3.2.12 \(\int \frac {(a+b \coth ^{-1}(c+d x))^2}{e+f x} \, dx\) [112]

Optimal. Leaf size=214 \[ -\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+c+d x}\right )}{f}+\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {b \left (a+b \coth ^{-1}(c+d x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{f}-\frac {b \left (a+b \coth ^{-1}(c+d x)\right ) \text {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {b^2 \text {PolyLog}\left (3,1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {b^2 \text {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f} \]

[Out]

-(a+b*arccoth(d*x+c))^2*ln(2/(d*x+c+1))/f+(a+b*arccoth(d*x+c))^2*ln(2*d*(f*x+e)/(-c*f+d*e+f)/(d*x+c+1))/f+b*(a
+b*arccoth(d*x+c))*polylog(2,1-2/(d*x+c+1))/f-b*(a+b*arccoth(d*x+c))*polylog(2,1-2*d*(f*x+e)/(-c*f+d*e+f)/(d*x
+c+1))/f+1/2*b^2*polylog(3,1-2/(d*x+c+1))/f-1/2*b^2*polylog(3,1-2*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 = 214, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6247, 6060} \begin {gather*} -\frac {b \left (a+b \coth ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{f}+\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}+\frac {b \text {Li}_2\left (1-\frac {2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{f}-\frac {\log \left (\frac {2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{f}-\frac {b^2 \text {Li}_3\left (1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{2 f}+\frac {b^2 \text {Li}_3\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])^2/(e + f*x),x]

[Out]

-(((a + b*ArcCoth[c + d*x])^2*Log[2/(1 + c + d*x)])/f) + ((a + b*ArcCoth[c + d*x])^2*Log[(2*d*(e + f*x))/((d*e
+ f - c*f)*(1 + c + d*x))])/f + (b*(a + b*ArcCoth[c + d*x])*PolyLog[2, 1 - 2/(1 + c + d*x)])/f - (b*(a + b*Ar
cCoth[c + d*x])*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/f + (b^2*PolyLog[3, 1 - 2/(1
+ c + d*x)])/(2*f) - (b^2*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f)

