\(\int \frac {\coth ^{-1}(c x)^2}{d+e x} \, dx\) [1]

Optimal result

Integrand size = 14, antiderivative size = 164 \[ \int \frac {\coth ^{-1}(c x)^2}{d+e x} \, dx=-\frac {\coth ^{-1}(c x)^2 \log \left (\frac {2}{1+c x}\right )}{e}+\frac {\coth ^{-1}(c x)^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e}+\frac {\coth ^{-1}(c x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{e}-\frac {\coth ^{-1}(c x) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 e}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e} \] Output:

-arccoth(c*x)^2*ln(2/(c*x+1))/e+arccoth(c*x)^2*ln(2*c*(e*x+d)/(c*d+e)/(c*x 
+1))/e+arccoth(c*x)*polylog(2,1-2/(c*x+1))/e-arccoth(c*x)*polylog(2,1-2*c* 
(e*x+d)/(c*d+e)/(c*x+1))/e+1/2*polylog(3,1-2/(c*x+1))/e-1/2*polylog(3,1-2* 
c*(e*x+d)/(c*d+e)/(c*x+1))/e
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 5.36 (sec) , antiderivative size = 864, normalized size of antiderivative = 5.27 \[ \int \frac {\coth ^{-1}(c x)^2}{d+e x} \, dx =\text {Too large to display} \] Input:

Integrate[ArcCoth[c*x]^2/(d + e*x),x]
 

Output:

((-I)*e*Pi^3 + 8*c*d*ArcCoth[c*x]^3 + 8*e*ArcCoth[c*x]^3 - 24*e*ArcCoth[c* 
x]^2*Log[1 - E^(2*ArcCoth[c*x])] - 24*e*ArcCoth[c*x]*PolyLog[2, E^(2*ArcCo 
th[c*x])] + 12*e*PolyLog[3, E^(2*ArcCoth[c*x])] + (24*(-(c*d) + e)*(c*d + 
e)*(-2*c*d*ArcCoth[c*x]^3 + 6*e*ArcCoth[c*x]^3 + (4*c*d*Sqrt[1 - e^2/(c^2* 
d^2)]*ArcCoth[c*x]^3)/E^ArcTanh[e/(c*d)] + (6*I)*e*Pi*ArcCoth[c*x]*Log[(E^ 
(-ArcCoth[c*x]) + E^ArcCoth[c*x])/2] + 6*e*ArcCoth[c*x]^2*Log[1 - (Sqrt[c* 
d + e]*E^ArcCoth[c*x])/Sqrt[c*d - e]] + 6*e*ArcCoth[c*x]^2*Log[1 + (Sqrt[c 
*d + e]*E^ArcCoth[c*x])/Sqrt[c*d - e]] - 6*e*ArcCoth[c*x]^2*Log[1 - E^(Arc 
Coth[c*x] + ArcTanh[e/(c*d)])] - 6*e*ArcCoth[c*x]^2*Log[1 + E^(ArcCoth[c*x 
] + ArcTanh[e/(c*d)])] - 6*e*ArcCoth[c*x]^2*Log[1 - E^(2*(ArcCoth[c*x] + A 
rcTanh[e/(c*d)]))] - 12*e*ArcCoth[c*x]*ArcTanh[e/(c*d)]*Log[(I/2)*E^(-ArcC 
oth[c*x] - ArcTanh[e/(c*d)])*(-1 + E^(2*(ArcCoth[c*x] + ArcTanh[e/(c*d)])) 
)] - 6*e*ArcCoth[c*x]^2*Log[(c*d*(-1 + E^(2*ArcCoth[c*x])) + e*(1 + E^(2*A 
rcCoth[c*x])))/(2*E^ArcCoth[c*x])] - (6*I)*e*Pi*ArcCoth[c*x]*Log[1/Sqrt[1 
- 1/(c^2*x^2)]] + 6*e*ArcCoth[c*x]^2*Log[(d + e*x)/(Sqrt[1 - 1/(c^2*x^2)]* 
x)] + 12*e*ArcCoth[c*x]*ArcTanh[e/(c*d)]*Log[I*Sinh[ArcCoth[c*x] + ArcTanh 
[e/(c*d)]]] + 12*e*ArcCoth[c*x]*PolyLog[2, -((Sqrt[c*d + e]*E^ArcCoth[c*x] 
)/Sqrt[c*d - e])] + 12*e*ArcCoth[c*x]*PolyLog[2, (Sqrt[c*d + e]*E^ArcCoth[ 
c*x])/Sqrt[c*d - e]] - 12*e*ArcCoth[c*x]*PolyLog[2, -E^(ArcCoth[c*x] + Arc 
Tanh[e/(c*d)])] - 12*e*ArcCoth[c*x]*PolyLog[2, E^(ArcCoth[c*x] + ArcTan...
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {6475}

