189 c^{2} a^{6} d^{2} x^{2}+\frac {35 d^{4} x^{4} a^{4}}{4}}{315 a^{8}}}{a}\) | \(301\) |
| default | \(\frac {\operatorname {arccoth}\left (x a \right ) c^{4} x a +\frac {4 a \,\operatorname {arccoth}\left (x a \right ) c^{3} d \,x^{3}}{3}+\frac {6 a \,\operatorname {arccoth}\left (x a \right ) c^{2} d^{2} x^{5}}{5}+\frac {4 a \,\operatorname {arccoth}\left (x a \right ) c \,d^{3} x^{7}}{7}+\frac {a \,\operatorname {arccoth}\left (x a \right ) d^{4} x^{9}}{9}+\frac {\frac {35 d^{4} x^{8} a^{8}}{8}+30 c \,a^{8} d^{3} x^{6}+\frac {189 c^{2} a^{8} d^{2} x^{4}}{2}+210 c^{3} a^{8} d \,x^{2}-\frac {\left (-315 a^{8} c^{4}-420 a^{6} c^{3} d -378 a^{4} c^{2} d^{2}-180 a^{2} c \,d^{3}-35 d^{4}\right ) \ln \left (x a +1\right )}{2}+\frac {\left (315 a^{8} c^{4}+420 a^{6} c^{3} d +378 a^{4} c^{2} d^{2}+180 a^{2} c \,d^{3}+35 d^{4}\right ) \ln \left (x a -1\right )}{2}+\frac {35 d^{4} x^{6} a^{6}}{6}+\frac {35 d^{4} x^{2} a^{2}}{2}+45 a^{6} c \,d^{3} x^{4}+90 a^{4} c \,d^{3} x^{2}+189 c^{2} a^{6} d^{2} x^{2}+\frac {35 d^{4} x^{4} a^{4}}{4}}{315 a^{8}}}{a}\) | \(301\) |
| parallelrisch | \(-\frac {-140 d^{4} x^{6} a^{6}-105 d^{4} x^{8} a^{8}-720 c \,a^{8} d^{3} x^{6}-2268 c^{2} a^{8} d^{2} x^{4}-210 d^{4} x^{4} a^{4}-420 d^{4} x^{2} a^{2}-4536 c^{2} a^{6} d^{2} x^{2}-2160 a^{4} c \,d^{3} x^{2}-1080 a^{6} c \,d^{3} x^{4}-5040 c^{3} a^{8} d \,x^{2}-10080 x^{3} \operatorname {arccoth}\left (x a \right ) a^{9} c^{3} d -9072 x^{5} \operatorname {arccoth}\left (x a \right ) a^{9} c^{2} d^{2}-4320 x^{7} \operatorname {arccoth}\left (x a \right ) a^{9} c \,d^{3}-10080 \ln \left (x a -1\right ) a^{6} c^{3} d -9072 \ln \left (x a -1\right ) a^{4} c^{2} d^{2}-4320 \ln \left (x a -1\right ) a^{2} c \,d^{3}-7560 c^{4} \operatorname {arccoth}\left (x a \right ) x \,a^{9}-840 x^{9} \operatorname {arccoth}\left (x a \right ) a^{9} d^{4}-10080 \,\operatorname {arccoth}\left (x a \right ) a^{6} c^{3} d -9072 \,\operatorname {arccoth}\left (x a \right ) a^{4} c^{2} d^{2}-4320 \,\operatorname {arccoth}\left (x a \right ) a^{2} c \,d^{3}-840 \,\operatorname {arccoth}\left (x a \right ) d^{4}-840 \ln \left (x a -1\right ) d^{4}-7560 \,\operatorname {arccoth}\left (x a \right ) a^{8} c^{4}-7560 \ln \left (x a -1\right ) a^{8} c^{4}}{7560 a^{9}}\) | \(339\) |
