\(\int \frac {a+b \text {arctanh}(c+d x)}{x^3 (e-f x^2)} \, dx\) [69]

Optimal result

Integrand size = 24, antiderivative size = 428 \[ \int \frac {a+b \text {arctanh}(c+d x)}{x^3 \left (e-f x^2\right )} \, dx=-\frac {b d}{2 \left (1-c^2\right ) e x}-\frac {a+b \text {arctanh}(c+d x)}{2 e x^2}+\frac {b c d^2 \log (x)}{\left (1-c^2\right )^2 e}-\frac {b d^2 \log (1-c-d x)}{4 (1-c)^2 e}+\frac {f (a+b \text {arctanh}(c+d x)) \log \left (\frac {2 d x}{(1-c) (1+c+d x)}\right )}{e^2}+\frac {b d^2 \log (1+c+d x)}{4 (1+c)^2 e}-\frac {f (a+b \text {arctanh}(c+d x)) \log \left (\frac {2 d \left (\sqrt {e}-\sqrt {f} x\right )}{\left (d \sqrt {e}-(1-c) \sqrt {f}\right ) (1+c+d x)}\right )}{2 e^2}-\frac {f (a+b \text {arctanh}(c+d x)) \log \left (\frac {2 d \left (\sqrt {e}+\sqrt {f} x\right )}{\left (d \sqrt {e}+(1-c) \sqrt {f}\right ) (1+c+d x)}\right )}{2 e^2}-\frac {b f \operatorname {PolyLog}\left (2,1-\frac {2 d x}{(1-c) (1+c+d x)}\right )}{2 e^2}+\frac {b f \operatorname {PolyLog}\left (2,1-\frac {2 d \left (\sqrt {e}-\sqrt {f} x\right )}{\left (d \sqrt {e}-(1-c) \sqrt {f}\right ) (1+c+d x)}\right )}{4 e^2}+\frac {b f \operatorname {PolyLog}\left (2,1-\frac {2 d \left (\sqrt {e}+\sqrt {f} x\right )}{\left (d \sqrt {e}+(1-c) \sqrt {f}\right ) (1+c+d x)}\right )}{4 e^2} \] Output:

-1/2*b*d/(-c^2+1)/e/x-1/2*(a+b*arctanh(d*x+c))/e/x^2+b*c*d^2*ln(x)/(-c^2+1 
)^2/e-1/4*b*d^2*ln(-d*x-c+1)/(1-c)^2/e+f*(a+b*arctanh(d*x+c))*ln(2*d*x/(1- 
c)/(d*x+c+1))/e^2+1/4*b*d^2*ln(d*x+c+1)/(1+c)^2/e-1/2*f*(a+b*arctanh(d*x+c 
))*ln(2*d*(e^(1/2)-f^(1/2)*x)/(d*e^(1/2)-(1-c)*f^(1/2))/(d*x+c+1))/e^2-1/2 
*f*(a+b*arctanh(d*x+c))*ln(2*d*(e^(1/2)+f^(1/2)*x)/(d*e^(1/2)+(1-c)*f^(1/2 
))/(d*x+c+1))/e^2-1/2*b*f*polylog(2,1-2*d*x/(1-c)/(d*x+c+1))/e^2+1/4*b*f*p 
olylog(2,1-2*d*(e^(1/2)-f^(1/2)*x)/(d*e^(1/2)-(1-c)*f^(1/2))/(d*x+c+1))/e^ 
2+1/4*b*f*polylog(2,1-2*d*(e^(1/2)+f^(1/2)*x)/(d*e^(1/2)+(1-c)*f^(1/2))/(d 
*x+c+1))/e^2
 

Mathematica [A] (verified)

