Integrand size = 12, antiderivative size = 168 \[ \int \text {arctanh}\left (a+b f^{c+d x}\right ) \, dx=-\frac {\text {arctanh}\left (a+b f^{c+d x}\right ) \log \left (\frac {2}{1+a+b f^{c+d x}}\right )}{d \log (f)}+\frac {\text {arctanh}\left (a+b f^{c+d x}\right ) \log \left (\frac {2 b f^{c+d x}}{(1-a) \left (1+a+b f^{c+d x}\right )}\right )}{d \log (f)}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+a+b f^{c+d x}}\right )}{2 d \log (f)}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 b f^{c+d x}}{(1-a) \left (1+a+b f^{c+d x}\right )}\right )}{2 d \log (f)} \]
-arctanh(a+b*f^(d*x+c))*ln(2/(1+a+b*f^(d*x+c)))/d/ln(f)+arctanh(a+b*f^(d*x +c))*ln(2*b*f^(d*x+c)/(1-a)/(1+a+b*f^(d*x+c)))/d/ln(f)+1/2*polylog(2,1-2/( 1+a+b*f^(d*x+c)))/d/ln(f)-1/2*polylog(2,1-2*b*f^(d*x+c)/(1-a)/(1+a+b*f^(d* x+c)))/d/ln(f)
Time = 0.49 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.64 \[ \int \text {arctanh}\left (a+b f^{c+d x}\right ) \, dx=\frac {d x \log (f) \left (2 \text {arctanh}\left (a+b f^{c+d x}\right )+\log \left (\frac {-1+a+b f^{c+d x}}{-1+a}\right )-\log \left (\frac {1+a+b f^{c+d x}}{1+a}\right )\right )+\operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{-1+a}\right )-\operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{1+a}\right )}{2 d \log (f)} \]
(d*x*Log[f]*(2*ArcTanh[a + b*f^(c + d*x)] + Log[(-1 + a + b*f^(c + d*x))/( -1 + a)] - Log[(1 + a + b*f^(c + d*x))/(1 + a)]) + PolyLog[2, -((b*f^(c + d*x))/(-1 + a))] - PolyLog[2, -((b*f^(c + d*x))/(1 + a))])/(2*d*Log[f])
Time = 0.53 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.89, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2720, 6661, 25, 27, 6472, 2849, 2752, 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.
| \(\displaystyle \int \text {arctanh}\left (a+b f^{c+d x}\right ) \, dx\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {\int f^{-c-d x} \text {arctanh}\left (b f^{c+d x}+a\right )df^{c+d x}}{d \log (f)}\) |
| \(\Big \downarrow \) 6661 |
| \(\displaystyle \frac {\int f^{-c-d x} \text {arctanh}\left (b f^{c+d x}+a\right )d\left (b f^{c+d x}+a\right )}{b d \log (f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -f^{-c-d x} \text {arctanh}\left (b f^{c+d x}+a\right )d\left (b f^{c+d x}+a\right )}{b d \log (f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int -\frac {f^{-c-d x} \text {arctanh}\left (b f^{c+d x}+a\right )}{b}d\left (b f^{c+d x}+a\right )}{d \log (f)}\) |
| \(\Big \downarrow \) 6472 |
| \(\displaystyle -\frac {-\int \frac {\log \left (\frac {2}{b f^{c+d x}+a+1}\right )}{1-f^{2 c+2 d x}}d\left (b f^{c+d x}+a\right )+\int \frac {\log \left (\frac {2 b f^{c+d x}}{(1-a) \left (b f^{c+d x}+a+1\right )}\right )}{1-f^{2 c+2 d x}}d\left (b f^{c+d x}+a\right )+\text {arctanh}\left (a+b f^{c+d x}\right ) \log \left (\frac {2}{a+b f^{c+d x}+1}\right )-\text {arctanh}\left (a+b f^{c+d x}\right ) \log \left (\frac {2 b f^{c+d x}}{(1-a) \left (a+b f^{c+d x}+1\right )}\right )}{d \log (f)}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle -\frac {-\int \frac {\log \left (\frac {2}{b f^{c+d x}+a+1}\right )}{1-\frac {2}{b f^{c+d x}+a+1}}d\frac {1}{b f^{c+d x}+a+1}+\int \frac {\log \left (\frac {2 b f^{c+d x}}{(1-a) \left (b f^{c+d x}+a+1\right )}\right )}{1-f^{2 c+2 d x}}d\left (b f^{c+d x}+a\right )+\text {arctanh}\left (a+b f^{c+d x}\right ) \log \left (\frac {2}{a+b f^{c+d x}+1}\right )-\text {arctanh}\left (a+b f^{c+d x}\right ) \log \left (\frac {2 b f^{c+d x}}{(1-a) \left (a+b f^{c+d x}+1\right )}\right )}{d \log (f)}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle -\frac {\int \frac {\log \left (\frac {2 b f^{c+d x}}{(1-a) \left (b f^{c+d