\(\int \frac {(a+b \text {arctanh}(c \sqrt {x}))^3}{x^3} \, dx\) [207]

Optimal result

Integrand size = 18, antiderivative size = 234 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x^3} \, dx=-\frac {b^3 c^3}{2 \sqrt {x}}+\frac {1}{2} b^3 c^4 \text {arctanh}\left (c \sqrt {x}\right )-\frac {b^2 c^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{2 x}+2 b c^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {b c \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 x^{3/2}}-\frac {3 b c^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} c^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{2 x^2}+4 b^2 c^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (2-\frac {2}{1+c \sqrt {x}}\right )-2 b^3 c^4 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c \sqrt {x}}\right ) \] Output:

-1/2*b^3*c^3/x^(1/2)+1/2*b^3*c^4*arctanh(c*x^(1/2))-1/2*b^2*c^2*(a+b*arcta 
nh(c*x^(1/2)))/x+2*b*c^4*(a+b*arctanh(c*x^(1/2)))^2-1/2*b*c*(a+b*arctanh(c 
*x^(1/2)))^2/x^(3/2)-3/2*b*c^3*(a+b*arctanh(c*x^(1/2)))^2/x^(1/2)+1/2*c^4* 
(a+b*arctanh(c*x^(1/2)))^3-1/2*(a+b*arctanh(c*x^(1/2)))^3/x^2+4*b^2*c^4*(a 
+b*arctanh(c*x^(1/2)))*ln(2-2/(1+c*x^(1/2)))-2*b^3*c^4*polylog(2,-1+2/(1+c 
*x^(1/2)))
 

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x^3} \, dx=-\frac {2 a^3+2 a^2 b c \sqrt {x}+2 a b^2 c^2 x+6 a^2 b c^3 x^{3/2}+2 b^3 c^3 x^{3/2}-2 a b^2 c^4 x^2-2 b^2 \left (b c \sqrt {x} \left (-1-3 c^2 x+4 c^3 x^{3/2}\right )+3 a \left (-1+c^4 x^2\right )\right ) \text {arctanh}\left (c \sqrt {x}\right )^2-2 b^3 \left (-1+c^4 x^2\right ) \text {arctanh}\left (c \sqrt {x}\right )^3+2 b \text {arctanh}\left (c \sqrt {x}\right ) \left (3 a^2+b^2 c^2 x \left (1-c^2 x\right )+2 a b c \sqrt {x} \left (1+3 c^2 x\right )-8 b^2 c^4 x^2 \log \left (1-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )+3 a^2 b c^4 x^2 \log \left (1-c \sqrt {x}\right )-3 a^2 b c^4 x^2 \log \left (1+c \sqrt {x}\right )-16 a b^2 c^4 x^2 \log \left (\frac {c \sqrt {x}}{\sqrt {1-c^2 x}}\right )+8 b^3 c^4 x^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )}{4 x^2} \] Input:

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

Output:

-1/4*(2*a^3 + 2*a^2*b*c*Sqrt[x] + 2*a*b^2*c^2*x + 6*a^2*b*c^3*x^(3/2) + 2* 
b^3*c^3*x^(3/2) - 2*a*b^2*c^4*x^2 - 2*b^2*(b*c*Sqrt[x]*(-1 - 3*c^2*x + 4*c 
^3*x^(3/2)) + 3*a*(-1 + c^4*x^2))*ArcTanh[c*Sqrt[x]]^2 - 2*b^3*(-1 + c^4*x 
^2)*ArcTanh[c*Sqrt[x]]^3 + 2*b*ArcTanh[c*Sqrt[x]]*(3*a^2 + b^2*c^2*x*(1 - 
c^2*x) + 2*a*b*c*Sqrt[x]*(1 + 3*c^2*x) - 8*b^2*c^4*x^2*Log[1 - E^(-2*ArcTa 
nh[c*Sqrt[x]])]) + 3*a^2*b*c^4*x^2*Log[1 - c*Sqrt[x]] - 3*a^2*b*c^4*x^2*Lo 
g[1 + c*Sqrt[x]] - 16*a*b^2*c^4*x^2*Log[(c*Sqrt[x])/Sqrt[1 - c^2*x]] + 8*b 
^3*c^4*x^2*PolyLog[2, E^(-2*ArcTanh[c*Sqrt[x]])])/x^2
 

Rubi [A] (verified)

Time = 2.01 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.31, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6454, 6452, 6544, 6452, 6544, 6452, 264, 219, 6510, 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.

