Integrand size = 38, antiderivative size = 89 \[ \int \frac {a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=-\frac {a \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+c x}}{\sqrt {1-c x}}\right )}{2 c}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {1+c x}}{\sqrt {1-c x}}\right )}{2 c} \]
-a*ln((-c*x+1)^(1/2)/(c*x+1)^(1/2))/c-1/2*b*polylog(2,-(c*x+1)^(1/2)/(-c*x +1)^(1/2))/c+1/2*b*polylog(2,(c*x+1)^(1/2)/(-c*x+1)^(1/2))/c
Time = 0.34 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\frac {a \text {arctanh}(c x)}{c}+\frac {b \left (\text {arctanh}(c x) \left (2 \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )+\log \left (1-e^{-\text {arctanh}(c x)}\right )-\log \left (1+e^{-\text {arctanh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arctanh}(c x)}\right )\right )}{2 c} \]
(a*ArcTanh[c*x])/c + (b*(ArcTanh[c*x]*(2*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c* x]] + Log[1 - E^(-ArcTanh[c*x])] - Log[1 + E^(-ArcTanh[c*x])]) + PolyLog[2 , -E^(-ArcTanh[c*x])] - PolyLog[2, E^(-ArcTanh[c*x])]))/(2*c)
Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {7232, 6447}
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.
| \(\displaystyle \int \frac {a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{1-c^2 x^2} \, dx\) |
| \(\Big \downarrow \) 7232 |
| \(\displaystyle -\frac {\int \frac {\sqrt {c x+1} \left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{\sqrt {1-c x}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}{c}\) |
| \(\Big \downarrow \) 6447 |
| \(\displaystyle -\frac {a \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )+\frac {1}{2} b \operatorname {PolyLog}\left (2,-\frac {\sqrt {c x+1}}{\sqrt {1-c x}}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {\sqrt {c x+1}}{\sqrt {1-c x}}\right )}{c}\) |
-((a*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]] + (b*PolyLog[2, -(Sqrt[1 + c*x]/Sqrt [1 - c*x])])/2 - (b*PolyLog[2, Sqrt[1 + c*x]/Sqrt[1 - c*x]])/2)/c)
3.2.25.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x ] + (Simp[(b/2)*PolyLog[2, -(c*x)^(-1)], x] - Simp[(b/2)*PolyLog[2, 1/(c*x) ], x]) /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(F_)[((c_.)*Sqrt[(d_.) + (e_.)*(x_)])/Sqrt[(f_.) + (g_.) *(x_)]])^(n_.)/((A_.) + (C_.)*(x_)^2), x_Symbol] :> Simp[2*e*(g/(C*(e*f - d *g))) Subst[Int[(a + b*F[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]], x] /; FreeQ[{a, b, c, d, e, f, g, A, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0] && IGtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(371\) vs. \(2(73)=146\).
Time = 0.67 (sec) , antiderivative size = 372, normalized size of antiderivative = 4.18
| method | result | size |
| default | \(-\frac {a \ln \left (c x -1\right )}{2 c}+\frac {a \ln \left (c x +1\right )}{2 c}-b \left (-\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}-\frac {\operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}+\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\right )}{c}+\frac {\operatorname {polylog}\left (2, -\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\right )}{2 c}-\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1-\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}-\frac {\operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}\right )\) | \(372\) |
| parts | \(-\frac {a \ln \left (c x -1\right )}{2 c}+\frac {a \ln \left (c x +1\right )}{2 c}-b \left (-\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}-\frac {\operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}+\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\right )}{c}+\frac {\operatorname {polylog}\left (2, -\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\right )}{2 c}-\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1-\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}-\frac {\operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}\right )\) | \(372\) |
-1/2*a/c*ln(c*x-1)+1/2*a/c*ln(c*x+1)-b*(-1/c*arccoth((-c*x+1)^(1/2)/(c*x+1 )^(1/2))*ln(1+1/(((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)/((-c*x+1)^(1/2)/(c*x+1)^ (1/2)+1))^(1/2))-1/c*polylog(2,-1/(((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)/((-c*x +1)^(1/2)/(c*x+1)^(1/2)+1))^(1/2))+1/c*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2 ))*ln(1+1/((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)*((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1 ))+1/2/c*polylog(2,-1/((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)*((-c*x+1)^(1/2)/(c* x+1)^(1/2)+1))-1/c*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2))*ln(1-1/(((-c*x+1) ^(1/2)/(c*x+1)^(1/2)-1)/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1))^(1/2))-1/c*polyl og(2,1/(((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)) ^(1/2)))
\[ \int \frac {a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\int { -\frac {b \operatorname {arcoth}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a}{c^{2} x^{2} - 1} \,d x } \]
\[ \int \frac {a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=- \int \frac {a}{c^{2} x^{2} - 1}\, dx - \int \frac {b \operatorname {acoth}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \]
-Integral(a/(c**2*x**2 - 1), x) - Integral(b*acoth(sqrt(-c*x + 1)/sqrt(c*x + 1))/(c**2*x**2 - 1), x)
\[ \int \frac {a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\int { -\frac {b \operatorname {arcoth}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a}{c^{2} x^{2} - 1} \,d x } \]
1/4*b*(((log(c*x + 1) - log(-c*x + 1))*log(sqrt(c*x + 1) + sqrt(-c*x + 1)) - (log(c*x + 1) - log(-c*x + 1))*log(-sqrt(c*x + 1) + sqrt(-c*x + 1)))/c - 2*integrate(-1/2*sqrt(c*x + 1)*(log(c*x + 1) - log(-c*x + 1))/((c^2*x^2 - 1)*sqrt(c*x + 1) + (c^2*x^2 - 1)*sqrt(-c*x + 1)), x) - 2*integrate(1/2*s qrt(c*x + 1)*(log(c*x + 1) - log(-c*x + 1))/((c^2*x^2 - 1)*sqrt(c*x + 1) - (c^2*x^2 - 1)*sqrt(-c*x + 1)), x)) + 1/2*a*(log(c*x + 1)/c - log(c*x - 1) /c)
\[ \int \frac {a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\int { -\frac {b \operatorname {arcoth}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a}{c^{2} x^{2} - 1} \,d x } \]
Timed out. \[ \int \frac {a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\int -\frac {a+b\,\mathrm {acoth}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )}{c^2\,x^2-1} \,d x \]