Integrand size = 14, antiderivative size = 429 \[ \int \frac {\text {arctanh}(a x)}{c+d x^2} \, dx=-\frac {\log (1-a x) \log \left (\frac {a \left (\sqrt {-c}-\sqrt {d} x\right )}{a \sqrt {-c}-\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log (1+a x) \log \left (\frac {a \left (\sqrt {-c}-\sqrt {d} x\right )}{a \sqrt {-c}+\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log (1+a x) \log \left (\frac {a \left (\sqrt {-c}+\sqrt {d} x\right )}{a \sqrt {-c}-\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log (1-a x) \log \left (\frac {a \left (\sqrt {-c}+\sqrt {d} x\right )}{a \sqrt {-c}+\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} (1-a x)}{a \sqrt {-c}-\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} (1-a x)}{a \sqrt {-c}+\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} (1+a x)}{a \sqrt {-c}-\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} (1+a x)}{a \sqrt {-c}+\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}} \]
-1/4*ln(-a*x+1)*ln(a*((-c)^(1/2)-x*d^(1/2))/(a*(-c)^(1/2)-d^(1/2)))/(-c)^( 1/2)/d^(1/2)+1/4*ln(a*x+1)*ln(a*((-c)^(1/2)-x*d^(1/2))/(a*(-c)^(1/2)+d^(1/ 2)))/(-c)^(1/2)/d^(1/2)-1/4*ln(a*x+1)*ln(a*((-c)^(1/2)+x*d^(1/2))/(a*(-c)^ (1/2)-d^(1/2)))/(-c)^(1/2)/d^(1/2)+1/4*ln(-a*x+1)*ln(a*((-c)^(1/2)+x*d^(1/ 2))/(a*(-c)^(1/2)+d^(1/2)))/(-c)^(1/2)/d^(1/2)-1/4*polylog(2,-(-a*x+1)*d^( 1/2)/(a*(-c)^(1/2)-d^(1/2)))/(-c)^(1/2)/d^(1/2)-1/4*polylog(2,-(a*x+1)*d^( 1/2)/(a*(-c)^(1/2)-d^(1/2)))/(-c)^(1/2)/d^(1/2)+1/4*polylog(2,(-a*x+1)*d^( 1/2)/(a*(-c)^(1/2)+d^(1/2)))/(-c)^(1/2)/d^(1/2)+1/4*polylog(2,(a*x+1)*d^(1 /2)/(a*(-c)^(1/2)+d^(1/2)))/(-c)^(1/2)/d^(1/2)
Result contains complex when optimal does not.
Time = 1.37 (sec) , antiderivative size = 662, normalized size of antiderivative = 1.54 \[ \int \frac {\text {arctanh}(a x)}{c+d x^2} \, dx=-\frac {a \left (-2 i \arccos \left (\frac {-a^2 c+d}{a^2 c+d}\right ) \arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )+4 \arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right ) \text {arctanh}(a x)-\left (\arccos \left (\frac {-a^2 c+d}{a^2 c+d}\right )+2 \arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right ) \log \left (\frac {2 i a c \left (i d+\sqrt {a^2 c d}\right ) (-1+a x)}{\left (a^2 c+d\right ) \left (a c+i \sqrt {a^2 c d} x\right )}\right )-\left (\arccos \left (\frac {-a^2 c+d}{a^2 c+d}\right )-2 \arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right ) \log \left (\frac {2 a c \left (d+i \sqrt {a^2 c d}\right ) (1+a x)}{\left (a^2 c+d\right ) \left (a c+i \sqrt {a^2 c d} x\right )}\right )+\left (\arccos \left (\frac {-a^2 c+d}{a^2 c+d}\right )+2 \left (\arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )+\arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{-\text {arctanh}(a x)}}{\sqrt {a^2 c+d} \sqrt {a^2 c-d+\left (a^2 c+d\right ) \cosh (2 \text {arctanh}(a x))}}\right )+\left (\arccos \left (\frac {-a^2 c+d}{a^2 c+d}\right )-2 \left (\arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )+\arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{\text {arctanh}(a