3.1.82 \(\int \frac {\coth ^{-1}(d+e x)}{a+b x+c x^2} \, dx\) [82]

Optimal. Leaf size=335 \[ \frac {\coth ^{-1}(d+e x) \log \left (\frac {2 e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c (1-d)+\left (b-\sqrt {b^2-4 a c}\right ) e\right ) (1+d+e x)}\right )}{\sqrt {b^2-4 a c}}-\frac {\coth ^{-1}(d+e x) \log \left (\frac {2 e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c (1-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1+d+e x)}\right )}{\sqrt {b^2-4 a c}}-\frac {\text {PolyLog}\left (2,1+\frac {2 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (2 c-2 c d+b e-\sqrt {b^2-4 a c} e\right ) (1+d+e x)}\right )}{2 \sqrt {b^2-4 a c}}+\frac {\text {PolyLog}\left (2,1+\frac {2 \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (2 c (1-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1+d+e x)}\right )}{2 \sqrt {b^2-4 a c}} \]

[Out]

arccoth(e*x+d)*ln(2*e*(b+2*c*x-(-4*a*c+b^2)^(1/2))/(e*x+d+1)/(2*c*(1-d)+e*(b-(-4*a*c+b^2)^(1/2))))/(-4*a*c+b^2
)^(1/2)-arccoth(e*x+d)*ln(2*e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(e*x+d+1)/(2*c*(1-d)+e*(b+(-4*a*c+b^2)^(1/2))))/(-4
*a*c+b^2)^(1/2)-1/2*polylog(2,1+2*(2*c*d-2*c*(e*x+d)-e*(b-(-4*a*c+b^2)^(1/2)))/(e*x+d+1)/(2*c-2*c*d+b*e-e*(-4*
a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(1/2)+1/2*polylog(2,1+2*(2*c*d-2*c*(e*x+d)-e*(b+(-4*a*c+b^2)^(1/2)))/(e*x+d+1)/(
2*c*(1-d)+e*(b+(-4*a*c+b^2)^(1/2))))/(-4*a*c+b^2)^(1/2)

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Rubi [A]
time = 0.48, antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {632, 212, 6860, 6247, 6058, 2449, 2352, 2497} \begin {gather*} -\frac {\text {Li}_2\left (\frac {2 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (-2 d c+2 c+b e-\sqrt {b^2-4 a c} e\right ) (d+e x+1)}+1\right )}{2 \sqrt {b^2-4 a c}}+\frac {\text {Li}_2\left (\frac {2 \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (2 c (1-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (d+e x+1)}+1\right )}{2 \sqrt {b^2-4 a c}}+\frac {\coth ^{-1}(d+e x) \log \left (\frac {2 e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{(d+e x+1) \left (e \left (b-\sqrt {b^2-4 a c}\right )+2 c (1-d)\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {\coth ^{-1}(d+e x) \log \left (\frac {2 e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{(d+e x+1) \left (e \left (\sqrt {b^2-4 a c}+b\right )+2 c (1-d)\right )}\right )}{\sqrt {b^2-4 a c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(ArcCoth[d + e*x]*Log[(2*e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*(1 - d) + (b - Sqrt[b^2 - 4*a*c])*e)*(1 + d
+ e*x))])/Sqrt[b^2 - 4*a*c] - (ArcCoth[d + e*x]*Log[(2*e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*(1 - d) + (b +
Sqrt[b^2 - 4*a*c])*e)*(1 + d + e*x))])/Sqrt[b^2 - 4*a*c] - PolyLog[2, 1 + (2*(2*c*d - (b - Sqrt[b^2 - 4*a*c])
*e - 2*c*(d + e*x)))/((2*c - 2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e)*(1 + d + e*x))]/(2*Sqrt[b^2 - 4*a*c]) + PolyLo
