Integrand size = 32, antiderivative size = 83 \[ \int \frac {(c e+d e x) (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2} \, dx=-\frac {e (a+b \text {arctanh}(c+d x))^2}{2 b d}+\frac {e (a+b \text {arctanh}(c+d x)) \log \left (\frac {2}{1-c-d x}\right )}{d}+\frac {b e \operatorname {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{2 d} \]
-1/2*e*(a+b*arctanh(d*x+c))^2/b/d+e*(a+b*arctanh(d*x+c))*ln(2/(-d*x-c+1))/ d+1/2*b*e*polylog(2,(-d*x-c-1)/(-d*x-c+1))/d
Time = 0.07 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.81 \[ \int \frac {(c e+d e x) (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2} \, dx=e \left (-\frac {a \log (1-c-d x)}{2 d}+\frac {b \log ^2(1-c-d x)}{8 d}-\frac {b \log (2) \log (-1+c+d x)}{4 d}-\frac {a \log (1+c+d x)}{2 d}+\frac {b \log (2) \log (1+c+d x)}{4 d}-\frac {b \log ^2(1+c+d x)}{8 d}+\frac {b \operatorname {PolyLog}\left (2,\frac {1}{2} (1-c-d x)\right )}{4 d}-\frac {b \operatorname {PolyLog}\left (2,\frac {1}{2} (1+c+d x)\right )}{4 d}\right ) \]
e*(-1/2*(a*Log[1 - c - d*x])/d + (b*Log[1 - c - d*x]^2)/(8*d) - (b*Log[2]* Log[-1 + c + d*x])/(4*d) - (a*Log[1 + c + d*x])/(2*d) + (b*Log[2]*Log[1 + c + d*x])/(4*d) - (b*Log[1 + c + d*x]^2)/(8*d) + (b*PolyLog[2, (1 - c - d* x)/2])/(4*d) - (b*PolyLog[2, (1 + c + d*x)/2])/(4*d))
Time = 0.59 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.87, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {7281, 27, 6546, 6470, 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.
| \(\displaystyle \int \frac {(c e+d e x) (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 7281 |
| \(\displaystyle \frac {\int \frac {e (c+d x) (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int \frac {(c+d x) (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {e \left (\int \frac {a+b \text {arctanh}(c+d x)}{-c-d x+1}d(c+d x)-\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}\right )}{d}\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {e \left (-b \int \frac {\log \left (\frac {2}{-c-d x+1}\right )}{1-(c+d x)^2}d(c+d x)-\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}+\log \left (\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))\right )}{d}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {e \left (b \int \frac {\log \left (\frac {2}{-c-d x+1}\right )}{1-\frac {2}{-c-d x+1}}d\frac {1}{-c-d x+1}-\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}+\log \left (\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))\right )}{d}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {e \left (-\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}+\log \left (\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))+\frac {1}{2} b \operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right )\right )}{d}\) |
(e*(-1/2*(a + b*ArcTanh[c + d*x])^2/b + (a + b*ArcTanh[c + d*x])*Log[2/(1 - c - d*x)] + (b*PolyLog[2, 1 - 2/(1 - c - d*x)])/2))/d
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[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_.) + 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[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ (c*d) Int[(a + b*ArcTanh[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[u_, x_Symbol] :> With[{lst = FunctionOfLinear[u, x]}, Simp[1/lst[[3]] Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x] /; !FalseQ[lst]]
Time = 0.19 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.80
| method | result | size |
| derivativedivides | \(\frac {-a e \left (\frac {\ln \left (d x +c -1\right )}{2}+\frac {\ln \left (d x +c +1\right )}{2}\right )-b e \left (\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{2}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{2}+\frac {\ln \left (d x +c -1\right )^{2}}{8}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (d x +c +1\right )^{2}}{8}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{4}\right )}{d}\) | \(149\) |