| risch | \(\left (\frac {1}{18} d^{4} x^{9}+\frac {2}{7} c \,d^{3} x^{7}+\frac {3}{5} c^{2} d^{2} x^{5}+\frac {2}{3} c^{3} d \,x^{3}+\frac {1}{2} c^{4} x \right ) \ln \left (x a +1\right )-\frac {d^{4} x^{9} \ln \left (x a -1\right )}{18}-\frac {2 c \,d^{3} x^{7} \ln \left (x a -1\right )}{7}+\frac {d^{4} x^{8}}{72 a}-\frac {3 c^{2} d^{2} x^{5} \ln \left (x a -1\right )}{5}+\frac {2 c \,d^{3} x^{6}}{21 a}-\frac {2 c^{3} d \,x^{3} \ln \left (x a -1\right )}{3}+\frac {3 c^{2} d^{2} x^{4}}{10 a}+\frac {d^{4} x^{6}}{54 a^{3}}-\frac {c^{4} x \ln \left (x a -1\right )}{2}+\frac {2 c^{3} d \,x^{2}}{3 a}+\frac {c \,d^{3} x^{4}}{7 a^{3}}+\frac {\ln \left (a^{2} x^{2}-1\right ) c^{4}}{2 a}+\frac {3 c^{2} d^{2} x^{2}}{5 a^{3}}+\frac {d^{4} x^{4}}{36 a^{5}}+\frac {2 \ln \left (a^{2} x^{2}-1\right ) c^{3} d}{3 a^{3}}+\frac {2 c \,d^{3} x^{2}}{7 a^{5}}+\frac {3 \ln \left (a^{2} x^{2}-1\right ) c^{2} d^{2}}{5 a^{5}}+\frac {d^{4} x^{2}}{18 a^{7}}+\frac {2 \ln \left (a^{2} x^{2}-1\right ) c \,d^{3}}{7 a^{7}}+\frac {\ln \left (a^{2} x^{2}-1\right ) d^{4}}{18 a^{9}}\) | \(341\) |
Input:
int((d*x^2+c)^4*arccoth(x*a),x,method=_RETURNVERBOSE)
Output:
1/9*d^4*x^9*arccoth(x*a)+4/7*c*d^3*x^7*arccoth(x*a)+6/5*c^2*d^2*x^5*arccot h(x*a)+4/3*c^3*d*x^3*arccoth(x*a)+c^4*x*arccoth(x*a)+1/315*a*(1/2*d/a^8*(3 5/4*a^6*d^3*x^8+60*a^6*c*d^2*x^6+189*a^6*c^2*d*x^4+420*a^6*c^3*x^2+35/3*a^ 4*d^3*x^6+90*a^4*c*d^2*x^4+378*a^4*c^2*d*x^2+35/2*a^2*d^3*x^4+180*a^2*c*d^ 2*x^2+35*d^3*x^2)+1/2*(315*a^8*c^4+420*a^6*c^3*d+378*a^4*c^2*d^2+180*a^2*c *d^3+35*d^4)/a^10*ln(a^2*x^2-1))
Time = 0.09 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.01 \[ \int \left (c+d x^2\right )^4 \coth ^{-1}(a x) \, dx=\frac {105 \, a^{8} d^{4} x^{8} + 20 \, {\left (36 \, a^{8} c d^{3} + 7 \, a^{6} d^{4}\right )} x^{6} + 6 \, {\left (378 \, a^{8} c^{2} d^{2} + 180 \, a^{6} c d^{3} + 35 \, a^{4} d^{4}\right )} x^{4} + 12 \, {\left (420 \, a^{8} c^{3} d + 378 \, a^{6} c^{2} d^{2} + 180 \, a^{4} c d^{3} + 35 \, a^{2} d^{4}\right )} x^{2} + 12 \, {\left (315 \, a^{8} c^{4} + 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a^{2} x^{2} - 1\right ) + 12 \, {\left (35 \, a^{9} d^{4} x^{9} + 180 \, a^{9} c d^{3} x^{7} + 378 \, a^{9} c^{2} d^{2} x^{5} + 420 \, a^{9} c^{3} d x^{3} + 315 \, a^{9} c^{4} x\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{7560 \, a^{9}} \] Input:
integrate((d*x^2+c)^4*arccoth(a*x),x, algorithm="fricas")
Output:
1/7560*(105*a^8*d^4*x^8 + 20*(36*a^8*c*d^3 + 7*a^6*d^4)*x^6 + 6*(378*a^8*c ^2*d^2 + 180*a^6*c*d^3 + 35*a^4*d^4)*x^4 + 12*(420*a^8*c^3*d + 378*a^6*c^2 *d^2 + 180*a^4*c*d^3 + 35*a^2*d^4)*x^2 + 12*(315*a^8*c^4 + 420*a^6*c^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*log(a^2*x^2 - 1) + 12*(35*a^9*d ^4*x^9 + 180*a^9*c*d^3*x^7 + 378*a^9*c^2*d^2*x^5 + 420*a^9*c^3*d*x^3 + 315 *a^9*c^4*x)*log((a*x + 1)/(a*x - 1)))/a^9
Result contains complex when optimal does not.
