3.39 \(\int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx\)

Optimal. Leaf size=390 \[ -\frac {i \text {Li}_2\left (\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (i a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}+1\right )}{4 \sqrt {c} \sqrt {d}}+\frac {i \text {Li}_2\left (1-\frac {2 \sqrt {c} \sqrt {d} (a x+1)}{\left (i \sqrt {c} a+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}-\frac {\log \left (1-\frac {1}{a x}\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\log \left (\frac {1}{a x}+1\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (-\sqrt {d}+i a \sqrt {c}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \sqrt {d} (a x+1)}{\left (\sqrt {d}+i a \sqrt {c}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}} \]

[Out]

-1/2*arctan(x*d^(1/2)/c^(1/2))*ln(1-1/a/x)/c^(1/2)/d^(1/2)+1/2*arctan(x*d^(1/2)/c^(1/2))*ln(1+1/a/x)/c^(1/2)/d
^(1/2)+1/2*arctan(x*d^(1/2)/c^(1/2))*ln(-2*(-a*x+1)*c^(1/2)*d^(1/2)/(I*a*c^(1/2)-d^(1/2))/(c^(1/2)-I*x*d^(1/2)
))/c^(1/2)/d^(1/2)-1/2*arctan(x*d^(1/2)/c^(1/2))*ln(2*(a*x+1)*c^(1/2)*d^(1/2)/(I*a*c^(1/2)+d^(1/2))/(c^(1/2)-I
*x*d^(1/2)))/c^(1/2)/d^(1/2)-1/4*I*polylog(2,1+2*(-a*x+1)*c^(1/2)*d^(1/2)/(I*a*c^(1/2)-d^(1/2))/(c^(1/2)-I*x*d
^(1/2)))/c^(1/2)/d^(1/2)+1/4*I*polylog(2,1-2*(a*x+1)*c^(1/2)*d^(1/2)/(I*a*c^(1/2)+d^(1/2))/(c^(1/2)-I*x*d^(1/2
)))/c^(1/2)/d^(1/2)

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Rubi [A]  time = 0.95, antiderivative size = 390, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 13, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {5973, 205, 2470, 12, 260, 6688, 4876, 4848, 2391, 4856, 2402, 2315, 2447} \[ -\frac {i \text {PolyLog}\left (2,1+\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (-\sqrt {d}+i a \sqrt {c}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {i \text {PolyLog}\left (2,1-\frac {2 \sqrt {c} \sqrt {d} (a x+1)}{\left (\sqrt {d}+i a \sqrt {c}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}-\frac {\log \left (1-\frac {1}{a x}\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\log \left (\frac {1}{a x}+1\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (-\sqrt {d}+i a \sqrt {c}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \sqrt {d} (a x+1)}{\left (\sqrt {d}+i a \sqrt {c}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[a*x]/(c + d*x^2),x]

[Out]

-(ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[1 - 1/(a*x)])/(2*Sqrt[c]*Sqrt[d]) + (ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[1 + 1/(
a*x)])/(2*Sqrt[c]*Sqrt[d]) + (ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[(-2*Sqrt[c]*Sqrt[d]*(1 - a*x))/((I*a*Sqrt[c] - S
qrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(2*Sqrt[c]*Sqrt[d]) - (ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[(2*Sqrt[c]*Sqrt[d]*(
1 + a*x))/((I*a*Sqrt[c] + Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(2*Sqrt[c]*Sqrt[d]) - ((I/4)*PolyLog[2, 1 + (2*S
qrt[c]*Sqrt[d]*(1 - a*x))/((I*a*Sqrt[c] - Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(Sqrt[c]*Sqrt[d]) + ((I/4)*PolyL
og[2, 1 - (2*Sqrt[c]*Sqrt[d]*(1 + a*x))/((I*a*Sqrt[c] + Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(Sqrt[c]*Sqrt[d])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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 2391

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

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 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]]

Rule 2470

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[(u*x^(n - 1))/(d + e*x^n
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 4848

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I*c*x
]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rule 4856

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])*Log[2/(1 -
I*c*x)])/e, x] + (Dist[(b*c)/e, Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] - Dist[(b*c)/e, Int[Log[(2*c*(d
+ e*x))/((c*d + I*e)*(1 - I*c*x))]/(1 + c^2*x^2), x], x] + Simp[((a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d
+ I*e)*(1 - I*c*x))])/e, x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]

Rule 4876

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[Ex
pandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p,
0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])

