\(\int \frac {a+b \text {arctanh}(c \sqrt {x})}{x (1-c^2 x)} \, dx\) [41]

Optimal result

Integrand size = 26, antiderivative size = 69 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x \left (1-c^2 x\right )} \, dx=\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{b}+2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (2-\frac {2}{1+c \sqrt {x}}\right )-b \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c \sqrt {x}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x \left (1-c^2 x\right )} \, dx=-b \text {arctanh}\left (c \sqrt {x}\right )^2+2 b \text {arctanh}\left (c \sqrt {x}\right ) \log \left (1-e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+a \left (\log (x)-\log \left (1-c^2 x\right )\right )+b \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right ) \] Input:

Integrate[(a + b*ArcTanh[c*Sqrt[x]])/(x*(1 - c^2*x)),x]
 

Output:

-(b*ArcTanh[c*Sqrt[x]]^2) + 2*b*ArcTanh[c*Sqrt[x]]*Log[1 - E^(2*ArcTanh[c* 
Sqrt[x]])] + a*(Log[x] - Log[1 - c^2*x]) + b*PolyLog[2, E^(2*ArcTanh[c*Sqr 
t[x]])]
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {7267, 2026, 6550, 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.

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

\(\Big \downarrow \) 7267

\(\displaystyle 2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x}-c^2 x^{3/2}}d\sqrt {x}\)

\(\Big \downarrow \) 2026

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 2897

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

Input:

Int[(a + b*ArcTanh[c*Sqrt[x]])/(x*(1 - c^2*x)),x]
 

Output:

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

Defintions of rubi rules used

rule 2026

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])
 

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 6494

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]
 

rule 6550

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), 
 x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)), x] + Simp[1/ 
d   Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(179\) vs. \(2(61)=122\).

Time = 0.09 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.61

method	result	size
parts	\(-a \left (-\ln \left (x \right )+\ln \left (c^{2} x -1\right )\right )-b \left (-2 \,\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}\right )+\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )+\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )+\operatorname {dilog}\left (c \sqrt {x}\right )+\operatorname {dilog}\left (1+c \sqrt {x}\right )+\ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )+\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{4}-\operatorname {dilog}\left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )-\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{4}+\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{2}\right )\)	\(180\)
derivativedivides	\(-2 a \left (-\ln \left (c \sqrt {x}\right )+\frac {\ln \left (c \sqrt {x}-1\right )}{2}+\frac {\ln \left (1+c \sqrt {x}\right )}{2}\right )-2 b \left (-\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}\right )+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}+\frac {\operatorname {dilog}\left (c \sqrt {x}\right )}{2}+\frac {\operatorname {dilog}\left (1+c \sqrt {x}\right )}{2}+\frac {\ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{8}-\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{8}+\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}\right )\)	\(203\)
default	\(-2 a \left (-\ln \left (c \sqrt {x}\right )+\frac {\ln \left (c \sqrt {x}-1\right )}{2}+\frac {\ln \left (1+c \sqrt {x}\right )}{2}\right )-2 b \left (-\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}\right )+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}+\frac {\operatorname {dilog}\left (c \sqrt {x}\right )}{2}+\frac {\operatorname {dilog}\left (1+c \sqrt {x}\right )}{2}+\frac {\ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{8}-\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{8}+\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}\right )\)	\(203\)

Input:

int((a+b*arctanh(c*x^(1/2)))/x/(-c^2*x+1),x,method=_RETURNVERBOSE)
 

Output:

-a*(-ln(x)+ln(c^2*x-1))-b*(-2*arctanh(c*x^(1/2))*ln(c*x^(1/2))+arctanh(c*x 
^(1/2))*ln(c*x^(1/2)-1)+arctanh(c*x^(1/2))*ln(1+c*x^(1/2))+dilog(c*x^(1/2) 
)+dilog(1+c*x^(1/2))+ln(c*x^(1/2))*ln(1+c*x^(1/2))+1/4*ln(c*x^(1/2)-1)^2-d 
ilog(1/2*c*x^(1/2)+1/2)-1/2*ln(c*x^(1/2)-1)*ln(1/2*c*x^(1/2)+1/2)-1/4*ln(1 
+c*x^(1/2))^2+1/2*(ln(1+c*x^(1/2))-ln(1/2*c*x^(1/2)+1/2))*ln(-1/2*c*x^(1/2 
)+1/2))
 

