\(\int \frac {a+b \text {arctanh}(c x)}{d+e x} \, dx\) [5]

Optimal result

Integrand size = 16, antiderivative size = 114 \[ \int \frac {a+b \text {arctanh}(c x)}{d+e x} \, dx=-\frac {(a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{e}+\frac {(a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 e}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e} \] Output:

-(a+b*arctanh(c*x))*ln(2/(c*x+1))/e+(a+b*arctanh(c*x))*ln(2*c*(e*x+d)/(c*d 
+e)/(c*x+1))/e+1/2*b*polylog(2,1-2/(c*x+1))/e-1/2*b*polylog(2,1-2*c*(e*x+d 
)/(c*d+e)/(c*x+1))/e
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.25 \[ \int \frac {a+b \text {arctanh}(c x)}{d+e x} \, dx=\frac {a \log (d+e x)+b \text {arctanh}(c x) \left (\frac {1}{2} \log \left (1-c^2 x^2\right )+\log \left (i \sinh \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )\right )\right )-\frac {1}{2} i b \left (-\frac {1}{4} i (\pi -2 i \text {arctanh}(c x))^2+i \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )^2+(\pi -2 i \text {arctanh}(c x)) \log \left (1+e^{2 \text {arctanh}(c x)}\right )+2 i \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right ) \log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )-(\pi -2 i \text {arctanh}(c x)) \log \left (\frac {2}{\sqrt {1-c^2 x^2}}\right )-2 i \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right ) \log \left (2 i \sinh \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )\right )-i \operatorname {PolyLog}\left (2,-e^{2 \text {arctanh}(c x)}\right )-i \operatorname {PolyLog}\left (2,e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )\right )}{e} \] Input:

Integrate[(a + b*ArcTanh[c*x])/(d + e*x),x]
 

Output:

(a*Log[d + e*x] + b*ArcTanh[c*x]*(Log[1 - c^2*x^2]/2 + Log[I*Sinh[ArcTanh[ 
(c*d)/e] + ArcTanh[c*x]]]) - (I/2)*b*((-1/4*I)*(Pi - (2*I)*ArcTanh[c*x])^2 
 + I*(ArcTanh[(c*d)/e] + ArcTanh[c*x])^2 + (Pi - (2*I)*ArcTanh[c*x])*Log[1 
 + E^(2*ArcTanh[c*x])] + (2*I)*(ArcTanh[(c*d)/e] + ArcTanh[c*x])*Log[1 - E 
^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - (Pi - (2*I)*ArcTanh[c*x])*Log[2 
/Sqrt[1 - c^2*x^2]] - (2*I)*(ArcTanh[(c*d)/e] + ArcTanh[c*x])*Log[(2*I)*Si 
nh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] - I*PolyLog[2, -E^(2*ArcTanh[c*x])] - 
 I*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/e
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {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.

\(\displaystyle \int \frac {a+b \text {arctanh}(c x)}{d+e x} \, dx\)

\(\Big \downarrow \) 6472

\(\displaystyle -\frac {b c \int \frac {\log \left (\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{1-c^2 x^2}dx}{e}+\frac {b c \int \frac {\log \left (\frac {2}{c x+1}\right )}{1-c^2 x^2}dx}{e}+\frac {(a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e}-\frac {\log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{e}\)

\(\Big \downarrow \) 2849

\(\displaystyle -\frac {b c \int \frac {\log \left (\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{1-c^2 x^2}dx}{e}+\frac {b \int \frac {\log \left (\frac {2}{c x+1}\right )}{1-\frac {2}{c x+1}}d\frac {1}{c x+1}}{e}+\frac {(a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e}-\frac {\log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{e}\)

\(\Big \downarrow \) 2752

\(\displaystyle -\frac {b c \int \frac {\log \left (\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{1-c^2 x^2}dx}{e}+\frac {(a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e}-\frac {\log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{e}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 e}\)

\(\Big \downarrow \) 2897

\(\displaystyle \frac {(a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e}-\frac {\log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{e}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 e}\)

Input:

Int[(a + b*ArcTanh[c*x])/(d + e*x),x]
 

Output:

-(((a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/e) + ((a + b*ArcTanh[c*x])*Log[( 
2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e + (b*PolyLog[2, 1 - 2/(1 + c*x)]) 
/(2*e) - (b*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e)
 

Defintions of rubi rules used

rule 2752

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]
 

rule 2849

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]
 

rule 2897

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]]
 

rule 6472

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]
 

