3.1.73 \(\int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx\) [73]

Optimal. Leaf size=148 \[ -\coth ^{-1}(a+b x)^2 \log \left (\frac {2}{1+a+b x}\right )+\coth ^{-1}(a+b x)^2 \log \left (\frac {2 b x}{(1-a) (1+a+b x)}\right )+\coth ^{-1}(a+b x) \text {PolyLog}\left (2,1-\frac {2}{1+a+b x}\right )-\coth ^{-1}(a+b x) \text {PolyLog}\left (2,1-\frac {2 b x}{(1-a) (1+a+b x)}\right )+\frac {1}{2} \text {PolyLog}\left (3,1-\frac {2}{1+a+b x}\right )-\frac {1}{2} \text {PolyLog}\left (3,1-\frac {2 b x}{(1-a) (1+a+b x)}\right ) \]

[Out]

-arccoth(b*x+a)^2*ln(2/(b*x+a+1))+arccoth(b*x+a)^2*ln(2*b*x/(1-a)/(b*x+a+1))+arccoth(b*x+a)*polylog(2,1-2/(b*x
+a+1))-arccoth(b*x+a)*polylog(2,1-2*b*x/(1-a)/(b*x+a+1))+1/2*polylog(3,1-2/(b*x+a+1))-1/2*polylog(3,1-2*b*x/(1
-a)/(b*x+a+1))

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Rubi [A]
time = 0.06, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6247, 6060} \begin {gather*} \frac {1}{2} \text {Li}_3\left (1-\frac {2}{a+b x+1}\right )-\frac {1}{2} \text {Li}_3\left (1-\frac {2 b x}{(1-a) (a+b x+1)}\right )+\text {Li}_2\left (1-\frac {2}{a+b x+1}\right ) \coth ^{-1}(a+b x)-\text {Li}_2\left (1-\frac {2 b x}{(1-a) (a+b x+1)}\right ) \coth ^{-1}(a+b x)-\log \left (\frac {2}{a+b x+1}\right ) \coth ^{-1}(a+b x)^2+\log \left (\frac {2 b x}{(1-a) (a+b x+1)}\right ) \coth ^{-1}(a+b x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCoth[a + b*x]^2/x,x]

[Out]

-(ArcCoth[a + b*x]^2*Log[2/(1 + a + b*x)]) + ArcCoth[a + b*x]^2*Log[(2*b*x)/((1 - a)*(1 + a + b*x))] + ArcCoth
[a + b*x]*PolyLog[2, 1 - 2/(1 + a + b*x)] - ArcCoth[a + b*x]*PolyLog[2, 1 - (2*b*x)/((1 - a)*(1 + a + b*x))] +
PolyLog[3, 1 - 2/(1 + a + b*x)]/2 - PolyLog[3, 1 - (2*b*x)/((1 - a)*(1 + a + b*x))]/2

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]

