\(\int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx\) [39]

Optimal result

Integrand size = 14, antiderivative size = 390 \[ \int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (i a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \sqrt {d} (1+a x)}{\left (i a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (i a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {c} \sqrt {d} (1+a x)}{\left (i a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}} \]

[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)

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {6120, 211, 2520, 12, 266, 6820, 4996, 4940, 2438, 4966, 2449, 2352, 2497} \[ \int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx=-\frac {\log \left (1-\frac {1}{a x}\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\log \left (\frac {1}{a x}+1\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (-\sqrt {d}+i a \sqrt {c}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \sqrt {d} (a x+1)}{\left (\sqrt {d}+i a \sqrt {c}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (i a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}+1\right )}{4 \sqrt {c} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {c} \sqrt {d} (a x+1)}{\left (i \sqrt {c} a+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}} \]

[In]

Int[ArcCoth[a*x]/(c + d*x^2),x]

[Out]

-1/2*(ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[1 - 1/(a*x)])/(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] -
Sqrt[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
*Sqrt[c]*Sqrt[d]*(1 - a*x))/((I*a*Sqrt[c] - Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(Sqrt[c]*Sqrt[d]) + ((I/4)*Pol
yLog[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 211

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

Rule 266

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 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 2438

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 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 2520

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 4940

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 4966

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 4996

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 6120

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 6820

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

Rubi steps \begin{align*} \text {integral}& = -\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 {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\arctan \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 {\arctan \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 {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\arctan \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 {\arctan \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 {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \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 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (-1+a x)} \, dx}{2 a \sqrt {c} \sqrt {d}}+\frac {\int \frac {a \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (1+a x)} \, dx}{2 a \sqrt {c} \sqrt {d}} \\ & = -\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (-1+a x)} \, dx}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (1+a x)} \, dx}{2 \sqrt {c} \sqrt {d}} \\ & = -\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \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 {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x}+\frac {a \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{-1+a x}\right ) \, dx}{2 \sqrt {c} \sqrt {d}}+\frac {\int \left (\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x}-\frac {a \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{1+a x}\right ) \, dx}{2 \sqrt {c} \sqrt {d}} \\ & = -\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \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 {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{-1+a x} \, dx}{2 \sqrt {c} \sqrt {d}}-\frac {a \int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{1+a x} \, dx}{2 \sqrt {c} \sqrt {d}} \\ & = -\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \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 {\arctan \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 {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \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 {\arctan \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 \operatorname {PolyLog}\left (2,1+\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 \operatorname {PolyLog}\left (2,1-\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*}

Mathematica [A] (verified)

Time = 1.29 (sec) , antiderivative size = 671, normalized size of antiderivative = 1.72 \[ \int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx=\frac {a \left (-2 i \arccos \left (\frac {a^2 c-d}{a^2 c+d}\right ) \arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )+4 \coth ^{-1}(a x) \arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )-\left (\arccos \left (\frac {a^2 c-d}{a^2 c+d}\right )+2 \arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )\right ) \log \left (\frac {2 d \left (a^2 c-i \sqrt {a^2 c d}\right ) (-1+a x)}{\left (a^2 c+d\right ) \left (i \sqrt {a^2 c d}+a d x\right )}\right )-\left (\arccos \left (\frac {a^2 c-d}{a^2 c+d}\right )-2 \arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )\right ) \log \left (\frac {2 d \left (a^2 c+i \sqrt {a^2 c d}\right ) (1+a x)}{\left (a^2 c+d\right ) \left (i \sqrt {a^2 c d}+a d x\right )}\right )+\left (\arccos \left (\frac {a^2 c-d}{a^2 c+d}\right )+2 \left (\arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )+\arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{-\coth ^{-1}(a x)}}{\sqrt {a^2 c+d} \sqrt {-a^2 c+d+\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )}}\right )+\left (\arccos \left (\frac {a^2 c-d}{a^2 c+d}\right )-2 \left (\arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )+\arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{\coth ^{-1}(a x)}}{\sqrt {a^2 c+d} \sqrt {-a^2 c+d+\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )}}\right )+i \left (-\operatorname {PolyLog}\left (2,\frac {\left (a^2 c-d-2 i \sqrt {a^2 c d}\right ) \left (\sqrt {a^2 c d}+i a d x\right )}{\left (a^2 c+d\right ) \left (\sqrt {a^2 c d}-i a d x\right )}\right )+\operatorname {PolyLog}\left (2,\frac {\left (a^2 c-d+2 i \sqrt {a^2 c d}\right ) \left (\sqrt {a^2 c d}+i a d x\right )}{\left (a^2 c+d\right ) \left (\sqrt {a^2 c d}-i a d x\right )}\right )\right )\right )}{4 \sqrt {a^2 c d}} \]

