3.2.1 \(\int \frac {\coth ^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx\) [101]

Optimal. Leaf size=35 \[ \frac {\text {PolyLog}\left (2,-\frac {1}{a+b x}\right )}{2 d}-\frac {\text {PolyLog}\left (2,\frac {1}{a+b x}\right )}{2 d} \]

[Out]

1/2*polylog(2,-1/(b*x+a))/d-1/2*polylog(2,1/(b*x+a))/d

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Rubi [A]
time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {6243, 12, 6032} \begin {gather*} \frac {\text {Li}_2\left (-\frac {1}{a+b x}\right )}{2 d}-\frac {\text {Li}_2\left (\frac {1}{a+b x}\right )}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCoth[a + b*x]/((a*d)/b + d*x),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6032

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Simp[(b/2)*PolyLog[2, -(c*x)^(
-1)], x] - Simp[(b/2)*PolyLog[2, 1/(c*x)], x]) /; FreeQ[{a, b, c}, x]

Rule 6243

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

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(312\) vs. \(2(35)=70\).
time = 0.02, size = 312, normalized size = 8.91 \begin {gather*} b \left (-\frac {\log \left (\frac {b (-1+a+b x)}{(-1+a) b-a b}\right ) \log \left (\frac {-((-1+a) b)+a b}{b (a+b x)}\right )}{2 b d}-\frac {\log ^2\left (\frac {-((-1+a) b)+a b}{b (a+b x)}\right )}{4 b d}+\frac {\log \left (\frac {b (-1-a-b x)}{(-1-a) b+a b}\right ) \log \left (\frac {a b-(1+a) b}{b (a+b x)}\right )}{2 b d}+\frac {\log ^2\left (\frac {a b-(1+a) b}{b (a+b x)}\right )}{4 b d}+\frac {\log \left (\frac {-((-1+a) b)+a b}{b (a+b x)}\right ) \log \left (\frac {-1+a+b x}{a+b x}\right )}{2 b d}-\frac {\log \left (\frac {a b-(1+a) b}{b (a+b x)}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{2 b d}-\frac {\text {PolyLog}(2,-a-b x)}{2 b d}+\frac {\text {PolyLog}(2,a+b x)}{2 b d}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[a + b*x]/((a*d)/b + d*x),x]

[Out]

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

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Maple [A]
time = 0.12, size = 61, normalized size = 1.74

method	result	size
risch	\(-\frac {\dilog \left (b x +a +1\right )}{2 d}-\frac {\ln \left (b x +a -1\right ) \ln \left (b x +a \right )}{2 d}-\frac {\dilog \left (b x +a \right )}{2 d}\)	\(43\)
derivativedivides	\(\frac {\frac {b \ln \left (b x +a \right ) \mathrm {arccoth}\left (b x +a \right )}{d}+\frac {b \left (-\frac {\dilog \left (b x +a \right )}{2}-\frac {\dilog \left (b x +a +1\right )}{2}-\frac {\ln \left (b x +a \right ) \ln \left (b x +a +1\right )}{2}\right )}{d}}{b}\)	\(61\)
default	\(\frac {\frac {b \ln \left (b x +a \right ) \mathrm {arccoth}\left (b x +a \right )}{d}+\frac {b \left (-\frac {\dilog \left (b x +a \right )}{2}-\frac {\dilog \left (b x +a +1\right )}{2}-\frac {\ln \left (b x +a \right ) \ln \left (b x +a +1\right )}{2}\right )}{d}}{b}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(b/d*ln(b*x+a)*arccoth(b*x+a)+b/d*(-1/2*dilog(b*x+a)-1/2*dilog(b*x+a+1)-1/2*ln(b*x+a)*ln(b*x+a+1)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (29) = 58\).
time = 0.28, size = 132, normalized size = 3.77 \begin {gather*} -\frac {1}{2} \, b {\left (\frac {\log \left (b x + a\right ) \log \left (b x + a - 1\right ) + {\rm Li}_2\left (-b x - a + 1\right )}{b d} - \frac {\log \left (b x + a + 1\right ) \log \left (-b x - a\right ) + {\rm Li}_2\left (b x + a + 1\right )}{b d}\right )} - \frac {b {\left (\frac {\log \left (b x + a + 1\right )}{b} - \frac {\log \left (b x + a - 1\right )}{b}\right )} \log \left (d x + \frac {a d}{b}\right )}{2 \, d} + \frac {\operatorname {arcoth}\left (b x + a\right ) \log \left (d x + \frac {a d}{b}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*((log(b*x + a)*log(b*x + a - 1) + dilog(-b*x - a + 1))/(b*d) - (log(b*x + a + 1)*log(-b*x - a) + dilog(
b*x + a + 1))/(b*d)) - 1/2*b*(log(b*x + a + 1)/b - log(b*x + a - 1)/b)*log(d*x + a*d/b)/d + arccoth(b*x + a)*l
og(d*x + a*d/b)/d

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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)/(a*d/b+d*x),x, algorithm="fricas")

[Out]

integral(b*arccoth(b*x + a)/(b*d*x + a*d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*Integral(acoth(a + b*x)/(a + b*x), x)/d

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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)/(a*d/b+d*x),x, algorithm="giac")

[Out]

integrate(arccoth(b*x + a)/(d*x + a*d/b), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\mathrm {acoth}\left (a+b\,x\right )}{d\,x+\frac {a\,d}{b}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acoth(a + b*x)/(d*x + (a*d)/b),x)

[Out]

int(acoth(a + b*x)/(d*x + (a*d)/b), x)

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