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.

\(\displaystyle \int \frac {\coth ^{-1}(c x)^2}{d+e x} \, dx\)

\(\Big \downarrow \) 6475

\(\displaystyle -\frac {\operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 e}-\frac {\coth ^{-1}(c x) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{e}+\frac {\coth ^{-1}(c x)^2 \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{2 e}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) \coth ^{-1}(c x)}{e}-\frac {\log \left (\frac {2}{c x+1}\right ) \coth ^{-1}(c x)^2}{e}\)

Input:

Int[ArcCoth[c*x]^2/(d + e*x),x]
 

Output:

-((ArcCoth[c*x]^2*Log[2/(1 + c*x)])/e) + (ArcCoth[c*x]^2*Log[(2*c*(d + e*x 
))/((c*d + e)*(1 + c*x))])/e + (ArcCoth[c*x]*PolyLog[2, 1 - 2/(1 + c*x)])/ 
e - (ArcCoth[c*x]*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e 
 + PolyLog[3, 1 - 2/(1 + c*x)]/(2*e) - PolyLog[3, 1 - (2*c*(d + e*x))/((c* 
d + e)*(1 + c*x))]/(2*e)
 

Defintions of rubi rules used

rule 6475

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> 
 Simp[(-(a + b*ArcCoth[c*x])^2)*(Log[2/(1 + c*x)]/e), x] + (Simp[(a + b*Arc 
Coth[c*x])^2*(Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp[b*(a 
 + b*ArcCoth[c*x])*(PolyLog[2, 1 - 2/(1 + c*x)]/e), x] - Simp[b*(a + b*ArcC 
oth[c*x])*(PolyLog[2, 1 - 2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + S 
imp[b^2*(PolyLog[3, 1 - 2/(1 + c*x)]/(2*e)), x] - Simp[b^2*(PolyLog[3, 1 - 
2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/(2*e)), x]) /; FreeQ[{a, b, c, d, e} 
, x] && NeQ[c^2*d^2 - e^2, 0]
 

Maple [C] (warning: unable to verify)

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

Time = 7.76 (sec) , antiderivative size = 869, normalized size of antiderivative = 5.30

