3.71 \(\int x \coth ^{-1}(a+b x)^2 \, dx\)

Optimal. Leaf size=136 \[ \frac{a \text{PolyLog}\left (2,-\frac{a+b x+1}{-a-b x+1}\right )}{b^2}-\frac{\left (a^2+1\right ) \coth ^{-1}(a+b x)^2}{2 b^2}+\frac{\log \left (1-(a+b x)^2\right )}{2 b^2}-\frac{a \coth ^{-1}(a+b x)^2}{b^2}+\frac{(a+b x) \coth ^{-1}(a+b x)}{b^2}+\frac{2 a \log \left (\frac{2}{-a-b x+1}\right ) \coth ^{-1}(a+b x)}{b^2}+\frac{1}{2} x^2 \coth ^{-1}(a+b x)^2 \]

[Out]

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

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Rubi [A]  time = 0.209095, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6112, 5929, 5911, 260, 6049, 5949, 5985, 5919, 2402, 2315} \[ \frac{a \text{PolyLog}\left (2,-\frac{a+b x+1}{-a-b x+1}\right )}{b^2}-\frac{\left (a^2+1\right ) \coth ^{-1}(a+b x)^2}{2 b^2}+\frac{\log \left (1-(a+b x)^2\right )}{2 b^2}-\frac{a \coth ^{-1}(a+b x)^2}{b^2}+\frac{(a+b x) \coth ^{-1}(a+b x)}{b^2}+\frac{2 a \log \left (\frac{2}{-a-b x+1}\right ) \coth ^{-1}(a+b x)}{b^2}+\frac{1}{2} x^2 \coth ^{-1}(a+b x)^2 \]

Antiderivative was successfully verified.

[In]

Int[x*ArcCoth[a + b*x]^2,x]

[Out]

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

Rule 6112

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]

Rule 5929

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1)*(
a + b*ArcCoth[c*x])^p)/(e*(q + 1)), x] - Dist[(b*c*p)/(e*(q + 1)), Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^(p
- 1), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] &
& NeQ[q, -1]

Rule 5911

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x])^p, x] - Dist[b*c*p, In
t[(x*(a + b*ArcCoth[c*x])^(p - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 6049

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :>
Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x]
&& IGtQ[p, 0] && EqQ[c^2*d + e, 0] && IGtQ[m, 0]

Rule 5949

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCoth[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 5985

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCoth[c
*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcCoth[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 5919

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcCoth[c*x])^p*
Log[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcCoth[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 - c^2
*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

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

Mathematica [A]  time = 0.246845, size = 106, normalized size = 0.78 \[ \frac{-2 a \text{PolyLog}\left (2,e^{-2 \coth ^{-1}(a+b x)}\right )+\left (-a^2+2 a+b^2 x^2-1\right ) \coth ^{-1}(a+b x)^2-2 \log \left (\frac{1}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}\right )+2 \coth ^{-1}(a+b x) \left (2 a \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right )+a+b x\right )}{2 b^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x*ArcCoth[a + b*x]^2,x]

[Out]

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

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Maple [B]  time = 0.053, size = 365, normalized size = 2.7 \begin{align*}{\frac{{x}^{2} \left ({\rm arccoth} \left (bx+a\right ) \right ) ^{2}}{2}}-{\frac{ \left ({\rm arccoth} \left (bx+a\right ) \right ) ^{2}{a}^{2}}{2\,{b}^{2}}}+{\frac{x{\rm arccoth} \left (bx+a\right )}{b}}+{\frac{{\rm arccoth} \left (bx+a\right )a}{{b}^{2}}}-{\frac{{\rm arccoth} \left (bx+a\right )\ln \left ( bx+a-1 \right ) a}{{b}^{2}}}+{\frac{{\rm arccoth} \left (bx+a\right )\ln \left ( bx+a-1 \right ) }{2\,{b}^{2}}}-{\frac{{\rm arccoth} \left (bx+a\right )\ln \left ( bx+a+1 \right ) a}{{b}^{2}}}-{\frac{{\rm arccoth} \left (bx+a\right )\ln \left ( bx+a+1 \right ) }{2\,{b}^{2}}}+{\frac{\ln \left ( bx+a-1 \right ) }{2\,{b}^{2}}}+{\frac{\ln \left ( bx+a+1 \right ) }{2\,{b}^{2}}}-{\frac{ \left ( \ln \left ( bx+a-1 \right ) \right ) ^{2}a}{4\,{b}^{2}}}+{\frac{a}{{b}^{2}}{\it dilog} \left ({\frac{1}{2}}+{\frac{bx}{2}}+{\frac{a}{2}} \right ) }+{\frac{\ln \left ( bx+a-1 \right ) a}{2\,{b}^{2}}\ln \left ({\frac{1}{2}}+{\frac{bx}{2}}+{\frac{a}{2}} \right ) }+{\frac{ \left ( \ln \left ( bx+a-1 \right ) \right ) ^{2}}{8\,{b}^{2}}}-{\frac{\ln \left ( bx+a-1 \right ) }{4\,{b}^{2}}\ln \left ({\frac{1}{2}}+{\frac{bx}{2}}+{\frac{a}{2}} \right ) }+{\frac{a}{2\,{b}^{2}}\ln \left ( -{\frac{bx}{2}}-{\frac{a}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{bx}{2}}+{\frac{a}{2}} \right ) }-{\frac{\ln \left ( bx+a+1 \right ) a}{2\,{b}^{2}}\ln \left ( -{\frac{bx}{2}}-{\frac{a}{2}}+{\frac{1}{2}} \right ) }+{\frac{ \left ( \ln \left ( bx+a+1 \right ) \right ) ^{2}a}{4\,{b}^{2}}}+{\frac{1}{4\,{b}^{2}}\ln \left ( -{\frac{bx}{2}}-{\frac{a}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{bx}{2}}+{\frac{a}{2}} \right ) }-{\frac{\ln \left ( bx+a+1 \right ) }{4\,{b}^{2}}\ln \left ( -{\frac{bx}{2}}-{\frac{a}{2}}+{\frac{1}{2}} \right ) }+{\frac{ \left ( \ln \left ( bx+a+1 \right ) \right ) ^{2}}{8\,{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arccoth(b*x+a)^2,x)