g[2, 1 + (2*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e*x)))/((2*c*(1 - d) + (b + Sqrt[b^2 - 4*a*c])*e)*(1
+ d + e*x))]/(2*Sqrt[b^2 - 4*a*c])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

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]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\coth ^{-1}(d+e x)}{a+b x+c x^2} \, dx &=\int \left (\frac {2 c \coth ^{-1}(d+e x)}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}-\frac {2 c \coth ^{-1}(d+e x)}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right ) \, dx\\ &=\frac {(2 c) \int \frac {\coth ^{-1}(d+e x)}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {\coth ^{-1}(d+e x)}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {(2 c) \text {Subst}\left (\int \frac {\coth ^{-1}(x)}{\frac {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}{e}+\frac {2 c x}{e}} \, dx,x,d+e x\right )}{\sqrt {b^2-4 a c} e}-\frac {(2 c) \text {Subst}\left (\int \frac {\coth ^{-1}(x)}{\frac {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}{e}+\frac {2 c x}{e}} \, dx,x,d+e x\right )}{\sqrt {b^2-4 a c} e}\\ &=\frac {\coth ^{-1}(d+e x) \log \left (\frac {2 e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c-2 c d+b e-\sqrt {b^2-4 a c} e\right ) (1+d+e x)}\right )}{\sqrt {b^2-4 a c}}-\frac {\coth ^{-1}(d+e x) \log \left (\frac {2 e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c (1-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1+d+e x)}\right )}{\sqrt {b^2-4 a c}}-\frac {\text {Subst}\left (\int \frac {\log \left (\frac {2 \left (\frac {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}{e}+\frac {2 c x}{e}\right )}{\left (\frac {2 c}{e}+\frac {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}{e}\right ) (1+x)}\right )}{1-x^2} \, dx,x,d+e x\right )}{\sqrt {b^2-4 a c}}+\frac {\text {Subst}\left (\int \frac {\log \left (\frac {2 \left (\frac {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}{e}+\frac {2 c x}{e}\right )}{\left (\frac {2 c}{e}+\frac {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}{e}\right ) (1+x)}\right )}{1-x^2} \, dx,x,d+e x\right )}{\sqrt {b^2-4 a c}}\\ &=\frac {\coth ^{-1}(d+e x) \log \left (\frac {2 e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c-2 c d+b e-\sqrt {b^2-4 a c} e\right ) (1+d+e x)}\right )}{\sqrt {b^2-4 a c}}-\frac {\coth ^{-1}(d+e x) \log \left (\frac {2 e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c (1-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1+d+e x)}\right )}{\sqrt {b^2-4 a c}}-\frac {\text {Li}_2\left (1-\frac {2 e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c-2 c d+b e-\sqrt {b^2-4 a c} e\right ) (1+d+e x)}\right )}{2 \sqrt {b^2-4 a c}}+\frac {\text {Li}_2\left (1-\frac {2 e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c (1-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1+d+e x)}\right )}{2 \sqrt {b^2-4 a c}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 24.72, size = 