Integrand size = 10, antiderivative size = 101 \[ \int x^3 \coth ^{-1}(a+b x) \, dx=\frac {\left (1+6 a^2\right ) x}{4 b^3}-\frac {a (a+b x)^2}{2 b^4}+\frac {(a+b x)^3}{12 b^4}+\frac {1}{4} x^4 \coth ^{-1}(a+b x)+\frac {(1-a)^4 \log (1-a-b x)}{8 b^4}-\frac {(1+a)^4 \log (1+a+b x)}{8 b^4} \]
1/4*(6*a^2+1)*x/b^3-1/2*a*(b*x+a)^2/b^4+1/12*(b*x+a)^3/b^4+1/4*x^4*arccoth (b*x+a)+1/8*(1-a)^4*ln(-b*x-a+1)/b^4-1/8*(1+a)^4*ln(b*x+a+1)/b^4
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.80 \[ \int x^3 \coth ^{-1}(a+b x) \, dx=\frac {6 \left (1+3 a^2\right ) b x-6 a b^2 x^2+2 b^3 x^3+6 b^4 x^4 \coth ^{-1}(a+b x)+3 (-1+a)^4 \log (1-a-b x)-3 (1+a)^4 \log (1+a+b x)}{24 b^4} \]
(6*(1 + 3*a^2)*b*x - 6*a*b^2*x^2 + 2*b^3*x^3 + 6*b^4*x^4*ArcCoth[a + b*x] + 3*(-1 + a)^4*Log[1 - a - b*x] - 3*(1 + a)^4*Log[1 + a + b*x])/(24*b^4)
Time = 0.34 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6662, 25, 27, 6479, 477, 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.
| \(\displaystyle \int x^3 \coth ^{-1}(a+b x) \, dx\) |
| \(\Big \downarrow \) 6662 |
| \(\displaystyle \frac {\int x^3 \coth ^{-1}(a+b x)d(a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -x^3 \coth ^{-1}(a+b x)d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int -b^3 x^3 \coth ^{-1}(a+b x)d(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 6479 |
| \(\displaystyle -\frac {\frac {1}{4} \int \frac {b^4 x^4}{1-(a+b x)^2}d(a+b x)-\frac {1}{4} b^4 x^4 \coth ^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 477 |
| \(\displaystyle -\frac {\frac {1}{4} \int \left (\frac {(1-a)^4}{2 (-a-b x+1)}-6 a^2-(a+b x)^2+4 a (a+b x)+\frac {(a+1)^4}{2 (a+b x+1)}-1\right )d(a+b x)-\frac {1}{4} b^4 x^4 \coth ^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {1}{4} \left (-\left (6 a^2+1\right ) (a+b x)-\frac {1}{3} (a+b x)^3+2 a (a+b x)^2-\frac {1}{2} (1-a)^4 \log (-a-b x+1)+\frac {1}{2} (a+1)^4 \log (a+b x+1)\right )-\frac {1}{4} b^4 x^4 \coth ^{-1}(a+b x)}{b^4}\) |
-((-1/4*(b^4*x^4*ArcCoth[a + b*x]) + (-((1 + 6*a^2)*(a + b*x)) + 2*a*(a + b*x)^2 - (a + b*x)^3/3 - ((1 - a)^4*Log[1 - a - b*x])/2 + ((1 + a)^4*Log[1 + a + b*x])/2)/4)/b^4)
3.1.62.3.1 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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ a^p Int[ExpandIntegrand[(c + d*x)^n*(1 - Rt[-b/a, 2]*x)^p*(1 + Rt[-b/a, 2 ]*x)^p, x], x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[n] & & NiceSqrtQ[-b/a] && !FractionalPowerFactorQ[Rt[-b/a, 2]]
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_))^(q_.), x_Symbol ] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcCoth[c*x])/(e*(q + 1))), x] - Simp[b *(c/(e*(q + 1))) Int[(d + e*x)^(q + 1)/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
Int[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcCoth[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IG tQ[p, 0]
Time = 0.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.05
| method | result | size |
