\(\int \frac {a+b \text {arctanh}(c \sqrt {x})}{x^3 (1-c^2 x)} \, dx\) [43]

Optimal result

Integrand size = 26, antiderivative size = 157 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^3 \left (1-c^2 x\right )} \, dx=-\frac {b c}{6 x^{3/2}}-\frac {3 b c^3}{2 \sqrt {x}}+\frac {3}{2} b c^4 \text {arctanh}\left (c \sqrt {x}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x^2}-\frac {c^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{x}+\frac {c^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{b}+2 c^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (2-\frac {2}{1+c \sqrt {x}}\right )-b c^4 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c \sqrt {x}}\right ) \] Output:

-1/6*b*c/x^(3/2)-3/2*b*c^3/x^(1/2)+3/2*b*c^4*arctanh(c*x^(1/2))-1/2*(a+b*a 
rctanh(c*x^(1/2)))/x^2-c^2*(a+b*arctanh(c*x^(1/2)))/x+c^4*(a+b*arctanh(c*x 
^(1/2)))^2/b+2*c^4*(a+b*arctanh(c*x^(1/2)))*ln(2-2/(1+c*x^(1/2)))-b*c^4*po 
lylog(2,-1+2/(1+c*x^(1/2)))
 

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.01 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^3 \left (1-c^2 x\right )} \, dx=-\frac {3 a+b c \sqrt {x}+6 a c^2 x+9 b c^3 x^{3/2}-6 b c^4 x^2 \text {arctanh}\left (c \sqrt {x}\right )^2-3 b \text {arctanh}\left (c \sqrt {x}\right ) \left (-1-2 c^2 x+3 c^4 x^2+4 c^4 x^2 \log \left (1-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )-6 a c^4 x^2 \log (x)+6 a c^4 x^2 \log \left (1-c^2 x\right )}{6 x^2}-b c^4 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right ) \] Input:

Integrate[(a + b*ArcTanh[c*Sqrt[x]])/(x^3*(1 - c^2*x)),x]
 

Output:

-1/6*(3*a + b*c*Sqrt[x] + 6*a*c^2*x + 9*b*c^3*x^(3/2) - 6*b*c^4*x^2*ArcTan 
h[c*Sqrt[x]]^2 - 3*b*ArcTanh[c*Sqrt[x]]*(-1 - 2*c^2*x + 3*c^4*x^2 + 4*c^4* 
x^2*Log[1 - E^(-2*ArcTanh[c*Sqrt[x]])]) - 6*a*c^4*x^2*Log[x] + 6*a*c^4*x^2 
*Log[1 - c^2*x])/x^2 - b*c^4*PolyLog[2, E^(-2*ArcTanh[c*Sqrt[x]])]
 

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.18, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {7267, 2026, 6544, 6452, 264, 264, 219, 6544, 6452, 264, 219, 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.

\(\displaystyle \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^3 \left (1-c^2 x\right )} \, dx\)

\(\Big \downarrow \) 7267

\(\displaystyle 2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^{5/2}-c^2 x^{7/2}}d\sqrt {x}\)

\(\Big \downarrow \) 2026

\(\displaystyle 2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^{5/2} \left (1-c^2 x\right )}d\sqrt {x}\)

\(\Big \downarrow \) 6544

\(\displaystyle 2 \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^{3/2} \left (1-c^2 x\right )}d\sqrt {x}+\int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^{5/2}}d\sqrt {x}\right )\)

\(\Big \downarrow \) 6452

\(\displaystyle 2 \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^{3/2} \left (1-c^2 x\right )}d\sqrt {x}+\frac {1}{4} b c \int \frac {1}{x^2 \left (1-c^2 x\right )}d\sqrt {x}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{4 x^2}\right )\)

\(\Big \downarrow \) 264

\(\displaystyle 2 \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^{3/2} \left (1-c^2 x\right )}d\sqrt {x}+\frac {1}{4} b c \left (c^2 \int \frac {1}{x \left (1-c^2 x\right )}d\sqrt {x}-\frac {1}{3 x^{3/2}}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{4 x^2}\right )\)

\(\Big \downarrow \) 264

\(\displaystyle 2 \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^{3/2} \left (1-c^2 x\right )}d\sqrt {x}+\frac {1}{4} b c \left (c^2 \left (c^2 \int \frac {1}{1-c^2 x}d\sqrt {x}-\frac {1}{\sqrt {x}}\right )-\frac {1}{3 x^{3/2}}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{4 x^2}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle 2 \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^{3/2} \left (1-c^2 x\right )}d\sqrt {x}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{4 x^2}+\frac {1}{4} b c \left (c^2 \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )-\frac {1}{3 x^{3/2}}\right )\right )\)

