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

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

[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

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Rubi [A]
time = 0.10, antiderivative size = 130, normalized size of antiderivative = 1.00, 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 = {6247, 6058, 2449, 2352, 2497} \begin {gather*} \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 \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{2 f}+\frac {b \text {Li}_2\left (1-\frac {2}{c+d x+1}\right )}{2 f} \end {gather*}

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.19, size = 352, normalized size = 2.71 \begin {gather*} \frac {a \log (e+f x)+b \left (\coth ^{-1}(c+d x)-\tanh ^{-1}(c+d x)\right ) \log (e+f x)+b \tanh ^{-1}(c+d x) \left (-\log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )+\log \left (i \sinh \left (\tanh ^{-1}\left (\frac {d e-c f}{f}\right )+\tanh ^{-1}(c+d x)\right )\right )\right )-\frac {1}{2} i b \left (-\frac {1}{4} i \left (\pi -2 i \tanh ^{-1}(c+d x)\right )^2+i \left (\tanh ^{-1}\left (\frac {d e-c f}{f}\right )+\tanh ^{-1}(c+d x)\right )^2+\left (\pi -2 i \tanh ^{-1}(c+d x)\right ) \log \left (1+e^{2 \tanh ^{-1}(c+d x)}\right )+2 i \left (\tanh ^{-1}\left (\frac {d e-c f}{f}\right )+\tanh ^{-1}(c+d x)\right ) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {d e-c f}{f}\right )+\tanh ^{-1}(c+d x)\right )}\right )-\left (\pi -2 i \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{\sqrt {1-(c+d x)^2}}\right )-2 i \left (\tanh ^{-1}\left (\frac {d e-c f}{f}\right )+\tanh ^{-1}(c+d x)\right ) \log \left (2 i \sinh \left (\tanh ^{-1}\left (\frac {d e-c f}{f}\right )+\tanh ^{-1}(c+d x)\right )\right )-i \text {PolyLog}\left (2,-e^{2 \tanh ^{-1}(c+d x)}\right )-i \text {PolyLog}\left (2,e^{-2 \left (\tanh ^{-1}\left (\frac {d e-c f}{f}\right )+\tanh ^{-1}(c+d x)\right )}\right )\right )}{f} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*Log[e + f*x] + b*(ArcCoth[c + d*x] - ArcTanh[c + d*x])*Log[e + f*x] + b*ArcTanh[c + d*x]*(-Log[1/Sqrt[1 - (
c + d*x)^2]] + Log[I*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]]) - (I/2)*b*((-1/4*I)*(Pi - (2*I)*ArcTanh
[c + d*x])^2 + I*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])^2 + (Pi - (2*I)*ArcTanh[c + d*x])*Log[1 + E^(2*Ar
cTanh[c + d*x])] + (2*I)*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])*Log[1 - E^(-2*(ArcTanh[(d*e - c*f)/f] + A
rcTanh[c + d*x]))] - (Pi - (2*I)*ArcTanh[c + d*x])*Log[2/Sqrt[1 - (c + d*x)^2]] - (2*I)*(ArcTanh[(d*e - c*f)/f
] + ArcTanh[c + d*x])*Log[(2*I)*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]] - I*PolyLog[2, -E^(2*ArcTanh[
c + d*x])] - I*PolyLog[2, E^(-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))]))/f

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Maple [A]
time = 0.96, size = 220, normalized size = 1.69

method	result	size
risch	\(\frac {a \ln \left (\left (d x +c -1\right ) f -c f +d e +f \right )}{f}-\frac {b \dilog \left (\frac {\left (d x +c -1\right ) f -c f +d e +f}{-c f +d e +f}\right )}{2 f}-\frac {b \ln \left (d x +c -1\right ) \ln \left (\frac {\left (d x +c -1\right ) f -c f +d e +f}{-c f +d e +f}\right )}{2 f}+\frac {b \dilog \left (\frac {\left (d x +c +1\right ) f -c f +d e -f}{-c f +d e -f}\right )}{2 f}+\frac {b \ln \left (d x +c +1\right ) \ln \left (\frac {\left (d x +c +1\right ) f -c f +d e -f}{-c f +d e -f}\right )}{2 f}\)	\(191\)
derivativedivides	\(\frac {\frac {a d \ln \left (c f -d e -f \left (d x +c \right )\right )}{f}+\frac {b d \ln \left (c f -d e -f \left (d x +c \right )\right ) \mathrm {arccoth}\left (d x +c \right )}{f}+\frac {b d \ln \left (c f -d e -f \left (d x +c \right )\right ) \ln \left (\frac {-f \left (d x +c \right )+f}{-c f +d e +f}\right )}{2 f}+\frac {b d \dilog \left (\frac {-f \left (d x +c \right )+f}{-c f +d e +f}\right )}{2 f}-\frac {b d \ln \left (c f -d e -f \left (d x +c \right )\right ) \ln \left (\frac {-f \left (d x +c \right )-f}{-c f +d e -f}\right )}{2 f}-\frac {b d \dilog \left (\frac {-f \left (d x +c \right )-f}{-c f +d e -f}\right )}{2 f}}{d}\)	\(220\)
default	\(\frac {\frac {a d \ln \left (c f -d e -f \left (d x +c \right )\right )}{f}+\frac {b d \ln \left (c f -d e -f \left (d x +c \right )\right ) \mathrm {arccoth}\left (d x +c \right )}{f}+\frac {b d \ln \left (c f -d e -f \left (d x +c \right )\right ) \ln \left (\frac {-f \left (d x +c \right )+f}{-c f +d e +f}\right )}{2 f}+\frac {b d \dilog \left (\frac {-f \left (d x +c \right )+f}{-c f +d e +f}\right )}{2 f}-\frac {b d \ln \left (c f -d e -f \left (d x +c \right )\right ) \ln \left (\frac {-f \left (d x +c \right )-f}{-c f +d e -f}\right )}{2 f}-\frac {b d \dilog \left (\frac {-f \left (d x +c \right )-f}{-c f +d e -f}\right )}{2 f}}{d}\)	\(220\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {acoth}\left (c+d\,x\right )}{e+f\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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