Time = 33.03 (sec) , antiderivative size = 608, normalized size of antiderivative = 1.42 \[ \int \frac {a+b \text {arctanh}(c+d x)}{x^3 \left (e-f x^2\right )} \, dx=\frac {-\frac {2 a e}{x^2}+4 a f \log (x)-2 a f \log \left (e-f x^2\right )+b \left (-\frac {2 d e}{x-c^2 x}+\frac {4 c d^2 e \log (x)}{\left (-1+c^2\right )^2}-\frac {d^2 e \log (1-c-d x)}{(-1+c)^2}+\frac {d^2 e \log (1+c+d x)}{(1+c)^2}-\frac {2 \text {arctanh}(c+d x) \left (e-2 f x^2 \log (x)+f x^2 \log \left (e-f x^2\right )\right )}{x^2}+2 f \left (\log (x) \left (\log \left (1+\frac {d x}{-1+c}\right )-\log \left (1+\frac {d x}{1+c}\right )\right )+\operatorname {PolyLog}\left (2,-\frac {d x}{-1+c}\right )-\operatorname {PolyLog}\left (2,-\frac {d x}{1+c}\right )\right )+f \left (-\log \left (\frac {\sqrt {e}}{\sqrt {f}}+x\right ) \log \left (-\frac {\sqrt {f} (-1+c+d x)}{d \sqrt {e}-(-1+c) \sqrt {f}}\right )-\log \left (-\frac {\sqrt {e}}{\sqrt {f}}+x\right ) \log \left (\frac {\sqrt {f} (-1+c+d x)}{d \sqrt {e}+(-1+c) \sqrt {f}}\right )+\log \left (\frac {\sqrt {e}}{\sqrt {f}}+x\right ) \log \left (-\frac {\sqrt {f} (1+c+d x)}{d \sqrt {e}-(1+c) \sqrt {f}}\right )+\log \left (-\frac {\sqrt {e}}{\sqrt {f}}+x\right ) \log \left (\frac {\sqrt {f} (1+c+d x)}{d \sqrt {e}+(1+c) \sqrt {f}}\right )+(\log (1-c-d x)-\log (1+c+d x)) \left (\log \left (-\frac {\sqrt {e}}{\sqrt {f}}+x\right )+\log \left (\frac {\sqrt {e}}{\sqrt {f}}+x\right )-\log \left (e-f x^2\right )\right )-\operatorname {PolyLog}\left (2,\frac {d \left (\frac {\sqrt {e}}{\sqrt {f}}+x\right )}{1-c+\frac {d \sqrt {e}}{\sqrt {f}}}\right )-\operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {e}-\sqrt {f} x\right )}{d \sqrt {e}+(-1+c) \sqrt {f}}\right )+\operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {e}-\sqrt {f} x\right )}{d \sqrt {e}+(1+c) \sqrt {f}}\right )+\operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {e}+\sqrt {f} x\right )}{d \sqrt {e}-(1+c) \sqrt {f}}\right )\right )\right )}{4 e^2} \] Input:

Integrate[(a + b*ArcTanh[c + d*x])/(x^3*(e - f*x^2)),x]
 

Output:

((-2*a*e)/x^2 + 4*a*f*Log[x] - 2*a*f*Log[e - f*x^2] + b*((-2*d*e)/(x - c^2 
*x) + (4*c*d^2*e*Log[x])/(-1 + c^2)^2 - (d^2*e*Log[1 - c - d*x])/(-1 + c)^ 
2 + (d^2*e*Log[1 + c + d*x])/(1 + c)^2 - (2*ArcTanh[c + d*x]*(e - 2*f*x^2* 
Log[x] + f*x^2*Log[e - f*x^2]))/x^2 + 2*f*(Log[x]*(Log[1 + (d*x)/(-1 + c)] 
 - Log[1 + (d*x)/(1 + c)]) + PolyLog[2, -((d*x)/(-1 + c))] - PolyLog[2, -( 
(d*x)/(1 + c))]) + f*(-(Log[Sqrt[e]/Sqrt[f] + x]*Log[-((Sqrt[f]*(-1 + c + 
d*x))/(d*Sqrt[e] - (-1 + c)*Sqrt[f]))]) - Log[-(Sqrt[e]/Sqrt[f]) + x]*Log[ 
(Sqrt[f]*(-1 + c + d*x))/(d*Sqrt[e] + (-1 + c)*Sqrt[f])] + Log[Sqrt[e]/Sqr 
t[f] + x]*Log[-((Sqrt[f]*(1 + c + d*x))/(d*Sqrt[e] - (1 + c)*Sqrt[f]))] + 
Log[-(Sqrt[e]/Sqrt[f]) + x]*Log[(Sqrt[f]*(1 + c + d*x))/(d*Sqrt[e] + (1 + 
c)*Sqrt[f])] + (Log[1 - c - d*x] - Log[1 + c + d*x])*(Log[-(Sqrt[e]/Sqrt[f 
]) + x] + Log[Sqrt[e]/Sqrt[f] + x] - Log[e - f*x^2]) - PolyLog[2, (d*(Sqrt 
[e]/Sqrt[f] + x))/(1 - c + (d*Sqrt[e])/Sqrt[f])] - PolyLog[2, (d*(Sqrt[e] 
- Sqrt[f]*x))/(d*Sqrt[e] + (-1 + c)*Sqrt[f])] + PolyLog[2, (d*(Sqrt[e] - S 
qrt[f]*x))/(d*Sqrt[e] + (1 + c)*Sqrt[f])] + PolyLog[2, (d*(Sqrt[e] + Sqrt[ 
f]*x))/(d*Sqrt[e] - (1 + c)*Sqrt[f])])))/(4*e^2)
 

