Integrand size = 20, antiderivative size = 46 \[ \int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)} \, dx=\frac {(a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )}{d}-\frac {b \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )}{2 d} \]
Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.20 \[ \int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)} \, dx=\frac {2 b \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )+2 a \log (x)-2 a \log (1+c x)-b \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )}{2 d} \]
(2*b*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] + 2*a*Log[x] - 2*a*Log[1 + c*x] - b*PolyLog[2, E^(-2*ArcTanh[c*x])])/(2*d)
Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6494, 2897}
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.
\(\displaystyle \int \frac {a+b \text {arctanh}(c x)}{x (c d x+d)} \, dx\) |
\(\Big \downarrow \) 6494 |
\(\displaystyle \frac {\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d}-\frac {b c \int \frac {\log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx}{d}\) |
\(\Big \downarrow \) 2897 |
\(\displaystyle \frac {\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d}-\frac {b \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )}{2 d}\) |
3.1.47.3.1 Defintions of rubi rules used
Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x _Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Simp[b*c*(p/d) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 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]
Leaf count of result is larger than twice the leaf count of optimal. \(114\) vs. \(2(44)=88\).
Time = 1.02 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.50
method | result | size |
parts | \(\frac {a \left (-\ln \left (c x +1\right )+\ln \left (x \right )\right )}{d}+\frac {b \left (-\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )+\ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (c x +1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}\right )}{d}\) | \(115\) |
derivativedivides | \(\frac {a \left (-\ln \left (c x +1\right )+\ln \left (c x \right )\right )}{d}+\frac {b \left (-\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )+\ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (c x +1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}\right )}{d}\) | \(117\) |
default | \(\frac {a \left (-\ln \left (c x +1\right )+\ln \left (c x \right )\right )}{d}+\frac {b \left (-\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )+\ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (c x +1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}\right )}{d}\) | \(117\) |
risch | \(\frac {b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{2 d}-\frac {b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d}-\frac {a \ln \left (-c x -1\right )}{d}-\frac {b \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d}+\frac {\ln \left (-c x \right ) a}{d}+\frac {\operatorname {dilog}\left (-c x +1\right ) b}{2 d}-\frac {b \operatorname {dilog}\left (c x +1\right )}{2 d}-\frac {b \ln \left (c x +1\right )^{2}}{4 d}\) | \(117\) |
a/d*(-ln(c*x+1)+ln(x))+b/d*(-arctanh(c*x)*ln(c*x+1)+ln(c*x)*arctanh(c*x)-1 /2*(ln(c*x+1)-ln(1/2*c*x+1/2))*ln(-1/2*c*x+1/2)+1/2*dilog(1/2*c*x+1/2)+1/4 *ln(c*x+1)^2-1/2*dilog(c*x+1)-1/2*ln(c*x)*ln(c*x+1)-1/2*dilog(c*x))
\[ \int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )} x} \,d x } \]
\[ \int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)} \, dx=\frac {\int \frac {a}{c x^{2} + x}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{c x^{2} + x}\, dx}{d} \]
\[ \int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )} x} \,d x } \]
-a*(log(c*x + 1)/d - log(x)/d) + 1/2*b*integrate((log(c*x + 1) - log(-c*x + 1))/(c*d*x^2 + d*x), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )} x} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{x\,\left (d+c\,d\,x\right )} \,d x \]