Rule 5973

Int[ArcCoth[(c_.)*(x_)]/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[Log[1 + 1/(c*x)]/(d + e*x^2), x], x
] - Dist[1/2, Int[Log[1 - 1/(c*x)]/(d + e*x^2), x], x] /; FreeQ[{c, d, e}, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

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

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Mathematica [A]  time = 1.39, size = 671, normalized size = 1.72 \[ \frac {a \left (i \left (\text {Li}_2\left (\frac {\left (c a^2-d+2 i \sqrt {a^2 c d}\right ) \left (i a d x+\sqrt {a^2 c d}\right )}{\left (c a^2+d\right ) \left (\sqrt {a^2 c d}-i a d x\right )}\right )-\text {Li}_2\left (\frac {\left (c a^2-d-2 i \sqrt {a^2 c d}\right ) \left (i a d x+\sqrt {a^2 c d}\right )}{\left (c a^2+d\right ) \left (\sqrt {a^2 c d}-i a d x\right )}\right )\right )-2 i \cos ^{-1}\left (\frac {a^2 c-d}{a^2 c+d}\right ) \tan ^{-1}\left (\frac {a c}{x \sqrt {a^2 c d}}\right )+4 \coth ^{-1}(a x) \tan ^{-1}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )-\log \left (\frac {2 d (a x-1) \left (a^2 c-i \sqrt {a^2 c d}\right )}{\left (a^2 c+d\right ) \left (a d x+i \sqrt {a^2 c d}\right )}\right ) \left (2 \tan ^{-1}\left (\frac {a c}{x \sqrt {a^2 c d}}\right )+\cos ^{-1}\left (\frac {a^2 c-d}{a^2 c+d}\right )\right )-\log \left (\frac {2 d (a x+1) \left (a^2 c+i \sqrt {a^2 c d}\right )}{\left (a^2 c+d\right ) \left (a d x+i \sqrt {a^2 c d}\right )}\right ) \left (\cos ^{-1}\left (\frac {a^2 c-d}{a^2 c+d}\right )-2 \tan ^{-1}\left (\frac {a c}{x \sqrt {a^2 c d}}\right )\right )+\left (2 \left (\tan ^{-1}\left (\frac {a c}{x \sqrt {a^2 c d}}\right )+\tan ^{-1}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )+\cos ^{-1}\left (\frac {a^2 c-d}{a^2 c+d}\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{-\coth ^{-1}(a x)}}{\sqrt {a^2 c+d} \sqrt {\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )+a^2 (-c)+d}}\right )+\left (\cos ^{-1}\left (\frac {a^2 c-d}{a^2 c+d}\right )-2 \left (\tan ^{-1}\left (\frac {a c}{x \sqrt {a^2 c d}}\right )+\tan ^{-1}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{\coth ^{-1}(a x)}}{\sqrt {a^2 c+d} \sqrt {\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )+a^2 (-c)+d}}\right )\right )}{4 \sqrt {a^2 c d}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[a*x]/(c + d*x^2),x]

[Out]

(a*((-2*I)*ArcCos[(a^2*c - d)/(a^2*c + d)]*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + 4*ArcCoth[a*x]*ArcTan[(a*d*x)/Sqr
t[a^2*c*d]] - (ArcCos[(a^2*c - d)/(a^2*c + d)] + 2*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)])*Log[(2*d*(a^2*c - I*Sqrt[a
^2*c*d])*(-1 + a*x))/((a^2*c + d)*(I*Sqrt[a^2*c*d] + a*d*x))] - (ArcCos[(a^2*c - d)/(a^2*c + d)] - 2*ArcTan[(a
*c)/(Sqrt[a^2*c*d]*x)])*Log[(2*d*(a^2*c + I*Sqrt[a^2*c*d])*(1 + a*x))/((a^2*c + d)*(I*Sqrt[a^2*c*d] + a*d*x))]
+ (ArcCos[(a^2*c - d)/(a^2*c + d)] + 2*(ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log
[(Sqrt[2]*Sqrt[a^2*c*d])/(Sqrt[a^2*c + d]*E^ArcCoth[a*x]*Sqrt[-(a^2*c) + d + (a^2*c + d)*Cosh[2*ArcCoth[a*x]]]
)] + (ArcCos[(a^2*c - d)/(a^2*c + d)] - 2*(ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*L
og[(Sqrt[2]*Sqrt[a^2*c*d]*E^ArcCoth[a*x])/(Sqrt[a^2*c + d]*Sqrt[-(a^2*c) + d + (a^2*c + d)*Cosh[2*ArcCoth[a*x]
]])] + I*(-PolyLog[2, ((a^2*c - d - (2*I)*Sqrt[a^2*c*d])*(Sqrt[a^2*c*d] + I*a*d*x))/((a^2*c + d)*(Sqrt[a^2*c*d
] - I*a*d*x))] + PolyLog[2, ((a^2*c - d + (2*I)*Sqrt[a^2*c*d])*(Sqrt[a^2*c*d] + I*a*d*x))/((a^2*c + d)*(Sqrt[a
^2*c*d] - I*a*d*x))])))/(4*Sqrt[a^2*c*d])