method	result	size
derivativedivides	\(\frac {\frac {c \ln \left (c e x +c d \right ) \operatorname {arccoth}\left (c x \right )^{2}}{e}+\frac {2 c \left (-\frac {\operatorname {arccoth}\left (c x \right )^{2} \ln \left (d c \left (\frac {c x +1}{c x -1}-1\right )+e \left (\frac {c x +1}{c x -1}+1\right )\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (d c \left (\frac {c x +1}{c x -1}-1\right )+e \left (\frac {c x +1}{c x -1}+1\right )\right )}{\frac {c x +1}{c x -1}-1}\right ) \left (\operatorname {csgn}\left (i \left (d c \left (\frac {c x +1}{c x -1}-1\right )+e \left (\frac {c x +1}{c x -1}+1\right )\right )\right ) \operatorname {csgn}\left (\frac {i}{\frac {c x +1}{c x -1}-1}\right )-\operatorname {csgn}\left (\frac {i \left (d c \left (\frac {c x +1}{c x -1}-1\right )+e \left (\frac {c x +1}{c x -1}+1\right )\right )}{\frac {c x +1}{c x -1}-1}\right ) \operatorname {csgn}\left (\frac {i}{\frac {c x +1}{c x -1}-1}\right )-\operatorname {csgn}\left (i \left (d c \left (\frac {c x +1}{c x -1}-1\right )+e \left (\frac {c x +1}{c x -1}+1\right )\right )\right ) \operatorname {csgn}\left (\frac {i \left (d c \left (\frac {c x +1}{c x -1}-1\right )+e \left (\frac {c x +1}{c x -1}+1\right )\right )}{\frac {c x +1}{c x -1}-1}\right )+\operatorname {csgn}\left (\frac {i \left (d c \left (\frac {c x +1}{c x -1}-1\right )+e \left (\frac {c x +1}{c x -1}+1\right )\right )}{\frac {c x +1}{c x -1}-1}\right )^{2}\right ) \operatorname {arccoth}\left (c x \right )^{2}}{4}+\frac {\operatorname {arccoth}\left (c x \right )^{2} \ln \left (\frac {c x +1}{c x -1}-1\right )}{2}-\frac {\operatorname {arccoth}\left (c x \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {c x -1}{c x +1}}}\right )}{2}-\operatorname {arccoth}\left (c x \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {c x -1}{c x +1}}}\right )+\operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {c x -1}{c x +1}}}\right )-\frac {\operatorname {arccoth}\left (c x \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {c x -1}{c x +1}}}\right )}{2}-\operatorname {arccoth}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {c x -1}{c x +1}}}\right )+\operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {c x -1}{c x +1}}}\right )+\frac {e \operatorname {arccoth}\left (c x \right )^{2} \ln \left (1-\frac {\left (c d +e \right ) \left (c x +1\right )}{\left (c x -1\right ) \left (c d -e \right )}\right )}{2 c d +2 e}+\frac {e \,\operatorname {arccoth}\left (c x \right ) \operatorname {polylog}\left (2, \frac {\left (c d +e \right ) \left (c x +1\right )}{\left (c x -1\right ) \left (c d -e \right )}\right )}{2 c d +2 e}-\frac {e \operatorname {polylog}\left (3, \frac {\left (c d +e \right ) \left (c x +1\right )}{\left (c x -1\right ) \left (c d -e \right )}\right )}{4 \left (c d +e \right )}+\frac {d c \operatorname {arccoth}\left (c x \right )^{2} \ln \left (1-\frac {\left (c d +e \right ) \left (c x +1\right )}{\left (c x -1\right ) \left (c d -e \right )}\right )}{2 c d +2 e}+\frac {d c \,\operatorname {arccoth}\left (c x \right ) \operatorname {polylog}\left (2, \frac {\left (c d +e \right ) \left (c x +1\right )}{\left (c x -1\right ) \left (c d -e \right )}\right )}{2 c d +2 e}-\frac {d c \operatorname {polylog}\left (3, \frac {\left (c d +e \right ) \left (c x +1\right )}{\left (c x -1\right ) \left (c d -e \right )}\right )}{4 \left (c d +e \right )}\right )}{e}}{c}\)	\(869\)
