Integrand size = 14, antiderivative size = 169 \[ \int \left (c+d x^2\right )^3 \coth ^{-1}(a x) \, dx=\frac {d \left (35 a^4 c^2+21 a^2 c d+5 d^2\right ) x^2}{70 a^5}+\frac {d^2 \left (21 a^2 c+5 d\right ) x^4}{140 a^3}+\frac {d^3 x^6}{42 a}+c^3 x \coth ^{-1}(a x)+c^2 d x^3 \coth ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \coth ^{-1}(a x)+\frac {1}{7} d^3 x^7 \coth ^{-1}(a x)+\frac {\left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right ) \log \left (1-a^2 x^2\right )}{70 a^7} \]
1/70*d*(35*a^4*c^2+21*a^2*c*d+5*d^2)*x^2/a^5+1/140*d^2*(21*a^2*c+5*d)*x^4/ a^3+1/42*d^3*x^6/a+c^3*x*arccoth(a*x)+c^2*d*x^3*arccoth(a*x)+3/5*c*d^2*x^5 *arccoth(a*x)+1/7*d^3*x^7*arccoth(a*x)+1/70*(35*a^6*c^3+35*a^4*c^2*d+21*a^ 2*c*d^2+5*d^3)*ln(-a^2*x^2+1)/a^7
Time = 0.04 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.89 \[ \int \left (c+d x^2\right )^3 \coth ^{-1}(a x) \, dx=\frac {a^2 d x^2 \left (30 d^2+3 a^2 d \left (42 c+5 d x^2\right )+a^4 \left (210 c^2+63 c d x^2+10 d^2 x^4\right )\right )+12 a^7 x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right ) \coth ^{-1}(a x)+6 \left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right ) \log \left (1-a^2 x^2\right )}{420 a^7} \]
(a^2*d*x^2*(30*d^2 + 3*a^2*d*(42*c + 5*d*x^2) + a^4*(210*c^2 + 63*c*d*x^2 + 10*d^2*x^4)) + 12*a^7*x*(35*c^3 + 35*c^2*d*x^2 + 21*c*d^2*x^4 + 5*d^3*x^ 6)*ArcCoth[a*x] + 6*(35*a^6*c^3 + 35*a^4*c^2*d + 21*a^2*c*d^2 + 5*d^3)*Log [1 - a^2*x^2])/(420*a^7)
Time = 0.57 (sec) , antiderivative size = 171, 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.
| \(\displaystyle \int \coth ^{-1}(a x) \left (c+d x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 6539 |
| \(\displaystyle -a \int \frac {x \left (5 d^3 x^6+21 c d^2 x^4+35 c^2 d x^2+35 c^3\right )}{35 \left (1-a^2 x^2\right )}dx+c^3 x \coth ^{-1}(a x)+c^2 d x^3 \coth ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \coth ^{-1}(a x)+\frac {1}{7} d^3 x^7 \coth ^{-1}(a x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{35} a \int \frac {x \left (5 d^3 x^6+21 c d^2 x^4+35 c^2 d x^2+35 c^3\right )}{1-a^2 x^2}dx+c^3 x \coth ^{-1}(a x)+c^2 d x^3 \coth ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \coth ^{-1}(a x)+\frac {1}{7} d^3 x^7 \coth ^{-1}(a x)\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle -\frac {1}{70} a \int \frac {5 d^3 x^6+21 c d^2 x^4+35 c^2 d x^2+35 c^3}{1-a^2 x^2}dx^2+c^3 x \coth ^{-1}(a x)+c^2 d x^3 \coth ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \coth ^{-1}(a x)+\frac {1}{7} d^3 x^7 \coth ^{-1}(a x)\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle -\frac {1}{70} a \int \left (-\frac {5 d^3 x^4}{a^2}-\frac {d^2 \left (21 c a^2+5 d\right ) x^2}{a^4}-\frac {d \left (35 c^2 a^4+21 c d a^2+5 d^2\right )}{a^6}+\frac {-35 c^3 a^6-35 c^2 d a^4-21 c d^2 a^2-5 d^3}{a^6 \left (a^2 x^2-1\right )}\right )dx^2+c^3 x \coth ^{-1}(a x)+c^2 d x^3 \coth ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \coth ^{-1}(a x)+\frac {1}{7} d^3 x^7 \coth ^{-1}(a x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{70} a \left (-\frac {5 d^3 x^6}{3 a^2}-\frac {d^2 x^4 \left (21 a^2 c+5 d\right )}{2 a^4}-\frac {d x^2 \left (35 a^4 c^2+21 a^2 c d+5 d^2\right )}{a^6}-\frac {\left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right ) \log \left (1-a^2 x^2\right )}{a^8}\right )+c^3 x \coth ^{-1}(a x)+c^2 d x^3 \coth ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \coth ^{-1}(a x)+\frac {1}{7} d^3 x^7 \coth ^{-1}(a x)\) |
c^3*x*ArcCoth[a*x] + c^2*d*x^3*ArcCoth[a*x] + (3*c*d^2*x^5*ArcCoth[a*x])/5 + (d^3*x^7*ArcCoth[a*x])/7 - (a*(-((d*(35*a^4*c^2 + 21*a^2*c*d + 5*d^2)*x ^2)/a^6) - (d^2*(21*a^2*c + 5*d)*x^4)/(2*a^4) - (5*d^3*x^6)/(3*a^2) - ((35 *a^6*c^3 + 35*a^4*c^2*d + 21*a^2*c*d^2 + 5*d^3)*Log[1 - a^2*x^2])/a^8))/70
3.1.36.3.1 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[(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 = 0.48 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.99
| method | result | size |
| parts | \(\frac {d^{3} x^{7} \operatorname {arccoth}\left (a x \right )}{7}+\frac {3 c \,d^{2} x^{5} \operatorname {arccoth}\left (a x \right )}{5}+c^{2} d \,x^{3} \operatorname {arccoth}\left (a x \right )+c^{3} x \,\operatorname {arccoth}\left (a x \right )+\frac {a \left (\frac {d \left (\frac {5}{3} a^{4} d^{2} x^{6}+\frac {21}{2} a^{4} c d \,x^{4}+35 a^{4} c^{2} x^{2}+\frac {5}{2} a^{2} d^{2} x^{4}+21 a^{2} c d \,x^{2}+5 d^{2} x^{2}\right )}{2 a^{6}}+\frac {\left (35 a^{6} c^{3}+35 a^{4} c^{2} d +21 a^{2} c \,d^{2}+5 d^{3}\right ) \ln \left (a^{2} x^{2}-1\right )}{2 a^{8}}\right )}{35}\) | \(167\) |
