\(\int \frac {\coth ^{-1}(c x)^2}{d+e x} \, dx\) [34]

Optimal result

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

[Out]

-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

Rubi [A] (verified)

Time = 0.03 (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 = {6060} \[ \int \frac {\coth ^{-1}(c x)^2}{d+e x} \, dx=-\frac {\operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 e}-\frac {\coth ^{-1}(c x) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{e}+\frac {\coth ^{-1}(c x)^2 \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{2 e}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) \coth ^{-1}(c x)}{e}-\frac {\log \left (\frac {2}{c x+1}\right ) \coth ^{-1}(c x)^2}{e} \]

[In]

Int[ArcCoth[c*x]^2/(d + e*x),x]

[Out]

-((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 + (ArcC
oth[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)

Rule 6060

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^2)*(Lo
g[2/(1 + c*x)]/e), x] + (Simp[(a + b*ArcCoth[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*ArcCoth[c*x])*(PolyLog[2, 1 - 2*c*
((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp[b^2*(PolyLog[3, 1 - 2/(1 + c*x)]/(2*e)), x] - Simp[b^2*(PolyL
og[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]

Rubi steps \begin{align*} \text {integral}& = -\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} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 5.73 (sec) , antiderivative size = 864, normalized size of antiderivative = 5.27 \[ \int \frac {\coth ^{-1}(c x)^2}{d+e x} \, dx=\frac {-i e \pi ^3+8 c d \coth ^{-1}(c x)^3+8 e \coth ^{-1}(c x)^3-24 e \coth ^{-1}(c x)^2 \log \left (1-e^{2 \coth ^{-1}(c x)}\right )-24 e \coth ^{-1}(c x) \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(c x)}\right )+12 e \operatorname {PolyLog}\left (3,e^{2 \coth ^{-1}(c x)}\right )+\frac {24 (-c d+e) (c d+e) \left (-2 c d \coth ^{-1}(c x)^3+6 e \coth ^{-1}(c x)^3+4 c d \sqrt {1-\frac {e^2}{c^2 d^2}} e^{-\text {arctanh}\left (\frac {e}{c d}\right )} \coth ^{-1}(c x)^3+6 i e \pi \coth ^{-1}(c x) \log \left (\frac {1}{2} \left (e^{-\coth ^{-1}(c x)}+e^{\coth ^{-1}(c x)}\right )\right )+6 e \coth ^{-1}(c x)^2 \log \left (1-\frac {\sqrt {c d+e} e^{\coth ^{-1}(c x)}}{\sqrt {c d-e}}\right )+6 e \coth ^{-1}(c x)^2 \log \left (1+\frac {\sqrt {c d+e} e^{\coth ^{-1}(c x)}}{\sqrt {c d-e}}\right )-6 e \coth ^{-1}(c x)^2 \log \left (1-e^{\coth ^{-1}(c x)+\text {arctanh}\left (\frac {e}{c d}\right )}\right )-6 e \coth ^{-1}(c x)^2 \log \left (1+e^{\coth ^{-1}(c x)+\text {arctanh}\left (\frac {e}{c d}\right )}\right )-6 e \coth ^{-1}(c x)^2 \log \left (1-e^{2 \left (\coth ^{-1}(c x)+\text {arctanh}\left (\frac {e}{c d}\right )\right )}\right )-12 e \coth ^{-1}(c x) \text {arctanh}\left (\frac {e}{c d}\right ) \log \left (\frac {1}{2} i e^{-\coth ^{-1}(c x)-\text {arctanh}\left (\frac {e}{c d}\right )} \left (-1+e^{2 \left (\coth ^{-1}(c x)+\text {arctanh}\left (\frac {e}{c d}\right )\right )}\right )\right )-6 e \coth ^{-1}(c x)^2 \log \left (\frac {1}{2} e^{-\coth ^{-1}(c x)} \left (c d \left (-1+e^{2 \coth ^{-1}(c x)}\right )+e \left (1+e^{2 \coth ^{-1}(c x)}\right )\right )\right )-6 i e \pi \coth ^{-1}(c x) \log \left (\frac {1}{\sqrt {1-\frac {1}{c^2 x^2}}}\right )+6 e \coth ^{-1}(c x)^2 \log \left (\frac {d+e x}{\sqrt {1-\frac {1}{c^2 x^2}} x}\right )+12 e \coth ^{-1}(c x) \text {arctanh}\left (\frac {e}{c d}\right ) \log \left (i \sinh \left (\coth ^{-1}(c x)+\text {arctanh}\left (\frac {e}{c d}\right )\right )\right )+12 e \coth ^{-1}(c x) \operatorname {PolyLog}\left (2,-\frac {\sqrt {c d+e} e^{\coth ^{-1}(c x)}}{\sqrt {c d-e}}\right )+12 e \coth ^{-1}(c x) \operatorname {PolyLog}\left (2,\frac {\sqrt {c d+e} e^{\coth ^{-1}(c x)}}{\sqrt {c d-e}}\right )-12 e \coth ^{-1}(c x) \operatorname {PolyLog}\left (2,-e^{\coth ^{-1}(c x)+\text {arctanh}\left (\frac {e}{c d}\right )}\right )-12 e \coth ^{-1}(c x) \operatorname {PolyLog}\left (2,e^{\coth ^{-1}(c x)+\text {arctanh}\left (\frac {e}{c d}\right )}\right )-6 e \coth ^{-1}(c x) \operatorname {PolyLog}\left (2,e^{2 \left (\coth ^{-1}(c x)+\text {arctanh}\left (\frac {e}{c d}\right )\right )}\right )-12 e \operatorname {PolyLog}\left (3,-\frac {\sqrt {c d+e} e^{\coth ^{-1}(c x)}}{\sqrt {c d-e}}\right )-12 e \operatorname {PolyLog}\left (3,\frac {\sqrt {c d+e} e^{\coth ^{-1}(c x)}}{\sqrt {c d-e}}\right )+12 e \operatorname {PolyLog}\left (3,-e^{\coth ^{-1}(c x)+\text {arctanh}\left (\frac {e}{c d}\right )}\right )+12 e \operatorname {PolyLog}\left (3,e^{\coth ^{-1}(c x)+\text {arctanh}\left (\frac {e}{c d}\right )}\right )+3 e \operatorname {PolyLog}\left (3,e^{2 \left (\coth ^{-1}(c x)+\text {arctanh}\left (\frac {e}{c d}\right )\right )}\right )\right )}{6 c^2 d^2-6 e^2}}{24 e^2} \]

