Integrand size = 19, antiderivative size = 84 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=-\frac {(a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{c d}+\frac {b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c d}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 c d} \]
-(a+b*arctanh(c*x))^2*ln(2/(c*x+1))/c/d+b*(a+b*arctanh(c*x))*polylog(2,1-2 /(c*x+1))/c/d+1/2*b^2*polylog(3,1-2/(c*x+1))/c/d
Time = 0.44 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\frac {-4 a b \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-2 b^2 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+2 a^2 \log (1+c x)+2 b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+b^2 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )}{2 c d} \]
(-4*a*b*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 2*b^2*ArcTanh[c*x]^2*L og[1 + E^(-2*ArcTanh[c*x])] + 2*a^2*Log[1 + c*x] + 2*b*(a + b*ArcTanh[c*x] )*PolyLog[2, -E^(-2*ArcTanh[c*x])] + b^2*PolyLog[3, -E^(-2*ArcTanh[c*x])]) /(2*c*d)
Time = 0.49 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {6470, 6618, 7164}
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.
| \(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^2}{c d x+d} \, dx\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {2 b \int \frac {(a+b \text {arctanh}(c x)) \log \left (\frac {2}{c x+1}\right )}{1-c^2 x^2}dx}{d}-\frac {\log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{c d}\) |
| \(\Big \downarrow \) 6618 |
| \(\displaystyle \frac {2 b \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{2 c}-\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx\right )}{d}-\frac {\log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{c d}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {2 b \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{2 c}+\frac {b \operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{4 c}\right )}{d}-\frac {\log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{c d}\) |
-(((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c*d)) + (2*b*(((a + b*ArcTanh [c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(2*c) + (b*PolyLog[3, 1 - 2/(1 + c*x)] )/(4*c)))/d
3.1.98.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol ] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c *(p/e) Int[(a + b*ArcTanh[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[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x ] - Simp[b*(p/2) Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[2, 1 - u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 + c*x))^2, 0]
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /; !FalseQ[w]] /; FreeQ[n, x]
Leaf count of result is larger than twice the leaf count of optimal. \(286\) vs. \(2(82)=164\).
Time = 1.84 (sec) , antiderivative size = 287, normalized size of antiderivative = 3.42
| method | result | size |
| risch | \(\frac {\ln \left (-c x +1\right )^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) b^{2}}{4 c d}-\frac {\ln \left (-c x +1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) a b}{c d}+\frac {\ln \left (-c x +1\right ) \operatorname {polylog}\left (2, -\frac {c x}{2}+\frac {1}{2}\right ) b^{2}}{2 c d}+\frac {a b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{c d}+\frac {a^{2} \ln \left (-c x -1\right )}{c d}+\frac {\operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right ) a b}{c d}-\frac {\operatorname {polylog}\left (3, -\frac {c x}{2}+\frac {1}{2}\right ) b^{2}}{2 c d}+\frac {b^{2} \ln \left (c x +1\right )^{3}}{12 d c}+\frac {b a \ln \left (c x +1\right )^{2}}{2 d c}-\frac {b^{2} \ln \left (c x +1\right )^{2} \ln \left (-c x +1\right )}{4 d c}+\frac {b^{2} \ln \left (c x +1\right )^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{4 d c}+\frac {b^{2} \ln \left (c x +1\right ) \operatorname {polylog}\left (2, \frac {c x}{2}+\frac {1}{2}\right )}{2 d c}-\frac {b^{2} \operatorname {polylog}\left (3, \frac {c x}{2}+\frac {1}{2}\right )}{2 d c}\) | \(287\) |