\(\Big \downarrow \) 6544

\(\displaystyle 2 \left (c^2 \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x} \left (1-c^2 x\right )}d\sqrt {x}+\int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^{3/2}}d\sqrt {x}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{4 x^2}+\frac {1}{4} b c \left (c^2 \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )-\frac {1}{3 x^{3/2}}\right )\right )\)

\(\Big \downarrow \) 6452

\(\displaystyle 2 \left (c^2 \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x} \left (1-c^2 x\right )}d\sqrt {x}+\frac {1}{2} b c \int \frac {1}{x \left (1-c^2 x\right )}d\sqrt {x}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{4 x^2}+\frac {1}{4} b c \left (c^2 \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )-\frac {1}{3 x^{3/2}}\right )\right )\)

\(\Big \downarrow \) 264

\(\displaystyle 2 \left (c^2 \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x} \left (1-c^2 x\right )}d\sqrt {x}+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-c^2 x}d\sqrt {x}-\frac {1}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{4 x^2}+\frac {1}{4} b c \left (c^2 \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )-\frac {1}{3 x^{3/2}}\right )\right )\)

\(\Big \downarrow \) 219

\(\displaystyle 2 \left (c^2 \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x} \left (1-c^2 x\right )}d\sqrt {x}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x}+\frac {1}{2} b c \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{4 x^2}+\frac {1}{4} b c \left (c^2 \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )-\frac {1}{3 x^{3/2}}\right )\right )\)

\(\Big \downarrow \) 6550

\(\displaystyle 2 \left (c^2 \left (c^2 \left (\int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\left (\sqrt {x} c+1\right ) \sqrt {x}}d\sqrt {x}+\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x}+\frac {1}{2} b c \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{4 x^2}+\frac {1}{4} b c \left (c^2 \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )-\frac {1}{3 x^{3/2}}\right )\right )\)

\(\Big \downarrow \) 6494

\(\displaystyle 2 \left (c^2 \left (c^2 \left (-b c \int \frac {\log \left (2-\frac {2}{\sqrt {x} c+1}\right )}{1-c^2 x}d\sqrt {x}+\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b}+\log \left (2-\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x}+\frac {1}{2} b c \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{4 x^2}+\frac {1}{4} b c \left (c^2 \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )-\frac {1}{3 x^{3/2}}\right )\right )\)

\(\Big \downarrow \) 2897

\(\displaystyle 2 \left (c^2 \left (c^2 \left (\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b}+\log \left (2-\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{\sqrt {x} c+1}-1\right )\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x}+\frac {1}{2} b c \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{4 x^2}+\frac {1}{4} b c \left (c^2 \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )-\frac {1}{3 x^{3/2}}\right )\right )\)

Input:

Int[(a + b*ArcTanh[c*Sqrt[x]])/(x^3*(1 - c^2*x)),x]
 

Output:

2*(-1/4*(a + b*ArcTanh[c*Sqrt[x]])/x^2 + (b*c*(-1/3*1/x^(3/2) + c^2*(-(1/S 
qrt[x]) + c*ArcTanh[c*Sqrt[x]])))/4 + c^2*(-1/2*(a + b*ArcTanh[c*Sqrt[x]]) 
/x + (b*c*(-(1/Sqrt[x]) + c*ArcTanh[c*Sqrt[x]]))/2 + c^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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

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 6452

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : 
> Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m 
+ 1))   Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x 
], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 
] && IntegerQ[m])) && NeQ[m, -1]
 

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 6544

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

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. \(272\) vs. \(2(131)=262\).