| parts | \(\frac {x^{4} \operatorname {arccoth}\left (b x +a \right )}{4}+\frac {b \left (\frac {\frac {1}{3} b^{2} x^{3}-a b \,x^{2}+3 a^{2} x +x}{b^{4}}+\frac {\left (a^{4}-4 a^{3}+6 a^{2}-4 a +1\right ) \ln \left (b x +a -1\right )}{2 b^{5}}+\frac {\left (-a^{4}-4 a^{3}-6 a^{2}-4 a -1\right ) \ln \left (b x +a +1\right )}{2 b^{5}}\right )}{4}\) | \(106\) |
| parallelrisch | \(-\frac {-3 \,\operatorname {arccoth}\left (b x +a \right ) x^{4} b^{4}-b^{3} x^{3}+3 a \,b^{2} x^{2}+3 \,\operatorname {arccoth}\left (b x +a \right ) a^{4}+12 \ln \left (b x +a -1\right ) a^{3}-9 a^{2} b x +12 \,\operatorname {arccoth}\left (b x +a \right ) a^{3}+18 \,\operatorname {arccoth}\left (b x +a \right ) a^{2}+15 a^{3}+12 \ln \left (b x +a -1\right ) a -3 b x +12 \,\operatorname {arccoth}\left (b x +a \right ) a +3 \,\operatorname {arccoth}\left (b x +a \right )+9 a}{12 b^{4}}\) | \(129\) |
| derivativedivides | \(\frac {\frac {\operatorname {arccoth}\left (b x +a \right ) a^{4}}{4}-\operatorname {arccoth}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\operatorname {arccoth}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arccoth}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\operatorname {arccoth}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}+\frac {3 a^{2} \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2} a}{2}+\frac {\left (b x +a \right )^{3}}{12}+\frac {b x}{4}+\frac {a}{4}+\frac {\left (a^{4}-4 a^{3}+6 a^{2}-4 a +1\right ) \ln \left (b x +a -1\right )}{8}-\frac {\left (a^{4}+4 a^{3}+6 a^{2}+4 a +1\right ) \ln \left (b x +a +1\right )}{8}}{b^{4}}\) | \(172\) |
| default | \(\frac {\frac {\operatorname {arccoth}\left (b x +a \right ) a^{4}}{4}-\operatorname {arccoth}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\operatorname {arccoth}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arccoth}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\operatorname {arccoth}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}+\frac {3 a^{2} \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2} a}{2}+\frac {\left (b x +a \right )^{3}}{12}+\frac {b x}{4}+\frac {a}{4}+\frac {\left (a^{4}-4 a^{3}+6 a^{2}-4 a +1\right ) \ln \left (b x +a -1\right )}{8}-\frac {\left (a^{4}+4 a^{3}+6 a^{2}+4 a +1\right ) \ln \left (b x +a +1\right )}{8}}{b^{4}}\) | \(172\) |
| risch | \(\frac {x^{4} \ln \left (b x +a +1\right )}{8}-\frac {x^{4} \ln \left (b x +a -1\right )}{8}+\frac {x^{3}}{12 b}-\frac {\ln \left (b x +a +1\right ) a^{4}}{8 b^{4}}+\frac {\ln \left (-b x -a +1\right ) a^{4}}{8 b^{4}}-\frac {x^{2} a}{4 b^{2}}-\frac {\ln \left (b x +a +1\right ) a^{3}}{2 b^{4}}-\frac {\ln \left (-b x -a +1\right ) a^{3}}{2 b^{4}}+\frac {3 x \,a^{2}}{4 b^{3}}-\frac {3 \ln \left (b x +a +1\right ) a^{2}}{4 b^{4}}+\frac {3 \ln \left (-b x -a +1\right ) a^{2}}{4 b^{4}}-\frac {\ln \left (b x +a +1\right ) a}{2 b^{4}}-\frac {\ln \left (-b x -a +1\right ) a}{2 b^{4}}+\frac {x}{4 b^{3}}-\frac {\ln \left (b x +a +1\right )}{8 b^{4}}+\frac {\ln \left (-b x -a +1\right )}{8 b^{4}}\) | \(213\) |