[Out]

1/2*x^2*arccoth(b*x+a)^2-1/2/b^2*arccoth(b*x+a)^2*a^2+1/b*arccoth(b*x+a)*x+1/b^2*arccoth(b*x+a)*a-1/b^2*arccot
h(b*x+a)*ln(b*x+a-1)*a+1/2/b^2*arccoth(b*x+a)*ln(b*x+a-1)-1/b^2*arccoth(b*x+a)*ln(b*x+a+1)*a-1/2/b^2*arccoth(b
*x+a)*ln(b*x+a+1)+1/2/b^2*ln(b*x+a-1)+1/2/b^2*ln(b*x+a+1)-1/4/b^2*ln(b*x+a-1)^2*a+1/b^2*dilog(1/2+1/2*b*x+1/2*
a)*a+1/2/b^2*ln(1/2+1/2*b*x+1/2*a)*ln(b*x+a-1)*a+1/8/b^2*ln(b*x+a-1)^2-1/4/b^2*ln(b*x+a-1)*ln(1/2+1/2*b*x+1/2*
a)+1/2/b^2*ln(-1/2*b*x-1/2*a+1/2)*ln(1/2+1/2*b*x+1/2*a)*a-1/2/b^2*ln(-1/2*b*x-1/2*a+1/2)*ln(b*x+a+1)*a+1/4/b^2
*ln(b*x+a+1)^2*a+1/4/b^2*ln(-1/2*b*x-1/2*a+1/2)*ln(1/2+1/2*b*x+1/2*a)-1/4/b^2*ln(-1/2*b*x-1/2*a+1/2)*ln(b*x+a+
1)+1/8/b^2*ln(b*x+a+1)^2

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Maxima [A]  time = 0.984672, size = 273, normalized size = 2.01 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{arcoth}\left (b x + a\right )^{2} + \frac{1}{8} \, b^{2}{\left (\frac{8 \,{\left (\log \left (b x + a - 1\right ) \log \left (\frac{1}{2} \, b x + \frac{1}{2} \, a + \frac{1}{2}\right ) +{\rm Li}_2\left (-\frac{1}{2} \, b x - \frac{1}{2} \, a + \frac{1}{2}\right )\right )} a}{b^{4}} + \frac{4 \,{\left (a + 1\right )} \log \left (b x + a + 1\right )}{b^{4}} + \frac{{\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a + 1\right )^{2} - 2 \,{\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a + 1\right ) \log \left (b x + a - 1\right ) +{\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a - 1\right )^{2} - 4 \,{\left (a - 1\right )} \log \left (b x + a - 1\right )}{b^{4}}\right )} + \frac{1}{2} \, b{\left (\frac{2 \, x}{b^{2}} - \frac{{\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{3}} + \frac{{\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{3}}\right )} \operatorname{arcoth}\left (b x + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*arccoth(b*x + a)^2 + 1/8*b^2*(8*(log(b*x + a - 1)*log(1/2*b*x + 1/2*a + 1/2) + dilog(-1/2*b*x - 1/2*a
+ 1/2))*a/b^4 + 4*(a + 1)*log(b*x + a + 1)/b^4 + ((a^2 + 2*a + 1)*log(b*x + a + 1)^2 - 2*(a^2 + 2*a + 1)*log(b
*x + a + 1)*log(b*x + a - 1) + (a^2 - 2*a + 1)*log(b*x + a - 1)^2 - 4*(a - 1)*log(b*x + a - 1))/b^4) + 1/2*b*(
2*x/b^2 - (a^2 + 2*a + 1)*log(b*x + a + 1)/b^3 + (a^2 - 2*a + 1)*log(b*x + a - 1)/b^3)*arccoth(b*x + a)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x \operatorname{arcoth}\left (b x + a\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x*arccoth(b*x + a)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*acoth(a + b*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{arcoth}\left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*arccoth(b*x + a)^2, x)