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fricas [F]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (a x\right )}{d x^{2} + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccoth(a*x)/(d*x^2 + c), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (a x\right )}{d x^{2} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.68, size = 785, normalized size = 2.01 \[ -\frac {a^{3} \ln \left (1-\frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c -2 \sqrt {-a^{2} c d}-d \right )}\right ) \mathrm {arccoth}\left (a x \right ) \sqrt {-a^{2} c d}\, c}{2 d \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {a \ln \left (1-\frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c -2 \sqrt {-a^{2} c d}-d \right )}\right ) \mathrm {arccoth}\left (a x \right ) \sqrt {-a^{2} c d}}{a^{4} c^{2}+2 a^{2} c d +d^{2}}+\frac {a^{3} \mathrm {arccoth}\left (a x \right )^{2} \sqrt {-a^{2} c d}\, c}{2 d \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}+\frac {a \mathrm {arccoth}\left (a x \right )^{2} \sqrt {-a^{2} c d}}{a^{4} c^{2}+2 a^{2} c d +d^{2}}-\frac {a^{3} \polylog \left (2, \frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c -2 \sqrt {-a^{2} c d}-d \right )}\right ) \sqrt {-a^{2} c d}\, c}{4 d \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {a \polylog \left (2, \frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c -2 \sqrt {-a^{2} c d}-d \right )}\right ) \sqrt {-a^{2} c d}}{2 \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\ln \left (1-\frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c -2 \sqrt {-a^{2} c d}-d \right )}\right ) \mathrm {arccoth}\left (a x \right ) \sqrt {-a^{2} c d}\, d}{2 a c \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}+\frac {\mathrm {arccoth}\left (a x \right )^{2} \sqrt {-a^{2} c d}\, d}{2 a c \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\polylog \left (2, \frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c -2 \sqrt {-a^{2} c d}-d \right )}\right ) \sqrt {-a^{2} c d}\, d}{4 a c \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}+\frac {\sqrt {-a^{2} c d}\, \mathrm {arccoth}\left (a x \right ) \ln \left (1-\frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c +2 \sqrt {-a^{2} c d}-d \right )}\right )}{2 a c d}-\frac {\sqrt {-a^{2} c d}\, \mathrm {arccoth}\left (a x \right )^{2}}{2 a c d}+\frac {\sqrt {-a^{2} c d}\, \polylog \left (2, \frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c +2 \sqrt {-a^{2} c d}-d \right )}\right )}{4 a c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(a*x)/(d*x^2+c),x)

[Out]