| default | \(\frac {-a e \left (\frac {\ln \left (d x +c -1\right )}{2}+\frac {\ln \left (d x +c +1\right )}{2}\right )-b e \left (\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{2}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{2}+\frac {\ln \left (d x +c -1\right )^{2}}{8}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (d x +c +1\right )^{2}}{8}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{4}\right )}{d}\) | \(149\) |
| risch | \(\frac {e b \ln \left (-d x -c +1\right )^{2}}{8 d}+\frac {e b \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right ) \ln \left (-d x -c +1\right )}{4 d}-\frac {e b \operatorname {dilog}\left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{4 d}-\frac {e a \ln \left (\left (-d x -c +1\right ) \left (-d x -c -1\right )\right )}{2 d}-\frac {b e \ln \left (d x +c +1\right )^{2}}{8 d}-\frac {b e \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right ) \ln \left (d x +c +1\right )}{4 d}+\frac {b e \operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{4 d}\) | \(149\) |
| parts | \(-\frac {a e \ln \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}{2 d}-\frac {b e \left (\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{2}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{2}+\frac {\ln \left (d x +c -1\right )^{2}}{8}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (d x +c +1\right )^{2}}{8}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{4}\right )}{d}\) | \(150\) |
1/d*(-a*e*(1/2*ln(d*x+c-1)+1/2*ln(d*x+c+1))-b*e*(1/2*arctanh(d*x+c)*ln(d*x +c-1)+1/2*arctanh(d*x+c)*ln(d*x+c+1)+1/8*ln(d*x+c-1)^2-1/2*dilog(1/2*d*x+1 /2*c+1/2)-1/4*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2)-1/8*ln(d*x+c+1)^2+1/4*(ln( d*x+c+1)-ln(1/2*d*x+1/2*c+1/2))*ln(-1/2*d*x-1/2*c+1/2)))
\[ \int \frac {(c e+d e x) (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2} \, dx=\int { -\frac {{\left (d e x + c e\right )} {\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}}{{\left (d x + c\right )}^{2} - 1} \,d x } \]
integral(-(a*d*e*x + a*c*e + (b*d*e*x + b*c*e)*arctanh(d*x + c))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x)
\[ \int \frac {(c e+d e x) (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2} \, dx=- e \left (\int \frac {a c}{c^{2} + 2 c d x + d^{2} x^{2} - 1}\, dx + \int \frac {a d x}{c^{2} + 2 c d x + d^{2} x^{2} - 1}\, dx + \int \frac {b c \operatorname {atanh}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2} - 1}\, dx + \int \frac {b d x \operatorname {atanh}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2} - 1}\, dx\right ) \]
-e*(Integral(a*c/(c**2 + 2*c*d*x + d**2*x**2 - 1), x) + Integral(a*d*x/(c* *2 + 2*c*d*x + d**2*x**2 - 1), x) + Integral(b*c*atanh(c + d*x)/(c**2 + 2* c*d*x + d**2*x**2 - 1), x) + Integral(b*d*x*atanh(c + d*x)/(c**2 + 2*c*d*x + d**2*x**2 - 1), x))
\[ \int \frac {(c e+d e x) (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2} \, dx=\int { -\frac {{\left (d e x + c e\right )} {\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}}{{\left (d x + c\right )}^{2} - 1} \,d x } \]
1/2*b*c*e*(log(d*x + c + 1)/d - log(d*x + c - 1)/d)*arctanh(d*x + c) - 1/2 *a*d*e*((c + 1)*log(d*x + c + 1)/d^2 - (c - 1)*log(d*x + c - 1)/d^2) + 1/2 *a*c*e*(log(d*x + c + 1)/d - log(d*x + c - 1)/d) + 1/8*b*d*e*((2*(c + 1)*l og(d*x + c + 1)*log(-d*x - c + 1) - (c - 1)*log(-d*x - c + 1)^2)/d^2 - 4*i ntegrate(1/2*(c^2 + (c*d + 3*d)*x + 2*c + 1)*log(d*x + c + 1)/(d^3*x^2 + 2 *c*d^2*x + c^2*d - d), x)) - 1/8*(log(d*x + c + 1)^2 - 2*log(d*x + c + 1)* log(d*x + c - 1) + log(d*x + c - 1)^2)*b*c*e/d
\[ \int \frac {(c e+d e x) (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2} \, dx=\int { -\frac {{\left (d e x + c e\right )} {\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}}{{\left (d x + c\right )}^{2} - 1} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x) (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2} \, dx=\int -\frac {\left (c\,e+d\,e\,x\right )\,\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}{{\left (c+d\,x\right )}^2-1} \,d x \]