[In]

Integrate[ArcCoth[c*x]^2/(d + e*x),x]

[Out]

((-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])] - 2
4*e*ArcCoth[c*x]*PolyLog[2, E^(2*ArcCoth[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 - (Sqr
t[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^(ArcCoth[c*x] + ArcTanh[e/(c*d)])] - 6*e*ArcCoth[c*x]^2*Log[1 + E^(ArcCo
th[c*x] + ArcTanh[e/(c*d)])] - 6*e*ArcCoth[c*x]^2*Log[1 - E^(2*(ArcCoth[c*x] + ArcTanh[e/(c*d)]))] - 12*e*ArcC
oth[c*x]*ArcTanh[e/(c*d)]*Log[(I/2)*E^(-ArcCoth[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*ArcCoth[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] + ArcTanh[e/(c*d)])] - 12*e*A
rcCoth[c*x]*PolyLog[2, E^(ArcCoth[c*x] + ArcTanh[e/(c*d)])] - 6*e*ArcCoth[c*x]*PolyLog[2, E^(2*(ArcCoth[c*x] +
ArcTanh[e/(c*d)]))] - 12*e*PolyLog[3, -((Sqrt[c*d + e]*E^ArcCoth[c*x])/Sqrt[c*d - e])] - 12*e*PolyLog[3, (Sqr
t[c*d + e]*E^ArcCoth[c*x])/Sqrt[c*d - e]] + 12*e*PolyLog[3, -E^(ArcCoth[c*x] + ArcTanh[e/(c*d)])] + 12*e*PolyL
og[3, E^(ArcCoth[c*x] + ArcTanh[e/(c*d)])] + 3*e*PolyLog[3, E^(2*(ArcCoth[c*x] + ArcTanh[e/(c*d)]))]))/(6*c^2*
d^2 - 6*e^2))/(24*e^2)

Maple [C] (warning: unable to verify)

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

Time = 8.68 (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	\(\text {Expression too large to display}\)	\(1303\)

[In]

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

[Out]

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*a
rccoth(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))-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*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)*arccoth(c*x)*polylog(2,(c*d+e)/(c*x-1)*(c*x+1)/(c*d-e))-1/4*d*c/(
c*d+e)*polylog(3,(c*d+e)/(c*x-1)*(c*x+1)/(c*d-e))))

Fricas [F]

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

[In]

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

[Out]

integral(arccoth(c*x)^2/(e*x + d), x)

Sympy [F]

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

[In]

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

[Out]

Integral(acoth(c*x)**2/(d + e*x), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(acoth(c*x)^2/(d + e*x),x)

[Out]

int(acoth(c*x)^2/(d + e*x), x)