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.06, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {7276, 2009}

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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*ArcTanh[c + d*x])/(x^3*(e - f*x^2)),x]
 

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 730, normalized size of antiderivative = 1.71

method	result	size
parts	\(a \left (-\frac {1}{2 e \,x^{2}}+\frac {f \ln \left (x \right )}{e^{2}}-\frac {f \ln \left (f \,x^{2}-e \right )}{2 e^{2}}\right )+b \,d^{2} \left (-\frac {\operatorname {arctanh}\left (d x +c \right ) f \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2 d^{2} e^{2}}-\frac {\operatorname {arctanh}\left (d x +c \right )}{2 e \,x^{2} d^{2}}+\frac {\operatorname {arctanh}\left (d x +c \right ) f \ln \left (d x \right )}{d^{2} e^{2}}+\frac {d^{2} \left (-\frac {-\frac {\ln \left (d x +c +1\right )}{2 \left (1+c \right )^{2}}-\frac {1}{\left (-1+c \right ) \left (1+c \right ) d x}-\frac {2 c \ln \left (d x \right )}{\left (-1+c \right )^{2} \left (1+c \right )^{2}}+\frac {\ln \left (d x +c -1\right )}{2 \left (-1+c \right )^{2}}}{e \,d^{2}}+\frac {2 f \left (\frac {\operatorname {dilog}\left (\frac {d x +c -1}{-1+c}\right )}{2}+\frac {\ln \left (d x \right ) \ln \left (\frac {d x +c -1}{-1+c}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {d x +c +1}{1+c}\right )}{2}-\frac {\ln \left (d x \right ) \ln \left (\frac {d x +c +1}{1+c}\right )}{2}\right )}{d^{4} e^{2}}-\frac {f \left (-\frac {\ln \left (d x +c +1\right ) \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2}+f \left (\frac {\ln \left (d x +c +1\right ) \left (\ln \left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )+\ln \left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )\right )}{2 f}+\frac {\operatorname {dilog}\left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )+\operatorname {dilog}\left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )}{2 f}\right )+\frac {\ln \left (d x +c -1\right ) \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2}-f \left (\frac {\ln \left (d x +c -1\right ) \left (\ln \left (\frac {d \sqrt {e f}+f c -f \left (d x +c -1\right )-f}{d \sqrt {e f}+f c -f}\right )+\ln \left (\frac {d \sqrt {e f}-f c +f \left (d x +c -1\right )+f}{d \sqrt {e f}-f c +f}\right )\right )}{2 f}+\frac {\operatorname {dilog}\left (\frac {d \sqrt {e f}+f c -f \left (d x +c -1\right )-f}{d \sqrt {e f}+f c -f}\right )+\operatorname {dilog}\left (\frac {d \sqrt {e f}-f c +f \left (d x +c -1\right )+f}{d \sqrt {e f}-f c +f}\right )}{2 f}\right )\right )}{d^{4} e^{2}}\right )}{2}\right )\)	\(730\)
derivativedivides	\(d^{2} \left (-\frac {a}{2 e \,d^{2} x^{2}}+\frac {a f \ln \left (-d x \right )}{d^{2} e^{2}}-\frac {a f \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2 d^{2} e^{2}}+b \,d^{2} \left (-\frac {\operatorname {arctanh}\left (d x +c \right )}{2 e \,d^{4} x^{2}}+\frac {\operatorname {arctanh}\left (d x +c \right ) f \ln \left (-d x \right )}{e^{2} d^{4}}-\frac {\operatorname {arctanh}\left (d x +c \right ) f \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2 e^{2} d^{4}}-\frac {\frac {\ln \left (d x +c -1\right )}{2 \left (-1+c \right )^{2}}-\frac {\ln \left (d x +c +1\right )}{2 \left (1+c \right )^{2}}-\frac {1}{\left (-1+c \right ) \left (1+c \right ) d x}-\frac {2 c \ln \left (-d x \right )}{\left (-1+c \right )^{2} \left (1+c \right )^{2}}}{2 e \,d^{2}}-\frac {f \left (\frac {\ln \left (d x +c -1\right ) \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2}+f \left (-\frac {\ln \left (d x +c -1\right ) \left (\ln \left (\frac {d \sqrt {e f}+f c -f \left (d x +c -1\right )-f}{d \sqrt {e f}+f c -f}\right )+\ln \left (\frac {d \sqrt {e f}-f c +f \left (d x +c -1\right )+f}{d \sqrt {e f}-f c +f}\right )\right )}{2 f}-\frac {\operatorname {dilog}\left (\frac {d \sqrt {e f}+f c -f \left (d x +c -1\right )-f}{d \sqrt {e f}+f c -f}\right )+\operatorname {dilog}\left (\frac {d \sqrt {e f}-f c +f \left (d x +c -1\right )+f}{d \sqrt {e f}-f c +f}\right )}{2 f}\right )-\frac {\ln \left (d x +c +1\right ) \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2}-f \left (-\frac {\ln \left (d x +c +1\right ) \left (\ln \left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )+\ln \left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )\right )}{2 f}-\frac {\operatorname {dilog}\left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )+\operatorname {dilog}\left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )}{2 f}\right )\right )}{2 e^{2} d^{4}}+\frac {f \left (-\frac {\operatorname {dilog}\left (\frac {-d x -c -1}{-1-c}\right )}{2}-\frac {\ln \left (-d x \right ) \ln \left (\frac {-d x -c -1}{-1-c}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {-d x -c +1}{1-c}\right )}{2}+\frac {\ln \left (-d x \right ) \ln \left (\frac {-d x -c +1}{1-c}\right )}{2}\right )}{e^{2} d^{4}}\right )\right )\)	\(784\)
default	\(d^{2} \left (-\frac {a}{2 e \,d^{2} x^{2}}+\frac {a f \ln \left (-d x \right )}{d^{2} e^{2}}-\frac {a f \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2 d^{2} e^{2}}+b \,d^{2} \left (-\frac {\operatorname {arctanh}\left (d x +c \right )}{2 e \,d^{4} x^{2}}+\frac {\operatorname {arctanh}\left (d x +c \right ) f \ln \left (-d x \right )}{e^{2} d^{4}}-\frac {\operatorname {arctanh}\left (d x +c \right ) f \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2 e^{2} d^{4}}-\frac {\frac {\ln \left (d x +c -1\right )}{2 \left (-1+c \right )^{2}}-\frac {\ln \left (d x +c +1\right )}{2 \left (1+c \right )^{2}}-\frac {1}{\left (-1+c \right ) \left (1+c \right ) d x}-\frac {2 c \ln \left (-d x \right )}{\left (-1+c \right )^{2} \left (1+c \right )^{2}}}{2 e \,d^{2}}-\frac {f \left (\frac {\ln \left (d x +c -1\right ) \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2}+f \left (-\frac {\ln \left (d x +c -1\right ) \left (\ln \left (\frac {d \sqrt {e f}+f c -f \left (d x +c -1\right )-f}{d \sqrt {e f}+f c -f}\right )+\ln \left (\frac {d \sqrt {e f}-f c +f \left (d x +c -1\right )+f}{d \sqrt {e f}-f c +f}\right )\right )}{2 f}-\frac {\operatorname {dilog}\left (\frac {d \sqrt {e f}+f c -f \left (d x +c -1\right )-f}{d \sqrt {e f}+f c -f}\right )+\operatorname {dilog}\left (\frac {d \sqrt {e f}-f c +f \left (d x +c -1\right )+f}{d \sqrt {e f}-f c +f}\right )}{2 f}\right )-\frac {\ln \left (d x +c +1\right ) \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2}-f \left (-\frac {\ln \left (d x +c +1\right ) \left (\ln \left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )+\ln \left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )\right )}{2 f}-\frac {\operatorname {dilog}\left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )+\operatorname {dilog}\left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )}{2 f}\right )\right )}{2 e^{2} d^{4}}+\frac {f \left (-\frac {\operatorname {dilog}\left (\frac {-d x -c -1}{-1-c}\right )}{2}-\frac {\ln \left (-d x \right ) \ln \left (\frac {-d x -c -1}{-1-c}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {-d x -c +1}{1-c}\right )}{2}+\frac {\ln \left (-d x \right ) \ln \left (\frac {-d x -c +1}{1-c}\right )}{2}\right )}{e^{2} d^{4}}\right )\right )\)	\(784\)
risch	\(-\frac {a}{2 e \,x^{2}}-\frac {d b}{4 e \left (-1+c \right )^{2} x}+\frac {b f \operatorname {dilog}\left (\frac {x d}{-1-c}\right )}{2 e^{2}}-\frac {b f \operatorname {dilog}\left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )}{4 e^{2}}-\frac {b f \operatorname {dilog}\left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )}{4 e^{2}}-\frac {b f \operatorname {dilog}\left (-\frac {x d}{-1+c}\right )}{2 e^{2}}+\frac {b f \operatorname {dilog}\left (\frac {d \sqrt {e f}-\left (-d x -c +1\right ) f -f c +f}{d \sqrt {e f}-f c +f}\right )}{4 e^{2}}+\frac {b f \operatorname {dilog}\left (\frac {d \sqrt {e f}+\left (-d x -c +1\right ) f +f c -f}{d \sqrt {e f}+f c -f}\right )}{4 e^{2}}+\frac {a f \ln \left (-d x \right )}{e^{2}}-\frac {a f \ln \left (f \left (-d x -c +1\right )^{2}+2 \left (-d x -c +1\right ) c f +c^{2} f -e \,d^{2}-2 \left (-d x -c +1\right ) f -2 f c +f \right )}{2 e^{2}}+\frac {b \,d^{2} \ln \left (d x +c +1\right )}{4 \left (1+c \right )^{2} e}-\frac {b d c}{4 e \left (1+c \right )^{2} x}-\frac {b \ln \left (d x +c +1\right ) c^{2}}{4 e \,x^{2} \left (1+c \right )^{2}}-\frac {b \ln \left (d x +c +1\right ) c}{2 e \,x^{2} \left (1+c \right )^{2}}+\frac {d b c}{4 e \left (-1+c \right )^{2} x}+\frac {b \ln \left (-d x -c +1\right ) c^{2}}{4 e \,x^{2} \left (-1+c \right )^{2}}-\frac {b \ln \left (-d x -c +1\right ) c}{2 e \,x^{2} \left (-1+c \right )^{2}}-\frac {b d}{4 e \left (1+c \right )^{2} x}+\frac {b f \ln \left (d x +c +1\right ) \ln \left (\frac {x d}{-1-c}\right )}{2 e^{2}}-\frac {b \,d^{2} \ln \left (d x \right )}{4 e \left (1+c \right )^{2}}-\frac {b \ln \left (d x +c +1\right )}{4 e \,x^{2} \left (1+c \right )^{2}}-\frac {b f \ln \left (d x +c +1\right ) \ln \left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )}{4 e^{2}}-\frac {b f \ln \left (d x +c +1\right ) \ln \left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )}{4 e^{2}}+\frac {b f \ln \left (-d x -c +1\right ) \ln \left (\frac {d \sqrt {e f}-\left (-d x -c +1\right ) f -f c +f}{d \sqrt {e f}-f c +f}\right )}{4 e^{2}}+\frac {b f \ln \left (-d x -c +1\right ) \ln \left (\frac {d \sqrt {e f}+\left (-d x -c +1\right ) f +f c -f}{d \sqrt {e f}+f c -f}\right )}{4 e^{2}}-\frac {d^{2} b \ln \left (-d x -c +1\right )}{4 e \left (-1+c \right )^{2}}+\frac {b \ln \left (-d x -c +1\right )}{4 e \,x^{2} \left (-1+c \right )^{2}}-\frac {b f \ln \left (-d x -c +1\right ) \ln \left (-\frac {x d}{-1+c}\right )}{2 e^{2}}+\frac {d^{2} b \ln \left (-d x \right )}{4 e \left (-1+c \right )^{2}}\)	\(877\)

