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

Optimal result

Integrand size = 19, antiderivative size = 35 \[ \int \frac {\coth ^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=\frac {\operatorname {PolyLog}\left (2,-\frac {1}{a+b x}\right )}{2 d}-\frac {\operatorname {PolyLog}\left (2,\frac {1}{a+b x}\right )}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {6243, 12, 6032} \[ \int \frac {\coth ^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=\frac {\operatorname {PolyLog}\left (2,-\frac {1}{a+b x}\right )}{2 d}-\frac {\operatorname {PolyLog}\left (2,\frac {1}{a+b x}\right )}{2 d} \]

[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*} \text {integral}& = \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 {\operatorname {PolyLog}\left (2,-\frac {1}{a+b x}\right )}{2 d}-\frac {\operatorname {PolyLog}\left (2,\frac {1}{a+b x}\right )}{2 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(144\) vs. \(2(35)=70\).

Time = 0.02 (sec) , antiderivative size = 144, normalized size of antiderivative = 4.11 \[ \int \frac {\coth ^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=\frac {\log ^2\left (-\frac {1}{a+b x}\right )-2 \log (1-a-b x) \log \left (\frac {1}{a+b x}\right )-\log ^2\left (\frac {1}{a+b x}\right )+2 \log \left (\frac {1}{a+b x}\right ) \log \left (\frac {-1+a+b x}{a+b x}\right )+2 \log \left (-\frac {1}{a+b x}\right ) \log (1+a+b x)-2 \log \left (-\frac {1}{a+b x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )-2 \operatorname {PolyLog}(2,-a-b x)+2 \operatorname {PolyLog}(2,a+b x)}{4 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23

method	result	size
risch	\(-\frac {\operatorname {dilog}\left (b x +a +1\right )}{2 d}-\frac {\ln \left (b x +a -1\right ) \ln \left (b x +a \right )}{2 d}-\frac {\operatorname {dilog}\left (b x +a \right )}{2 d}\)	\(43\)
parts	\(\frac {\ln \left (b x +a \right ) \operatorname {arccoth}\left (b x +a \right )}{d}+\frac {-\frac {\operatorname {dilog}\left (b x +a \right )}{2}-\frac {\operatorname {dilog}\left (b x +a +1\right )}{2}-\frac {\ln \left (b x +a \right ) \ln \left (b x +a +1\right )}{2}}{d}\)	\(55\)
derivativedivides	\(\frac {\frac {b \ln \left (b x +a \right ) \operatorname {arccoth}\left (b x +a \right )}{d}+\frac {b \left (-\frac {\operatorname {dilog}\left (b x +a \right )}{2}-\frac {\operatorname {dilog}\left (b x +a +1\right )}{2}-\frac {\ln \left (b x +a \right ) \ln \left (b x +a +1\right )}{2}\right )}{d}}{b}\)	\(61\)
default	\(\frac {\frac {b \ln \left (b x +a \right ) \operatorname {arccoth}\left (b x +a \right )}{d}+\frac {b \left (-\frac {\operatorname {dilog}\left (b x +a \right )}{2}-\frac {\operatorname {dilog}\left (b x +a +1\right )}{2}-\frac {\ln \left (b x +a \right ) \ln \left (b x +a +1\right )}{2}\right )}{d}}{b}\)	\(61\)

[In]

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

[Out]

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

Fricas [F]

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

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (29) = 58\).

Time = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 3.77 \[ \int \frac {\coth ^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=-\frac {1}{2} \, b {\left (\frac {\log \left (b x + a\right ) \log \left (b x + a - 1\right ) + {\rm Li}_2\left (-b x - a + 1\right )}{b d} - \frac {\log \left (b x + a + 1\right ) \log \left (-b x - a\right ) + {\rm Li}_2\left (b x + a + 1\right )}{b d}\right )} - \frac {b {\left (\frac {\log \left (b x + a + 1\right )}{b} - \frac {\log \left (b x + a - 1\right )}{b}\right )} \log \left (d x + \frac {a d}{b}\right )}{2 \, d} + \frac {\operatorname {arcoth}\left (b x + a\right ) \log \left (d x + \frac {a d}{b}\right )}{d} \]

[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

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

int(acoth(a + b*x)/(d*x + (a*d)/b),x)

[Out]

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