| derivativedivides | \(\frac {\frac {a^{2} \ln \left (c x +1\right )}{d}+\frac {b^{2} \left (\operatorname {arctanh}\left (c x \right )^{2} \ln \left (c x +1\right )-2 \operatorname {arctanh}\left (c x \right )^{2} \ln \left (\frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )+\frac {2 \operatorname {arctanh}\left (c x \right )^{3}}{3}-\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )^{2}+i \pi {\operatorname {csgn}\left (\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )^{3}+2 \ln \left (2\right )\right ) \operatorname {arctanh}\left (c x \right )^{2}}{2}-\operatorname {arctanh}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )}{2}\right )}{d}+\frac {2 a b \left (\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c x +1\right )^{2}}{4}\right )}{d}}{c}\) | \(637\) |
| default | \(\frac {\frac {a^{2} \ln \left (c x +1\right )}{d}+\frac {b^{2} \left (\operatorname {arctanh}\left (c x \right )^{2} \ln \left (c x +1\right )-2 \operatorname {arctanh}\left (c x \right )^{2} \ln \left (\frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )+\frac {2 \operatorname {arctanh}\left (c x \right )^{3}}{3}-\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )^{2}+i \pi {\operatorname {csgn}\left (\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )^{3}+2 \ln \left (2\right )\right ) \operatorname {arctanh}\left (c x \right )^{2}}{2}-\operatorname {arctanh}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )}{2}\right )}{d}+\frac {2 a b \left (\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c x +1\right )^{2}}{4}\right )}{d}}{c}\) | \(637\) |
| parts | \(\frac {a^{2} \ln \left (c x +1\right )}{d c}+\frac {b^{2} \left (\operatorname {arctanh}\left (c x \right )^{2} \ln \left (c x +1\right )-2 \operatorname {arctanh}\left (c x \right )^{2} \ln \left (\frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )+\frac {2 \operatorname {arctanh}\left (c x \right )^{3}}{3}-\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )^{2}+i \pi {\operatorname {csgn}\left (\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )^{3}+2 \ln \left (2\right )\right ) \operatorname {arctanh}\left (c x \right )^{2}}{2}-\operatorname {arctanh}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )}{2}\right )}{d c}+\frac {2 a b \left (\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c x +1\right )^{2}}{4}\right )}{d c}\) | \(642\) |
1/4/c/d*ln(-c*x+1)^2*ln(1/2*c*x+1/2)*b^2-1/c/d*ln(-c*x+1)*ln(1/2*c*x+1/2)* a*b+1/2/c/d*ln(-c*x+1)*polylog(2,-1/2*c*x+1/2)*b^2+1/c/d*a*b*ln(1/2*c*x+1/ 2)*ln(-1/2*c*x+1/2)+1/c/d*a^2*ln(-c*x-1)+1/c/d*dilog(-1/2*c*x+1/2)*a*b-1/2 /c/d*polylog(3,-1/2*c*x+1/2)*b^2+1/12*b^2/d/c*ln(c*x+1)^3+1/2*b/d*a*ln(c*x +1)^2/c-1/4*b^2/d/c*ln(c*x+1)^2*ln(-c*x+1)+1/4*b^2/d/c*ln(c*x+1)^2*ln(-1/2 *c*x+1/2)+1/2*b^2/d/c*ln(c*x+1)*polylog(2,1/2*c*x+1/2)-1/2*b^2/d/c*polylog (3,1/2*c*x+1/2)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{c d x + d} \,d x } \]
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\frac {\int \frac {a^{2}}{c x + 1}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c x + 1}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{c x + 1}\, dx}{d} \]
(Integral(a**2/(c*x + 1), x) + Integral(b**2*atanh(c*x)**2/(c*x + 1), x) + Integral(2*a*b*atanh(c*x)/(c*x + 1), x))/d
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{c d x + d} \,d x } \]
1/4*b^2*log(c*x + 1)*log(-c*x + 1)^2/(c*d) + a^2*log(c*d*x + d)/(c*d) - in tegrate(-1/4*((b^2*c*x - b^2)*log(c*x + 1)^2 + 4*(a*b*c*x - a*b)*log(c*x + 1) - 4*(b^2*c*x*log(c*x + 1) + a*b*c*x - a*b)*log(-c*x + 1))/(c^2*d*x^2 - d), x)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{c d x + d} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{d+c\,d\,x} \,d x \]