x)}}{\sqrt {a^2 c+d} \sqrt {a^2 c-d+\left (a^2 c+d\right ) \cosh (2 \text {arctanh}(a x))}}\right )+i \left (-\operatorname {PolyLog}\left (2,\frac {\left (-a^2 c+d-2 i \sqrt {a^2 c d}\right ) \left (i a c+\sqrt {a^2 c d} x\right )}{\left (a^2 c+d\right ) \left (-i a c+\sqrt {a^2 c d} x\right )}\right )+\operatorname {PolyLog}\left (2,\frac {\left (-a^2 c+d+2 i \sqrt {a^2 c d}\right ) \left (i a c+\sqrt {a^2 c d} x\right )}{\left (a^2 c+d\right ) \left (-i a c+\sqrt {a^2 c d} x\right )}\right )\right )\right )}{4 \sqrt {a^2 c d}} \]
-1/4*(a*((-2*I)*ArcCos[(-(a^2*c) + d)/(a^2*c + d)]*ArcTan[(a*d*x)/Sqrt[a^2 *c*d]] + 4*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)]*ArcTanh[a*x] - (ArcCos[(-(a^2*c ) + d)/(a^2*c + d)] + 2*ArcTan[(a*d*x)/Sqrt[a^2*c*d]])*Log[((2*I)*a*c*(I*d + Sqrt[a^2*c*d])*(-1 + a*x))/((a^2*c + d)*(a*c + I*Sqrt[a^2*c*d]*x))] - ( ArcCos[(-(a^2*c) + d)/(a^2*c + d)] - 2*ArcTan[(a*d*x)/Sqrt[a^2*c*d]])*Log[ (2*a*c*(d + I*Sqrt[a^2*c*d])*(1 + a*x))/((a^2*c + d)*(a*c + I*Sqrt[a^2*c*d ]*x))] + (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] + 2*(ArcTan[(a*c)/(Sqrt[a^2*c *d]*x)] + ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d])/(Sqr t[a^2*c + d]*E^ArcTanh[a*x]*Sqrt[a^2*c - d + (a^2*c + d)*Cosh[2*ArcTanh[a* x]]])] + (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] - 2*(ArcTan[(a*c)/(Sqrt[a^2*c *d]*x)] + ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d]*E^Arc Tanh[a*x])/(Sqrt[a^2*c + d]*Sqrt[a^2*c - d + (a^2*c + d)*Cosh[2*ArcTanh[a* x]]])] + I*(-PolyLog[2, ((-(a^2*c) + d - (2*I)*Sqrt[a^2*c*d])*(I*a*c + Sqr t[a^2*c*d]*x))/((a^2*c + d)*((-I)*a*c + Sqrt[a^2*c*d]*x))] + PolyLog[2, (( -(a^2*c) + d + (2*I)*Sqrt[a^2*c*d])*(I*a*c + Sqrt[a^2*c*d]*x))/((a^2*c + d )*((-I)*a*c + Sqrt[a^2*c*d]*x))])))/Sqrt[a^2*c*d]
Time = 0.73 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6534, 2856, 2009}
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.
\(\displaystyle \int \frac {\text {arctanh}(a x)}{c+d x^2} \, dx\) |
\(\Big \downarrow \) 6534 |
\(\displaystyle \frac {1}{2} \int \frac {\log (a x+1)}{d x^2+c}dx-\frac {1}{2} \int \frac {\log (1-a x)}{d x^2+c}dx\) |
\(\Big \downarrow \) 2856 |
\(\displaystyle \frac {1}{2} \int \left (\frac {\sqrt {-c} \log (a x+1)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \log (a x+1)}{2 c \left (\sqrt {d} x+\sqrt {-c}\right )}\right )dx-\frac {1}{2} \int \left (\frac {\sqrt {-c} \log (1-a x)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \log (1-a x)}{2 c \left (\sqrt {d} x+\sqrt {-c}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} (1-a x)}{a \sqrt {-c}-\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} (1-a x)}{\sqrt {-c} a+\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\log (1-a x) \log \left (\frac {a \left (\sqrt {-c}-\sqrt {d} x\right )}{a \sqrt {-c}-\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\log (1-a x) \log \left (\frac {a \left (\sqrt {-c}+\sqrt {d} x\right )}{a \sqrt {-c}+\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}\right )+\frac {1}{2} \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} (a