[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])

Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.92

method	result	size
risch	\(-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a \sqrt {-c d}-\left (a x -1\right ) d -d}{a \sqrt {-c d}-d}\right )}{4 \sqrt {-c d}}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a \sqrt {-c d}+\left (a x -1\right ) d +d}{a \sqrt {-c d}+d}\right )}{4 \sqrt {-c d}}-\frac {\operatorname {dilog}\left (\frac {a \sqrt {-c d}-\left (a x -1\right ) d -d}{a \sqrt {-c d}-d}\right )}{4 \sqrt {-c d}}+\frac {\operatorname {dilog}\left (\frac {a \sqrt {-c d}+\left (a x -1\right ) d +d}{a \sqrt {-c d}+d}\right )}{4 \sqrt {-c d}}+\frac {\ln \left (a x +1\right ) \ln \left (\frac {a \sqrt {-c d}-\left (a x +1\right ) d +d}{a \sqrt {-c d}+d}\right )}{4 \sqrt {-c d}}-\frac {\ln \left (a x +1\right ) \ln \left (\frac {a \sqrt {-c d}+\left (a x +1\right ) d -d}{a \sqrt {-c d}-d}\right )}{4 \sqrt {-c d}}+\frac {\operatorname {dilog}\left (\frac {a \sqrt {-c d}-\left (a x +1\right ) d +d}{a \sqrt {-c d}+d}\right )}{4 \sqrt {-c d}}-\frac {\operatorname {dilog}\left (\frac {a \sqrt {-c d}+\left (a x +1\right ) d -d}{a \sqrt {-c d}-d}\right )}{4 \sqrt {-c d}}\)	\(358\)
derivativedivides	\(\frac {\frac {\sqrt {-a^{2} c d}\, \operatorname {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 c d}-\frac {\sqrt {-a^{2} c d}\, \operatorname {arccoth}\left (a x \right )^{2}}{2 c d}+\frac {\sqrt {-a^{2} c d}\, \operatorname {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 c d}+\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) a^{2} \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 ) \operatorname {arccoth}\left (a x \right )}{2 d \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) a^{2} \operatorname {arccoth}\left (a x \right )^{2}}{2 d \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}+\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) a^{2} \operatorname {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 d \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\left (a^{2} c +2 \sqrt {-a^{2} c d}-d \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 ) \operatorname {arccoth}\left (a x \right ) a^{2}}{a^{4} c^{2}+2 a^{2} c d +d^{2}}-\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \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 ) \operatorname {arccoth}\left (a x \right )}{2 c \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}+\frac {\left (a^{2} c +2 \sqrt {-a^{2} c d}-d \right ) a^{2} \operatorname {arccoth}\left (a x \right )^{2}}{a^{4} c^{2}+2 a^{2} c d +d^{2}}+\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) \operatorname {arccoth}\left (a x \right )^{2}}{2 c \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\left (a^{2} c +2 \sqrt {-a^{2} c d}-d \right ) \operatorname {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 ) a^{2}}{2 \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) \operatorname {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 c \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}}{a}\)	\(954\)
default	\(\frac {\frac {\sqrt {-a^{2} c d}\, \operatorname {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 c d}-\frac {\sqrt {-a^{2} c d}\, \operatorname {arccoth}\left (a x \right )^{2}}{2 c d}+\frac {\sqrt {-a^{2} c d}\, \operatorname {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 c d}+\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) a^{2} \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 ) \operatorname {arccoth}\left (a x \right )}{2 d \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) a^{2} \operatorname {arccoth}\left (a x \right )^{2}}{2 d \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}+\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) a^{2} \operatorname {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 d \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\left (a^{2} c +2 \sqrt {-a^{2} c d}-d \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 ) \operatorname {arccoth}\left (a x \right ) a^{2}}{a^{4} c^{2}+2 a^{2} c d +d^{2}}-\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \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 ) \operatorname {arccoth}\left (a x \right )}{2 c \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}+\frac {\left (a^{2} c +2 \sqrt {-a^{2} c d}-d \right ) a^{2} \operatorname {arccoth}\left (a x \right )^{2}}{a^{4} c^{2}+2 a^{2} c d +d^{2}}+\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) \operatorname {arccoth}\left (a x \right )^{2}}{2 c \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\left (a^{2} c +2 \sqrt {-a^{2} c d}-d \right ) \operatorname {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 ) a^{2}}{2 \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) \operatorname {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 c \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}}{a}\)	\(954\)