default	\(\frac {\frac {c \ln \left (c e x +c d \right ) \operatorname {arccoth}\left (c x \right )^{2}}{e}+\frac {2 c \left (-\frac {\operatorname {arccoth}\left (c x \right )^{2} \ln \left (d c \left (\frac {c x +1}{c x -1}-1\right )+e \left (\frac {c x +1}{c x -1}+1\right )\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (d c \left (\frac {c x +1}{c x -1}-1\right )+e \left (\frac {c x +1}{c x -1}+1\right )\right )}{\frac {c x +1}{c x -1}-1}\right ) \left (\operatorname {csgn}\left (i \left (d c \left (\frac {c x +1}{c x -1}-1\right )+e \left (\frac {c x +1}{c x -1}+1\right )\right )\right ) \operatorname {csgn}\left (\frac {i}{\frac {c x +1}{c x -1}-1}\right )-\operatorname {csgn}\left (\frac {i \left (d c \left (\frac {c x +1}{c x -1}-1\right )+e \left (\frac {c x +1}{c x -1}+1\right )\right )}{\frac {c x +1}{c x -1}-1}\right ) \operatorname {csgn}\left (\frac {i}{\frac {c x +1}{c x -1}-1}\right )-\operatorname {csgn}\left (i \left (d c \left (\frac {c x +1}{c x -1}-1\right )+e \left (\frac {c x +1}{c x -1}+1\right )\right )\right ) \operatorname {csgn}\left (\frac {i \left (d c \left (\frac {c x +1}{c x -1}-1\right )+e \left (\frac {c x +1}{c x -1}+1\right )\right )}{\frac {c x +1}{c x -1}-1}\right )+\operatorname {csgn}\left (\frac {i \left (d c \left (\frac {c x +1}{c x -1}-1\right )+e \left (\frac {c x +1}{c x -1}+1\right )\right )}{\frac {c x +1}{c x -1}-1}\right )^{2}\right ) \operatorname {arccoth}\left (c x \right )^{2}}{4}+\frac {\operatorname {arccoth}\left (c x \right )^{2} \ln \left (\frac {c x +1}{c x -1}-1\right )}{2}-\frac {\operatorname {arccoth}\left (c x \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {c x -1}{c x +1}}}\right )}{2}-\operatorname {arccoth}\left (c x \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {c x -1}{c x +1}}}\right )+\operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {c x -1}{c x +1}}}\right )-\frac {\operatorname {arccoth}\left (c x \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {c x -1}{c x +1}}}\right )}{2}-\operatorname {arccoth}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {c x -1}{c x +1}}}\right )+\operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {c x -1}{c x +1}}}\right )+\frac {e \operatorname {arccoth}\left (c x \right )^{2} \ln \left (1-\frac {\left (c d +e \right ) \left (c x +1\right )}{\left (c x -1\right ) \left (c d -e \right )}\right )}{2 c d +2 e}+\frac {e \,\operatorname {arccoth}\left (c x \right ) \operatorname {polylog}\left (2, \frac {\left (c d +e \right ) \left (c x +1\right )}{\left (c x -1\right ) \left (c d -e \right )}\right )}{2 c d +2 e}-\frac {e \operatorname {polylog}\left (3, \frac {\left (c d +e \right ) \left (c x +1\right )}{\left (c x -1\right ) \left (c d -e \right )}\right )}{4 \left (c d +e \right )}+\frac {d c \operatorname {arccoth}\left (c x \right )^{2} \ln \left (1-\frac {\left (c d +e \right ) \left (c x +1\right )}{\left (c x -1\right ) \left (c d -e \right )}\right )}{2 c d +2 e}+\frac {d c \,\operatorname {arccoth}\left (c x \right ) \operatorname {polylog}\left (2, \frac {\left (c d +e \right ) \left (c x +1\right )}{\left (c x -1\right ) \left (c d -e \right )}\right )}{2 c d +2 e}-\frac {d c \operatorname {polylog}\left (3, \frac {\left (c d +e \right ) \left (c x +1\right )}{\left (c x -1\right ) \left (c d -e \right )}\right )}{4 \left (c d +e \right )}\right )}{e}}{c}\)	\(869\)
parts	\(\text {Expression too large to display}\)	\(1303\)