| derivativedivides | \(\frac {\operatorname {arccoth}\left (a x \right ) c^{3} a x +a \,\operatorname {arccoth}\left (a x \right ) c^{2} d \,x^{3}+\frac {3 a \,\operatorname {arccoth}\left (a x \right ) c \,d^{2} x^{5}}{5}+\frac {a \,\operatorname {arccoth}\left (a x \right ) d^{3} x^{7}}{7}+\frac {\frac {21 c \,a^{4} d^{2} x^{2}}{2}+\frac {21 c \,a^{6} d^{2} x^{4}}{4}+\frac {35 c^{2} a^{6} d \,x^{2}}{2}+\frac {\left (35 a^{6} c^{3}+35 a^{4} c^{2} d +21 a^{2} c \,d^{2}+5 d^{3}\right ) \ln \left (a x -1\right )}{2}+\frac {5 d^{3} a^{4} x^{4}}{4}+\frac {5 d^{3} a^{2} x^{2}}{2}+\frac {5 d^{3} a^{6} x^{6}}{6}-\frac {\left (-35 a^{6} c^{3}-35 a^{4} c^{2} d -21 a^{2} c \,d^{2}-5 d^{3}\right ) \ln \left (a x +1\right )}{2}}{35 a^{6}}}{a}\) | \(211\) |
| default | \(\frac {\operatorname {arccoth}\left (a x \right ) c^{3} a x +a \,\operatorname {arccoth}\left (a x \right ) c^{2} d \,x^{3}+\frac {3 a \,\operatorname {arccoth}\left (a x \right ) c \,d^{2} x^{5}}{5}+\frac {a \,\operatorname {arccoth}\left (a x \right ) d^{3} x^{7}}{7}+\frac {\frac {21 c \,a^{4} d^{2} x^{2}}{2}+\frac {21 c \,a^{6} d^{2} x^{4}}{4}+\frac {35 c^{2} a^{6} d \,x^{2}}{2}+\frac {\left (35 a^{6} c^{3}+35 a^{4} c^{2} d +21 a^{2} c \,d^{2}+5 d^{3}\right ) \ln \left (a x -1\right )}{2}+\frac {5 d^{3} a^{4} x^{4}}{4}+\frac {5 d^{3} a^{2} x^{2}}{2}+\frac {5 d^{3} a^{6} x^{6}}{6}-\frac {\left (-35 a^{6} c^{3}-35 a^{4} c^{2} d -21 a^{2} c \,d^{2}-5 d^{3}\right ) \ln \left (a x +1\right )}{2}}{35 a^{6}}}{a}\) | \(211\) |
| parallelrisch | \(-\frac {-60 x^{7} \operatorname {arccoth}\left (a x \right ) a^{7} d^{3}-252 x^{5} \operatorname {arccoth}\left (a x \right ) a^{7} c \,d^{2}-10 d^{3} a^{6} x^{6}-420 x^{3} \operatorname {arccoth}\left (a x \right ) a^{7} c^{2} d -63 c \,a^{6} d^{2} x^{4}-420 c^{3} x \,\operatorname {arccoth}\left (a x \right ) a^{7}-15 d^{3} a^{4} x^{4}-210 c^{2} a^{6} d \,x^{2}-420 \ln \left (a x -1\right ) a^{6} c^{3}-420 \,\operatorname {arccoth}\left (a x \right ) a^{6} c^{3}-126 c \,a^{4} d^{2} x^{2}-420 \ln \left (a x -1\right ) a^{4} c^{2} d -420 \,\operatorname {arccoth}\left (a x \right ) a^{4} c^{2} d -30 d^{3} a^{2} x^{2}-252 \ln \left (a x -1\right ) a^{2} c \,d^{2}-252 \,\operatorname {arccoth}\left (a x \right ) a^{2} c \,d^{2}-60 \ln \left (a x -1\right ) d^{3}-60 \,\operatorname {arccoth}\left (a x \right ) d^{3}}{420 a^{7}}\) | \(238\) |
| risch | \(\left (\frac {1}{14} d^{3} x^{7}+\frac {3}{10} c \,d^{2} x^{5}+\frac {1}{2} c^{2} d \,x^{3}+\frac {1}{2} c^{3} x \right ) \ln \left (a x +1\right )-\frac {d^{3} x^{7} \ln \left (a x -1\right )}{14}-\frac {3 c \,d^{2} x^{5} \ln \left (a x -1\right )}{10}+\frac {d^{3} x^{6}}{42 a}-\frac {c^{2} d \,x^{3} \ln \left (a x -1\right )}{2}+\frac {3 c \,d^{2} x^{4}}{20 a}-\frac {c^{3} x \ln \left (a x -1\right )}{2}+\frac {c^{2} d \,x^{2}}{2 a}+\frac {d^{3} x^{4}}{28 a^{3}}+\frac {\ln \left (a^{2} x^{2}-1\right ) c^{3}}{2 a}+\frac {3 c \,d^{2} x^{2}}{10 a^{3}}+\frac {\ln \left (a^{2} x^{2}-1\right ) c^{2} d}{2 a^{3}}+\frac {d^{3} x^{2}}{14 a^{5}}+\frac {3 \ln \left (a^{2} x^{2}-1\right ) c \,d^{2}}{10 a^{5}}+\frac {\ln \left (a^{2} x^{2}-1\right ) d^{3}}{14 a^{7}}\) | \(241\) |