1239, normalized size = 3.70 \begin {gather*} \frac {\left (1-(d+e x)^2\right ) \left (-2 \sqrt {b^2-4 a c} e e^{-\tanh ^{-1}\left (\frac {2 c (-1+d)-b e}{\sqrt {b^2-4 a c} e}\right )-\tanh ^{-1}\left (\frac {2 c (1+d)-b e}{\sqrt {b^2-4 a c} e}\right )} \left (b e \left (\sqrt {-\frac {c \left (c (1+d)^2+e (-b (1+d)+a e)\right )}{\left (b^2-4 a c\right ) e^2}} e^{\tanh ^{-1}\left (\frac {2 c (-1+d)-b e}{\sqrt {b^2-4 a c} e}\right )}-\sqrt {-\frac {c \left (c (-1+d)^2+e (b-b d+a e)\right )}{\left (b^2-4 a c\right ) e^2}} e^{\tanh ^{-1}\left (\frac {2 c (1+d)-b e}{\sqrt {b^2-4 a c} e}\right )}\right )-2 c \left ((-1+d) \sqrt {-\frac {c \left (c (1+d)^2+e (-b (1+d)+a e)\right )}{\left (b^2-4 a c\right ) e^2}} e^{\tanh ^{-1}\left (\frac {2 c (-1+d)-b e}{\sqrt {b^2-4 a c} e}\right )}-(1+d) \sqrt {-\frac {c \left (c (-1+d)^2+e (b-b d+a e)\right )}{\left (b^2-4 a c\right ) e^2}} e^{\tanh ^{-1}\left (\frac {2 c (1+d)-b e}{\sqrt {b^2-4 a c} e}\right )}+e^{\tanh ^{-1}\left (\frac {2 c (-1+d)-b e}{\sqrt {b^2-4 a c} e}\right )+\tanh ^{-1}\left (\frac {2 c (1+d)-b e}{\sqrt {b^2-4 a c} e}\right )}\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )^2+2 \left (4 c^2 \left (-1+d^2\right )-4 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (2 \coth ^{-1}(d+e x)+\tanh ^{-1}\left (\frac {2 c (-1+d)-b e}{\sqrt {b^2-4 a c} e}\right )-\tanh ^{-1}\left (\frac {2 c (1+d)-b e}{\sqrt {b^2-4 a c} e}\right )+\log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {2 c (-1+d)-b e}{\sqrt {b^2-4 a c} e}\right )+\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right )}\right )-\log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {2 c (1+d)-b e}{\sqrt {b^2-4 a c} e}\right )+\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right )}\right )\right )+2 \left (4 c^2 \left (-1+d^2\right )-4 b c d e+b^2 e^2\right ) \left (\tanh ^{-1}\left (\frac {2 c (-1+d)-b e}{\sqrt {b^2-4 a c} e}\right ) \left (\log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {2 c (-1+d)-b e}{\sqrt {b^2-4 a c} e}\right )+\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right )}\right )-\log \left (i \sinh \left (\tanh ^{-1}\left (\frac {2 c (-1+d)-b e}{\sqrt {b^2-4 a c} e}\right )+\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right )\right )\right )+\tanh ^{-1}\left (\frac {2 c (1+d)-b e}{\sqrt {b^2-4 a c} e}\right ) \left (-\log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {2 c (1+d)-b e}{\sqrt {b^2-4 a c} e}\right )+\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right )}\right )+\log \left (i \sinh \left (\tanh ^{-1}\left (\frac {2 c (1+d)-b e}{\sqrt {b^2-4 a c} e}\right )+\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right )\right )\right )\right )-\left (4 c^2 \left (-1+d^2\right )-4 b c d e+b^2 e^2\right ) \text {PolyLog}\left (2,e^{-2 \left (\tanh ^{-1}\left (\frac {2 c (-1+d)-b e}{\sqrt {b^2-4 a c} e}\right )+\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right )}\right )+\left (4 c^2 \left (-1+d^2\right )-4 b c d e+b^2 e^2\right ) \text {PolyLog}\left (2,e^{-2 \left (\tanh ^{-1}\left (\frac {2 c (1+d)-b e}{\sqrt {b^2-4 a c} e}\right )+\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right )}\right )\right )}{2 \sqrt {b^2-4 a c} (-2 c (-1+d)+b e) (-2 c (1+d)+b e) (d+e x)^2 \left (1-\frac {1}{(d+e x)^2}\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 - (d + e*x)^2)*(-2*Sqrt[b^2 - 4*a*c]*e*E^(-ArcTanh[(2*c*(-1 + d) - b*e)/(Sqrt[b^2 - 4*a*c]*e)] - ArcTanh[(
2*c*(1 + d) - b*e)/(Sqrt[b^2 - 4*a*c]*e)])*(b*e*(Sqrt[-((c*(c*(1 + d)^2 + e*(-(b*(1 + d)) + a*e)))/((b^2 - 4*a
*c)*e^2))]*E^ArcTanh[(2*c*(-1 + d) - b*e)/(Sqrt[b^2 - 4*a*c]*e)] - Sqrt[-((c*(c*(-1 + d)^2 + e*(b - b*d + a*e)
))/((b^2 - 4*a*c)*e^2))]*E^ArcTanh[(2*c*(1 + d) - b*e)/(Sqrt[b^2 - 4*a*c]*e)]) - 2*c*((-1 + d)*Sqrt[-((c*(c*(1
+ d)^2 + e*(-(b*(1 + d)) + a*e)))/((b^2 - 4*a*c)*e^2))]*E^ArcTanh[(2*c*(-1 + d) - b*e)/(Sqrt[b^2 - 4*a*c]*e)]
- (1 + d)*Sqrt[-((c*(c*(-1 + d)^2 + e*(b - b*d + a*e)))/((b^2 - 4*a*c)*e^2))]*E^ArcTanh[(2*c*(1 + d) - b*e)/(
Sqrt[b^2 - 4*a*c]*e)] + E^(ArcTanh[(2*c*(-1 + d) - b*e)/(Sqrt[b^2 - 4*a*c]*e)] + ArcTanh[(2*c*(1 + d) - b*e)/(
Sqrt[b^2 - 4*a*c]*e)])))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]^2 + 2*(4*c^2*(-1 + d^2) - 4*b*c*d*e + b^2*e^2)
*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]*(2*ArcCoth[d + e*x] + ArcTanh[(2*c*(-1 + d) - b*e)/(Sqrt[b^2 - 4*a*c]*
e)] - ArcTanh[(2*c*(1 + d) - b*e)/(Sqrt[b^2 - 4*a*c]*e)] + Log[1 - E^(-2*(ArcTanh[(2*c*(-1 + d) - b*e)/(Sqrt[b
^2 - 4*a*c]*e)] + ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]))] - Log[1 - E^(-2*(ArcTanh[(2*c*(1 + d) - b*e)/(Sqrt
[b^2 - 4*a*c]*e)] + ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]))]) + 2*(4*c^2*(-1 + d^2) - 4*b*c*d*e + b^2*e^2)*(A
rcTanh[(2*c*(-1 + d) - b*e)/(Sqrt[b^2 - 4*a*c]*e)]*(Log[1 - E^(-2*(ArcTanh[(2*c*(-1 + d) - b*e)/(Sqrt[b^2 - 4*
a*c]*e)] + ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]))] - Log[I*Sinh[ArcTanh[(2*c*(-1 + d) - b*e)/(Sqrt[b^2 - 4*a
*c]*e)] + ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]]]) + ArcTanh[(2*c*(1 + d) - b*e)/(Sqrt[b^2 - 4*a*c]*e)]*(-Log
[1 - E^(-2*(ArcTanh[(2*c*(1 + d) - b*e)/(Sqrt[b^2 - 4*a*c]*e)] + ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]))] + L
og[I*Sinh[ArcTanh[(2*c*(1 + d) - b*e)/(Sqrt[b^2 - 4*a*c]*e)] + ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]]])) - (4
*c^2*(-1 + d^2) - 4*b*c*d*e + b^2*e^2)*PolyLog[2, E^(-2*(ArcTanh[(2*c*(-1 + d) - b*e)/(Sqrt[b^2 - 4*a*c]*e)] +
ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]))] + (4*c^2*(-1 + d^2) - 4*b*c*d*e + b^2*e^2)*PolyLog[2, E^(-2*(ArcTan
h[(2*c*(1 + d) - b*e)/(Sqrt[b^2 - 4*a*c]*e)] + ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]))]))/(2*Sqrt[b^2 - 4*a*c
]*(-2*c*(-1 + d) + b*e)*(-2*c*(1 + d) + b*e)*(d + e*x)^2*(1 - (d + e*x)^(-2)))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2089\) vs. \(2(307)=614\).
