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

Optimal. Leaf size=130 \[ -\frac{b \text{PolyLog}\left (2,1-\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{2 f}+\frac{b \text{PolyLog}\left (2,1-\frac{2}{c+d x+1}\right )}{2 f}+\frac{\left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}-\frac{\log \left (\frac{2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{f} \]

[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)

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Rubi [A]  time = 0.14231, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6112, 5921, 2402, 2315, 2447} \[ -\frac{b \text{PolyLog}\left (2,1-\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{2 f}+\frac{b \text{PolyLog}\left (2,1-\frac{2}{c+d x+1}\right )}{2 f}+\frac{\left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}-\frac{\log \left (\frac{2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{f} \]

Antiderivative was successfully verified.

[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 6112

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]

Rule 5921

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 2402

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 2315

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

Rule 2447

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]]

Rubi steps

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

Mathematica [A]  time = 0.11257, size = 206, normalized size = 1.58 \[ -\frac{b \text{PolyLog}\left (2,\frac{d (e+f x)}{-c f+d e-f}\right )}{2 f}+\frac{b \text{PolyLog}\left (2,\frac{d (e+f x)}{-c f+d e+f}\right )}{2 f}+\frac{a \log (e+f x)}{f}+\frac{b \log (e+f x) \log \left (\frac{f (-c-d x+1)}{-c f+d e+f}\right )}{2 f}-\frac{b \log \left (-\frac{-c-d x+1}{c+d x}\right ) \log (e+f x)}{2 f}-\frac{b \log (e+f x) \log \left (-\frac{f (c+d x+1)}{-c f+d e-f}\right )}{2 f}+\frac{b \log \left (\frac{c+d x+1}{c+d x}\right ) \log (e+f x)}{2 f} \]

Warning: Unable to verify antiderivative.

[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)

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Maple [A]  time = 0.153, size = 202, normalized size = 1.6 \begin{align*}{\frac{a\ln \left ( \left ( dx+c \right ) f-cf+de \right ) }{f}}+{\frac{b\ln \left ( \left ( dx+c \right ) f-cf+de \right ){\rm arccoth} \left (dx+c\right )}{f}}-{\frac{b\ln \left ( \left ( dx+c \right ) f-cf+de \right ) }{2\,f}\ln \left ({\frac{ \left ( dx+c \right ) f+f}{cf-de+f}} \right ) }-{\frac{b}{2\,f}{\it dilog} \left ({\frac{ \left ( dx+c \right ) f+f}{cf-de+f}} \right ) }+{\frac{b\ln \left ( \left ( dx+c \right ) f-cf+de \right ) }{2\,f}\ln \left ({\frac{ \left ( dx+c \right ) f-f}{cf-de-f}} \right ) }+{\frac{b}{2\,f}{\it dilog} \left ({\frac{ \left ( dx+c \right ) f-f}{cf-de-f}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, b \int \frac{\log \left (\frac{1}{d x + c} + 1\right ) - \log \left (-\frac{1}{d x + c} + 1\right )}{f x + e}\,{d x} + \frac{a \log \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{arcoth}\left (d x + c\right ) + a}{f x + e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[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)