1/7*d^3*x^7*arccoth(a*x)+3/5*c*d^2*x^5*arccoth(a*x)+c^2*d*x^3*arccoth(a*x) +c^3*x*arccoth(a*x)+1/35*a*(1/2*d/a^6*(5/3*a^4*d^2*x^6+21/2*a^4*c*d*x^4+35 *a^4*c^2*x^2+5/2*a^2*d^2*x^4+21*a^2*c*d*x^2+5*d^2*x^2)+1/2*(35*a^6*c^3+35* a^4*c^2*d+21*a^2*c*d^2+5*d^3)/a^8*ln(a^2*x^2-1))
Time = 0.25 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.05 \[ \int \left (c+d x^2\right )^3 \coth ^{-1}(a x) \, dx=\frac {10 \, a^{6} d^{3} x^{6} + 3 \, {\left (21 \, a^{6} c d^{2} + 5 \, a^{4} d^{3}\right )} x^{4} + 6 \, {\left (35 \, a^{6} c^{2} d + 21 \, a^{4} c d^{2} + 5 \, a^{2} d^{3}\right )} x^{2} + 6 \, {\left (35 \, a^{6} c^{3} + 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} + 5 \, d^{3}\right )} \log \left (a^{2} x^{2} - 1\right ) + 6 \, {\left (5 \, a^{7} d^{3} x^{7} + 21 \, a^{7} c d^{2} x^{5} + 35 \, a^{7} c^{2} d x^{3} + 35 \, a^{7} c^{3} x\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{420 \, a^{7}} \]
1/420*(10*a^6*d^3*x^6 + 3*(21*a^6*c*d^2 + 5*a^4*d^3)*x^4 + 6*(35*a^6*c^2*d + 21*a^4*c*d^2 + 5*a^2*d^3)*x^2 + 6*(35*a^6*c^3 + 35*a^4*c^2*d + 21*a^2*c *d^2 + 5*d^3)*log(a^2*x^2 - 1) + 6*(5*a^7*d^3*x^7 + 21*a^7*c*d^2*x^5 + 35* a^7*c^2*d*x^3 + 35*a^7*c^3*x)*log((a*x + 1)/(a*x - 1)))/a^7
Result contains complex when optimal does not.
Time = 0.47 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.67 \[ \int \left (c+d x^2\right )^3 \coth ^{-1}(a x) \, dx=\begin {cases} c^{3} x \operatorname {acoth}{\left (a x \right )} + c^{2} d x^{3} \operatorname {acoth}{\left (a x \right )} + \frac {3 c d^{2} x^{5} \operatorname {acoth}{\left (a x \right )}}{5} + \frac {d^{3} x^{7} \operatorname {acoth}{\left (a x \right )}}{7} + \frac {c^{3} \log {\left (x - \frac {1}{a} \right )}}{a} + \frac {c^{3} \operatorname {acoth}{\left (a x \right )}}{a} + \frac {c^{2} d x^{2}}{2 a} + \frac {3 c d^{2} x^{4}}{20 a} + \frac {d^{3} x^{6}}{42 a} + \frac {c^{2} d \log {\left (x - \frac {1}{a} \right )}}{a^{3}} + \frac {c^{2} d \operatorname {acoth}{\left (a x \right )}}{a^{3}} + \frac {3 c d^{2} x^{2}}{10 a^{3}} + \frac {d^{3} x^{4}}{28 a^{3}} + \frac {3 c d^{2} \log {\left (x - \frac {1}{a} \right )}}{5 a^{5}} + \frac {3 c