1/4*x^4*arccoth(b*x+a)+1/4*b*(1/b^4*(1/3*b^2*x^3-a*b*x^2+3*a^2*x+x)+1/2*(a ^4-4*a^3+6*a^2-4*a+1)/b^5*ln(b*x+a-1)+1/2*(-a^4-4*a^3-6*a^2-4*a-1)/b^5*ln( b*x+a+1))
Time = 0.25 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.11 \[ \int x^3 \coth ^{-1}(a+b x) \, dx=\frac {3 \, b^{4} x^{4} \log \left (\frac {b x + a + 1}{b x + a - 1}\right ) + 2 \, b^{3} x^{3} - 6 \, a b^{2} x^{2} + 6 \, {\left (3 \, a^{2} + 1\right )} b x - 3 \, {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right ) + 3 \, {\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (b x + a - 1\right )}{24 \, b^{4}} \]
1/24*(3*b^4*x^4*log((b*x + a + 1)/(b*x + a - 1)) + 2*b^3*x^3 - 6*a*b^2*x^2 + 6*(3*a^2 + 1)*b*x - 3*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*log(b*x + a + 1) + 3*(a^4 - 4*a^3 + 6*a^2 - 4*a + 1)*log(b*x + a - 1))/b^4
Time = 0.44 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.51 \[ \int x^3 \coth ^{-1}(a+b x) \, dx=\begin {cases} - \frac {a^{4} \operatorname {acoth}{\left (a + b x \right )}}{4 b^{4}} - \frac {a^{3} \log {\left (a + b x + 1 \right )}}{b^{4}} + \frac {a^{3} \operatorname {acoth}{\left (a + b x \right )}}{b^{4}} + \frac {3 a^{2} x}{4 b^{3}} - \frac {3 a^{2} \operatorname {acoth}{\left (a + b x \right )}}{2 b^{4}} - \frac {a x^{2}}{4 b^{2}} - \frac {a \log {\left (a + b x + 1 \right )}}{b^{4}} + \frac {a \operatorname {acoth}{\left (a + b x \right )}}{b^{4}} + \frac {x^{4} \operatorname {acoth}{\left (a + b x \right )}}{4} + \frac {x^{3}}{12 b} + \frac {x}{4 b^{3}} - \frac {\operatorname {acoth}{\left (a + b x \right )}}{4 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {acoth}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \]
Piecewise((-a**4*acoth(a + b*x)/(4*b**4) - a**3*log(a + b*x + 1)/b**4 + a* *3*acoth(a + b*x)/b**4 + 3*a**2*x/(4*b**3) - 3*a**2*acoth(a + b*x)/(2*b**4 ) - a*x**2/(4*b**2) - a*log(a + b*x + 1)/b**4 + a*acoth(a + b*x)/b**4 + x* *4*acoth(a + b*x)/4 + x**3/(12*b) + x/(4*b**3) - acoth(a + b*x)/(4*b**4), Ne(b, 0)), (x**4*acoth(a)/4, True))
Time = 0.20 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.05 \[ \int x^3 \coth ^{-1}(a+b x) \, dx=\frac {1}{4} \, x^{4} \operatorname {arcoth}\left (b x + a\right ) + \frac {1}{24} \, b {\left (\frac {2 \, {\left (b^{2} x^{3} - 3 \, a b x^{2} + 3 \, {\left (3 \, a^{2} + 1\right )} x\right )}}{b^{4}} - \frac {3 \, {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{5}} + \frac {3 \, {\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{5}}\right )} \]
1/4*x^4*arccoth(b*x + a) + 1/24*b*(2*(b^2*x^3 - 3*a*b*x^2 + 3*(3*a^2 + 1)* x)/b^4 - 3*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*log(b*x + a + 1)/b^5 + 3*(a^4 - 4*a^3 + 6*a^2 - 4*a + 1)*log(b*x + a - 1)/b^5)
Leaf count of result is larger than twice the leaf count of optimal. 512 vs. \(2 (87) = 174\).