x}+a+1\right )}\right )}{1-f^{2 c+2 d x}}d\left (b f^{c+d x}+a\right )+\text {arctanh}\left (a+b f^{c+d x}\right ) \log \left (\frac {2}{a+b f^{c+d x}+1}\right )-\text {arctanh}\left (a+b f^{c+d x}\right ) \log \left (\frac {2 b f^{c+d x}}{(1-a) \left (a+b f^{c+d x}+1\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2}{b f^{c+d x}+a+1}\right )}{d \log (f)}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle -\frac {\text {arctanh}\left (a+b f^{c+d x}\right ) \log \left (\frac {2}{a+b f^{c+d x}+1}\right )-\text {arctanh}\left (a+b f^{c+d x}\right ) \log \left (\frac {2 b f^{c+d x}}{(1-a) \left (a+b f^{c+d x}+1\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2}{b f^{c+d x}+a+1}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2 b f^{c+d x}}{(1-a) \left (b f^{c+d x}+a+1\right )}\right )}{d \log (f)}\) |
-((ArcTanh[a + b*f^(c + d*x)]*Log[2/(1 + a + b*f^(c + d*x))] - ArcTanh[a + b*f^(c + d*x)]*Log[(2*b*f^(c + d*x))/((1 - a)*(1 + a + b*f^(c + d*x)))] - PolyLog[2, 1 - 2/(1 + a + b*f^(c + d*x))]/2 + PolyLog[2, 1 - (2*b*f^(c + d*x))/((1 - a)*(1 + a + b*f^(c + d*x)))]/2)/(d*Log[f]))
3.4.52.3.1 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[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[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[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_.))/((d_) + (e_.)*(x_)), x_Symbol] :> S imp[(-(a + b*ArcTanh[c*x]))*(Log[2/(1 + c*x)]/e), x] + (Simp[(a + b*ArcTanh [c*x])*(Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp[b*(c/e) Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x], x] - Simp[b*(c/e) Int[Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/(1 - c^2*x^2), x], x]) /; FreeQ[{a, b, c, d , e}, x] && NeQ[c^2*d^2 - e^2, 0]
Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcTanh[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IG tQ[p, 0]
Time = 0.67 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.95
| method | result | size |
| derivativedivides | \(\frac {\ln \left (-b \,f^{d x +c}\right ) \operatorname {arctanh}\left (a +b \,f^{d x +c}\right )+\frac {\operatorname {dilog}\left (\frac {1-a -b \,f^{d x +c}}{1-a}\right )}{2}+\frac {\ln \left (-b \,f^{d x +c}\right ) \ln \left (\frac {1-a -b \,f^{d x +c}}{1-a}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {-b \,f^{d x +c}-a -1}{-1-a}\right )}{2}-\frac {\ln \left (-b \,f^{d x +c}\right ) \ln \left (\frac {-b \,f^{d x +c}-a -1}{-1-a}\right )}{2}}{d \ln \left (f \right )}\) | \(160\) |
| default | \(\frac {\ln \left (-b \,f^{d x +c}\right ) \operatorname {arctanh}\left (a +b \,f^{d x +c}\right )+\frac {\operatorname {dilog}\left (\frac {1-a -b \,f^{d x +c}}{1-a}\right )}{2}+\frac {\ln \left (-b \,f^{d x +c}\right ) \ln \left (\frac {1-a -b \,f^{d x +c}}{1-a}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {-b \,f^{d x +c}-a -1}{-1-a}\right )}{2}-\frac {\ln \left (-b \,f^{d x +c}\right ) \ln \left (\frac {-b \,f^{d x +c}-a -1}{-1-a}\right )}{2}}{d \ln \left (f \right )}\) | \(160\) |
| risch | \(\frac {x \ln \left (1+a +b \,f^{d x +c}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {1+a +f^{d x} f^{c} b}{1+a}\right )}{2 \ln \left (f \right ) d}-\frac {\ln \left (\frac {1+a +f^{d x} f^{c} b}{1+a}\right ) x}{2}-\frac {\ln \left (\frac {1+a +f^{d x} f^{c} b}{1+a}\right ) c}{2 d}+\frac {c \ln \left (1+a +f^{d x} f^{c} b \right )}{2 d}-\frac {\ln \left (1-a -b \,f^{d x +c}\right ) \ln \left (-\frac {f^{d x +c} b}{-1+a}\right )}{2 d \ln \left (f \right )}-\frac {\operatorname {dilog}\left (-\frac {f^{d x +c} b}{-1+a}\right )}{2 d \ln \left (f \right )}\) | \(182\) |