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.27

method	result	size
parts	\(\frac {a \ln \left (e x +d \right )}{e}+\frac {b \ln \left (c e x +c d \right ) \operatorname {arctanh}\left (c x \right )}{e}+\frac {b \ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )}{2 e}+\frac {b \operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )}{2 e}-\frac {b \ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )}{2 e}-\frac {b \operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )}{2 e}\)	\(145\)
derivativedivides	\(\frac {\frac {a c \ln \left (c e x +c d \right )}{e}+b c \left (\frac {\ln \left (c e x +c d \right ) \operatorname {arctanh}\left (c x \right )}{e}-\frac {-\frac {e \left (\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )\right )}{2}+\frac {e \left (\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )\right )}{2}}{e^{2}}\right )}{c}\)	\(148\)
default	\(\frac {\frac {a c \ln \left (c e x +c d \right )}{e}+b c \left (\frac {\ln \left (c e x +c d \right ) \operatorname {arctanh}\left (c x \right )}{e}-\frac {-\frac {e \left (\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )\right )}{2}+\frac {e \left (\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )\right )}{2}}{e^{2}}\right )}{c}\)	\(148\)
risch	\(-\frac {b \operatorname {dilog}\left (\frac {\left (-c x +1\right ) e -c d -e}{-c d -e}\right )}{2 e}-\frac {b \ln \left (-c x +1\right ) \ln \left (\frac {\left (-c x +1\right ) e -c d -e}{-c d -e}\right )}{2 e}+\frac {a \ln \left (\left (-c x +1\right ) e -c d -e \right )}{e}+\frac {b \operatorname {dilog}\left (\frac {\left (c x +1\right ) e +c d -e}{c d -e}\right )}{2 e}+\frac {b \ln \left (c x +1\right ) \ln \left (\frac {\left (c x +1\right ) e +c d -e}{c d -e}\right )}{2 e}\)	\(167\)

Input:

int((a+b*arctanh(c*x))/(e*x+d),x,method=_RETURNVERBOSE)
 

Output:

a*ln(e*x+d)/e+b*ln(c*e*x+c*d)/e*arctanh(c*x)+1/2*b/e*ln(c*e*x+c*d)*ln((c*e 
*x-e)/(-c*d-e))+1/2*b/e*dilog((c*e*x-e)/(-c*d-e))-1/2*b/e*ln(c*e*x+c*d)*ln 
((c*e*x+e)/(-c*d+e))-1/2*b/e*dilog((c*e*x+e)/(-c*d+e))
 

Fricas [F]

\[ \int \frac {a+b \text {arctanh}(c x)}{d+e x} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{e x + d} \,d x } \] Input:

integrate((a+b*arctanh(c*x))/(e*x+d),x, algorithm="fricas")
 

Output:

integral((b*arctanh(c*x) + a)/(e*x + d), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arctanh}(c x)}{d+e x} \, dx=\int \frac {a + b \operatorname {atanh}{\left (c x \right )}}{d + e x}\, dx \] Input:

integrate((a+b*atanh(c*x))/(e*x+d),x)
 

Output:

Integral((a + b*atanh(c*x))/(d + e*x), x)
 

Maxima [F]

\[ \int \frac {a+b \text {arctanh}(c x)}{d+e x} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{e x + d} \,d x } \] Input:

integrate((a+b*arctanh(c*x))/(e*x+d),x, algorithm="maxima")
 

Output:

1/2*b*integrate((log(c*x + 1) - log(-c*x + 1))/(e*x + d), x) + a*log(e*x + 
 d)/e
 

Giac [F]

\[ \int \frac {a+b \text {arctanh}(c x)}{d+e x} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{e x + d} \,d x } \] Input:

integrate((a+b*arctanh(c*x))/(e*x+d),x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*x) + a)/(e*x + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arctanh}(c x)}{d+e x} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{d+e\,x} \,d x \] Input:

int((a + b*atanh(c*x))/(d + e*x),x)
 

Output:

int((a + b*atanh(c*x))/(d + e*x), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arctanh}(c x)}{d+e x} \, dx=\frac {\left (\int \frac {\mathit {atanh} \left (c x \right )}{e x +d}d x \right ) b e +\mathrm {log}\left (e x +d \right ) a}{e} \] Input:

int((a+b*atanh(c*x))/(e*x+d),x)
                                                                                    
                                                                                    
 

Output:

(int(atanh(c*x)/(d + e*x),x)*b*e + log(d + e*x)*a)/e