Time = 0.30 (sec) , antiderivative size = 512, normalized size of antiderivative = 5.07 \[ \int x^3 \coth ^{-1}(a+b x) \, dx=-\frac {1}{6} \, {\left ({\left (a + 1\right )} b - {\left (a - 1\right )} b\right )} {\left (\frac {3 \, {\left (a^{3} + a\right )} \log \left (\frac {{\left | b x + a + 1 \right |}}{{\left | b x + a - 1 \right |}}\right )}{b^{5}} - \frac {3 \, {\left (a^{3} + a\right )} \log \left ({\left | \frac {b x + a + 1}{b x + a - 1} - 1 \right |}\right )}{b^{5}} - \frac {9 \, a^{2} + \frac {3 \, {\left (3 \, a^{2} - 2 \, a + 1\right )} {\left (b x + a + 1\right )}^{2}}{{\left (b x + a - 1\right )}^{2}} - \frac {3 \, {\left (6 \, a^{2} - 2 \, a + 1\right )} {\left (b x + a + 1\right )}}{b x + a - 1} + 2}{b^{5} {\left (\frac {b x + a + 1}{b x + a - 1} - 1\right )}^{3}} + \frac {3 \, {\left (\frac {{\left (b x + a + 1\right )}^{3} a^{3}}{{\left (b x + a - 1\right )}^{3}} - \frac {3 \, {\left (b x + a + 1\right )}^{2} a^{3}}{{\left (b x + a - 1\right )}^{2}} + \frac {3 \, {\left (b x + a + 1\right )} a^{3}}{b x + a - 1} - a^{3} - \frac {3 \, {\left (b x + a + 1\right )}^{3} a^{2}}{{\left (b x + a - 1\right )}^{3}} + \frac {6 \, {\left (b x + a + 1\right )}^{2} a^{2}}{{\left (b x + a - 1\right )}^{2}} - \frac {3 \, {\left (b x + a + 1\right )} a^{2}}{b x + a - 1} + \frac {3 \, {\left (b x + a + 1\right )}^{3} a}{{\left (b x + a - 1\right )}^{3}} - \frac {3 \, {\left (b x + a + 1\right )}^{2} a}{{\left (b x + a - 1\right )}^{2}} + \frac {{\left (b x + a + 1\right )} a}{b x + a - 1} - a - \frac {{\left (b x + a + 1\right )}^{3}}{{\left (b x + a - 1\right )}^{3}} - \frac {b x + a + 1}{b x + a - 1}\right )} \log \left (-\frac {\frac {1}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} {\left (a - 1\right )}}{b x + a - 1} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a - 1} - b}} + 1}{\frac {1}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} {\left (a - 1\right )}}{b x + a - 1} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a - 1} - b}} - 1}\right )}{b^{5} {\left (\frac {b x + a + 1}{b x + a - 1} - 1\right )}^{4}}\right )} \]
-1/6*((a + 1)*b - (a - 1)*b)*(3*(a^3 + a)*log(abs(b*x + a + 1)/abs(b*x + a - 1))/b^5 - 3*(a^3 + a)*log(abs((b*x + a + 1)/(b*x + a - 1) - 1))/b^5 - ( 9*a^2 + 3*(3*a^2 - 2*a + 1)*(b*x + a + 1)^2/(b*x + a - 1)^2 - 3*(6*a^2 - 2 *a + 1)*(b*x + a + 1)/(b*x + a - 1) + 2)/(b^5*((b*x + a + 1)/(b*x + a - 1) - 1)^3) + 3*((b*x + a + 1)^3*a^3/(b*x + a - 1)^3 - 3*(b*x + a + 1)^2*a^3/ (b*x + a - 1)^2 + 3*(b*x + a + 1)*a^3/(b*x + a - 1) - a^3 - 3*(b*x + a + 1 )^3*a^2/(b*x + a - 1)^3 + 6*(b*x + a + 1)^2*a^2/(b*x + a - 1)^2 - 3*(b*x + a + 1)*a^2/(b*x + a - 1) + 3*(b*x + a + 1)^3*a/(b*x + a - 1)^3 - 3*(b*x + a + 1)^2*a/(b*x + a - 1)^2 + (b*x + a + 1)*a/(b*x + a - 1) - a - (b*x + a + 1)^3/(b*x + a - 1)^3 - (b*x + a + 1)/(b*x + a - 1))*log(-(1/(a - ((b*x + a + 1)*(a - 1)/(b*x + a - 1) - a - 1)*b/((b*x + a + 1)*b/(b*x + a - 1) - b)) + 1)/(1/(a - ((b*x + a + 1)*(a - 1)/(b*x + a - 1) - a - 1)*b/((b*x + a + 1)*b/(b*x + a - 1) - b)) - 1))/(b^5*((b*x + a + 1)/(b*x + a - 1) - 1)^ 4))
Time = 4.52 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.33 \[ \int x^3 \coth ^{-1}(a+b x) \, dx=\frac {x^4\,\ln \left (\frac {1}{a+b\,x}+1\right )}{8}-x\,\left (\frac {4\,a^2-4}{16\,b^3}-\frac {a^2}{b^3}\right )-\frac {x^4\,\ln \left (1-\frac {1}{a+b\,x}\right )}{8}+\frac {x^3}{12\,b}-\frac {a\,x^2}{4\,b^2}+\frac {\ln \left (a+b\,x-1\right )\,\left (a^4-4\,a^3+6\,a^2-4\,a+1\right )}{8\,b^4}-\frac {\ln \left (a+b\,x+1\right )\,\left (a^4+4\,a^3+6\,a^2+4\,a+1\right )}{8\,b^4} \]