\(\int \frac {(a+b \text {arctanh}(c x)) (d+e \log (1-c^2 x^2))}{x^4} \, dx\) [529]

Optimal result

Integrand size = 27, antiderivative size = 197 \[ \int \frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^4} \, dx=\frac {2 c^2 e (a+b \text {arctanh}(c x))}{3 x}-\frac {c^3 e (a+b \text {arctanh}(c x))^2}{3 b}-b c^3 e \log (x)+\frac {1}{3} b c^3 e \log \left (1-c^2 x^2\right )-\frac {b c \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{6 x^2}-\frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right )}{3 x^3}+\frac {1}{6} b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right ) \log \left (1-\frac {1}{1-c^2 x^2}\right )-\frac {1}{6} b c^3 e \operatorname {PolyLog}\left (2,\frac {1}{1-c^2 x^2}\right ) \] Output:

2/3*c^2*e*(a+b*arctanh(c*x))/x-1/3*c^3*e*(a+b*arctanh(c*x))^2/b-b*c^3*e*ln 
(x)+1/3*b*c^3*e*ln(-c^2*x^2+1)-1/6*b*c*(-c^2*x^2+1)*(d+e*ln(-c^2*x^2+1))/x 
^2-1/3*(a+b*arctanh(c*x))*(d+e*ln(-c^2*x^2+1))/x^3+1/6*b*c^3*(d+e*ln(-c^2* 
x^2+1))*ln(1-1/(-c^2*x^2+1))-1/6*b*c^3*e*polylog(2,1/(-c^2*x^2+1))
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(460\) vs. \(2(197)=394\).

Time = 0.21 (sec) , antiderivative size = 460, normalized size of antiderivative = 2.34 \[ \int \frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^4} \, dx=\frac {1}{6} \left (-\frac {2 a d}{x^3}-\frac {b c d}{x^2}+\frac {4 a c^2 e}{x}-4 a c^3 e \text {arctanh}(c x)-\frac {2 b d \text {arctanh}(c x)}{x^3}+\frac {4 b c^2 e \text {arctanh}(c x)}{x}-2 b c^3 e \text {arctanh}(c x)^2+2 b c^3 d \log (x)-2 b c^3 e \log (x)+\frac {1}{2} b c^3 e \log ^2\left (-\frac {1}{c}+x\right )+\frac {1}{2} b c^3 e \log ^2\left (\frac {1}{c}+x\right )+b c^3 e \log \left (\frac {1}{c}+x\right ) \log \left (\frac {1}{2} (1-c x)\right )-2 b c^3 e \log (x) \log (1-c x)+b c^3 e \log \left (-\frac {1}{c}+x\right ) \log \left (\frac {1}{2} (1+c x)\right )-2 b c^3 e \log (x) \log (1+c x)-4 b c^3 e \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )-b c^3 d \log \left (1-c^2 x^2\right )+b c^3 e \log \left (1-c^2 x^2\right )-\frac {2 a e \log \left (1-c^2 x^2\right )}{x^3}-\frac {b c e \log \left (1-c^2 x^2\right )}{x^2}-\frac {2 b e \text {arctanh}(c x) \log \left (1-c^2 x^2\right )}{x^3}+2 b c^3 e \log (x) \log \left (1-c^2 x^2\right )-b c^3 e \log \left (-\frac {1}{c}+x\right ) \log \left (1-c^2 x^2\right )-b c^3 e \log \left (\frac {1}{c}+x\right ) \log \left (1-c^2 x^2\right )-2 b c^3 e \operatorname {PolyLog}(2,-c x)-2 b c^3 e \operatorname {PolyLog}(2,c x)+b c^3 e \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {c x}{2}\right )+b c^3 e \operatorname {PolyLog}\left (2,\frac {1}{2} (1+c x)\right )\right ) \] Input:

Integrate[((a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]))/x^4,x]
 

Output:

