Integrand size = 12, antiderivative size = 97 \[ \int \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d}-\frac {2 b \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d}-\frac {b^2 \operatorname {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{d} \]
(a+b*arccoth(d*x+c))^2/d+(d*x+c)*(a+b*arccoth(d*x+c))^2/d-2*b*(a+b*arccoth (d*x+c))*ln(2/(-d*x-c+1))/d-b^2*polylog(2,(-d*x-c-1)/(-d*x-c+1))/d
Time = 0.14 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.14 \[ \int \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\frac {b^2 (-1+c+d x) \coth ^{-1}(c+d x)^2+2 b \coth ^{-1}(c+d x) \left (a c+a d x-b \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )+a \left (a c+a d x-2 b \log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )\right )+b^2 \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )}{d} \]
(b^2*(-1 + c + d*x)*ArcCoth[c + d*x]^2 + 2*b*ArcCoth[c + d*x]*(a*c + a*d*x - b*Log[1 - E^(-2*ArcCoth[c + d*x])]) + a*(a*c + a*d*x - 2*b*Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])]) + b^2*PolyLog[2, E^(-2*ArcCoth[c + d*x])] )/d
Time = 0.50 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6654, 6437, 6547, 6471, 2849, 2752}
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.
| \(\displaystyle \int \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 6654 |
| \(\displaystyle \frac {\int \left (a+b \coth ^{-1}(c+d x)\right )^2d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6437 |
| \(\displaystyle \frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2-2 b \int \frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )}{1-(c+d x)^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6547 |
| \(\displaystyle \frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2-2 b \left (\int \frac {a+b \coth ^{-1}(c+d x)}{-c-d x+1}d(c+d x)-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{2 b}\right )}{d}\) |
| \(\Big \downarrow \) 6471 |
| \(\displaystyle \frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2-2 b \left (-b \int \frac {\log \left (\frac {2}{-c-d x+1}\right )}{1-(c+d x)^2}d(c+d x)-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{2 b}+\log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )\right )}{d}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2-2 b \left (b \int \frac {\log \left (\frac {2}{-c-d x+1}\right )}{1-\frac {2}{-c-d x+1}}d\frac {1}{-c-d x+1}-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{2 b}+\log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )\right )}{d}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2-2 b \left (-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{2 b}+\log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )+\frac {1}{2} b \operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right )\right )}{d}\) |
((c + d*x)*(a + b*ArcCoth[c + d*x])^2 - 2*b*(-1/2*(a + b*ArcCoth[c + d*x]) ^2/b + (a + b*ArcCoth[c + d*x])*Log[2/(1 - c - d*x)] + (b*PolyLog[2, 1 - 2 /(1 - c - d*x)])/2))/d
3.2.11.3.1 Defintions of rubi rules used
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcCoth[c*x^n]) ^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol ] :> Simp[(-(a + b*ArcCoth[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c *(p/e) Int[(a + b*ArcCoth[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^2*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2 , 0]
Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCoth[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ (c*d) Int[(a + b*ArcCoth[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Int[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcCoth[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d }, x] && IGtQ[p, 0]
Time = 0.44 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.79
| method | result | size |
| parts | \(a^{2} x +\frac {b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{2}-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )}{d}+\frac {2 a b \left (\left (d x +c \right ) \operatorname {arccoth}\left (d x +c \right )+\frac {\ln \left (\left (d x +c \right )^{2}-1\right )}{2}\right )}{d}\) | \(174\) |
| derivativedivides | \(\frac {\left (d x +c \right ) a^{2}+b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{2}-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )+2 a b \left (d x +c \right ) \operatorname {arccoth}\left (d x +c \right )+a b \ln \left (\left (d x +c \right )^{2}-1\right )}{d}\) | \(175\) |
| default | \(\frac {\left (d x +c \right ) a^{2}+b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{2}-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )+2 a b \left (d x +c \right ) \operatorname {arccoth}\left (d x +c \right )+a b \ln \left (\left (d x +c \right )^{2}-1\right )}{d}\) | \(175\) |
| risch | \(a^{2} x -\frac {b^{2} \ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{d}+\frac {\ln \left (d x +c -1\right )^{2} b^{2} c}{4 d}-\ln \left (d x +c -1\right ) a b x +\frac {\ln \left (d x +c -1\right ) a b}{d}+\frac {a b \ln \left (d x +c +1\right )}{d}+\frac {a^{2} c}{d}+\left (-\frac {b^{2} x \ln \left (d x +c -1\right )}{2}+\frac {b \left (2 a d x -b \ln \left (d x +c -1\right ) c +b \ln \left (d x +c -1\right )\right )}{2 d}\right ) \ln \left (d x +c +1\right )+\frac {\left (d x +c +1\right ) b^{2} \ln \left (d x +c +1\right )^{2}}{4 d}+\frac {\ln \left (d x +c -1\right )^{2} b^{2} x}{4}-\frac {\ln \left (d x +c -1\right )^{2} b^{2}}{4 d}-\frac {b^{2} \operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{d}-\frac {a^{2}}{d}+\frac {a b \ln \left (d x +c +1\right ) c}{d}-\frac {\ln \left (d x +c -1\right ) a b c}{d}\) | \(260\) |
a^2*x+b^2/d*(arccoth(d*x+c)^2*(d*x+c-1)+2*arccoth(d*x+c)^2-2*arccoth(d*x+c )*ln(1-1/((d*x+c-1)/(d*x+c+1))^(1/2))-2*polylog(2,1/((d*x+c-1)/(d*x+c+1))^ (1/2))-2*arccoth(d*x+c)*ln(1+1/((d*x+c-1)/(d*x+c+1))^(1/2))-2*polylog(2,-1 /((d*x+c-1)/(d*x+c+1))^(1/2)))+2*a*b/d*((d*x+c)*arccoth(d*x+c)+1/2*ln((d*x +c)^2-1))
\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{2}\, dx \]
\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
a^2*x + 1/4*b^2*((d*x*log(d*x + c - 1)^2 + (d*x + c + 1)*log(d*x + c + 1)^ 2 - 2*(d*x + c - 1)*log(d*x + c + 1)*log(d*x + c - 1))/d + integrate(2*(c^ 2 + (c*d - 3*d)*x - 2*c + 1)*log(d*x + c - 1)/(d^2*x^2 + 2*c*d*x + c^2 - 1 ), x)) + (2*(d*x + c)*arccoth(d*x + c) + log(-(d*x + c)^2 + 1))*a*b/d
\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int {\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^2 \,d x \]