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

\(\Big \downarrow \) 6454

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 264

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 6510

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 2897

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

Input:

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

Output:

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

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 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 6454

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

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 6510

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

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]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 16.69 (sec) , antiderivative size = 1219, normalized size of antiderivative = 5.21

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1219\)
default	\(\text {Expression too large to display}\)	\(1219\)
parts	\(\text {Expression too large to display}\)	\(1275\)

Input:

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

Output:

2*c^4*(-1/4*b^3*arctanh(c*x^(1/2))^2/c^3/x^(3/2)-1/4*b^3/c^4/x^2*arctanh(c 
*x^(1/2))^3-1/4*b^3*arctanh(c*x^(1/2))/c^2/x+3/8*I*b^3*Pi*arctanh(c*x^(1/2 
))^2-3/8*I*b^3*Pi*csgn(I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))*csgn(I*(1+c*x^(1/ 
2))^2/(c^2*x-1))^2*arctanh(c*x^(1/2))^2+3/16*I*b^3*Pi*csgn(I*(1+c*x^(1/2)) 
^2/(c^2*x-1))*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1)/(1-(1+c*x^(1/2))^2/(c^2*x-1 
)))^2*arctanh(c*x^(1/2))^2-3/16*I*b^3*Pi*csgn(I/(1-(1+c*x^(1/2))^2/(c^2*x- 
1)))*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1)/(1-(1+c*x^(1/2))^2/(c^2*x-1)))^2*arc 
tanh(c*x^(1/2))^2-3/16*I*b^3*Pi*csgn(I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))^2*c 
sgn(I*(1+c*x^(1/2))^2/(c^2*x-1))*arctanh(c*x^(1/2))^2+3/16*I*b^3*Pi*csgn(I 
/(1-(1+c*x^(1/2))^2/(c^2*x-1)))*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1))*csgn(I*( 
1+c*x^(1/2))^2/(c^2*x-1)/(1-(1+c*x^(1/2))^2/(c^2*x-1)))*arctanh(c*x^(1/2)) 
^2+3*a*b^2*(-1/4/c^4/x^2*arctanh(c*x^(1/2))^2-1/6*arctanh(c*x^(1/2))/c^3/x 
^(3/2)-1/2*arctanh(c*x^(1/2))/c/x^(1/2)+1/4*arctanh(c*x^(1/2))*ln(1+c*x^(1 
/2))-1/4*arctanh(c*x^(1/2))*ln(c*x^(1/2)-1)-1/16*ln(c*x^(1/2)-1)^2+1/8*ln( 
c*x^(1/2)-1)*ln(1/2*c*x^(1/2)+1/2)+1/8*(ln(1+c*x^(1/2))-ln(1/2*c*x^(1/2)+1 
/2))*ln(-1/2*c*x^(1/2)+1/2)-1/16*ln(1+c*x^(1/2))^2-1/12/c^2/x+2/3*ln(c*x^( 
1/2))-1/3*ln(1+c*x^(1/2))-1/3*ln(c*x^(1/2)-1))-1/4*b^3/(c*x^(1/2)-(-c^2*x+ 
1)^(1/2)+1)*(-c^2*x+1)^(1/2)+1/4*b^3/(c*x^(1/2)+(-c^2*x+1)^(1/2)+1)*(-c^2* 
x+1)^(1/2)+3*a^2*b*(-1/4/c^4/x^2*arctanh(c*x^(1/2))-1/12/c^3/x^(3/2)-1/4/c 
/x^(1/2)+1/8*ln(1+c*x^(1/2))-1/8*ln(c*x^(1/2)-1))-1/4*a^3/c^4/x^2-3/4*b...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((a + b*atanh(c*sqrt(x)))**3/x**3, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (193) = 386\).