((-2*a*d)/x^3 - (b*c*d)/x^2 + (4*a*c^2*e)/x - 4*a*c^3*e*ArcTanh[c*x] - (2* 
b*d*ArcTanh[c*x])/x^3 + (4*b*c^2*e*ArcTanh[c*x])/x - 2*b*c^3*e*ArcTanh[c*x 
]^2 + 2*b*c^3*d*Log[x] - 2*b*c^3*e*Log[x] + (b*c^3*e*Log[-c^(-1) + x]^2)/2 
 + (b*c^3*e*Log[c^(-1) + x]^2)/2 + b*c^3*e*Log[c^(-1) + x]*Log[(1 - c*x)/2 
] - 2*b*c^3*e*Log[x]*Log[1 - c*x] + b*c^3*e*Log[-c^(-1) + x]*Log[(1 + c*x) 
/2] - 2*b*c^3*e*Log[x]*Log[1 + c*x] - 4*b*c^3*e*Log[(c*x)/Sqrt[1 - c^2*x^2 
]] - b*c^3*d*Log[1 - c^2*x^2] + b*c^3*e*Log[1 - c^2*x^2] - (2*a*e*Log[1 - 
c^2*x^2])/x^3 - (b*c*e*Log[1 - c^2*x^2])/x^2 - (2*b*e*ArcTanh[c*x]*Log[1 - 
 c^2*x^2])/x^3 + 2*b*c^3*e*Log[x]*Log[1 - c^2*x^2] - b*c^3*e*Log[-c^(-1) + 
 x]*Log[1 - c^2*x^2] - b*c^3*e*Log[c^(-1) + x]*Log[1 - c^2*x^2] - 2*b*c^3* 
e*PolyLog[2, -(c*x)] - 2*b*c^3*e*PolyLog[2, c*x] + b*c^3*e*PolyLog[2, 1/2 
- (c*x)/2] + b*c^3*e*PolyLog[2, (1 + c*x)/2])/6
 

Rubi [A] (warning: unable to verify)

Time = 1.60 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.90, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {6643, 2925, 2858, 27, 2789, 2751, 16, 2779, 2838, 6544, 6452, 243, 47, 14, 16, 6510}

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.

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

\(\Big \downarrow \) 6643

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

\(\Big \downarrow \) 2925

\(\displaystyle -\frac {2}{3} c^2 e \int \frac {a+b \text {arctanh}(c x)}{x^2 \left (1-c^2 x^2\right )}dx+\frac {1}{6} b c \int \frac {d+e \log \left (1-c^2 x^2\right )}{x^4 \left (1-c^2 x^2\right )}dx^2-\frac {(a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}\)

\(\Big \downarrow \) 2858

\(\displaystyle -\frac {2}{3} c^2 e \int \frac {a+b \text {arctanh}(c x)}{x^2 \left (1-c^2 x^2\right )}dx-\frac {b \int \frac {d+e \log \left (1-c^2 x^2\right )}{x^6}d\left (1-c^2 x^2\right )}{6 c}-\frac {(a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{3} c^2 e \int \frac {a+b \text {arctanh}(c x)}{x^2 \left (1-c^2 x^2\right )}dx-\frac {1}{6} b c^3 \int \frac {d+e \log \left (1-c^2 x^2\right )}{c^4 x^6}d\left (1-c^2 x^2\right )-\frac {(a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}\)

\(\Big \downarrow \) 2789

\(\displaystyle -\frac {2}{3} c^2 e \int \frac {a+b \text {arctanh}(c x)}{x^2 \left (1-c^2 x^2\right )}dx-\frac {1}{6} b c^3 \left (\int \frac {d+e \log \left (1-c^2 x^2\right )}{c^2 x^4}d\left (1-c^2 x^2\right )+\int \frac {d+e \log \left (1-c^2 x^2\right )}{c^4 x^4}d\left (1-c^2 x^2\right )\right )-\frac {(a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}\)

\(\Big \downarrow \) 2751

\(\displaystyle -\frac {2}{3} c^2 e \int \frac {a+b \text {arctanh}(c x)}{x^2 \left (1-c^2 x^2\right )}dx-\frac {1}{6} b c^3 \left (\int \frac {d+e \log \left (1-c^2 x^2\right )}{c^2 x^4}d\left (1-c^2 x^2\right )-e \int \frac {1}{c^2 x^2}d\left (1-c^2 x^2\right )+\frac {\left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{c^2 x^2}\right )-\frac {(a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}\)