Rule 6060

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^2)*(Lo
g[2/(1 + c*x)]/e), x] + (Simp[(a + b*ArcCoth[c*x])^2*(Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp
[b*(a + b*ArcCoth[c*x])*(PolyLog[2, 1 - 2/(1 + c*x)]/e), x] - Simp[b*(a + b*ArcCoth[c*x])*(PolyLog[2, 1 - 2*c*
((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp[b^2*(PolyLog[3, 1 - 2/(1 + c*x)]/(2*e)), x] - Simp[b^2*(PolyL
og[3, 1 - 2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/(2*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 {\left (a+b \coth ^{-1}(c+d x)\right )^2}{e+f x} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \coth ^{-1}(x)\right )^2}{\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 )^2 \log \left (\frac {2}{1+c+d x}\right )}{f}+\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {b \left (a+b \coth ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2}{1+c+d x}\right )}{f}-\frac {b \left (a+b \coth ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {b^2 \text {Li}_3\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 = 12.52, size = 1376, normalized size = 6.43 \begin {gather*} \frac {24 a^2 f \log (e+f x)+48 a b f \left (\left (\coth ^{-1}(c+d x)-\tanh ^{-1}(c+d x)\right ) \log (e+f x)+\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}{8} \left (-\left (\pi -2 i \tanh ^{-1}(c+d x)\right )^2+4 \left (\tanh ^{-1}\left (\frac {d e-c f}{f}\right )+\tanh ^{-1}(c+d x)\right )^2-4 i \left (\pi -2 i \tanh ^{-1}(c+d x)\right ) \log \left (1+e^{2 \tanh ^{-1}(c+d x)}\right )+8 \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 )+4 \left (i \pi +2 \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{\sqrt {1-(c+d x)^2}}\right )-8 \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 )-4 \text {PolyLog}\left (2,-e^{2 \tanh ^{-1}(c+d x)}\right )-4 \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 )\right )-b^2 \left (i f \pi ^3-16 d e \coth ^{-1}(c+d x)^3+16 f \coth ^{-1}(c+d x)^3+16 c f \coth ^{-1}(c+d x)^3+16 d e e^{-\tanh ^{-1}\left (\frac {f}{d e-c f}\right )} \sqrt {\frac {d^2 e^2-2 c d e f+\left (-1+c^2\right ) f^2}{(d e-c f)^2}} \coth ^{-1}(c+d x)^3-16 c e^{-\tanh ^{-1}\left (\frac {f}{d e-c f}\right )} f \sqrt {\frac {d^2 e^2-2 c d e f+\left (-1+c^2\right ) f^2}{(d e-c f)^2}} \coth ^{-1}(c+d x)^3-24 i f \pi \coth ^{-1}(c+d x) \log (2)+4 f \coth ^{-1}(c+d x)^2 \log (64)+24 i f \pi \coth ^{-1}(c+d x) \log \left (e^{-\coth ^{-1}(c+d x)}+e^{\coth ^{-1}(c+d x)}\right )+24 f \coth ^{-1}(c+d x)^2 \log \left (1-e^{2 \coth ^{-1}(c+d x)}\right )-24 f \coth ^{-1}(c+d x)^2 \log \left (1-e^{\coth ^{-1}(c+d x)+\tanh ^{-1}\left (\frac {f}{d e-c f}\right )}\right )-24 f \coth ^{-1}(c+d x)^2 \log \left (1+e^{\coth ^{-1}(c+d x)+\tanh ^{-1}\left (\frac {f}{d e-c f}\right )}\right )-24 f \coth ^{-1}(c+d x)^2 \log \left (1-e^{2 \left (\coth ^{-1}(c+d x)+\tanh ^{-1}\left (\frac {f}{d e-c f}\right )\right )}\right )-48 f \coth ^{-1}(c+d x) \tanh ^{-1}\left (\frac {f}{d e-c f}\right ) \log \left (\frac {1}{2} i e^{-\coth ^{-1}(c+d x)-\tanh ^{-1}\left (\frac {f}{d e-c f}\right )} \left (-1+e^{2 \left (\coth ^{-1}(c+d x)+\tanh ^{-1}\left (\frac {f}{d e-c f}\right )\right )}\right )\right )-24 f \coth ^{-1}(c+d x)^2 \log \left (-e^{-\coth ^{-1}(c+d x)} \left (d e \left (-1+e^{2 \coth ^{-1}(c+d x)}\right )+\left (1+c+e^{2 \coth ^{-1}(c+d x)}-c e^{2 \coth ^{-1}(c+d x)}\right ) f\right )\right )+24 f \coth ^{-1}(c+d x)^2 \log \left (\frac {-d e \left (-1+e^{2 \coth ^{-1}(c+d x)}\right )+\left (-1-e^{2 \coth ^{-1}(c+d x)}+c \left (-1+e^{2 \coth ^{-1}(c+d x)}\right )\right ) f}{d e-(1+c) f}\right )-24 i f \pi \coth ^{-1}(c+d x) \log \left (\frac {1}{\sqrt {1-\frac {1}{(c+d x)^2}}}\right )+24 f \coth ^{-1}(c+d x)^2 \log \left (-\frac {d (e+f x)}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )+48 f \coth ^{-1}(c+d x) \tanh ^{-1}\left (\frac {f}{d e-c f}\right ) \log \left (i \sinh \left (\coth ^{-1}(c+d x)+\tanh ^{-1}\left (\frac {f}{d e-c f}\right )\right )\right )+24 f \coth ^{-1}(c+d x) \text {PolyLog}\left (2,e^{2 \coth ^{-1}(c+d x)}\right )-48 f \coth ^{-1}(c+d x) \text {PolyLog}\left (2,-e^{\coth ^{-1}(c+d x)+\tanh ^{-1}\left (\frac {f}{d e-c f}\right )}\right )-48 f \coth ^{-1}(c+d x) \text {PolyLog}\left (2,e^{\coth ^{-1}(c+d x)+\tanh ^{-1}\left (\frac {f}{d e-c f}\right )}\right )-24 f \coth ^{-1}(c+d x) \text {PolyLog}\left (2,e^{2 \left (\coth ^{-1}(c+d x)+\tanh ^{-1}\left (\frac {f}{d e-c f}\right )\right )}\right )+24 f \coth ^{-1}(c+d x) \text {PolyLog}\left (2,\frac {e^{2 \coth ^{-1}(c+d x)} (d e+f-c f)}{d e-(1+c) f}\right )-12 f \text {PolyLog}\left (3,e^{2 \coth ^{-1}(c+d x)}\right )+48 f \text {PolyLog}\left (3,-e^{\coth ^{-1}(c+d x)+\tanh ^{-1}\left (\frac {f}{d e-c f}\right )}\right )+48 f \text {PolyLog}\left (3,e^{\coth ^{-1}(c+d x)+\tanh ^{-1}\left (\frac {f}{d e-c f}\right )}\right )+12 f \text {PolyLog}\left (3,e^{2 \left (\coth ^{-1}(c+d x)+\tanh ^{-1}\left (\frac {f}{d e-c f}\right )\right )}\right )-12 f \text {PolyLog}\left (3,\frac {e^{2 \coth ^{-1}(c+d x)} (d e+f-c f)}{d e-(1+c) f}\right )\right )}{24 f^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCoth[c + d*x])^2/(e + f*x),x]