time = 1.22, size = 2090, normalized size = 6.24

method	result	size
risch	\(-\frac {e \ln \left (e x +d -1\right ) \ln \left (\frac {-2 \left (e x +d -1\right ) c -b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}{-b e +2 c d -2 c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {e \ln \left (e x +d -1\right ) \ln \left (\frac {2 \left (e x +d -1\right ) c +b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+2 c}{b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+2 c}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \dilog \left (\frac {-2 \left (e x +d -1\right ) c -b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}{-b e +2 c d -2 c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {e \dilog \left (\frac {2 \left (e x +d -1\right ) c +b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+2 c}{b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+2 c}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {e \ln \left (e x +d +1\right ) \ln \left (\frac {-2 \left (e x +d +1\right ) c -b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+2 c}{-b e +2 c d +2 c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \ln \left (e x +d +1\right ) \ln \left (\frac {2 \left (e x +d +1\right ) c +b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}{b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {e \dilog \left (\frac {-2 \left (e x +d +1\right ) c -b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+2 c}{-b e +2 c d +2 c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \dilog \left (\frac {2 \left (e x +d +1\right ) c +b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}{b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\)	\(750\)
derivativedivides	\(\text {Expression too large to display}\)	\(2090\)
default	\(\text {Expression too large to display}\)	\(2090\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e*((e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*arccoth(e*x+d)*ln(1-(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)/(e*x+d-1)*(e*x+d
+1)/(a*e^2-b*e*d+c*d^2+(-e^2*(4*a*c-b^2))^(1/2)-c))-(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*arccoth(e*x+d)^2+1/2*
(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*polylog(2,(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)/(e*x+d-1)*(e*x+d+1)/(a*e^2-b*e*
d+c*d^2+(-e^2*(4*a*c-b^2))^(1/2)-c))-(e^2*(-4*a*c+b^2))^(1/2)*e^2/(4*a*c-b^2)*ln(1-(a*e^2-b*d*e+c*d^2+b*e-2*c*
d+c)/(e*x+d-1)*(e*x+d+1)/(a*e^2-b*e*d+c*d^2-(-e^2*(4*a*c-b^2))^(1/2)-c))*a*arccoth(e*x+d)/(a*e^2-b*e*d+c*d^2-(
e^2*(-4*a*c+b^2))^(1/2)-c)+(e^2*(-4*a*c+b^2))^(1/2)*e/(4*a*c-b^2)*ln(1-(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)/(e*x+d-
1)*(e*x+d+1)/(a*e^2-b*e*d+c*d^2-(-e^2*(4*a*c-b^2))^(1/2)-c))*b*d*arccoth(e*x+d)/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*
c+b^2))^(1/2)-c)-(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*ln(1-(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)/(e*x+d-1)*(e*x+d+1)
/(a*e^2-b*e*d+c*d^2-(-e^2*(4*a*c-b^2))^(1/2)-c))*c*d^2*arccoth(e*x+d)/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1
/2)-c)+(e^2*(-4*a*c+b^2))^(1/2)*e^2/(4*a*c-b^2)*a*arccoth(e*x+d)^2/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)
-c)-(e^2*(-4*a*c+b^2))^(1/2)*e/(4*a*c-b^2)*b*d*arccoth(e*x+d)^2/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c)
+(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*c*d^2*arccoth(e*x+d)^2/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c)-1/
2*(e^2*(-4*a*c+b^2))^(1/2)*e^2/(4*a*c-b^2)*polylog(2,(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)/(e*x+d-1)*(e*x+d+1)/(a*e^
2-b*e*d+c*d^2-(-e^2*(4*a*c-b^2))^(1/2)-c))*a/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c)+1/2*(e^2*(-4*a*c+b
^2))^(1/2)*e/(4*a*c-b^2)*polylog(2,(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)/(e*x+d-1)*(e*x+d+1)/(a*e^2-b*e*d+c*d^2-(-e^
2*(4*a*c-b^2))^(1/2)-c))*b*d/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c)-1/2*(e^2*(-4*a*c+b^2))^(1/2)/(4*a*
c-b^2)*polylog(2,(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)/(e*x+d-1)*(e*x+d+1)/(a*e^2-b*e*d+c*d^2-(-e^2*(4*a*c-b^2))^(1/
2)-c))*c*d^2/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c)-e^2/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c)
*ln(1-(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)/(e*x+d-1)*(e*x+d+1)/(a*e^2-b*e*d+c*d^2-(-e^2*(4*a*c-b^2))^(1/2)-c))*arcc
oth(e*x+d)+e^2/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c)*arccoth(e*x+d)^2+(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c
-b^2)*ln(1-(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)/(e*x+d-1)*(e*x+d+1)/(a*e^2-b*e*d+c*d^2-(-e^2*(4*a*c-b^2))^(1/2)-c))
*c*arccoth(e*x+d)/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c)-(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*c*arccot
h(e*x+d)^2/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c)-1/2*e^2/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-
c)*polylog(2,(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)/(e*x+d-1)*(e*x+d+1)/(a*e^2-b*e*d+c*d^2-(-e^2*(4*a*c-b^2))^(1/2)-c
))+1/2*(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*polylog(2,(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)/(e*x+d-1)*(e*x+d+1)/(a*e
^2-b*e*d+c*d^2-(-e^2*(4*a*c-b^2))^(1/2)-c))*c/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(e*x+d)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

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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(arccoth(e*x+d)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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