Time = 1.17 (sec) , antiderivative size = 703, normalized size of antiderivative = 3.00 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-2*(log(c*sqrt(x) + 1)*log(-1/2*c*sqrt(x) + 1/2) + dilog(1/2*c*sqrt(x) + 1 
/2))*b^3*c^4 - 2*(log(c*sqrt(x))*log(-c*sqrt(x) + 1) + dilog(-c*sqrt(x) + 
1))*b^3*c^4 + 2*(log(c*sqrt(x) + 1)*log(-c*sqrt(x)) + dilog(c*sqrt(x) + 1) 
)*b^3*c^4 - 1/8*((6*c^3*log(c*sqrt(x) - 1) - 3*c^3*log(x) + (6*c^2*x + 3*c 
*sqrt(x) + 2)/x^(3/2))*c - 6*log(-c*sqrt(x) + 1)/x^2)*a^2*b + 1/4*(3*a^2*b 
*c^4 - 8*a*b^2*c^4 + b^3*c^4)*log(c*sqrt(x) + 1) - 1/4*(8*a*b^2*c^4 + b^3* 
c^4)*log(c*sqrt(x) - 1) - 1/8*(3*a^2*b*c^4 - 16*a*b^2*c^4)*log(x) - 1/2*a^ 
3/x^2 - 1/16*(4*a^2*b*c*sqrt(x) - (b^3*c^4*x^2 - b^3)*log(c*sqrt(x) + 1)^3 
 + (b^3*c^4*x^2 - b^3)*log(-c*sqrt(x) + 1)^3 + 2*(3*b^3*c^3*x^(3/2) + b^3* 
c*sqrt(x) + 3*a*b^2 - (3*a*b^2*c^4 - 4*b^3*c^4)*x^2)*log(c*sqrt(x) + 1)^2 
+ (6*b^3*c^3*x^(3/2) + 2*b^3*c*sqrt(x) + 6*a*b^2 - 2*(3*a*b^2*c^4 + 4*b^3* 
c^4)*x^2 - 3*(b^3*c^4*x^2 - b^3)*log(c*sqrt(x) + 1))*log(-c*sqrt(x) + 1)^2 
 + 4*(3*a^2*b*c^3 + 2*b^3*c^3)*x^(3/2) - 2*(3*a^2*b*c^2 - 4*a*b^2*c^2)*x + 
 4*(6*a*b^2*c^3*x^(3/2) + b^3*c^2*x + 2*a*b^2*c*sqrt(x) + 3*a^2*b)*log(c*s 
qrt(x) + 1) - (24*a*b^2*c^3*x^(3/2) + 4*b^3*c^2*x + 8*a*b^2*c*sqrt(x) - 3* 
(b^3*c^4*x^2 - b^3)*log(c*sqrt(x) + 1)^2 + 4*(3*b^3*c^3*x^(3/2) + b^3*c*sq 
rt(x) + 3*a*b^2 - (3*a*b^2*c^4 - 4*b^3*c^4)*x^2)*log(c*sqrt(x) + 1))*log(- 
c*sqrt(x) + 1))/x^2
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x^3} \, dx=\frac {\mathit {atanh} \left (\sqrt {x}\, c \right )^{3} b^{3} c^{4} x^{2}-\mathit {atanh} \left (\sqrt {x}\, c \right )^{3} b^{3}-3 \sqrt {x}\, \mathit {atanh} \left (\sqrt {x}\, c \right )^{2} b^{3} c^{3} x -\sqrt {x}\, \mathit {atanh} \left (\sqrt {x}\, c \right )^{2} b^{3} c +3 \mathit {atanh} \left (\sqrt {x}\, c \right )^{2} a \,b^{2} c^{4} x^{2}-3 \mathit {atanh} \left (\sqrt {x}\, c \right )^{2} a \,b^{2}-6 \sqrt {x}\, \mathit {atanh} \left (\sqrt {x}\, c \right ) a \,b^{2} c^{3} x -2 \sqrt {x}\, \mathit {atanh} \left (\sqrt {x}\, c \right ) a \,b^{2} c +3 \mathit {atanh} \left (\sqrt {x}\, c \right ) a^{2} b \,c^{4} x^{2}-3 \mathit {atanh} \left (\sqrt {x}\, c \right ) a^{2} b -8 \mathit {atanh} \left (\sqrt {x}\, c \right ) a \,b^{2} c^{4} x^{2}+\mathit {atanh} \left (\sqrt {x}\, c \right ) b^{3} c^{4} x^{2}-\mathit {atanh} \left (\sqrt {x}\, c \right ) b^{3} c^{2} x -3 \sqrt {x}\, a^{2} b \,c^{3} x -\sqrt {x}\, a^{2} b c -\sqrt {x}\, b^{3} c^{3} x -4 \left (\int \frac {\mathit {atanh} \left (\sqrt {x}\, c \right )}{c^{2} x^{2}-x}d x \right ) b^{3} c^{4} x^{2}-8 \,\mathrm {log}\left (\sqrt {x}\, c -1\right ) a \,b^{2} c^{4} x^{2}+8 \,\mathrm {log}\left (\sqrt {x}\right ) a \,b^{2} c^{4} x^{2}-a^{3}-a \,b^{2} c^{2} x}{2 x^{2}} \] Input:

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

Output:

(atanh(sqrt(x)*c)**3*b**3*c**4*x**2 - atanh(sqrt(x)*c)**3*b**3 - 3*sqrt(x) 
*atanh(sqrt(x)*c)**2*b**3*c**3*x - sqrt(x)*atanh(sqrt(x)*c)**2*b**3*c + 3* 
atanh(sqrt(x)*c)**2*a*b**2*c**4*x**2 - 3*atanh(sqrt(x)*c)**2*a*b**2 - 6*sq 
rt(x)*atanh(sqrt(x)*c)*a*b**2*c**3*x - 2*sqrt(x)*atanh(sqrt(x)*c)*a*b**2*c 
 + 3*atanh(sqrt(x)*c)*a**2*b*c**4*x**2 - 3*atanh(sqrt(x)*c)*a**2*b - 8*ata 
nh(sqrt(x)*c)*a*b**2*c**4*x**2 + atanh(sqrt(x)*c)*b**3*c**4*x**2 - atanh(s 
qrt(x)*c)*b**3*c**2*x - 3*sqrt(x)*a**2*b*c**3*x - sqrt(x)*a**2*b*c - sqrt( 
x)*b**3*c**3*x - 4*int(atanh(sqrt(x)*c)/(c**2*x**2 - x),x)*b**3*c**4*x**2 
- 8*log(sqrt(x)*c - 1)*a*b**2*c**4*x**2 + 8*log(sqrt(x))*a*b**2*c**4*x**2 
- a**3 - a*b**2*c**2*x)/(2*x**2)