Input:

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

Output:

a*(-1/2/e/x^2+f/e^2*ln(x)-1/2*f/e^2*ln(f*x^2-e))+b*d^2*(-1/2/d^2*arctanh(d 
*x+c)*f/e^2*ln(c^2*f-2*c*f*(d*x+c)-e*d^2+f*(d*x+c)^2)-1/2*arctanh(d*x+c)/e 
/x^2/d^2+1/d^2*arctanh(d*x+c)*f/e^2*ln(d*x)+1/2*d^2*(-1/e/d^2*(-1/2/(1+c)^ 
2*ln(d*x+c+1)-1/(-1+c)/(1+c)/d/x-2*c/(-1+c)^2/(1+c)^2*ln(d*x)+1/2/(-1+c)^2 
*ln(d*x+c-1))+2*f/d^4/e^2*(1/2*dilog((d*x+c-1)/(-1+c))+1/2*ln(d*x)*ln((d*x 
+c-1)/(-1+c))-1/2*dilog((d*x+c+1)/(1+c))-1/2*ln(d*x)*ln((d*x+c+1)/(1+c)))- 
f/d^4/e^2*(-1/2*ln(d*x+c+1)*ln(c^2*f-2*c*f*(d*x+c)-e*d^2+f*(d*x+c)^2)+f*(1 
/2*ln(d*x+c+1)*(ln((d*(e*f)^(1/2)+f*c-f*(d*x+c+1)+f)/(d*(e*f)^(1/2)+f*c+f) 
)+ln((d*(e*f)^(1/2)-f*c+f*(d*x+c+1)-f)/(d*(e*f)^(1/2)-f*c-f)))/f+1/2*(dilo 
g((d*(e*f)^(1/2)+f*c-f*(d*x+c+1)+f)/(d*(e*f)^(1/2)+f*c+f))+dilog((d*(e*f)^ 
(1/2)-f*c+f*(d*x+c+1)-f)/(d*(e*f)^(1/2)-f*c-f)))/f)+1/2*ln(d*x+c-1)*ln(c^2 
*f-2*c*f*(d*x+c)-e*d^2+f*(d*x+c)^2)-f*(1/2*ln(d*x+c-1)*(ln((d*(e*f)^(1/2)+ 
f*c-f*(d*x+c-1)-f)/(d*(e*f)^(1/2)+f*c-f))+ln((d*(e*f)^(1/2)-f*c+f*(d*x+c-1 
)+f)/(d*(e*f)^(1/2)-f*c+f)))/f+1/2*(dilog((d*(e*f)^(1/2)+f*c-f*(d*x+c-1)-f 
)/(d*(e*f)^(1/2)+f*c-f))+dilog((d*(e*f)^(1/2)-f*c+f*(d*x+c-1)+f)/(d*(e*f)^ 
(1/2)-f*c+f)))/f))))
 

Fricas [F]

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

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

Output:

integral(-(b*arctanh(d*x + c) + a)/(f*x^5 - e*x^3), x)
 

Sympy [F(-1)]

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

integrate((a+b*atanh(d*x+c))/x**3/(-f*x**2+e),x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/2*a*(f*log(f*x^2 - e)/e^2 - 2*f*log(x)/e^2 + 1/(e*x^2)) - 1/2*b*integra 
te((log(d*x + c + 1) - log(-d*x - c + 1))/(f*x^5 - e*x^3), x)
 

Giac [F]

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

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

Output:

integrate(-(b*arctanh(d*x + c) + a)/((f*x^2 - e)*x^3), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \text {arctanh}(c+d x)}{x^3 \left (e-f x^2\right )} \, dx=\frac {2 \left (\int \frac {\mathit {atanh} \left (d x +c \right )}{-f \,x^{5}+e \,x^{3}}d x \right ) b \,e^{2} x^{2}-\mathrm {log}\left (-\sqrt {f}\, \sqrt {e}-f x \right ) a f \,x^{2}-\mathrm {log}\left (\sqrt {f}\, \sqrt {e}-f x \right ) a f \,x^{2}+2 \,\mathrm {log}\left (x \right ) a f \,x^{2}-a e}{2 e^{2} x^{2}} \] Input:

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

Output:

(2*int(atanh(c + d*x)/(e*x**3 - f*x**5),x)*b*e**2*x**2 - log( - sqrt(f)*sq 
rt(e) - f*x)*a*f*x**2 - log(sqrt(f)*sqrt(e) - f*x)*a*f*x**2 + 2*log(x)*a*f 
*x**2 - a*e)/(2*e**2*x**2)