x+1)}{a \sqrt {-c}-\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} (a x+1)}{\sqrt {-c} a+\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\log (a x+1) \log \left (\frac {a \left (\sqrt {-c}-\sqrt {d} x\right )}{a \sqrt {-c}+\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\log (a x+1) \log \left (\frac {a \left (\sqrt {-c}+\sqrt {d} x\right )}{a \sqrt {-c}-\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}\right )\) |
(-1/2*(Log[1 - a*x]*Log[(a*(Sqrt[-c] - Sqrt[d]*x))/(a*Sqrt[-c] - Sqrt[d])] )/(Sqrt[-c]*Sqrt[d]) + (Log[1 - a*x]*Log[(a*(Sqrt[-c] + Sqrt[d]*x))/(a*Sqr t[-c] + Sqrt[d])])/(2*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt[d]*(1 - a*x)) /(a*Sqrt[-c] - Sqrt[d]))]/(2*Sqrt[-c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*(1 - a*x))/(a*Sqrt[-c] + Sqrt[d])]/(2*Sqrt[-c]*Sqrt[d]))/2 + ((Log[1 + a*x]*Log [(a*(Sqrt[-c] - Sqrt[d]*x))/(a*Sqrt[-c] + Sqrt[d])])/(2*Sqrt[-c]*Sqrt[d]) - (Log[1 + a*x]*Log[(a*(Sqrt[-c] + Sqrt[d]*x))/(a*Sqrt[-c] - Sqrt[d])])/(2 *Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt[d]*(1 + a*x))/(a*Sqrt[-c] - Sqrt[d ]))]/(2*Sqrt[-c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*(1 + a*x))/(a*Sqrt[-c] + S qrt[d])]/(2*Sqrt[-c]*Sqrt[d]))/2
3.6.2.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_. )*(x_)^(r_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x) ^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x] && I GtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))
Int[ArcTanh[(c_.)*(x_)]/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Simp[1/2 Int [Log[1 + c*x]/(d + e*x^2), x], x] - Simp[1/2 Int[Log[1 - c*x]/(d + e*x^2) , x], x] /; FreeQ[{c, d, e}, x]
Time = 0.41 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.85
method | result | size |
risch | \(\frac {\ln \left (-a x +1\right ) \ln \left (\frac {a \sqrt {-c d}-\left (-a x +1\right ) d +d}{a \sqrt {-c d}+d}\right )}{4 \sqrt {-c d}}-\frac {\ln \left (-a x +1\right ) \ln \left (\frac {a \sqrt {-c d}+\left (-a x +1\right ) d -d}{a \sqrt {-c d}-d}\right )}{4 \sqrt {-c d}}+\frac {\operatorname {dilog}\left (\frac {a \sqrt {-c d}-\left (-a x +1\right ) d +d}{a \sqrt {-c d}+d}\right )}{4 \sqrt {-c d}}-\frac {\operatorname {dilog}\left (\frac {a \sqrt {-c d}+\left (-a x +1\right ) d -d}{a \sqrt {-c d}-d}\right )}{4 \sqrt {-c d}}+\frac {\ln \left (a x +1\right ) \ln \left (\frac {a \sqrt {-c d}-\left (a x +1\right ) d +d}{a \sqrt {-c d}+d}\right )}{4 \sqrt {-c d}}-\frac {\ln \left (a x +1\right ) \ln \left (\frac {a \sqrt {-c d}+\left (a x +1\right ) d -d}{a \sqrt {-c d}-d}\right )}{4 \sqrt {-c d}}+\frac {\operatorname {dilog}\left (\frac {a \sqrt {-c d}-\left (a x +1\right ) d +d}{a \sqrt {-c d}+d}\right )}{4 \sqrt {-c d}}-\frac {\operatorname {dilog}\left (\frac {a \sqrt {-c d}+\left (a x +1\right ) d -d}{a \sqrt {-c d}-d}\right )}{4 \sqrt {-c d}}\) | \(364\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1002\) |
default | \(\text {Expression too large to display}\) | \(1002\) |