Time = 0.70 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.74 \[ \int \left (c+d x^2\right )^4 \coth ^{-1}(a x) \, dx=\begin {cases} c^{4} x \operatorname {acoth}{\left (a x \right )} + \frac {4 c^{3} d x^{3} \operatorname {acoth}{\left (a x \right )}}{3} + \frac {6 c^{2} d^{2} x^{5} \operatorname {acoth}{\left (a x \right )}}{5} + \frac {4 c d^{3} x^{7} \operatorname {acoth}{\left (a x \right )}}{7} + \frac {d^{4} x^{9} \operatorname {acoth}{\left (a x \right )}}{9} + \frac {c^{4} \log {\left (x - \frac {1}{a} \right )}}{a} + \frac {c^{4} \operatorname {acoth}{\left (a x \right )}}{a} + \frac {2 c^{3} d x^{2}}{3 a} + \frac {3 c^{2} d^{2} x^{4}}{10 a} + \frac {2 c d^{3} x^{6}}{21 a} + \frac {d^{4} x^{8}}{72 a} + \frac {4 c^{3} d \log {\left (x - \frac {1}{a} \right )}}{3 a^{3}} + \frac {4 c^{3} d \operatorname {acoth}{\left (a x \right )}}{3 a^{3}} + \frac {3 c^{2} d^{2} x^{2}}{5 a^{3}} + \frac {c d^{3} x^{4}}{7 a^{3}} + \frac {d^{4} x^{6}}{54 a^{3}} + \frac {6 c^{2} d^{2} \log {\left (x - \frac {1}{a} \right )}}{5 a^{5}} + \frac {6 c^{2} d^{2} \operatorname {acoth}{\left (a x \right )}}{5 a^{5}} + \frac {2 c d^{3} x^{2}}{7 a^{5}} + \frac {d^{4} x^{4}}{36 a^{5}} + \frac {4 c d^{3} \log {\left (x - \frac {1}{a} \right )}}{7 a^{7}} + \frac {4 c d^{3} \operatorname {acoth}{\left (a x \right )}}{7 a^{7}} + \frac {d^{4} x^{2}}{18 a^{7}} + \frac {d^{4} \log {\left (x - \frac {1}{a} \right )}}{9 a^{9}} + \frac {d^{4} \operatorname {acoth}{\left (a x \right )}}{9 a^{9}} & \text {for}\: a \neq 0 \\\frac {i \pi \left (c^{4} x + \frac {4 c^{3} d x^{3}}{3} + \frac {6 c^{2} d^{2} x^{5}}{5} + \frac {4 c d^{3} x^{7}}{7} + \frac {d^{4} x^{9}}{9}\right )}{2} & \text {otherwise} \end {cases} \] Input:
integrate((d*x**2+c)**4*acoth(a*x),x)
Output:
Piecewise((c**4*x*acoth(a*x) + 4*c**3*d*x**3*acoth(a*x)/3 + 6*c**2*d**2*x* *5*acoth(a*x)/5 + 4*c*d**3*x**7*acoth(a*x)/7 + d**4*x**9*acoth(a*x)/9 + c* *4*log(x - 1/a)/a + c**4*acoth(a*x)/a + 2*c**3*d*x**2/(3*a) + 3*c**2*d**2* x**4/(10*a) + 2*c*d**3*x**6/(21*a) + d**4*x**8/(72*a) + 4*c**3*d*log(x - 1 /a)/(3*a**3) + 4*c**3*d*acoth(a*x)/(3*a**3) + 3*c**2*d**2*x**2/(5*a**3) + c*d**3*x**4/(7*a**3) + d**4*x**6/(54*a**3) + 6*c**2*d**2*log(x - 1/a)/(5*a **5) + 6*c**2*d**2*acoth(a*x)/(5*a**5) + 2*c*d**3*x**2/(7*a**5) + d**4*x** 4/(36*a**5) + 4*c*d**3*log(x - 1/a)/(7*a**7) + 4*c*d**3*acoth(a*x)/(7*a**7 ) + d**4*x**2/(18*a**7) + d**4*log(x - 1/a)/(9*a**9) + d**4*acoth(a*x)/(9* a**9), Ne(a, 0)), (I*pi*(c**4*x + 4*c**3*d*x**3/3 + 6*c**2*d**2*x**5/5 + 4 *c*d**3*x**7/7 + d**4*x**9/9)/2, True))