Rule 6247

Int[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcCoth[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &
& IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx &=\frac {\text {Subst}\left (\int \frac {\coth ^{-1}(x)^2}{-\frac {a}{b}+\frac {x}{b}} \, dx,x,a+b x\right )}{b}\\ &=-\coth ^{-1}(a+b x)^2 \log \left (\frac {2}{1+a+b x}\right )+\coth ^{-1}(a+b x)^2 \log \left (\frac {2 b x}{(1-a) (1+a+b x)}\right )+\coth ^{-1}(a+b x) \text {Li}_2\left (1-\frac {2}{1+a+b x}\right )-\coth ^{-1}(a+b x) \text {Li}_2\left (1-\frac {2 b x}{(1-a) (1+a+b x)}\right )+\frac {1}{2} \text {Li}_3\left (1-\frac {2}{1+a+b x}\right )-\frac {1}{2} \text {Li}_3\left (1-\frac {2 b x}{(1-a) (1+a+b x)}\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.64, size = 714, normalized size = 4.82 \begin {gather*} -\frac {i \pi ^3}{24}-\frac {2}{3} \coth ^{-1}(a+b x)^3-\frac {2}{3} a \coth ^{-1}(a+b x)^3+\frac {2}{3} \sqrt {1-\frac {1}{a^2}} a e^{\tanh ^{-1}\left (\frac {1}{a}\right )} \coth ^{-1}(a+b x)^3-i \pi \coth ^{-1}(a+b x) \log \left (\frac {1}{2} \left (e^{-\coth ^{-1}(a+b x)}+e^{\coth ^{-1}(a+b x)}\right )\right )+\coth ^{-1}(a+b x)^2 \log \left (1-\sqrt {\frac {-1+a}{1+a}} e^{\coth ^{-1}(a+b x)}\right )+\coth ^{-1}(a+b x)^2 \log \left (1+\sqrt {\frac {-1+a}{1+a}} e^{\coth ^{-1}(a+b x)}\right )-\coth ^{-1}(a+b x)^2 \log \left (1-e^{2 \coth ^{-1}(a+b x)}\right )-\coth ^{-1}(a+b x)^2 \log \left (1-\frac {(-1+a) e^{2 \coth ^{-1}(a+b x)}}{1+a}\right )+\coth ^{-1}(a+b x)^2 \log \left (1-e^{2 \coth ^{-1}(a+b x)-2 \tanh ^{-1}\left (\frac {1}{a}\right )}\right )-2 \coth ^{-1}(a+b x) \tanh ^{-1}\left (\frac {1}{a}\right ) \log \left (\frac {1}{2} i \left (e^{\coth ^{-1}(a+b x)-\tanh ^{-1}\left (\frac {1}{a}\right )}-e^{-\coth ^{-1}(a+b x)+\tanh ^{-1}\left (\frac {1}{a}\right )}\right )\right )+\coth ^{-1}(a+b x)^2 \log \left (\frac {1}{2} e^{-\coth ^{-1}(a+b x)} \left (-1-e^{2 \coth ^{-1}(a+b x)}+a \left (-1+e^{2 \coth ^{-1}(a+b x)}\right )\right )\right )+i \pi \coth ^{-1}(a+b x) \log \left (\frac {1}{\sqrt {1-\frac {1}{(a+b x)^2}}}\right )-\coth ^{-1}(a+b x)^2 \log \left (-\frac {b x}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}\right )+2 \coth ^{-1}(a+b x) \tanh ^{-1}\left (\frac {1}{a}\right ) \log \left (i \sinh \left (\coth ^{-1}(a+b x)-\tanh ^{-1}\left (\frac {1}{a}\right )\right )\right )+2 \coth ^{-1}(a+b x) \text {PolyLog}\left (2,-\sqrt {\frac {-1+a}{1+a}} e^{\coth ^{-1}(a+b x)}\right )+2 \coth ^{-1}(a+b x) \text {PolyLog}\left (2,\sqrt {\frac {-1+a}{1+a}} e^{\coth ^{-1}(a+b x)}\right )-\coth ^{-1}(a+b x) \text {PolyLog}\left (2,e^{2 \coth ^{-1}(a+b x)}\right )-\coth ^{-1}(a+b x) \text {PolyLog}\left (2,\frac {(-1+a) e^{2 \coth ^{-1}(a+b x)}}{1+a}\right )+\coth ^{-1}(a+b x) \text {PolyLog}\left (2,e^{2 \coth ^{-1}(a+b x)-2 \tanh ^{-1}\left (\frac {1}{a}\right )}\right )-2 \text {PolyLog}\left (3,-\sqrt {\frac {-1+a}{1+a}} e^{\coth ^{-1}(a+b x)}\right )-2 \text {PolyLog}\left (3,\sqrt {\frac {-1+a}{1+a}} e^{\coth ^{-1}(a+b x)}\right )+\frac {1}{2} \text {PolyLog}\left (3,e^{2 \coth ^{-1}(a+b x)}\right )+\frac {1}{2} \text {PolyLog}\left (3,\frac {(-1+a) e^{2 \coth ^{-1}(a+b x)}}{1+a}\right )-\frac {1}{2} \text {PolyLog}\left (3,e^{2 \coth ^{-1}(a+b x)-2 \tanh ^{-1}\left (\frac {1}{a}\right )}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCoth[a + b*x]^2/x,x]