1/d/ln(f)*(ln(-b*f^(d*x+c))*arctanh(a+b*f^(d*x+c))+1/2*dilog((1-a-b*f^(d*x +c))/(1-a))+1/2*ln(-b*f^(d*x+c))*ln((1-a-b*f^(d*x+c))/(1-a))-1/2*dilog((-b *f^(d*x+c)-a-1)/(-1-a))-1/2*ln(-b*f^(d*x+c))*ln((-b*f^(d*x+c)-a-1)/(-1-a)) )
Time = 0.26 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.69 \[ \int \text {arctanh}\left (a+b f^{c+d x}\right ) \, dx=\frac {d x \log \left (f\right ) \log \left (-\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a + 1}{b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a - 1}\right ) + c \log \left (b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a + 1\right ) \log \left (f\right ) - c \log \left (b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a - 1\right ) \log \left (f\right ) - {\left (d x + c\right )} \log \left (f\right ) \log \left (\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a + 1}{a + 1}\right ) + {\left (d x + c\right )} \log \left (f\right ) \log \left (\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a - 1}{a - 1}\right ) - {\rm Li}_2\left (-\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a + 1}{a + 1} + 1\right ) + {\rm Li}_2\left (-\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a - 1}{a - 1} + 1\right )}{2 \, d \log \left (f\right )} \]
1/2*(d*x*log(f)*log(-(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f)) + a + 1)/(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f)) + a - 1)) + c*log(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f)) + a + 1)*log(f) - c*log(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f)) + a - 1)*log(f ) - (d*x + c)*log(f)*log((b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log( f)) + a + 1)/(a + 1)) + (d*x + c)*log(f)*log((b*cosh((d*x + c)*log(f)) + b *sinh((d*x + c)*log(f)) + a - 1)/(a - 1)) - dilog(-(b*cosh((d*x + c)*log(f )) + b*sinh((d*x + c)*log(f)) + a + 1)/(a + 1) + 1) + dilog(-(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f)) + a - 1)/(a - 1) + 1))/(d*log(f))
\[ \int \text {arctanh}\left (a+b f^{c+d x}\right ) \, dx=\int \operatorname {atanh}{\left (a + b f^{c + d x} \right )}\, dx \]
Time = 0.20 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.20 \[ \int \text {arctanh}\left (a+b f^{c+d x}\right ) \, dx=\frac {{\left (d x + c\right )} \operatorname {artanh}\left (b f^{d x + c} + a\right )}{d} - \frac {{\left (d x + c\right )} b {\left (\frac {\log \left (b f^{d x + c} + a + 1\right )}{b} - \frac {\log \left (b f^{d x + c} + a - 1\right )}{b}\right )} \log \left (f\right ) - b {\left (\frac {\log \left (b f^{d x + c} + a + 1\right ) \log \left (-\frac {b f^{d x + c} + a + 1}{a + 1} + 1\right ) + {\rm Li}_2\left (\frac {b f^{d x + c} + a + 1}{a + 1}\right )}{b} - \frac {\log \left (b f^{d x + c} + a - 1\right ) \log \left (-\frac {b f^{d x + c} + a - 1}{a - 1} + 1\right ) + {\rm Li}_2\left (\frac {b f^{d x + c} + a - 1}{a - 1}\right )}{b}\right )}}{2 \, d \log \left (f\right )} \]
(d*x + c)*arctanh(b*f^(d*x + c) + a)/d - 1/2*((d*x + c)*b*(log(b*f^(d*x + c) + a + 1)/b - log(b*f^(d*x + c) + a - 1)/b)*log(f) - b*((log(b*f^(d*x + c) + a + 1)*log(-(b*f^(d*x + c) + a + 1)/(a + 1) + 1) + dilog((b*f^(d*x + c) + a + 1)/(a + 1)))/b - (log(b*f^(d*x + c) + a - 1)*log(-(b*f^(d*x + c) + a - 1)/(a - 1) + 1) + dilog((b*f^(d*x + c) + a - 1)/(a - 1)))/b))/(d*log (f))
Exception generated. \[ \int \text {arctanh}\left (a+b f^{c+d x}\right ) \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Unable to divide, perhaps due to rounding error%%%{1,[0 ,1,2,0,0,0]%%%}+%%%{2,[0,1,1,1,1,0]%%%}+%%%{-2,[0,1,1,0,0,0]%%%}+%%%{1,[0, 1,0,2,0,1]%%
Timed out. \[ \int \text {arctanh}\left (a+b f^{c+d x}\right ) \, dx=\int \mathrm {atanh}\left (a+b\,f^{c+d\,x}\right ) \,d x \]