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

Optimal. Leaf size=69 \[ -\frac{\text{PolyLog}\left (2,(d+1) e^{2 a+2 b x}\right )}{4 b}-\frac{1}{2} x \log \left (1-(d+1) e^{2 a+2 b x}\right )+x \coth ^{-1}(d \coth (a+b x)+d+1)+\frac{b x^2}{2} \]

[Out]

(b*x^2)/2 + x*ArcCoth[1 + d + d*Coth[a + b*x]] - (x*Log[1 - (1 + d)*E^(2*a + 2*b*x)])/2 - PolyLog[2, (1 + d)*E
^(2*a + 2*b*x)]/(4*b)

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Rubi [A]  time = 0.144776, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6234, 2184, 2190, 2279, 2391} \[ -\frac{\text{PolyLog}\left (2,(d+1) e^{2 a+2 b x}\right )}{4 b}-\frac{1}{2} x \log \left (1-(d+1) e^{2 a+2 b x}\right )+x \coth ^{-1}(d \coth (a+b x)+d+1)+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[1 + d + d*Coth[a + b*x]],x]

[Out]

(b*x^2)/2 + x*ArcCoth[1 + d + d*Coth[a + b*x]] - (x*Log[1 - (1 + d)*E^(2*a + 2*b*x)])/2 - PolyLog[2, (1 + d)*E
^(2*a + 2*b*x)]/(4*b)

Rule 6234

Int[ArcCoth[(c_.) + Coth[(a_.) + (b_.)*(x_)]*(d_.)], x_Symbol] :> Simp[x*ArcCoth[c + d*Coth[a + b*x]], x] + Di
st[b, Int[x/(c - d - c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, 1]

Rule 2184

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[(c
+ d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[((c + d*x)^m*(F^(g*(e + f*x)))^n)/(a + b*(F^(g*(e + f*x)))^n)
, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

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

Rubi steps

\begin{align*} \int \coth ^{-1}(1+d+d \coth (a+b x)) \, dx &=x \coth ^{-1}(1+d+d \coth (a+b x))+b \int \frac{x}{1+(-1-d) e^{2 a+2 b x}} \, dx\\ &=\frac{b x^2}{2}+x \coth ^{-1}(1+d+d \coth (a+b x))+(b (1+d)) \int \frac{e^{2 a+2 b x} x}{1+(-1-d) e^{2 a+2 b x}} \, dx\\ &=\frac{b x^2}{2}+x \coth ^{-1}(1+d+d \coth (a+b x))-\frac{1}{2} x \log \left (1-(1+d) e^{2 a+2 b x}\right )+\frac{1}{2} \int \log \left (1+(-1-d) e^{2 a+2 b x}\right ) \, dx\\ &=\frac{b x^2}{2}+x \coth ^{-1}(1+d+d \coth (a+b x))-\frac{1}{2} x \log \left (1-(1+d) e^{2 a+2 b x}\right )+\frac{\operatorname{Subst}\left (\int \frac{\log (1+(-1-d) x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{4 b}\\ &=\frac{b x^2}{2}+x \coth ^{-1}(1+d+d \coth (a+b x))-\frac{1}{2} x \log \left (1-(1+d) e^{2 a+2 b x}\right )-\frac{\text{Li}_2\left ((1+d) e^{2 a+2 b x}\right )}{4 b}\\ \end{align*}

Mathematica [B]  time = 0.786928, size = 197, normalized size = 2.86 \[ \frac{-2 \text{PolyLog}\left (2,-\sqrt{d+1} e^{a+b x}\right )-2 \text{PolyLog}\left (2,\sqrt{d+1} e^{a+b x}\right )-2 \log \left (e^{a+b x}\right ) \log \left (1-\sqrt{d+1} e^{a+b x}\right )-2 \log \left (e^{a+b x}\right ) \log \left (\sqrt{d+1} e^{a+b x}+1\right )+2 \log \left (e^{a+b x}\right ) \log \left (e^{-a-b x} \left ((d+1) e^{2 (a+b x)}-1\right )\right )-2 b x \log ((d+2) \sinh (a+b x)+d \cosh (a+b x))+\log ^2\left (e^{a+b x}\right )+b^2 x^2}{4 b}+x \coth ^{-1}(d \coth (a+b x)+d+1) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCoth[1 + d + d*Coth[a + b*x]],x]

[Out]

x*ArcCoth[1 + d + d*Coth[a + b*x]] + (b^2*x^2 + Log[E^(a + b*x)]^2 - 2*Log[E^(a + b*x)]*Log[1 - Sqrt[1 + d]*E^
(a + b*x)] - 2*Log[E^(a + b*x)]*Log[1 + Sqrt[1 + d]*E^(a + b*x)] + 2*Log[E^(a + b*x)]*Log[E^(-a - b*x)*(-1 + (
1 + d)*E^(2*(a + b*x)))] - 2*b*x*Log[d*Cosh[a + b*x] + (2 + d)*Sinh[a + b*x]] - 2*PolyLog[2, -(Sqrt[1 + d]*E^(
a + b*x))] - 2*PolyLog[2, Sqrt[1 + d]*E^(a + b*x)])/(4*b)