[Out]

(24*a^2*f*Log[e + f*x] + 48*a*b*f*((ArcCoth[c + d*x] - ArcTanh[c + d*x])*Log[e + f*x] + 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]]]) + (-(Pi - (2*I)*ArcTanh[c
+ d*x])^2 + 4*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])^2 - (4*I)*(Pi - (2*I)*ArcTanh[c + d*x])*Log[1 + E^(2
*ArcTanh[c + d*x])] + 8*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])*Log[1 - E^(-2*(ArcTanh[(d*e - c*f)/f] + Ar
cTanh[c + d*x]))] + 4*(I*Pi + 2*ArcTanh[c + d*x])*Log[2/Sqrt[1 - (c + d*x)^2]] - 8*(ArcTanh[(d*e - c*f)/f] + A
rcTanh[c + d*x])*Log[(2*I)*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]] - 4*PolyLog[2, -E^(2*ArcTanh[c + d
*x])] - 4*PolyLog[2, E^(-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))])/8) - b^2*(I*f*Pi^3 - 16*d*e*ArcCoth[
c + d*x]^3 + 16*f*ArcCoth[c + d*x]^3 + 16*c*f*ArcCoth[c + d*x]^3 + (16*d*e*Sqrt[(d^2*e^2 - 2*c*d*e*f + (-1 + c
^2)*f^2)/(d*e - c*f)^2]*ArcCoth[c + d*x]^3)/E^ArcTanh[f/(d*e - c*f)] - (16*c*f*Sqrt[(d^2*e^2 - 2*c*d*e*f + (-1
+ c^2)*f^2)/(d*e - c*f)^2]*ArcCoth[c + d*x]^3)/E^ArcTanh[f/(d*e - c*f)] - (24*I)*f*Pi*ArcCoth[c + d*x]*Log[2]
+ 4*f*ArcCoth[c + d*x]^2*Log[64] + (24*I)*f*Pi*ArcCoth[c + d*x]*Log[E^(-ArcCoth[c + d*x]) + E^ArcCoth[c + d*x
]] + 24*f*ArcCoth[c + d*x]^2*Log[1 - E^(2*ArcCoth[c + d*x])] - 24*f*ArcCoth[c + d*x]^2*Log[1 - E^(ArcCoth[c +
d*x] + ArcTanh[f/(d*e - c*f)])] - 24*f*ArcCoth[c + d*x]^2*Log[1 + E^(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]
)] - 24*f*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 48*f*ArcCoth[c + d*x
]*ArcTanh[f/(d*e - c*f)]*Log[(I/2)*E^(-ArcCoth[c + d*x] - ArcTanh[f/(d*e - c*f)])*(-1 + E^(2*(ArcCoth[c + d*x]
+ ArcTanh[f/(d*e - c*f)])))] - 24*f*ArcCoth[c + d*x]^2*Log[-((d*e*(-1 + E^(2*ArcCoth[c + d*x])) + (1 + c + E^
(2*ArcCoth[c + d*x]) - c*E^(2*ArcCoth[c + d*x]))*f)/E^ArcCoth[c + d*x])] + 24*f*ArcCoth[c + d*x]^2*Log[(-(d*e*
(-1 + E^(2*ArcCoth[c + d*x]))) + (-1 - E^(2*ArcCoth[c + d*x]) + c*(-1 + E^(2*ArcCoth[c + d*x])))*f)/(d*e - (1
+ c)*f)] - (24*I)*f*Pi*ArcCoth[c + d*x]*Log[1/Sqrt[1 - (c + d*x)^(-2)]] + 24*f*ArcCoth[c + d*x]^2*Log[-((d*(e
+ f*x))/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]))] + 48*f*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[I*Sinh[ArcCo
th[c + d*x] + ArcTanh[f/(d*e - c*f)]]] + 24*f*ArcCoth[c + d*x]*PolyLog[2, E^(2*ArcCoth[c + d*x])] - 48*f*ArcCo
th[c + d*x]*PolyLog[2, -E^(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])] - 48*f*ArcCoth[c + d*x]*PolyLog[2, E^(A
rcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])] - 24*f*ArcCoth[c + d*x]*PolyLog[2, E^(2*(ArcCoth[c + d*x] + ArcTanh
[f/(d*e - c*f)]))] + 24*f*ArcCoth[c + d*x]*PolyLog[2, (E^(2*ArcCoth[c + d*x])*(d*e + f - c*f))/(d*e - (1 + c)*
f)] - 12*f*PolyLog[3, E^(2*ArcCoth[c + d*x])] + 48*f*PolyLog[3, -E^(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])
] + 48*f*PolyLog[3, E^(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])] + 12*f*PolyLog[3, E^(2*(ArcCoth[c + d*x] +
ArcTanh[f/(d*e - c*f)]))] - 12*f*PolyLog[3, (E^(2*ArcCoth[c + d*x])*(d*e + f - c*f))/(d*e - (1 + c)*f)]))/(24*
f^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 5.34, size = 1766, normalized size = 8.25

