Integrand size = 14, antiderivative size = 30 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x} \, dx=a \log (x)+\frac {1}{4} b \operatorname {PolyLog}\left (2,-\frac {c}{x^2}\right )-\frac {1}{4} b \operatorname {PolyLog}\left (2,\frac {c}{x^2}\right ) \]
Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x} \, dx=a \log (x)+\frac {1}{4} b \left (\operatorname {PolyLog}\left (2,-\frac {c}{x^2}\right )-\operatorname {PolyLog}\left (2,\frac {c}{x^2}\right )\right ) \]
Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6450, 6446}
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.
\(\displaystyle \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x} \, dx\) |
\(\Big \downarrow \) 6450 |
\(\displaystyle -\frac {1}{2} \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )d\frac {1}{x^2}\) |
\(\Big \downarrow \) 6446 |
\(\displaystyle \frac {1}{2} \left (-a \log \left (\frac {1}{x^2}\right )+\frac {1}{2} b \operatorname {PolyLog}\left (2,-\frac {c}{x^2}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {c}{x^2}\right )\right )\) |
3.2.61.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x ] + (-Simp[(b/2)*PolyLog[2, (-c)*x], x] + Simp[(b/2)*PolyLog[2, c*x], x]) / ; FreeQ[{a, b, c}, x]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[ 1/n Subst[Int[(a + b*ArcTanh[c*x])^p/x, x], x, x^n], x] /; FreeQ[{a, b, c , n}, x] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(148\) vs. \(2(26)=52\).
Time = 0.67 (sec) , antiderivative size = 149, normalized size of antiderivative = 4.97
method | result | size |
parts | \(a \ln \left (x \right )+b \left (-\ln \left (\frac {1}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x^{2}}\right )+2 c \left (-\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (1-\frac {\sqrt {c}}{x}\right )+\ln \left (1+\frac {\sqrt {c}}{x}\right )\right )}{4 c}-\frac {\operatorname {dilog}\left (1-\frac {\sqrt {c}}{x}\right )+\operatorname {dilog}\left (1+\frac {\sqrt {c}}{x}\right )}{4 c}+\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (1+\frac {\sqrt {-c}}{x}\right )+\ln \left (1-\frac {\sqrt {-c}}{x}\right )\right )}{4 c}+\frac {\operatorname {dilog}\left (1+\frac {\sqrt {-c}}{x}\right )+\operatorname {dilog}\left (1-\frac {\sqrt {-c}}{x}\right )}{4 c}\right )\right )\) | \(149\) |
derivativedivides | \(-a \ln \left (\frac {1}{x}\right )-b \left (\ln \left (\frac {1}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x^{2}}\right )-2 c \left (-\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (1-\frac {\sqrt {c}}{x}\right )+\ln \left (1+\frac {\sqrt {c}}{x}\right )\right )}{4 c}-\frac {\operatorname {dilog}\left (1-\frac {\sqrt {c}}{x}\right )+\operatorname {dilog}\left (1+\frac {\sqrt {c}}{x}\right )}{4 c}+\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (1+\frac {\sqrt {-c}}{x}\right )+\ln \left (1-\frac {\sqrt {-c}}{x}\right )\right )}{4 c}+\frac {\operatorname {dilog}\left (1+\frac {\sqrt {-c}}{x}\right )+\operatorname {dilog}\left (1-\frac {\sqrt {-c}}{x}\right )}{4 c}\right )\right )\) | \(152\) |
default | \(-a \ln \left (\frac {1}{x}\right )-b \left (\ln \left (\frac {1}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x^{2}}\right )-2 c \left (-\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (1-\frac {\sqrt {c}}{x}\right )+\ln \left (1+\frac {\sqrt {c}}{x}\right )\right )}{4 c}-\frac {\operatorname {dilog}\left (1-\frac {\sqrt {c}}{x}\right )+\operatorname {dilog}\left (1+\frac {\sqrt {c}}{x}\right )}{4 c}+\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (1+\frac {\sqrt {-c}}{x}\right )+\ln \left (1-\frac {\sqrt {-c}}{x}\right )\right )}{4 c}+\frac {\operatorname {dilog}\left (1+\frac {\sqrt {-c}}{x}\right )+\operatorname {dilog}\left (1-\frac {\sqrt {-c}}{x}\right )}{4 c}\right )\right )\) | \(152\) |
risch | \(\frac {b \ln \left (x \right ) \ln \left (x^{2}+c \right )}{2}+\frac {\left (-i b \pi {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{3}+4 a +2 i b \pi {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}+i b \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (-x^{2}+c \right )\right ) \operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )-i b \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (x^{2}+c \right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )-i b \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}+i b \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{2}-i b \pi \,\operatorname {csgn}\left (i \left (-x^{2}+c \right )\right ) {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}+i b \pi \,\operatorname {csgn}\left (i \left (x^{2}+c \right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{2}-i b \pi {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{3}-2 i b \pi \right ) \ln \left (x \right )}{4}-\frac {b \ln \left (x \right ) \ln \left (-x^{2}+c \right )}{2}+\frac {\ln \left (x \right ) \ln \left (\frac {-x +\sqrt {c}}{\sqrt {c}}\right ) b}{2}+\frac {\ln \left (x \right ) \ln \left (\frac {x +\sqrt {c}}{\sqrt {c}}\right ) b}{2}+\frac {\operatorname {dilog}\left (\frac {-x +\sqrt {c}}{\sqrt {c}}\right ) b}{2}+\frac {\operatorname {dilog}\left (\frac {x +\sqrt {c}}{\sqrt {c}}\right ) b}{2}-\frac {b \ln \left (x \right ) \ln \left (\frac {-x +\sqrt {-c}}{\sqrt {-c}}\right )}{2}-\frac {b \ln \left (x \right ) \ln \left (\frac {x +\sqrt {-c}}{\sqrt {-c}}\right )}{2}-\frac {b \operatorname {dilog}\left (\frac {-x +\sqrt {-c}}{\sqrt {-c}}\right )}{2}-\frac {b \operatorname {dilog}\left (\frac {x +\sqrt {-c}}{\sqrt {-c}}\right )}{2}\) | \(420\) |
a*ln(x)+b*(-ln(1/x)*arctanh(c/x^2)+2*c*(-1/4*ln(1/x)*(ln(1-1/x*c^(1/2))+ln (1+1/x*c^(1/2)))/c-1/4*(dilog(1-1/x*c^(1/2))+dilog(1+1/x*c^(1/2)))/c+1/4*l n(1/x)*(ln(1+(-c)^(1/2)/x)+ln(1-(-c)^(1/2)/x))/c+1/4*(dilog(1+(-c)^(1/2)/x )+dilog(1-(-c)^(1/2)/x))/c))
\[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x} \, dx=\int { \frac {b \operatorname {artanh}\left (\frac {c}{x^{2}}\right ) + a}{x} \,d x } \]
\[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x} \, dx=\int \frac {a + b \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{x}\, dx \]
\[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x} \, dx=\int { \frac {b \operatorname {artanh}\left (\frac {c}{x^{2}}\right ) + a}{x} \,d x } \]
\[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x} \, dx=\int { \frac {b \operatorname {artanh}\left (\frac {c}{x^{2}}\right ) + a}{x} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (\frac {c}{x^2}\right )}{x} \,d x \]