Time = 0.10 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.74

method	result	size
parts	\(-a \left (\frac {1}{2 x^{2}}+\frac {c^{2}}{x}-c^{4} \ln \left (x \right )+c^{4} \ln \left (c^{2} x -1\right )\right )-2 b \,c^{4} \left (\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{4 c^{4} x^{2}}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{2 c^{2} x}-\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}\right )+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}+\frac {\operatorname {dilog}\left (c \sqrt {x}\right )}{2}+\frac {\operatorname {dilog}\left (1+c \sqrt {x}\right )}{2}+\frac {\ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{8}-\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{8}+\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {1}{12 c^{3} x^{\frac {3}{2}}}+\frac {3}{4 c \sqrt {x}}+\frac {3 \ln \left (c \sqrt {x}-1\right )}{8}-\frac {3 \ln \left (1+c \sqrt {x}\right )}{8}\right )\)	\(273\)
derivativedivides	\(-2 c^{4} \left (a \left (\frac {1}{4 c^{4} x^{2}}+\frac {1}{2 c^{2} x}-\ln \left (c \sqrt {x}\right )+\frac {\ln \left (c \sqrt {x}-1\right )}{2}+\frac {\ln \left (1+c \sqrt {x}\right )}{2}\right )+b \left (\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{4 c^{4} x^{2}}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{2 c^{2} x}-\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}\right )+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}+\frac {\operatorname {dilog}\left (c \sqrt {x}\right )}{2}+\frac {\operatorname {dilog}\left (1+c \sqrt {x}\right )}{2}+\frac {\ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{8}-\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{8}+\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {1}{12 c^{3} x^{\frac {3}{2}}}+\frac {3}{4 c \sqrt {x}}+\frac {3 \ln \left (c \sqrt {x}-1\right )}{8}-\frac {3 \ln \left (1+c \sqrt {x}\right )}{8}\right )\right )\)	\(286\)
default	\(-2 c^{4} \left (a \left (\frac {1}{4 c^{4} x^{2}}+\frac {1}{2 c^{2} x}-\ln \left (c \sqrt {x}\right )+\frac {\ln \left (c \sqrt {x}-1\right )}{2}+\frac {\ln \left (1+c \sqrt {x}\right )}{2}\right )+b \left (\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{4 c^{4} x^{2}}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{2 c^{2} x}-\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}\right )+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}+\frac {\operatorname {dilog}\left (c \sqrt {x}\right )}{2}+\frac {\operatorname {dilog}\left (1+c \sqrt {x}\right )}{2}+\frac {\ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{8}-\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{8}+\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {1}{12 c^{3} x^{\frac {3}{2}}}+\frac {3}{4 c \sqrt {x}}+\frac {3 \ln \left (c \sqrt {x}-1\right )}{8}-\frac {3 \ln \left (1+c \sqrt {x}\right )}{8}\right )\right )\)	\(286\)

Input:

int((a+b*arctanh(c*x^(1/2)))/x^3/(-c^2*x+1),x,method=_RETURNVERBOSE)
 

Output:

-a*(1/2/x^2+c^2/x-c^4*ln(x)+c^4*ln(c^2*x-1))-2*b*c^4*(1/4*arctanh(c*x^(1/2 
))/c^4/x^2+1/2*arctanh(c*x^(1/2))/c^2/x-arctanh(c*x^(1/2))*ln(c*x^(1/2))+1 
/2*arctanh(c*x^(1/2))*ln(c*x^(1/2)-1)+1/2*arctanh(c*x^(1/2))*ln(1+c*x^(1/2 
))+1/2*dilog(c*x^(1/2))+1/2*dilog(1+c*x^(1/2))+1/2*ln(c*x^(1/2))*ln(1+c*x^ 
(1/2))-1/2*dilog(1/2*c*x^(1/2)+1/2)-1/4*ln(c*x^(1/2)-1)*ln(1/2*c*x^(1/2)+1 
/2)+1/8*ln(c*x^(1/2)-1)^2-1/8*ln(1+c*x^(1/2))^2+1/4*(ln(1+c*x^(1/2))-ln(1/ 
2*c*x^(1/2)+1/2))*ln(-1/2*c*x^(1/2)+1/2)+1/12/c^3/x^(3/2)+3/4/c/x^(1/2)+3/ 
8*ln(c*x^(1/2)-1)-3/8*ln(1+c*x^(1/2)))
 

Fricas [F]

\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^3 \left (1-c^2 x\right )} \, dx=\int { -\frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{{\left (c^{2} x - 1\right )} x^{3}} \,d x } \] Input:

integrate((a+b*arctanh(c*x^(1/2)))/x^3/(-c^2*x+1),x, algorithm="fricas")
 

Output:

integral(-(b*arctanh(c*sqrt(x)) + a)/(c^2*x^4 - x^3), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^3 \left (1-c^2 x\right )} \, dx=- \int \frac {a}{c^{2} x^{4} - x^{3}}\, dx - \int \frac {b \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{4} - x^{3}}\, dx \] Input:

integrate((a+b*atanh(c*x**(1/2)))/x**3/(-c**2*x+1),x)
 

Output:

-Integral(a/(c**2*x**4 - x**3), x) - Integral(b*atanh(c*sqrt(x))/(c**2*x** 
4 - x**3), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (130) = 260\).