d^{2} \operatorname {acoth}{\left (a x \right )}}{5 a^{5}} + \frac {d^{3} x^{2}}{14 a^{5}} + \frac {d^{3} \log {\left (x - \frac {1}{a} \right )}}{7 a^{7}} + \frac {d^{3} \operatorname {acoth}{\left (a x \right )}}{7 a^{7}} & \text {for}\: a \neq 0 \\\frac {i \pi \left (c^{3} x + c^{2} d x^{3} + \frac {3 c d^{2} x^{5}}{5} + \frac {d^{3} x^{7}}{7}\right )}{2} & \text {otherwise} \end {cases} \]
Piecewise((c**3*x*acoth(a*x) + c**2*d*x**3*acoth(a*x) + 3*c*d**2*x**5*acot h(a*x)/5 + d**3*x**7*acoth(a*x)/7 + c**3*log(x - 1/a)/a + c**3*acoth(a*x)/ a + c**2*d*x**2/(2*a) + 3*c*d**2*x**4/(20*a) + d**3*x**6/(42*a) + c**2*d*l og(x - 1/a)/a**3 + c**2*d*acoth(a*x)/a**3 + 3*c*d**2*x**2/(10*a**3) + d**3 *x**4/(28*a**3) + 3*c*d**2*log(x - 1/a)/(5*a**5) + 3*c*d**2*acoth(a*x)/(5* a**5) + d**3*x**2/(14*a**5) + d**3*log(x - 1/a)/(7*a**7) + d**3*acoth(a*x) /(7*a**7), Ne(a, 0)), (I*pi*(c**3*x + c**2*d*x**3 + 3*c*d**2*x**5/5 + d**3 *x**7/7)/2, True))
Time = 0.20 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.17 \[ \int \left (c+d x^2\right )^3 \coth ^{-1}(a x) \, dx=\frac {1}{420} \, a {\left (\frac {10 \, a^{4} d^{3} x^{6} + 3 \, {\left (21 \, a^{4} c d^{2} + 5 \, a^{2} d^{3}\right )} x^{4} + 6 \, {\left (35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} + 5 \, d^{3}\right )} x^{2}}{a^{6}} + \frac {6 \, {\left (35 \, a^{6} c^{3} + 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} + 5 \, d^{3}\right )} \log \left (a x + 1\right )}{a^{8}} + \frac {6 \, {\left (35 \, a^{6} c^{3} + 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} + 5 \, d^{3}\right )} \log \left (a x - 1\right )}{a^{8}}\right )} + \frac {1}{35} \, {\left (5 \, d^{3} x^{7} + 21 \, c d^{2} x^{5} + 35 \, c^{2} d x^{3} + 35 \, c^{3} x\right )} \operatorname {arcoth}\left (a x\right ) \]
1/420*a*((10*a^4*d^3*x^6 + 3*(21*a^4*c*d^2 + 5*a^2*d^3)*x^4 + 6*(35*a^4*c^ 2*d + 21*a^2*c*d^2 + 5*d^3)*x^2)/a^6 + 6*(35*a^6*c^3 + 35*a^4*c^2*d + 21*a ^2*c*d^2 + 5*d^3)*log(a*x + 1)/a^8 + 6*(35*a^6*c^3 + 35*a^4*c^2*d + 21*a^2 *c*d^2 + 5*d^3)*log(a*x - 1)/a^8) + 1/35*(5*d^3*x^7 + 21*c*d^2*x^5 + 35*c^ 2*d*x^3 + 35*c^3*x)*arccoth(a*x)
Leaf count of result is larger than twice the leaf count of optimal. 932 vs. \(2 (157) = 314\).
Time = 0.31 (sec) , antiderivative size = 932, normalized size of antiderivative = 5.51 \[ \int \left (c+d x^2\right )^3 \coth ^{-1}(a x) \, dx=\text {Too large to display} \]
