\(\int \frac {a+b \coth ^{-1}(c+d x)}{e+f x} \, dx\) [106]

Optimal result

Integrand size = 18, antiderivative size = 130 \[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f x} \, dx=-\frac {\left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+c+d x}\right )}{f}+\frac {\left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6247, 6058, 2449, 2352, 2497} \[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f x} \, dx=\frac {\left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}-\frac {\log \left (\frac {2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{f}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{2 f}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right )}{2 f} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.58 \[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f x} \, dx=\frac {a \log (e+f x)}{f}+\frac {b \log \left (\frac {f (1-c-d x)}{d e+f-c f}\right ) \log (e+f x)}{2 f}-\frac {b \log \left (-\frac {1-c-d x}{c+d x}\right ) \log (e+f x)}{2 f}-\frac {b \log \left (-\frac {f (1+c+d x)}{d e-f-c f}\right ) \log (e+f x)}{2 f}+\frac {b \log \left (\frac {1+c+d x}{c+d x}\right ) \log (e+f x)}{2 f}-\frac {b \operatorname {PolyLog}\left (2,\frac {d (e+f x)}{d e-f-c f}\right )}{2 f}+\frac {b \operatorname {PolyLog}\left (2,\frac {d (e+f x)}{d e+f-c f}\right )}{2 f} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.47

method	result	size
risch	\(\frac {a \ln \left (\left (d x +c -1\right ) f -c f +d e +f \right )}{f}-\frac {b \operatorname {dilog}\left (\frac {\left (d x +c -1\right ) f -c f +d e +f}{-c f +d e +f}\right )}{2 f}-\frac {b \ln \left (d x +c -1\right ) \ln \left (\frac {\left (d x +c -1\right ) f -c f +d e +f}{-c f +d e +f}\right )}{2 f}+\frac {b \operatorname {dilog}\left (\frac {\left (d x +c +1\right ) f -c f +d e -f}{-c f +d e -f}\right )}{2 f}+\frac {b \ln \left (d x +c +1\right ) \ln \left (\frac {\left (d x +c +1\right ) f -c f +d e -f}{-c f +d e -f}\right )}{2 f}\)	\(191\)
parts	\(\frac {a \ln \left (f x +e \right )}{f}+\frac {b \ln \left (f \left (d x +c \right )-c f +d e \right ) \operatorname {arccoth}\left (d x +c \right )}{f}+\frac {b \ln \left (f \left (d x +c \right )-c f +d e \right ) \ln \left (\frac {f \left (d x +c \right )-f}{c f -d e -f}\right )}{2 f}+\frac {b \operatorname {dilog}\left (\frac {f \left (d x +c \right )-f}{c f -d e -f}\right )}{2 f}-\frac {b \ln \left (f \left (d x +c \right )-c f +d e \right ) \ln \left (\frac {f \left (d x +c \right )+f}{c f -d e +f}\right )}{2 f}-\frac {b \operatorname {dilog}\left (\frac {f \left (d x +c \right )+f}{c f -d e +f}\right )}{2 f}\)	\(192\)
derivativedivides	\(\frac {\frac {a d \ln \left (c f -d e -f \left (d x +c \right )\right )}{f}-b d \left (-\frac {\ln \left (c f -d e -f \left (d x +c \right )\right ) \operatorname {arccoth}\left (d x +c \right )}{f}+\frac {-\frac {f \left (\operatorname {dilog}\left (\frac {-f \left (d x +c \right )+f}{-c f +d e +f}\right )+\ln \left (c f -d e -f \left (d x +c \right )\right ) \ln \left (\frac {-f \left (d x +c \right )+f}{-c f +d e +f}\right )\right )}{2}+\frac {f \left (\operatorname {dilog}\left (\frac {-f \left (d x +c \right )-f}{-c f +d e -f}\right )+\ln \left (c f -d e -f \left (d x +c \right )\right ) \ln \left (\frac {-f \left (d x +c \right )-f}{-c f +d e -f}\right )\right )}{2}}{f^{2}}\right )}{d}\)	\(211\)
default	\(\frac {\frac {a d \ln \left (c f -d e -f \left (d x +c \right )\right )}{f}-b d \left (-\frac {\ln \left (c f -d e -f \left (d x +c \right )\right ) \operatorname {arccoth}\left (d x +c \right )}{f}+\frac {-\frac {f \left (\operatorname {dilog}\left (\frac {-f \left (d x +c \right )+f}{-c f +d e +f}\right )+\ln \left (c f -d e -f \left (d x +c \right )\right ) \ln \left (\frac {-f \left (d x +c \right )+f}{-c f +d e +f}\right )\right )}{2}+\frac {f \left (\operatorname {dilog}\left (\frac {-f \left (d x +c \right )-f}{-c f +d e -f}\right )+\ln \left (c f -d e -f \left (d x +c \right )\right ) \ln \left (\frac {-f \left (d x +c \right )-f}{-c f +d e -f}\right )\right )}{2}}{f^{2}}\right )}{d}\)	\(211\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/2*b*integrate((log(1/(d*x + c) + 1) - log(-1/(d*x + c) + 1))/(f*x + e), x) + a*log(f*x + e)/f

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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