[Out]

(-1/24*I)*Pi^3 - (2*ArcCoth[a + b*x]^3)/3 - (2*a*ArcCoth[a + b*x]^3)/3 + (2*Sqrt[1 - a^(-2)]*a*E^ArcTanh[a^(-1
)]*ArcCoth[a + b*x]^3)/3 - I*Pi*ArcCoth[a + b*x]*Log[(E^(-ArcCoth[a + b*x]) + E^ArcCoth[a + b*x])/2] + ArcCoth
[a + b*x]^2*Log[1 - Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x]] + ArcCoth[a + b*x]^2*Log[1 + Sqrt[(-1 + a)/(1 +
a)]*E^ArcCoth[a + b*x]] - ArcCoth[a + b*x]^2*Log[1 - E^(2*ArcCoth[a + b*x])] - ArcCoth[a + b*x]^2*Log[1 - ((-
1 + a)*E^(2*ArcCoth[a + b*x]))/(1 + a)] + ArcCoth[a + b*x]^2*Log[1 - E^(2*ArcCoth[a + b*x] - 2*ArcTanh[a^(-1)]
)] - 2*ArcCoth[a + b*x]*ArcTanh[a^(-1)]*Log[(I/2)*(E^(ArcCoth[a + b*x] - ArcTanh[a^(-1)]) - E^(-ArcCoth[a + b*
x] + ArcTanh[a^(-1)]))] + ArcCoth[a + b*x]^2*Log[(-1 - E^(2*ArcCoth[a + b*x]) + a*(-1 + E^(2*ArcCoth[a + b*x])
))/(2*E^ArcCoth[a + b*x])] + I*Pi*ArcCoth[a + b*x]*Log[1/Sqrt[1 - (a + b*x)^(-2)]] - ArcCoth[a + b*x]^2*Log[-(
(b*x)/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]))] + 2*ArcCoth[a + b*x]*ArcTanh[a^(-1)]*Log[I*Sinh[ArcCoth[a + b*x]
- ArcTanh[a^(-1)]]] + 2*ArcCoth[a + b*x]*PolyLog[2, -(Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x])] + 2*ArcCoth[
a + b*x]*PolyLog[2, Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x]] - ArcCoth[a + b*x]*PolyLog[2, E^(2*ArcCoth[a +
b*x])] - ArcCoth[a + b*x]*PolyLog[2, ((-1 + a)*E^(2*ArcCoth[a + b*x]))/(1 + a)] + ArcCoth[a + b*x]*PolyLog[2,
E^(2*ArcCoth[a + b*x] - 2*ArcTanh[a^(-1)])] - 2*PolyLog[3, -(Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x])] - 2*P
olyLog[3, Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x]] + PolyLog[3, E^(2*ArcCoth[a + b*x])]/2 + PolyLog[3, ((-1
+ a)*E^(2*ArcCoth[a + b*x]))/(1 + a)]/2 - PolyLog[3, E^(2*ArcCoth[a + b*x] - 2*ArcTanh[a^(-1)])]/2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 2.92, size = 900, normalized size = 6.08 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(b*x+a)^2/x,x,method=_RETURNVERBOSE)