Input:

int(arccoth(c*x)^2/(e*x+d),x,method=_RETURNVERBOSE)
 

Output:

1/c*(c*ln(c*e*x+c*d)/e*arccoth(c*x)^2+2*c/e*(-1/2*arccoth(c*x)^2*ln(d*c*(1 
/(c*x-1)*(c*x+1)-1)+e*(1/(c*x-1)*(c*x+1)+1))+1/4*I*Pi*csgn(I*(d*c*(1/(c*x- 
1)*(c*x+1)-1)+e*(1/(c*x-1)*(c*x+1)+1))/(1/(c*x-1)*(c*x+1)-1))*(csgn(I*(d*c 
*(1/(c*x-1)*(c*x+1)-1)+e*(1/(c*x-1)*(c*x+1)+1)))*csgn(I/(1/(c*x-1)*(c*x+1) 
-1))-csgn(I*(d*c*(1/(c*x-1)*(c*x+1)-1)+e*(1/(c*x-1)*(c*x+1)+1))/(1/(c*x-1) 
*(c*x+1)-1))*csgn(I/(1/(c*x-1)*(c*x+1)-1))-csgn(I*(d*c*(1/(c*x-1)*(c*x+1)- 
1)+e*(1/(c*x-1)*(c*x+1)+1)))*csgn(I*(d*c*(1/(c*x-1)*(c*x+1)-1)+e*(1/(c*x-1 
)*(c*x+1)+1))/(1/(c*x-1)*(c*x+1)-1))+csgn(I*(d*c*(1/(c*x-1)*(c*x+1)-1)+e*( 
1/(c*x-1)*(c*x+1)+1))/(1/(c*x-1)*(c*x+1)-1))^2)*arccoth(c*x)^2+1/2*arccoth 
(c*x)^2*ln(1/(c*x-1)*(c*x+1)-1)-1/2*arccoth(c*x)^2*ln(1-1/((c*x-1)/(c*x+1) 
)^(1/2))-arccoth(c*x)*polylog(2,1/((c*x-1)/(c*x+1))^(1/2))+polylog(3,1/((c 
*x-1)/(c*x+1))^(1/2))-1/2*arccoth(c*x)^2*ln(1+1/((c*x-1)/(c*x+1))^(1/2))-a 
rccoth(c*x)*polylog(2,-1/((c*x-1)/(c*x+1))^(1/2))+polylog(3,-1/((c*x-1)/(c 
*x+1))^(1/2))+1/2*e/(c*d+e)*arccoth(c*x)^2*ln(1-(c*d+e)/(c*x-1)*(c*x+1)/(c 
*d-e))+1/2*e/(c*d+e)*arccoth(c*x)*polylog(2,(c*d+e)/(c*x-1)*(c*x+1)/(c*d-e 
))-1/4*e/(c*d+e)*polylog(3,(c*d+e)/(c*x-1)*(c*x+1)/(c*d-e))+1/2*d*c/(c*d+e 
)*arccoth(c*x)^2*ln(1-(c*d+e)/(c*x-1)*(c*x+1)/(c*d-e))+1/2*d*c/(c*d+e)*arc 
coth(c*x)*polylog(2,(c*d+e)/(c*x-1)*(c*x+1)/(c*d-e))-1/4*d*c/(c*d+e)*polyl 
og(3,(c*d+e)/(c*x-1)*(c*x+1)/(c*d-e))))
 

Fricas [F]

\[ \int \frac {\coth ^{-1}(c x)^2}{d+e x} \, dx=\int { \frac {\operatorname {arcoth}\left (c x\right )^{2}}{e x + d} \,d x } \] Input:

integrate(arccoth(c*x)^2/(e*x+d),x, algorithm="fricas")
 

Output:

integral(arccoth(c*x)^2/(e*x + d), x)
 

Sympy [F]

\[ \int \frac {\coth ^{-1}(c x)^2}{d+e x} \, dx=\int \frac {\operatorname {acoth}^{2}{\left (c x \right )}}{d + e x}\, dx \] Input:

integrate(acoth(c*x)**2/(e*x+d),x)
 

Output:

Integral(acoth(c*x)**2/(d + e*x), x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {\coth ^{-1}(c x)^2}{d+e x} \, dx=\int { \frac {\operatorname {arcoth}\left (c x\right )^{2}}{e x + d} \,d x } \] Input:

integrate(arccoth(c*x)^2/(e*x+d),x, algorithm="maxima")
 

Output:

integrate(arccoth(c*x)^2/(e*x + d), x)
 

Giac [F]

\[ \int \frac {\coth ^{-1}(c x)^2}{d+e x} \, dx=\int { \frac {\operatorname {arcoth}\left (c x\right )^{2}}{e x + d} \,d x } \] Input:

integrate(arccoth(c*x)^2/(e*x+d),x, algorithm="giac")
 

Output:

integrate(arccoth(c*x)^2/(e*x + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^{-1}(c x)^2}{d+e x} \, dx=\int \frac {{\mathrm {acoth}\left (c\,x\right )}^2}{d+e\,x} \,d x \] Input:

int(acoth(c*x)^2/(d + e*x),x)
 

Output:

int(acoth(c*x)^2/(d + e*x), x)
 

Reduce [F]

\[ \int \frac {\coth ^{-1}(c x)^2}{d+e x} \, dx=\int \frac {\mathit {acoth} \left (c x \right )^{2}}{e x +d}d x \] Input:

int(acoth(c*x)^2/(e*x+d),x)
 

Output:

int(acoth(c*x)**2/(d + e*x),x)