1/4*ln(-a*x+1)/(-c*d)^(1/2)*ln((a*(-c*d)^(1/2)-(-a*x+1)*d+d)/(a*(-c*d)^(1/ 2)+d))-1/4*ln(-a*x+1)/(-c*d)^(1/2)*ln((a*(-c*d)^(1/2)+(-a*x+1)*d-d)/(a*(-c *d)^(1/2)-d))+1/4/(-c*d)^(1/2)*dilog((a*(-c*d)^(1/2)-(-a*x+1)*d+d)/(a*(-c* d)^(1/2)+d))-1/4/(-c*d)^(1/2)*dilog((a*(-c*d)^(1/2)+(-a*x+1)*d-d)/(a*(-c*d )^(1/2)-d))+1/4*ln(a*x+1)/(-c*d)^(1/2)*ln((a*(-c*d)^(1/2)-(a*x+1)*d+d)/(a* (-c*d)^(1/2)+d))-1/4*ln(a*x+1)/(-c*d)^(1/2)*ln((a*(-c*d)^(1/2)+(a*x+1)*d-d )/(a*(-c*d)^(1/2)-d))+1/4/(-c*d)^(1/2)*dilog((a*(-c*d)^(1/2)-(a*x+1)*d+d)/ (a*(-c*d)^(1/2)+d))-1/4/(-c*d)^(1/2)*dilog((a*(-c*d)^(1/2)+(a*x+1)*d-d)/(a *(-c*d)^(1/2)-d))
\[ \int \frac {\text {arctanh}(a x)}{c+d x^2} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )}{d x^{2} + c} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)}{c+d x^2} \, dx=\int \frac {\operatorname {atanh}{\left (a x \right )}}{c + d x^{2}}\, dx \]
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 406, normalized size of antiderivative = 0.95 \[ \int \frac {\text {arctanh}(a x)}{c+d x^2} \, dx=\frac {\arctan \left (\frac {d x}{\sqrt {c d}}\right ) \operatorname {artanh}\left (a x\right )}{\sqrt {c d}} + \frac {{\left (\arctan \left (\frac {{\left (a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, \frac {a d x + d}{a^{2} c + d}\right ) - \arctan \left (\frac {{\left (a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, -\frac {a d x - d}{a^{2} c + d}\right )\right )} \log \left (d x^{2} + c\right ) - \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} + 2 \, a d x + d}{a^{2} c + d}\right ) + \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} - 2 \, a d x + d}{a^{2} c + d}\right ) - i \, {\rm Li}_2\left (\frac {a^{2} c + a d x - {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) - i \, {\rm Li}_2\left (\frac {a^{2} c - a d x + {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + i \, {\rm Li}_2\left (\frac {a^{2} c + a d x + {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + i \, {\rm Li}_2\left (\frac {a^{2} c - a d x - {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right )}{4 \, \sqrt {c d}} \]
arctan(d*x/sqrt(c*d))*arctanh(a*x)/sqrt(c*d) + 1/4*((arctan2((a^2*x + a)*s qrt(c)*sqrt(d)/(a^2*c + d), (a*d*x + d)/(a^2*c + d)) - arctan2((a^2*x - a) *sqrt(c)*sqrt(d)/(a^2*c + d), -(a*d*x - d)/(a^2*c + d)))*log(d*x^2 + c) - arctan(sqrt(d)*x/sqrt(c))*log((a^2*d*x^2 + 2*a*d*x + d)/(a^2*c + d)) + arc tan(sqrt(d)*x/sqrt(c))*log((a^2*d*x^2 - 2*a*d*x + d)/(a^2*c + d)) - I*dilo g((a^2*c + a*d*x - (I*a^2*x - I*a)*sqrt(c)*sqrt(d))/(a^2*c + 2*I*a*sqrt(c) *sqrt(d) - d)) - I*dilog((a^2*c - a*d*x + (I*a^2*x + I*a)*sqrt(c)*sqrt(d)) /(a^2*c + 2*I*a*sqrt(c)*sqrt(d) - d)) + I*dilog((a^2*c + a*d*x + (I*a^2*x - I*a)*sqrt(c)*sqrt(d))/(a^2*c - 2*I*a*sqrt(c)*sqrt(d) - d)) + I*dilog((a^ 2*c - a*d*x - (I*a^2*x + I*a)*sqrt(c)*sqrt(d))/(a^2*c - 2*I*a*sqrt(c)*sqrt (d) - d)))/sqrt(c*d)
\[ \int \frac {\text {arctanh}(a x)}{c+d x^2} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )}{d x^{2} + c} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a x)}{c+d x^2} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )}{d\,x^2+c} \,d x \]