Fricas [F]

\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x \left (1-c^2 x\right )} \, dx=\int { -\frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{{\left (c^{2} x - 1\right )} x} \,d x } \] Input:

integrate((a+b*arctanh(c*x^(1/2)))/x/(-c^2*x+1),x, algorithm="fricas")
 

Output:

integral(-(b*arctanh(c*sqrt(x)) + a)/(c^2*x^2 - x), x)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x \left (1-c^2 x\right )} \, dx=- \int \frac {a}{c^{2} x^{2} - x}\, dx - \int \frac {b \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{2} - x}\, dx \] Input:

integrate((a+b*atanh(c*x**(1/2)))/x/(-c**2*x+1),x)
 

Output:

-Integral(a/(c**2*x**2 - x), x) - Integral(b*atanh(c*sqrt(x))/(c**2*x**2 - 
 x), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (60) = 120\).

Time = 0.17 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.30 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x \left (1-c^2 x\right )} \, dx=-\frac {1}{4} \, b \log \left (c \sqrt {x} + 1\right )^{2} + \frac {1}{2} \, b \log \left (c \sqrt {x} + 1\right ) \log \left (-c \sqrt {x} + 1\right ) + \frac {1}{4} \, b \log \left (-c \sqrt {x} + 1\right )^{2} - {\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right )\right )} b - {\left (\log \left (c \sqrt {x}\right ) \log \left (-c \sqrt {x} + 1\right ) + {\rm Li}_2\left (-c \sqrt {x} + 1\right )\right )} b + {\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-c \sqrt {x}\right ) + {\rm Li}_2\left (c \sqrt {x} + 1\right )\right )} b - a {\left (\log \left (c \sqrt {x} + 1\right ) + \log \left (c \sqrt {x} - 1\right ) - \log \left (x\right )\right )} \] Input:

integrate((a+b*arctanh(c*x^(1/2)))/x/(-c^2*x+1),x, algorithm="maxima")
 

Output:

-1/4*b*log(c*sqrt(x) + 1)^2 + 1/2*b*log(c*sqrt(x) + 1)*log(-c*sqrt(x) + 1) 
 + 1/4*b*log(-c*sqrt(x) + 1)^2 - (log(c*sqrt(x) + 1)*log(-1/2*c*sqrt(x) + 
1/2) + dilog(1/2*c*sqrt(x) + 1/2))*b - (log(c*sqrt(x))*log(-c*sqrt(x) + 1) 
 + dilog(-c*sqrt(x) + 1))*b + (log(c*sqrt(x) + 1)*log(-c*sqrt(x)) + dilog( 
c*sqrt(x) + 1))*b - a*(log(c*sqrt(x) + 1) + log(c*sqrt(x) - 1) - log(x))
 

Giac [F]

\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x \left (1-c^2 x\right )} \, dx=\int { -\frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{{\left (c^{2} x - 1\right )} x} \,d x } \] Input:

integrate((a+b*arctanh(c*x^(1/2)))/x/(-c^2*x+1),x, algorithm="giac")
 

Output:

integrate(-(b*arctanh(c*sqrt(x)) + a)/((c^2*x - 1)*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x \left (1-c^2 x\right )} \, dx=-\int \frac {a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{x\,\left (c^2\,x-1\right )} \,d x \] Input:

int(-(a + b*atanh(c*x^(1/2)))/(x*(c^2*x - 1)),x)
 

Output:

-int((a + b*atanh(c*x^(1/2)))/(x*(c^2*x - 1)), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x \left (1-c^2 x\right )} \, dx=-\left (\int \frac {\mathit {atanh} \left (\sqrt {x}\, c \right )}{c^{2} x^{2}-x}d x \right ) b -\mathrm {log}\left (c^{2} x -1\right ) a +\mathrm {log}\left (x \right ) a \] Input:

int((a+b*atanh(c*x^(1/2)))/x/(-c^2*x+1),x)
 

Output:

 - int(atanh(sqrt(x)*c)/(c**2*x**2 - x),x)*b - log(c**2*x - 1)*a + log(x)* 
a