-1/2*a^3/d/(a^4*c^2+2*a^2*c*d+d^2)*ln(1-(a^2*c+d)*(a*x+1)/(a*x-1)/(a^2*c-2*(-a^2*c*d)^(1/2)-d))*arccoth(a*x)*(
-a^2*c*d)^(1/2)*c-a/(a^4*c^2+2*a^2*c*d+d^2)*ln(1-(a^2*c+d)*(a*x+1)/(a*x-1)/(a^2*c-2*(-a^2*c*d)^(1/2)-d))*arcco
th(a*x)*(-a^2*c*d)^(1/2)+1/2*a^3/d/(a^4*c^2+2*a^2*c*d+d^2)*arccoth(a*x)^2*(-a^2*c*d)^(1/2)*c+a/(a^4*c^2+2*a^2*
c*d+d^2)*arccoth(a*x)^2*(-a^2*c*d)^(1/2)-1/4*a^3/d/(a^4*c^2+2*a^2*c*d+d^2)*polylog(2,(a^2*c+d)*(a*x+1)/(a*x-1)
/(a^2*c-2*(-a^2*c*d)^(1/2)-d))*(-a^2*c*d)^(1/2)*c-1/2*a/(a^4*c^2+2*a^2*c*d+d^2)*polylog(2,(a^2*c+d)*(a*x+1)/(a
*x-1)/(a^2*c-2*(-a^2*c*d)^(1/2)-d))*(-a^2*c*d)^(1/2)-1/2/a/c/(a^4*c^2+2*a^2*c*d+d^2)*ln(1-(a^2*c+d)*(a*x+1)/(a
*x-1)/(a^2*c-2*(-a^2*c*d)^(1/2)-d))*arccoth(a*x)*(-a^2*c*d)^(1/2)*d+1/2/a/c/(a^4*c^2+2*a^2*c*d+d^2)*arccoth(a*
x)^2*(-a^2*c*d)^(1/2)*d-1/4/a/c/(a^4*c^2+2*a^2*c*d+d^2)*polylog(2,(a^2*c+d)*(a*x+1)/(a*x-1)/(a^2*c-2*(-a^2*c*d
)^(1/2)-d))*(-a^2*c*d)^(1/2)*d+1/2/a*(-a^2*c*d)^(1/2)/c/d*arccoth(a*x)*ln(1-(a^2*c+d)*(a*x+1)/(a*x-1)/(a^2*c+2
*(-a^2*c*d)^(1/2)-d))-1/2/a*(-a^2*c*d)^(1/2)/c/d*arccoth(a*x)^2+1/4/a*(-a^2*c*d)^(1/2)/c/d*polylog(2,(a^2*c+d)
*(a*x+1)/(a*x-1)/(a^2*c+2*(-a^2*c*d)^(1/2)-d))

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maxima [A]  time = 0.54, size = 406, normalized size = 1.04 \[ \frac {\operatorname {arcoth}\left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}} + \frac {{\left (\arctan \left (\frac {{\left (a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, \frac {a d x + d}{a^{2} c + d}\right ) - \arctan \left (\frac {{\left (a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, -\frac {a d x - d}{a^{2} c + d}\right )\right )} \log \left (d x^{2} + c\right ) - \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} + 2 \, a d x + d}{a^{2} c + d}\right ) + \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} - 2 \, a d x + d}{a^{2} c + d}\right ) - i \, {\rm Li}_2\left (\frac {a^{2} c + a d x - {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) - i \, {\rm Li}_2\left (\frac {a^{2} c - a d x + {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + i \, {\rm Li}_2\left (\frac {a^{2} c + a d x + {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + i \, {\rm Li}_2\left (\frac {a^{2} c - a d x - {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right )}{4 \, \sqrt {c d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arccoth(a*x)*arctan(d*x/sqrt(c*d))/sqrt(c*d) + 1/4*((arctan2((a^2*x + a)*sqrt(c)*sqrt(d)/(a^2*c + d), (a*d*x +
d)/(a^2*c + d)) - arctan2((a^2*x - a)*sqrt(c)*sqrt(d)/(a^2*c + d), -(a*d*x - d)/(a^2*c + d)))*log(d*x^2 + c)
- arctan(sqrt(d)*x/sqrt(c))*log((a^2*d*x^2 + 2*a*d*x + d)/(a^2*c + d)) + arctan(sqrt(d)*x/sqrt(c))*log((a^2*d*
x^2 - 2*a*d*x + d)/(a^2*c + d)) - I*dilog((a^2*c + a*d*x - (I*a^2*x - I*a)*sqrt(c)*sqrt(d))/(a^2*c + 2*I*a*sqr
t(c)*sqrt(d) - d)) - I*dilog((a^2*c - a*d*x + (I*a^2*x + I*a)*sqrt(c)*sqrt(d))/(a^2*c + 2*I*a*sqrt(c)*sqrt(d)
- d)) + I*dilog((a^2*c + a*d*x + (I*a^2*x - I*a)*sqrt(c)*sqrt(d))/(a^2*c - 2*I*a*sqrt(c)*sqrt(d) - d)) + I*dil
og((a^2*c - a*d*x - (I*a^2*x + I*a)*sqrt(c)*sqrt(d))/(a^2*c - 2*I*a*sqrt(c)*sqrt(d) - d)))/sqrt(c*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {acoth}\left (a\,x\right )}{d\,x^2+c} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acoth(a*x)/(c + d*x^2),x)

[Out]

int(acoth(a*x)/(c + d*x^2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acoth}{\left (a x \right )}}{c + d x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acoth(a*x)/(c + d*x**2), x)

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