Time = 0.03 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.13 \[ \int \left (c+d x^2\right )^4 \coth ^{-1}(a x) \, dx=\frac {1}{7560} \, a {\left (\frac {105 \, a^{6} d^{4} x^{8} + 20 \, {\left (36 \, a^{6} c d^{3} + 7 \, a^{4} d^{4}\right )} x^{6} + 6 \, {\left (378 \, a^{6} c^{2} d^{2} + 180 \, a^{4} c d^{3} + 35 \, a^{2} d^{4}\right )} x^{4} + 12 \, {\left (420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} x^{2}}{a^{8}} + \frac {12 \, {\left (315 \, a^{8} c^{4} + 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a x + 1\right )}{a^{10}} + \frac {12 \, {\left (315 \, a^{8} c^{4} + 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a x - 1\right )}{a^{10}}\right )} + \frac {1}{315} \, {\left (35 \, d^{4} x^{9} + 180 \, c d^{3} x^{7} + 378 \, c^{2} d^{2} x^{5} + 420 \, c^{3} d x^{3} + 315 \, c^{4} x\right )} \operatorname {arcoth}\left (a x\right ) \] Input:
integrate((d*x^2+c)^4*arccoth(a*x),x, algorithm="maxima")
Output:
1/7560*a*((105*a^6*d^4*x^8 + 20*(36*a^6*c*d^3 + 7*a^4*d^4)*x^6 + 6*(378*a^ 6*c^2*d^2 + 180*a^4*c*d^3 + 35*a^2*d^4)*x^4 + 12*(420*a^6*c^3*d + 378*a^4* c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*x^2)/a^8 + 12*(315*a^8*c^4 + 420*a^6*c^3 *d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*log(a*x + 1)/a^10 + 12*(315 *a^8*c^4 + 420*a^6*c^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*log(a *x - 1)/a^10) + 1/315*(35*d^4*x^9 + 180*c*d^3*x^7 + 378*c^2*d^2*x^5 + 420* c^3*d*x^3 + 315*c^4*x)*arccoth(a*x)
Leaf count of result is larger than twice the leaf count of optimal. 1473 vs. \(2 (227) = 454\).
Time = 0.16 (sec) , antiderivative size = 1473, normalized size of antiderivative = 6.01 \[ \int \left (c+d x^2\right )^4 \coth ^{-1}(a x) \, dx=\text {Too large to display} \] Input:
integrate((d*x^2+c)^4*arccoth(a*x),x, algorithm="giac")
Output:
1/945*a*(3*(315*a^8*c^4 + 420*a^6*c^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*log(abs(a*x + 1)/abs(a*x - 1))/a^10 - 3*(315*a^8*c^4 + 420*a^6*c ^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*log(abs((a*x + 1)/(a*x - 1) - 1))/a^10 + 8*(3*(105*a^6*c^3*d + 189*a^4*c^2*d^2 + 135*a^2*c*d^3 + 35 *d^4)*(a*x + 1)^7/(a*x - 1)^7 - 45*(42*a^6*c^3*d + 63*a^4*c^2*d^2 + 36*a^2 *c*d^3 + 7*d^4)*(a*x + 1)^6/(a*x - 1)^6 + (4725*a^6*c^3*d + 6237*a^4*c^2*d ^2 + 3555*a^2*c*d^3 + 875*d^4)*(a*x + 1)^5/(a*x - 1)^5 - 2*(3150*a^6*c^3*d + 3969*a^4*c^2*d^2 + 2340*a^2*c*d^3 + 455*d^4)*(a*x + 1)^4/(a*x - 1)^4 + (4725*a^6*c^3*d + 6237*a^4*c^2*d^2 + 3555*a^2*c*d^3 + 875*d^4)*(a*x + 1)^3 /(a*x - 1)^3 - 45*(42*a^6*c^3*d + 63*a^4*c^2*d^2 + 36*a^2*c*d^3 + 7*d^4)*( a*x + 1)^2/(a*x - 1)^2 + 3*(105*a^6*c^3*d + 189*a^4*c^2*d^2 + 135*a^2*c*d^ 3 + 35*d^4)*(a*x + 1)/(a*x - 1))/(a^10*((a*x + 1)/(a*x - 1) - 1)^8) + 3*(3 15*(a*x + 1)^8*a^8*c^4/(a*x - 1)^8 - 2520*(a*x + 1)^7*a^8*c^4/(a*x - 1)^7 + 8820*(a*x + 1)^6*a^8*c^4/(a*x - 1)^6 - 17640*(a*x + 1)^5*a^8*c^4/(a*x - 1)^5 + 22050*(a*x + 1)^4*a^8*c^4/(a*x - 1)^4 - 17640*(a*x + 1)^3*a^8*c^4/( a*x - 1)^3 + 8820*(a*x + 1)^2*a^8*c^4/(a*x - 1)^2 - 2520*(a*x + 1)*a^8*c^4 /(a*x - 1) + 315*a^8*c^4 + 1260*(a*x + 1)^8*a^6*c^3*d/(a*x - 1)^8 - 7560*( a*x + 1)^7*a^6*c^3*d/(a*x - 1)^7 + 19320*(a*x + 1)^6*a^6*c^3*d/(a*x - 1)^6 - 27720*(a*x + 1)^5*a^6*c^3*d/(a*x - 1)^5 + 25200*(a*x + 1)^4*a^6*c^3*d/( a*x - 1)^4 - 15960*(a*x + 1)^3*a^6*c^3*d/(a*x - 1)^3 + 7560*(a*x + 1)^2...
Time = 4.05 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.21 \[ \int \left (c+d x^2\right )^4 \coth ^{-1}(a x) \, dx=\ln \left (\frac {1}{a\,x}+1\right )\,\left (\frac {c^4\,x}{2}+\frac {2\,c^3\,d\,x^3}{3}+\frac {3\,c^2\,d^2\,x^5}{5}+\frac {2\,c\,d^3\,x^7}{7}+\frac {d^4\,x^9}{18}\right )-\ln \left (1-\frac {1}{a\,x}\right )\,\left (\frac {c^4\,x}{2}+\frac {2\,c^3\,d\,x^3}{3}+\frac {3\,c^2\,d^2\,x^5}{5}+\frac {2\,c\,d^3\,x^7}{7}+\frac {d^4\,x^9}{18}\right )+x^2\,\left (\frac {\frac {\frac {d^4}{9\,a^3}+\frac {4\,c\,d^3}{7\,a}}{a^2}+\frac {6\,c^2\,d^2}{5\,a}}{2\,a^2}+\frac {2\,c^3\,d}{3\,a}\right )+x^6\,\left (\frac {d^4}{54\,a^3}+\frac {2\,c\,d^3}{21\,a}\right )+x^4\,\left (\frac {\frac {d^4}{9\,a^3}+\frac {4\,c\,d^3}{7\,a}}{4\,a^2}+\frac {3\,c^2\,d^2}{10\,a}\right )+\frac {\ln \left (a^2\,x^2-1\right )\,\left (315\,a^8\,c^4+420\,a^6\,c^3\,d+378\,a^4\,c^2\,d^2+180\,a^2\,c\,d^3+35\,d^4\right )}{630\,a^9}+\frac {d^4\,x^8}{72\,a} \] Input:
int(acoth(a*x)*(c + d*x^2)^4,x)
Output:
log(1/(a*x) + 1)*((c^4*x)/2 + (d^4*x^9)/18 + (2*c^3*d*x^3)/3 + (2*c*d^3*x^ 7)/7 + (3*c^2*d^2*x^5)/5) - log(1 - 1/(a*x))*((c^4*x)/2 + (d^4*x^9)/18 + ( 2*c^3*d*x^3)/3 + (2*c*d^3*x^7)/7 + (3*c^2*d^2*x^5)/5) + x^2*(((d^4/(9*a^3) + (4*c*d^3)/(7*a))/a^2 + (6*c^2*d^2)/(5*a))/(2*a^2) + (2*c^3*d)/(3*a)) + x^6*(d^4/(54*a^3) + (2*c*d^3)/(21*a)) + x^4*((d^4/(9*a^3) + (4*c*d^3)/(7*a ))/(4*a^2) + (3*c^2*d^2)/(10*a)) + (log(a^2*x^2 - 1)*(35*d^4 + 315*a^8*c^4 + 180*a^2*c*d^3 + 420*a^6*c^3*d + 378*a^4*c^2*d^2))/(630*a^9) + (d^4*x^8) /(72*a)
Time = 0.17 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.46 \[ \int \left (c+d x^2\right )^4 \coth ^{-1}(a x) \, dx=\frac {-7560 \,\mathrm {log}\left (a^{2} x -a \right ) a^{8} c^{4}-105 a^{8} d^{4} x^{8}-140 a^{6} d^{4} x^{6}-210 a^{4} d^{4} x^{4}-420 a^{2} d^{4} x^{2}-840 \,\mathrm {log}\left (a^{2} x -a \right ) d^{4}+7560 \mathit {acoth} \left (a x \right ) a^{9} c^{4} x +840 \mathit {acoth} \left (a x \right ) a^{9} d^{4} x^{9}+10080 \mathit {acoth} \left (a x \right ) a^{6} c^{3} d +9072 \mathit {acoth} \left (a x \right ) a^{4} c^{2} d^{2}+4320 \mathit {acoth} \left (a x \right ) a^{2} c \,d^{3}+840 \mathit {acoth} \left (a x \right ) d^{4}-10080 \,\mathrm {log}\left (a^{2} x -a \right ) a^{6} c^{3} d -9072 \,\mathrm {log}\left (a^{2} x -a \right ) a^{4} c^{2} d^{2}-4320 \,\mathrm {log}\left (a^{2} x -a \right ) a^{2} c \,d^{3}-5040 a^{8} c^{3} d \,x^{2}-2268 a^{8} c^{2} d^{2} x^{4}-720 a^{8} c \,d^{3} x^{6}-4536 a^{6} c^{2} d^{2} x^{2}-1080 a^{6} c \,d^{3} x^{4}-2160 a^{4} c \,d^{3} x^{2}+10080 \mathit {acoth} \left (a x \right ) a^{9} c^{3} d \,x^{3}+9072 \mathit {acoth} \left (a x \right ) a^{9} c^{2} d^{2} x^{5}+4320 \mathit {acoth} \left (a x \right ) a^{9} c \,d^{3} x^{7}+7560 \mathit {acoth} \left (a x \right ) a^{8} c^{4}}{7560 a^{9}} \] Input:
int((d*x^2+c)^4*acoth(a*x),x)
Output:
(7560*acoth(a*x)*a**9*c**4*x + 10080*acoth(a*x)*a**9*c**3*d*x**3 + 9072*ac oth(a*x)*a**9*c**2*d**2*x**5 + 4320*acoth(a*x)*a**9*c*d**3*x**7 + 840*acot h(a*x)*a**9*d**4*x**9 + 7560*acoth(a*x)*a**8*c**4 + 10080*acoth(a*x)*a**6* c**3*d + 9072*acoth(a*x)*a**4*c**2*d**2 + 4320*acoth(a*x)*a**2*c*d**3 + 84 0*acoth(a*x)*d**4 - 7560*log(a**2*x - a)*a**8*c**4 - 10080*log(a**2*x - a) *a**6*c**3*d - 9072*log(a**2*x - a)*a**4*c**2*d**2 - 4320*log(a**2*x - a)* a**2*c*d**3 - 840*log(a**2*x - a)*d**4 - 5040*a**8*c**3*d*x**2 - 2268*a**8 *c**2*d**2*x**4 - 720*a**8*c*d**3*x**6 - 105*a**8*d**4*x**8 - 4536*a**6*c* *2*d**2*x**2 - 1080*a**6*c*d**3*x**4 - 140*a**6*d**4*x**6 - 2160*a**4*c*d* *3*x**2 - 210*a**4*d**4*x**4 - 420*a**2*d**4*x**2)/(7560*a**9)