[In]

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

[Out]

-1/4*ln(a*x-1)/(-c*d)^(1/2)*ln((a*(-c*d)^(1/2)-(a*x-1)*d-d)/(a*(-c*d)^(1/2)-d))+1/4*ln(a*x-1)/(-c*d)^(1/2)*ln(
(a*(-c*d)^(1/2)+(a*x-1)*d+d)/(a*(-c*d)^(1/2)+d))-1/4/(-c*d)^(1/2)*dilog((a*(-c*d)^(1/2)-(a*x-1)*d-d)/(a*(-c*d)
^(1/2)-d))+1/4/(-c*d)^(1/2)*dilog((a*(-c*d)^(1/2)+(a*x-1)*d+d)/(a*(-c*d)^(1/2)+d))+1/4*ln(a*x+1)/(-c*d)^(1/2)*
ln((a*(-c*d)^(1/2)-(a*x+1)*d+d)/(a*(-c*d)^(1/2)+d))-1/4*ln(a*x+1)/(-c*d)^(1/2)*ln((a*(-c*d)^(1/2)+(a*x+1)*d-d)
/(a*(-c*d)^(1/2)-d))+1/4/(-c*d)^(1/2)*dilog((a*(-c*d)^(1/2)-(a*x+1)*d+d)/(a*(-c*d)^(1/2)+d))-1/4/(-c*d)^(1/2)*
dilog((a*(-c*d)^(1/2)+(a*x+1)*d-d)/(a*(-c*d)^(1/2)-d))

Fricas [F]

\[ \int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )}{d x^{2} + c} \,d x } \]

[In]

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

[Out]

integral(arccoth(a*x)/(d*x^2 + c), x)

Sympy [F]

\[ \int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx=\int \frac {\operatorname {acoth}{\left (a x \right )}}{c + d x^{2}}\, dx \]

[In]

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

[Out]

Integral(acoth(a*x)/(c + d*x**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.04 \[ \int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx=\frac {\operatorname {arcoth}\left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}} + \frac {{\left (\arctan \left (\frac {{\left (a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, \frac {a d x + d}{a^{2} c + d}\right ) - \arctan \left (\frac {{\left (a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, -\frac {a d x - d}{a^{2} c + d}\right )\right )} \log \left (d x^{2} + c\right ) - \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} + 2 \, a d x + d}{a^{2} c + d}\right ) + \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} - 2 \, a d x + d}{a^{2} c + d}\right ) - i \, {\rm Li}_2\left (\frac {a^{2} c + a d x - {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) - i \, {\rm Li}_2\left (\frac {a^{2} c - a d x + {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + i \, {\rm Li}_2\left (\frac {a^{2} c + a d x + {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + i \, {\rm Li}_2\left (\frac {a^{2} c - a d x - {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right )}{4 \, \sqrt {c d}} \]

[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)

Giac [F]

\[ \int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )}{d x^{2} + c} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx=\int \frac {\mathrm {acoth}\left (a\,x\right )}{d\,x^2+c} \,d x \]

[In]

int(acoth(a*x)/(c + d*x^2),x)

[Out]

int(acoth(a*x)/(c + d*x^2), x)