Integrand size = 17, antiderivative size = 41 \[ \int \frac {\text {arctanh}(a x)}{c x+a c x^2} \, dx=\frac {\text {arctanh}(a x) \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )}{2 c} \] Output:
arctanh(a*x)*ln(2-2/(a*x+1))/c-1/2*polylog(2,-1+2/(a*x+1))/c
Time = 0.14 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.95 \[ \int \frac {\text {arctanh}(a x)}{c x+a c x^2} \, dx=\frac {\text {arctanh}(a x) \log \left (1-e^{-2 \text {arctanh}(a x)}\right )}{c}-\frac {\operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(a x)}\right )}{2 c} \] Input:
Integrate[ArcTanh[a*x]/(c*x + a*c*x^2),x]
Output:
(ArcTanh[a*x]*Log[1 - E^(-2*ArcTanh[a*x])])/c - PolyLog[2, E^(-2*ArcTanh[a *x])]/(2*c)
Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2026, 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.
\(\displaystyle \int \frac {\text {arctanh}(a x)}{a c x^2+c x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\text {arctanh}(a x)}{x (a c x+c)}dx\) |
\(\Big \downarrow \) 6494 |
\(\displaystyle \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )}{c}-\frac {a \int \frac {\log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx}{c}\) |
\(\Big \downarrow \) 2897 |
\(\displaystyle \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )}{c}-\frac {\operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 c}\) |
Input:
Int[ArcTanh[a*x]/(c*x + a*c*x^2),x]
Output:
(ArcTanh[a*x]*Log[2 - 2/(1 + a*x)])/c - PolyLog[2, -1 + 2/(1 + a*x)]/(2*c)
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
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. \(87\) vs. \(2(39)=78\).
Time = 0.23 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.15
method | result | size |
risch | \(-\frac {\ln \left (a x +1\right )^{2}}{4 c}-\frac {\operatorname {dilog}\left (a x +1\right )}{2 c}-\frac {\ln \left (\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2 c}+\frac {\ln \left (\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (-a x +1\right )}{2 c}+\frac {\operatorname {dilog}\left (-a x +1\right )}{2 c}-\frac {\operatorname {dilog}\left (-\frac {a x}{2}+\frac {1}{2}\right )}{2 c}\) | \(88\) |
derivativedivides | \(\frac {\frac {a \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )}{c}-\frac {a \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{c}-\frac {a \left (-\frac {\ln \left (a x +1\right )^{2}}{4}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\operatorname {dilog}\left (a x \right )}{2}+\frac {\operatorname {dilog}\left (a x +1\right )}{2}+\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}\right )}{c}}{a}\) | \(112\) |
default | \(\frac {\frac {a \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )}{c}-\frac {a \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{c}-\frac {a \left (-\frac {\ln \left (a x +1\right )^{2}}{4}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\operatorname {dilog}\left (a x \right )}{2}+\frac {\operatorname {dilog}\left (a x +1\right )}{2}+\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}\right )}{c}}{a}\) | \(112\) |
parts | \(\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (x \right )}{c}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{c}-\frac {a \left (\frac {-\frac {\ln \left (a x +1\right )^{2}}{4}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}}{a}-\frac {\left (\ln \left (x \right )-\ln \left (a x \right )\right ) \ln \left (-a x +1\right )}{2 a}+\frac {\operatorname {dilog}\left (a x \right )}{2 a}+\frac {\operatorname {dilog}\left (a x +1\right )}{2 a}+\frac {\ln \left (x \right ) \ln \left (a x +1\right )}{2 a}\right )}{c}\) | \(137\) |
Input:
int(arctanh(a*x)/(a*c*x^2+c*x),x,method=_RETURNVERBOSE)
Output:
-1/4/c*ln(a*x+1)^2-1/2/c*dilog(a*x+1)-1/2/c*ln(1/2*a*x+1/2)*ln(-1/2*a*x+1/ 2)+1/2/c*ln(1/2*a*x+1/2)*ln(-a*x+1)+1/2/c*dilog(-a*x+1)-1/2/c*dilog(-1/2*a *x+1/2)
\[ \int \frac {\text {arctanh}(a x)}{c x+a c x^2} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )}{a c x^{2} + c x} \,d x } \] Input:
integrate(arctanh(a*x)/(a*c*x^2+c*x),x, algorithm="fricas")
Output:
integral(arctanh(a*x)/(a*c*x^2 + c*x), x)
\[ \int \frac {\text {arctanh}(a x)}{c x+a c x^2} \, dx=\frac {\int \frac {\operatorname {atanh}{\left (a x \right )}}{a x^{2} + x}\, dx}{c} \] Input:
integrate(atanh(a*x)/(a*c*x**2+c*x),x)
Output:
Integral(atanh(a*x)/(a*x**2 + x), x)/c
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (38) = 76\).
Time = 0.03 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.93 \[ \int \frac {\text {arctanh}(a x)}{c x+a c x^2} \, dx=\frac {1}{4} \, a {\left (\frac {\log \left (a x + 1\right )^{2}}{a c} - \frac {2 \, {\left (\log \left (a x + 1\right ) \log \left (-\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a c} - \frac {2 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a c} + \frac {2 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )}}{a c}\right )} - {\left (\frac {\log \left (a x + 1\right )}{c} - \frac {\log \left (x\right )}{c}\right )} \operatorname {artanh}\left (a x\right ) \] Input:
integrate(arctanh(a*x)/(a*c*x^2+c*x),x, algorithm="maxima")
Output:
1/4*a*(log(a*x + 1)^2/(a*c) - 2*(log(a*x + 1)*log(-1/2*a*x + 1/2) + dilog( 1/2*a*x + 1/2))/(a*c) - 2*(log(a*x + 1)*log(x) + dilog(-a*x))/(a*c) + 2*(l og(-a*x + 1)*log(x) + dilog(a*x))/(a*c)) - (log(a*x + 1)/c - log(x)/c)*arc tanh(a*x)
\[ \int \frac {\text {arctanh}(a x)}{c x+a c x^2} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )}{a c x^{2} + c x} \,d x } \] Input:
integrate(arctanh(a*x)/(a*c*x^2+c*x),x, algorithm="giac")
Output:
integrate(arctanh(a*x)/(a*c*x^2 + c*x), x)
Timed out. \[ \int \frac {\text {arctanh}(a x)}{c x+a c x^2} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )}{a\,c\,x^2+c\,x} \,d x \] Input:
int(atanh(a*x)/(c*x + a*c*x^2),x)
Output:
int(atanh(a*x)/(c*x + a*c*x^2), x)
\[ \int \frac {\text {arctanh}(a x)}{c x+a c x^2} \, dx=\frac {-\mathit {atanh} \left (a x \right )^{2}-2 \left (\int \frac {\mathit {atanh} \left (a x \right )}{a^{2} x^{3}-x}d x \right )}{2 c} \] Input:
int(atanh(a*x)/(a*c*x^2+c*x),x)
Output:
( - atanh(a*x)**2 - 2*int(atanh(a*x)/(a**2*x**3 - x),x))/(2*c)