[Out]

ln(-b*x)*arccoth(b*x+a)^2-arccoth(b*x+a)^2*ln(a*(1/(b*x+a-1)*(b*x+a+1)-1)-1/(b*x+a-1)*(b*x+a+1)-1)+1/2*I*Pi*ar
ccoth(b*x+a)^2*csgn(I/(1/(b*x+a-1)*(b*x+a+1)-1))*csgn(I*(a*(1/(b*x+a-1)*(b*x+a+1)-1)-1/(b*x+a-1)*(b*x+a+1)-1))
*csgn(I*(a*(1/(b*x+a-1)*(b*x+a+1)-1)-1/(b*x+a-1)*(b*x+a+1)-1)/(1/(b*x+a-1)*(b*x+a+1)-1))-1/2*I*Pi*arccoth(b*x+
a)^2*csgn(I*(a*(1/(b*x+a-1)*(b*x+a+1)-1)-1/(b*x+a-1)*(b*x+a+1)-1))*csgn(I*(a*(1/(b*x+a-1)*(b*x+a+1)-1)-1/(b*x+
a-1)*(b*x+a+1)-1)/(1/(b*x+a-1)*(b*x+a+1)-1))^2-1/2*I*Pi*arccoth(b*x+a)^2*csgn(I/(1/(b*x+a-1)*(b*x+a+1)-1))*csg
n(I*(a*(1/(b*x+a-1)*(b*x+a+1)-1)-1/(b*x+a-1)*(b*x+a+1)-1)/(1/(b*x+a-1)*(b*x+a+1)-1))^2+1/2*I*Pi*arccoth(b*x+a)
^2*csgn(I*(a*(1/(b*x+a-1)*(b*x+a+1)-1)-1/(b*x+a-1)*(b*x+a+1)-1)/(1/(b*x+a-1)*(b*x+a+1)-1))^3+arccoth(b*x+a)^2*
ln(1/(b*x+a-1)*(b*x+a+1)-1)-arccoth(b*x+a)^2*ln(1-1/((b*x+a-1)/(b*x+a+1))^(1/2))-2*arccoth(b*x+a)*polylog(2,1/
((b*x+a-1)/(b*x+a+1))^(1/2))+2*polylog(3,1/((b*x+a-1)/(b*x+a+1))^(1/2))-arccoth(b*x+a)^2*ln(1+1/((b*x+a-1)/(b*
x+a+1))^(1/2))-2*arccoth(b*x+a)*polylog(2,-1/((b*x+a-1)/(b*x+a+1))^(1/2))+2*polylog(3,-1/((b*x+a-1)/(b*x+a+1))
^(1/2))+a/(-1+a)*arccoth(b*x+a)^2*ln(1-(-1+a)/(b*x+a-1)*(b*x+a+1)/(1+a))+a/(-1+a)*arccoth(b*x+a)*polylog(2,(-1
+a)/(b*x+a-1)*(b*x+a+1)/(1+a))-1/2*a/(-1+a)*polylog(3,(-1+a)/(b*x+a-1)*(b*x+a+1)/(1+a))-1/(-1+a)*arccoth(b*x+a
)^2*ln(1-(-1+a)/(b*x+a-1)*(b*x+a+1)/(1+a))-1/(-1+a)*arccoth(b*x+a)*polylog(2,(-1+a)/(b*x+a-1)*(b*x+a+1)/(1+a))
+1/2/(-1+a)*polylog(3,(-1+a)/(b*x+a-1)*(b*x+a+1)/(1+a))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(b*x+a)^2/x,x, algorithm="maxima")

[Out]

integrate(arccoth(b*x + a)^2/x, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(b*x+a)^2/x,x, algorithm="fricas")

[Out]

integral(arccoth(b*x + a)^2/x, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acoth}^{2}{\left (a + b x \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(b*x+a)**2/x,x)

[Out]

Integral(acoth(a + b*x)**2/x, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(b*x+a)^2/x,x, algorithm="giac")

[Out]

integrate(arccoth(b*x + a)^2/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {acoth}\left (a+b\,x\right )}^2}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acoth(a + b*x)^2/x,x)

[Out]

int(acoth(a + b*x)^2/x, x)

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