1/105*a*(3*(35*a^6*c^3 + 35*a^4*c^2*d + 21*a^2*c*d^2 + 5*d^3)*log(abs(a*x + 1)/abs(a*x - 1))/a^8 - 3*(35*a^6*c^3 + 35*a^4*c^2*d + 21*a^2*c*d^2 + 5*d ^3)*log(abs((a*x + 1)/(a*x - 1) - 1))/a^8 + 2*(3*(35*a^4*c^2*d + 42*a^2*c* d^2 + 15*d^3)*(a*x + 1)^5/(a*x - 1)^5 - 6*(70*a^4*c^2*d + 63*a^2*c*d^2 + 1 5*d^3)*(a*x + 1)^4/(a*x - 1)^4 + 2*(315*a^4*c^2*d + 252*a^2*c*d^2 + 85*d^3 )*(a*x + 1)^3/(a*x - 1)^3 - 6*(70*a^4*c^2*d + 63*a^2*c*d^2 + 15*d^3)*(a*x + 1)^2/(a*x - 1)^2 + 3*(35*a^4*c^2*d + 42*a^2*c*d^2 + 15*d^3)*(a*x + 1)/(a *x - 1))/(a^8*((a*x + 1)/(a*x - 1) - 1)^6) + 3*(35*(a*x + 1)^6*a^6*c^3/(a* x - 1)^6 - 210*(a*x + 1)^5*a^6*c^3/(a*x - 1)^5 + 525*(a*x + 1)^4*a^6*c^3/( a*x - 1)^4 - 700*(a*x + 1)^3*a^6*c^3/(a*x - 1)^3 + 525*(a*x + 1)^2*a^6*c^3 /(a*x - 1)^2 - 210*(a*x + 1)*a^6*c^3/(a*x - 1) + 35*a^6*c^3 + 105*(a*x + 1 )^6*a^4*c^2*d/(a*x - 1)^6 - 420*(a*x + 1)^5*a^4*c^2*d/(a*x - 1)^5 + 665*(a *x + 1)^4*a^4*c^2*d/(a*x - 1)^4 - 560*(a*x + 1)^3*a^4*c^2*d/(a*x - 1)^3 + 315*(a*x + 1)^2*a^4*c^2*d/(a*x - 1)^2 - 140*(a*x + 1)*a^4*c^2*d/(a*x - 1) + 35*a^4*c^2*d + 105*(a*x + 1)^6*a^2*c*d^2/(a*x - 1)^6 - 210*(a*x + 1)^5*a ^2*c*d^2/(a*x - 1)^5 + 315*(a*x + 1)^4*a^2*c*d^2/(a*x - 1)^4 - 420*(a*x + 1)^3*a^2*c*d^2/(a*x - 1)^3 + 231*(a*x + 1)^2*a^2*c*d^2/(a*x - 1)^2 - 42*(a *x + 1)*a^2*c*d^2/(a*x - 1) + 21*a^2*c*d^2 + 35*(a*x + 1)^6*d^3/(a*x - 1)^ 6 + 175*(a*x + 1)^4*d^3/(a*x - 1)^4 + 105*(a*x + 1)^2*d^3/(a*x - 1)^2 + 5* d^3)*log(-(((a*x + 1)*a/(a*x - 1) - a)/(a*((a*x + 1)/(a*x - 1) + 1)) + ...
Time = 4.57 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.12 \[ \int \left (c+d x^2\right )^3 \coth ^{-1}(a x) \, dx=c^3\,x\,\mathrm {acoth}\left (a\,x\right )+\frac {d^3\,x^7\,\mathrm {acoth}\left (a\,x\right )}{7}+\frac {c^3\,\ln \left (a^2\,x^2-1\right )}{2\,a}+\frac {d^3\,\ln \left (a^2\,x^2-1\right )}{14\,a^7}+\frac {d^3\,x^6}{42\,a}+\frac {d^3\,x^4}{28\,a^3}+\frac {d^3\,x^2}{14\,a^5}+\frac {c^2\,d\,\ln \left (a^2\,x^2-1\right )}{2\,a^3}+\frac {3\,c\,d^2\,\ln \left (a^2\,x^2-1\right )}{10\,a^5}+\frac {c^2\,d\,x^2}{2\,a}+\frac {3\,c\,d^2\,x^4}{20\,a}+\frac {3\,c\,d^2\,x^2}{10\,a^3}+c^2\,d\,x^3\,\mathrm {acoth}\left (a\,x\right )+\frac {3\,c\,d^2\,x^5\,\mathrm {acoth}\left (a\,x\right )}{5} \]
c^3*x*acoth(a*x) + (d^3*x^7*acoth(a*x))/7 + (c^3*log(a^2*x^2 - 1))/(2*a) + (d^3*log(a^2*x^2 - 1))/(14*a^7) + (d^3*x^6)/(42*a) + (d^3*x^4)/(28*a^3) + (d^3*x^2)/(14*a^5) + (c^2*d*log(a^2*x^2 - 1))/(2*a^3) + (3*c*d^2*log(a^2* x^2 - 1))/(10*a^5) + (c^2*d*x^2)/(2*a) + (3*c*d^2*x^4)/(20*a) + (3*c*d^2*x ^2)/(10*a^3) + c^2*d*x^3*acoth(a*x) + (3*c*d^2*x^5*acoth(a*x))/5