\(\Big \downarrow \) 16

\(\displaystyle -\frac {2}{3} c^2 e \int \frac {a+b \text {arctanh}(c x)}{x^2 \left (1-c^2 x^2\right )}dx-\frac {1}{6} b c^3 \left (\int \frac {d+e \log \left (1-c^2 x^2\right )}{c^2 x^4}d\left (1-c^2 x^2\right )+\frac {\left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{c^2 x^2}+e \log \left (c^2 x^2\right )\right )-\frac {(a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}\)

\(\Big \downarrow \) 2779

\(\displaystyle -\frac {2}{3} c^2 e \int \frac {a+b \text {arctanh}(c x)}{x^2 \left (1-c^2 x^2\right )}dx-\frac {1}{6} b c^3 \left (e \int \frac {\log \left (1-\frac {1}{x^2}\right )}{x^2}d\left (1-c^2 x^2\right )+\frac {\left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )+e \log \left (c^2 x^2\right )\right )-\frac {(a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {2}{3} c^2 e \int \frac {a+b \text {arctanh}(c x)}{x^2 \left (1-c^2 x^2\right )}dx-\frac {(a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}-\frac {1}{6} b c^3 \left (\frac {\left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )+e \log \left (c^2 x^2\right )+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\)

\(\Big \downarrow \) 6544

\(\displaystyle -\frac {2}{3} c^2 e \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx+\int \frac {a+b \text {arctanh}(c x)}{x^2}dx\right )-\frac {(a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}-\frac {1}{6} b c^3 \left (\frac {\left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )+e \log \left (c^2 x^2\right )+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\)

\(\Big \downarrow \) 6452

\(\displaystyle -\frac {2}{3} c^2 e \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx+b c \int \frac {1}{x \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arctanh}(c x)}{x}\right )-\frac {(a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}-\frac {1}{6} b c^3 \left (\frac {\left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )+e \log \left (c^2 x^2\right )+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {2}{3} c^2 e \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx+\frac {1}{2} b c \int \frac {1}{x^2 \left (1-c^2 x^2\right )}dx^2-\frac {a+b \text {arctanh}(c x)}{x}\right )-\frac {(a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}-\frac {1}{6} b c^3 \left (\frac {\left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )+e \log \left (c^2 x^2\right )+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\)

\(\Big \downarrow \) 47

\(\displaystyle -\frac {2}{3} c^2 e \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-c^2 x^2}dx^2+\int \frac {1}{x^2}dx^2\right )-\frac {a+b \text {arctanh}(c x)}{x}\right )-\frac {(a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}-\frac {1}{6} b c^3 \left (\frac {\left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )+e \log \left (c^2 x^2\right )+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\)

\(\Big \downarrow \) 14

\(\displaystyle -\frac {2}{3} c^2 e \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-c^2 x^2}dx^2+\log \left (x^2\right )\right )-\frac {a+b \text {arctanh}(c x)}{x}\right )-\frac {(a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}-\frac {1}{6} b c^3 \left (\frac {\left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )+e \log \left (c^2 x^2\right )+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\)

\(\Big \downarrow \) 16

\(\displaystyle -\frac {2}{3} c^2 e \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx-\frac {a+b \text {arctanh}(c x)}{x}+\frac {1}{2} b c \left (\log \left (x^2\right )-\log \left (1-c^2 x^2\right )\right )\right )-\frac {(a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}-\frac {1}{6} b c^3 \left (\frac {\left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )+e \log \left (c^2 x^2\right )+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\)

\(\Big \downarrow \) 6510

\(\displaystyle -\frac {(a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}-\frac {2}{3} c^2 e \left (\frac {c (a+b \text {arctanh}(c x))^2}{2 b}-\frac {a+b \text {arctanh}(c x)}{x}+\frac {1}{2} b c \left (\log \left (x^2\right )-\log \left (1-c^2 x^2\right )\right )\right )-\frac {1}{6} b c^3 \left (\frac {\left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )+e \log \left (c^2 x^2\right )+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\)

Input:

Int[((a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]))/x^4,x]
 

Output:

(-2*c^2*e*(-((a + b*ArcTanh[c*x])/x) + (c*(a + b*ArcTanh[c*x])^2)/(2*b) + 
(b*c*(Log[x^2] - Log[1 - c^2*x^2]))/2))/3 - ((a + b*ArcTanh[c*x])*(d + e*L 
og[1 - c^2*x^2]))/(3*x^3) - (b*c^3*(e*Log[c^2*x^2] + ((1 - c^2*x^2)*(d + e 
*Log[1 - c^2*x^2]))/(c^2*x^2) - Log[1 - x^(-2)]*(d + e*Log[1 - c^2*x^2]) + 
 e*PolyLog[2, x^(-2)]))/6
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2751

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x 
_Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* 
(n/d)   Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, 
x] && EqQ[r*(q + 1) + 1, 0]
 

rule 2779

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r 
_.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) 
, x] + Simp[b*n*(p/(d*r))   Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 
 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
 

rule 2789

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ 
(x_), x_Symbol] :> Simp[1/d   Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x 
), x], x] - Simp[e/d   Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free 
Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 2858

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ 
.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e   Subst[In 
t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + 
e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - 
d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
 

rule 2925

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Si 
mplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q, x], 
x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && Integer 
Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 
] || IGtQ[q, 0])
 

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 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 6643

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

Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )}{x^{4}}d x\]

Input:

int((a+b*arctanh(c*x))*(d+e*ln(-c^2*x^2+1))/x^4,x)
 

Output:

int((a+b*arctanh(c*x))*(d+e*ln(-c^2*x^2+1))/x^4,x)
 

Fricas [F]

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

integrate((a+b*arctanh(c*x))*(d+e*log(-c^2*x^2+1))/x^4,x, algorithm="frica 
s")
 

Output:

integral((b*d*arctanh(c*x) + a*d + (b*e*arctanh(c*x) + a*e)*log(-c^2*x^2 + 
 1))/x^4, x)
 

Sympy [F]

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

integrate((a+b*atanh(c*x))*(d+e*ln(-c**2*x**2+1))/x**4,x)
 

Output:

Integral((a + b*atanh(c*x))*(d + e*log(-c**2*x**2 + 1))/x**4, x)
 

Maxima [F]

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

integrate((a+b*arctanh(c*x))*(d+e*log(-c^2*x^2+1))/x^4,x, algorithm="maxim 
a")
 

Output:

-1/6*((c^2*log(c^2*x^2 - 1) - c^2*log(x^2) + 1/x^2)*c + 2*arctanh(c*x)/x^3 
)*b*d - 1/3*((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c^2 + log(-c^2*x^2 + 
1)/x^3)*a*e + 1/6*b*e*(log(-c*x + 1)^2/x^3 - 3*integrate(-1/3*(3*(c*x - 1) 
*log(c*x + 1)^2 - 2*c*x*log(-c*x + 1))/(c*x^5 - x^4), x)) - 1/3*a*d/x^3
 

Giac [F]

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

integrate((a+b*arctanh(c*x))*(d+e*log(-c^2*x^2+1))/x^4,x, algorithm="giac" 
)
 

Output:

integrate((b*arctanh(c*x) + a)*(e*log(-c^2*x^2 + 1) + d)/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (1-c^2\,x^2\right )\right )}{x^4} \,d x \] Input:

int(((a + b*atanh(c*x))*(d + e*log(1 - c^2*x^2)))/x^4,x)
 

Output:

int(((a + b*atanh(c*x))*(d + e*log(1 - c^2*x^2)))/x^4, x)
 

Reduce [F]

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

int((a+b*atanh(c*x))*(d+e*log(-c^2*x^2+1))/x^4,x)
                                                                                    
                                                                                    
 

Output:

( - 2*atanh(c*x)*b*c**3*d*x**3 - 2*atanh(c*x)*b*d + 6*int((atanh(c*x)*log( 
 - c**2*x**2 + 1))/x**4,x)*b*e*x**3 - 2*log( - c**2*x**2 + 1)*a*c**3*e*x** 
3 - 2*log( - c**2*x**2 + 1)*a*e + 4*log(c**2*x - c)*a*c**3*e*x**3 - 2*log( 
c**2*x - c)*b*c**3*d*x**3 + 2*log(x)*b*c**3*d*x**3 + 4*a*c**2*e*x**2 - 2*a 
*d - b*c*d*x)/(6*x**3)