Time = 0.20 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.85 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^3 \left (1-c^2 x\right )} \, dx=-{\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right )\right )} b c^{4} - {\left (\log \left (c \sqrt {x}\right ) \log \left (-c \sqrt {x} + 1\right ) + {\rm Li}_2\left (-c \sqrt {x} + 1\right )\right )} b c^{4} + {\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-c \sqrt {x}\right ) + {\rm Li}_2\left (c \sqrt {x} + 1\right )\right )} b c^{4} + \frac {3}{4} \, b c^{4} \log \left (c \sqrt {x} + 1\right ) - \frac {3}{4} \, b c^{4} \log \left (c \sqrt {x} - 1\right ) - \frac {1}{2} \, {\left (2 \, c^{4} \log \left (c \sqrt {x} + 1\right ) + 2 \, c^{4} \log \left (c \sqrt {x} - 1\right ) - 2 \, c^{4} \log \left (x\right ) + \frac {2 \, c^{2} x + 1}{x^{2}}\right )} a - \frac {3 \, b c^{4} x^{2} \log \left (c \sqrt {x} + 1\right )^{2} - 3 \, b c^{4} x^{2} \log \left (-c \sqrt {x} + 1\right )^{2} + 18 \, b c^{3} x^{\frac {3}{2}} + 2 \, b c \sqrt {x} + 3 \, {\left (2 \, b c^{2} x + b\right )} \log \left (c \sqrt {x} + 1\right ) - 3 \, {\left (2 \, b c^{4} x^{2} \log \left (c \sqrt {x} + 1\right ) + 2 \, b c^{2} x + b\right )} \log \left (-c \sqrt {x} + 1\right )}{12 \, x^{2}} \] Input:

integrate((a+b*arctanh(c*x^(1/2)))/x^3/(-c^2*x+1),x, algorithm="maxima")
 

Output:

-(log(c*sqrt(x) + 1)*log(-1/2*c*sqrt(x) + 1/2) + dilog(1/2*c*sqrt(x) + 1/2 
))*b*c^4 - (log(c*sqrt(x))*log(-c*sqrt(x) + 1) + dilog(-c*sqrt(x) + 1))*b* 
c^4 + (log(c*sqrt(x) + 1)*log(-c*sqrt(x)) + dilog(c*sqrt(x) + 1))*b*c^4 + 
3/4*b*c^4*log(c*sqrt(x) + 1) - 3/4*b*c^4*log(c*sqrt(x) - 1) - 1/2*(2*c^4*l 
og(c*sqrt(x) + 1) + 2*c^4*log(c*sqrt(x) - 1) - 2*c^4*log(x) + (2*c^2*x + 1 
)/x^2)*a - 1/12*(3*b*c^4*x^2*log(c*sqrt(x) + 1)^2 - 3*b*c^4*x^2*log(-c*sqr 
t(x) + 1)^2 + 18*b*c^3*x^(3/2) + 2*b*c*sqrt(x) + 3*(2*b*c^2*x + b)*log(c*s 
qrt(x) + 1) - 3*(2*b*c^4*x^2*log(c*sqrt(x) + 1) + 2*b*c^2*x + b)*log(-c*sq 
rt(x) + 1))/x^2
 

Giac [F]

\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^3 \left (1-c^2 x\right )} \, dx=\int { -\frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{{\left (c^{2} x - 1\right )} x^{3}} \,d x } \] Input:

integrate((a+b*arctanh(c*x^(1/2)))/x^3/(-c^2*x+1),x, algorithm="giac")
 

Output:

integrate(-(b*arctanh(c*sqrt(x)) + a)/((c^2*x - 1)*x^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^3 \left (1-c^2 x\right )} \, dx=-\int \frac {a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{x^3\,\left (c^2\,x-1\right )} \,d x \] Input:

int(-(a + b*atanh(c*x^(1/2)))/(x^3*(c^2*x - 1)),x)
 

Output:

-int((a + b*atanh(c*x^(1/2)))/(x^3*(c^2*x - 1)), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^3 \left (1-c^2 x\right )} \, dx=\frac {9 \mathit {atanh} \left (\sqrt {x}\, c \right ) b \,c^{4} x^{2}-6 \mathit {atanh} \left (\sqrt {x}\, c \right ) b \,c^{2} x -3 \mathit {atanh} \left (\sqrt {x}\, c \right ) b -9 \sqrt {x}\, b \,c^{3} x -\sqrt {x}\, b c -6 \left (\int \frac {\mathit {atanh} \left (\sqrt {x}\, c \right )}{c^{2} x^{2}-x}d x \right ) b \,c^{4} x^{2}-6 \,\mathrm {log}\left (c^{2} x -1\right ) a \,c^{4} x^{2}+6 \,\mathrm {log}\left (x \right ) a \,c^{4} x^{2}-6 a \,c^{2} x -3 a}{6 x^{2}} \] Input:

int((a+b*atanh(c*x^(1/2)))/x^3/(-c^2*x+1),x)
 

Output:

(9*atanh(sqrt(x)*c)*b*c**4*x**2 - 6*atanh(sqrt(x)*c)*b*c**2*x - 3*atanh(sq 
rt(x)*c)*b - 9*sqrt(x)*b*c**3*x - sqrt(x)*b*c - 6*int(atanh(sqrt(x)*c)/(c* 
*2*x**2 - x),x)*b*c**4*x**2 - 6*log(c**2*x - 1)*a*c**4*x**2 + 6*log(x)*a*c 
**4*x**2 - 6*a*c**2*x - 3*a)/(6*x**2)