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1766\)
default	\(\text {Expression too large to display}\)	\(1766\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccoth(d*x+c))^2/(f*x+e),x,method=_RETURNVERBOSE)

[Out]

1/d*(2*a*b*d*ln(c*f-d*e-f*(d*x+c))/f*arccoth(d*x+c)-1/2*I*b^2*d/f*Pi*arccoth(d*x+c)^2*csgn(I*(f*c*(1/(d*x+c-1)
*(d*x+c+1)-1)+(-1/(d*x+c-1)*(d*x+c+1)+1)*e*d+(-1/(d*x+c-1)*(d*x+c+1)-1)*f))*csgn(I*(f*c*(1/(d*x+c-1)*(d*x+c+1)
-1)+(-1/(d*x+c-1)*(d*x+c+1)+1)*e*d+(-1/(d*x+c-1)*(d*x+c+1)-1)*f)/(1/(d*x+c-1)*(d*x+c+1)-1))^2+a*b*d/f*ln(c*f-d
*e-f*(d*x+c))*ln((-f*(d*x+c)+f)/(-c*f+d*e+f))-a*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*I*b^2*d/f*Pi*arccoth(d*x+c)^2*csgn(I/(1/(d*x+c-1)*(d*x+c+1)-1))*csgn(I*(f*c*(1/(d*x+c-1)*(d*x+c+1)-1)+(-1/(d
*x+c-1)*(d*x+c+1)+1)*e*d+(-1/(d*x+c-1)*(d*x+c+1)-1)*f)/(1/(d*x+c-1)*(d*x+c+1)-1))^2+1/2*I*b^2*d/f*Pi*arccoth(d
*x+c)^2*csgn(I/(1/(d*x+c-1)*(d*x+c+1)-1))*csgn(I*(f*c*(1/(d*x+c-1)*(d*x+c+1)-1)+(-1/(d*x+c-1)*(d*x+c+1)+1)*e*d
+(-1/(d*x+c-1)*(d*x+c+1)-1)*f))*csgn(I*(f*c*(1/(d*x+c-1)*(d*x+c+1)-1)+(-1/(d*x+c-1)*(d*x+c+1)+1)*e*d+(-1/(d*x+
c-1)*(d*x+c+1)-1)*f)/(1/(d*x+c-1)*(d*x+c+1)-1))-b^2*d^2/f*e/(c*f-d*e-f)*arccoth(d*x+c)^2*ln(1-(c*f-d*e-f)/(d*x
+c-1)*(d*x+c+1)/(c*f-d*e+f))-b^2*d^2/f*e/(c*f-d*e-f)*arccoth(d*x+c)*polylog(2,(c*f-d*e-f)/(d*x+c-1)*(d*x+c+1)/
(c*f-d*e+f))+1/2*I*b^2*d/f*Pi*arccoth(d*x+c)^2*csgn(I*(f*c*(1/(d*x+c-1)*(d*x+c+1)-1)+(-1/(d*x+c-1)*(d*x+c+1)+1
)*e*d+(-1/(d*x+c-1)*(d*x+c+1)-1)*f)/(1/(d*x+c-1)*(d*x+c+1)-1))^3+b^2*d*c/(c*f-d*e-f)*arccoth(d*x+c)^2*ln(1-(c*
f-d*e-f)/(d*x+c-1)*(d*x+c+1)/(c*f-d*e+f))+b^2*d*c/(c*f-d*e-f)*arccoth(d*x+c)*polylog(2,(c*f-d*e-f)/(d*x+c-1)*(
d*x+c+1)/(c*f-d*e+f))+1/2*b^2*d^2/f*e/(c*f-d*e-f)*polylog(3,(c*f-d*e-f)/(d*x+c-1)*(d*x+c+1)/(c*f-d*e+f))+b^2*d
*ln(c*f-d*e-f*(d*x+c))/f*arccoth(d*x+c)^2-b^2*d/f*arccoth(d*x+c)^2*ln(f*c*(1/(d*x+c-1)*(d*x+c+1)-1)+(-1/(d*x+c
-1)*(d*x+c+1)+1)*e*d+(-1/(d*x+c-1)*(d*x+c+1)-1)*f)+b^2*d/f*arccoth(d*x+c)^2*ln(1/(d*x+c-1)*(d*x+c+1)-1)-b^2*d/
f*arccoth(d*x+c)^2*ln(1-1/((d*x+c-1)/(d*x+c+1))^(1/2))-2*b^2*d/f*arccoth(d*x+c)*polylog(2,1/((d*x+c-1)/(d*x+c+
1))^(1/2))-b^2*d/f*arccoth(d*x+c)^2*ln(1+1/((d*x+c-1)/(d*x+c+1))^(1/2))-2*b^2*d/f*arccoth(d*x+c)*polylog(2,-1/
((d*x+c-1)/(d*x+c+1))^(1/2))-1/2*b^2*d*c/(c*f-d*e-f)*polylog(3,(c*f-d*e-f)/(d*x+c-1)*(d*x+c+1)/(c*f-d*e+f))-b^
2*d/(c*f-d*e-f)*arccoth(d*x+c)^2*ln(1-(c*f-d*e-f)/(d*x+c-1)*(d*x+c+1)/(c*f-d*e+f))-b^2*d/(c*f-d*e-f)*arccoth(d
*x+c)*polylog(2,(c*f-d*e-f)/(d*x+c-1)*(d*x+c+1)/(c*f-d*e+f))+a*b*d/f*dilog((-f*(d*x+c)+f)/(-c*f+d*e+f))-a*b*d/
f*dilog((-f*(d*x+c)-f)/(-c*f+d*e-f))+a^2*d*ln(c*f-d*e-f*(d*x+c))/f+2*b^2*d/f*polylog(3,1/((d*x+c-1)/(d*x+c+1))
^(1/2))+2*b^2*d/f*polylog(3,-1/((d*x+c-1)/(d*x+c+1))^(1/2))+1/2*b^2*d/(c*f-d*e-f)*polylog(3,(c*f-d*e-f)/(d*x+c
-1)*(d*x+c+1)/(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))^2/(f*x+e),x, algorithm="maxima")

[Out]

a^2*log(f*x + e)/f + integrate(1/4*b^2*(log(1/(d*x + c) + 1) - log(-1/(d*x + c) + 1))^2/(f*x + e) + a*b*(log(1
/(d*x + c) + 1) - log(-1/(d*x + c) + 1))/(f*x + e), x)

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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))^2/(f*x+e),x, algorithm="fricas")

[Out]

integral((b^2*arccoth(d*x + c)^2 + 2*a*b*arccoth(d*x + c) + a^2)/(f*x + e), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{2}}{e + f x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acoth(d*x+c))**2/(f*x+e),x)

[Out]

Integral((a + b*acoth(c + d*x))**2/(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))^2/(f*x+e),x, algorithm="giac")

[Out]

integrate((b*arccoth(d*x + c) + a)^2/(f*x + e), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^2}{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))^2/(e + f*x),x)

[Out]

int((a + b*acoth(c + d*x))^2/(e + f*x), x)

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