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Maple [B]  time = 0.161, size = 247, normalized size = 3.6 \begin{align*}{\frac{{\rm arccoth} \left (1+d+d{\rm coth} \left (bx+a\right )\right )\ln \left ( d{\rm coth} \left (bx+a\right )+d \right ) }{2\,b}}-{\frac{{\rm arccoth} \left (1+d+d{\rm coth} \left (bx+a\right )\right )\ln \left ( d{\rm coth} \left (bx+a\right )-d \right ) }{2\,b}}+{\frac{1}{4\,b}{\it dilog} \left ({\frac{d{\rm coth} \left (bx+a\right )+d+2}{2\,d+2}} \right ) }+{\frac{\ln \left ( d{\rm coth} \left (bx+a\right )-d \right ) }{4\,b}\ln \left ({\frac{d{\rm coth} \left (bx+a\right )+d+2}{2\,d+2}} \right ) }-{\frac{1}{4\,b}{\it dilog} \left ({\frac{d{\rm coth} \left (bx+a\right )+d}{2\,d}} \right ) }-{\frac{\ln \left ( d{\rm coth} \left (bx+a\right )-d \right ) }{4\,b}\ln \left ({\frac{d{\rm coth} \left (bx+a\right )+d}{2\,d}} \right ) }+{\frac{ \left ( \ln \left ( d{\rm coth} \left (bx+a\right )+d \right ) \right ) ^{2}}{8\,b}}-{\frac{1}{4\,b}{\it dilog} \left ( 1+{\frac{d{\rm coth} \left (bx+a\right )}{2}}+{\frac{d}{2}} \right ) }-{\frac{\ln \left ( d{\rm coth} \left (bx+a\right )+d \right ) }{4\,b}\ln \left ( 1+{\frac{d{\rm coth} \left (bx+a\right )}{2}}+{\frac{d}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(1+d+d*coth(b*x+a)),x)

[Out]

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

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Maxima [A]  time = 3.59745, size = 97, normalized size = 1.41 \begin{align*} \frac{1}{4} \, b d{\left (\frac{2 \, x^{2}}{d} - \frac{2 \, b x \log \left (-{\left (d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) +{\rm Li}_2\left ({\left (d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}\right )}{b^{2} d}\right )} + x \operatorname{arcoth}\left (d \coth \left (b x + a\right ) + d + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(1+d+d*coth(b*x+a)),x, algorithm="maxima")

[Out]

1/4*b*d*(2*x^2/d - (2*b*x*log(-(d + 1)*e^(2*b*x + 2*a) + 1) + dilog((d + 1)*e^(2*b*x + 2*a)))/(b^2*d)) + x*arc
coth(d*coth(b*x + a) + d + 1)

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Fricas [B]  time = 2.05268, size = 667, normalized size = 9.67 \begin{align*} \frac{b^{2} x^{2} + b x \log \left (\frac{d \cosh \left (b x + a\right ) +{\left (d + 2\right )} \sinh \left (b x + a\right )}{d \cosh \left (b x + a\right ) + d \sinh \left (b x + a\right )}\right ) + a \log \left (2 \,{\left (d + 1\right )} \cosh \left (b x + a\right ) + 2 \,{\left (d + 1\right )} \sinh \left (b x + a\right ) + 2 \, \sqrt{d + 1}\right ) + a \log \left (2 \,{\left (d + 1\right )} \cosh \left (b x + a\right ) + 2 \,{\left (d + 1\right )} \sinh \left (b x + a\right ) - 2 \, \sqrt{d + 1}\right ) -{\left (b x + a\right )} \log \left (\sqrt{d + 1}{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) -{\left (b x + a\right )} \log \left (-\sqrt{d + 1}{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) -{\rm Li}_2\left (\sqrt{d + 1}{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) -{\rm Li}_2\left (-\sqrt{d + 1}{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(1+d+d*coth(b*x+a)),x, algorithm="fricas")

[Out]

1/2*(b^2*x^2 + b*x*log((d*cosh(b*x + a) + (d + 2)*sinh(b*x + a))/(d*cosh(b*x + a) + d*sinh(b*x + a))) + a*log(
2*(d + 1)*cosh(b*x + a) + 2*(d + 1)*sinh(b*x + a) + 2*sqrt(d + 1)) + a*log(2*(d + 1)*cosh(b*x + a) + 2*(d + 1)
*sinh(b*x + a) - 2*sqrt(d + 1)) - (b*x + a)*log(sqrt(d + 1)*(cosh(b*x + a) + sinh(b*x + a)) + 1) - (b*x + a)*l
og(-sqrt(d + 1)*(cosh(b*x + a) + sinh(b*x + a)) + 1) - dilog(sqrt(d + 1)*(cosh(b*x + a) + sinh(b*x + a))) - di
log(-sqrt(d + 1)*(cosh(b*x + a) + sinh(b*x + a))))/b

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{acoth}{\left (d \coth{\left (a + b x \right )} + d + 1 \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(1+d+d*coth(b*x+a)),x)

[Out]

Integral(acoth(d*coth(a + b*x) + d + 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{arcoth}\left (d \coth \left (b x + a\right ) + d + 1\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(1+d+d*coth(b*x+a)),x, algorithm="giac")

[Out]

integrate(arccoth(d*coth(b*x + a) + d + 1), x)