\(\int \frac {\coth ^{-1}(a+b x)}{c+d x} \, dx\) [77]

Optimal result

Integrand size = 14, antiderivative size = 120 \[ \int \frac {\coth ^{-1}(a+b x)}{c+d x} \, dx=-\frac {\coth ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{d}+\frac {\coth ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+a+b x}\right )}{2 d}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6247, 6058, 2449, 2352, 2497} \[ \int \frac {\coth ^{-1}(a+b x)}{c+d x} \, dx=-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 b (c+d x)}{(b c-a d+d) (a+b x+1)}\right )}{2 d}+\frac {\coth ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(a+b x+1) (-a d+b c+d)}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{a+b x+1}\right )}{2 d}-\frac {\log \left (\frac {2}{a+b x+1}\right ) \coth ^{-1}(a+b x)}{d} \]

[In]

Int[ArcCoth[a + b*x]/(c + d*x),x]

[Out]

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

Rule 2352

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

Rule 2449

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 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 6058

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x]))*(Log[2/
(1 + c*x)]/e), x] + (Dist[b*(c/e), Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x], x] - Dist[b*(c/e), Int[Log[2*c*((d
+ e*x)/((c*d + e)*(1 + c*x)))]/(1 - c^2*x^2), x], x] + Simp[(a + b*ArcCoth[c*x])*(Log[2*c*((d + e*x)/((c*d + e
)*(1 + c*x)))]/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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\coth ^{-1}(x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\coth ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{d}+\frac {\coth ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{d}+\frac {\text {Subst}\left (\int \frac {\log \left (\frac {2}{1+x}\right )}{1-x^2} \, dx,x,a+b x\right )}{d}-\frac {\text {Subst}\left (\int \frac {\log \left (\frac {2 \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )}{\left (\frac {d}{b}+\frac {b c-a d}{b}\right ) (1+x)}\right )}{1-x^2} \, dx,x,a+b x\right )}{d} \\ & = -\frac {\coth ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{d}+\frac {\coth ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{d}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{2 d}+\frac {\text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+a+b x}\right )}{d} \\ & = -\frac {\coth ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{d}+\frac {\coth ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+a+b x}\right )}{2 d}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.54 \[ \int \frac {\coth ^{-1}(a+b x)}{c+d x} \, dx=\frac {\log \left (\frac {d (1-a-b x)}{b c+d-a d}\right ) \log (c+d x)}{2 d}-\frac {\log \left (\frac {-1+a+b x}{a+b x}\right ) \log (c+d x)}{2 d}-\frac {\log \left (-\frac {d (1+a+b x)}{b c-d-a d}\right ) \log (c+d x)}{2 d}+\frac {\log \left (\frac {1+a+b x}{a+b x}\right ) \log (c+d x)}{2 d}-\frac {\operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-d-a d}\right )}{2 d}+\frac {\operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c+d-a d}\right )}{2 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.30 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.37

method	result	size
risch	\(-\frac {\operatorname {dilog}\left (\frac {\left (b x +a -1\right ) d -a d +b c +d}{-a d +b c +d}\right )}{2 d}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {\left (b x +a -1\right ) d -a d +b c +d}{-a d +b c +d}\right )}{2 d}+\frac {\operatorname {dilog}\left (\frac {\left (b x +a +1\right ) d -a d +b c -d}{-a d +b c -d}\right )}{2 d}+\frac {\ln \left (b x +a +1\right ) \ln \left (\frac {\left (b x +a +1\right ) d -a d +b c -d}{-a d +b c -d}\right )}{2 d}\)	\(164\)
parts	\(\frac {\ln \left (d x +c \right ) \operatorname {arccoth}\left (b x +a \right )}{d}+\frac {b \left (\frac {d \left (\frac {\operatorname {dilog}\left (\frac {a d -b c +b \left (d x +c \right )-d}{a d -b c -d}\right )}{b}+\frac {\ln \left (d x +c \right ) \ln \left (\frac {a d -b c +b \left (d x +c \right )-d}{a d -b c -d}\right )}{b}\right )}{2}-\frac {d \left (\frac {\operatorname {dilog}\left (\frac {a d -b c +b \left (d x +c \right )+d}{a d -b c +d}\right )}{b}+\frac {\ln \left (d x +c \right ) \ln \left (\frac {a d -b c +b \left (d x +c \right )+d}{a d -b c +d}\right )}{b}\right )}{2}\right )}{d^{2}}\)	\(184\)
derivativedivides	\(\frac {\frac {b \ln \left (a d -b c -d \left (b x +a \right )\right ) \operatorname {arccoth}\left (b x +a \right )}{d}-\frac {b \left (\frac {d \left (\operatorname {dilog}\left (\frac {-d \left (b x +a \right )-d}{-a d +b c -d}\right )+\ln \left (a d -b c -d \left (b x +a \right )\right ) \ln \left (\frac {-d \left (b x +a \right )-d}{-a d +b c -d}\right )\right )}{2}-\frac {d \left (\operatorname {dilog}\left (\frac {-d \left (b x +a \right )+d}{-a d +b c +d}\right )+\ln \left (a d -b c -d \left (b x +a \right )\right ) \ln \left (\frac {-d \left (b x +a \right )+d}{-a d +b c +d}\right )\right )}{2}\right )}{d^{2}}}{b}\)	\(185\)
default	\(\frac {\frac {b \ln \left (a d -b c -d \left (b x +a \right )\right ) \operatorname {arccoth}\left (b x +a \right )}{d}-\frac {b \left (\frac {d \left (\operatorname {dilog}\left (\frac {-d \left (b x +a \right )-d}{-a d +b c -d}\right )+\ln \left (a d -b c -d \left (b x +a \right )\right ) \ln \left (\frac {-d \left (b x +a \right )-d}{-a d +b c -d}\right )\right )}{2}-\frac {d \left (\operatorname {dilog}\left (\frac {-d \left (b x +a \right )+d}{-a d +b c +d}\right )+\ln \left (a d -b c -d \left (b x +a \right )\right ) \ln \left (\frac {-d \left (b x +a \right )+d}{-a d +b c +d}\right )\right )}{2}\right )}{d^{2}}}{b}\)	\(185\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(acoth(a + b*x)/(c + d*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.60 \[ \int \frac {\coth ^{-1}(a+b x)}{c+d x} \, dx=-\frac {1}{2} \, b {\left (\frac {\log \left (b x + a - 1\right ) \log \left (\frac {b d x + a d - d}{b c - a d + d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d - d}{b c - a d + d}\right )}{b d} - \frac {\log \left (b x + a + 1\right ) \log \left (\frac {b d x + a d + d}{b c - a d - d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d + d}{b c - a d - d}\right )}{b d}\right )} - \frac {b {\left (\frac {\log \left (b x + a + 1\right )}{b} - \frac {\log \left (b x + a - 1\right )}{b}\right )} \log \left (d x + c\right )}{2 \, d} + \frac {\operatorname {arcoth}\left (b x + a\right ) \log \left (d x + c\right )}{d} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(acoth(a + b*x)/(c + d*x),x)

[Out]

int(acoth(a + b*x)/(c + d*x), x)