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

Optimal. Leaf size=738 \[ -\frac {\sqrt {d} \left (\log (a+b x-1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )}{2 c^{3/2}}-\frac {\sqrt {d} \left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )}{2 c^{3/2}}+\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (-a-b x+1)}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (-a-b x+1)}{\sqrt {-c} (1-a)+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x+1)}{(a+1) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x+1)}{\sqrt {-c} (a+1)+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x-1) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (a+b x-1) \log \left (\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (a+b x+1) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(a+1) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x+1) \log \left (-\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{(a+1) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {(-a-b x+1) \log (a+b x-1)}{2 b c}+\frac {x \left (\log (a+b x-1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right )}{2 c}+\frac {(a+b x+1) \log (a+b x+1)}{2 b c}+\frac {x \left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right )}{2 c} \]

[Out]

1/2*(-b*x-a+1)*ln(b*x+a-1)/b/c+1/2*x*(ln(b*x+a-1)-ln((b*x+a-1)/(b*x+a))-ln(b*x+a))/c+1/2*(b*x+a+1)*ln(b*x+a+1)
/b/c+1/2*x*(ln(b*x+a)-ln(b*x+a+1)+ln((b*x+a+1)/(b*x+a)))/c-1/2*arctan(x*c^(1/2)/d^(1/2))*(ln(b*x+a-1)-ln((b*x+
a-1)/(b*x+a))-ln(b*x+a))*d^(1/2)/c^(3/2)-1/2*arctan(x*c^(1/2)/d^(1/2))*(ln(b*x+a)-ln(b*x+a+1)+ln((b*x+a+1)/(b*
x+a)))*d^(1/2)/c^(3/2)+1/4*ln(b*x+a-1)*ln(-b*(-x*(-c)^(1/2)+d^(1/2))/((1-a)*(-c)^(1/2)-b*d^(1/2)))*d^(1/2)/(-c
)^(3/2)+1/4*ln(b*x+a+1)*ln(-b*(x*(-c)^(1/2)+d^(1/2))/((1+a)*(-c)^(1/2)-b*d^(1/2)))*d^(1/2)/(-c)^(3/2)-1/4*ln(b
*x+a-1)*ln(b*(x*(-c)^(1/2)+d^(1/2))/((1-a)*(-c)^(1/2)+b*d^(1/2)))*d^(1/2)/(-c)^(3/2)-1/4*ln(b*x+a+1)*ln(b*(-x*
(-c)^(1/2)+d^(1/2))/((1+a)*(-c)^(1/2)+b*d^(1/2)))*d^(1/2)/(-c)^(3/2)+1/4*polylog(2,(-b*x-a+1)*(-c)^(1/2)/((1-a
)*(-c)^(1/2)-b*d^(1/2)))*d^(1/2)/(-c)^(3/2)+1/4*polylog(2,(b*x+a+1)*(-c)^(1/2)/((1+a)*(-c)^(1/2)-b*d^(1/2)))*d
^(1/2)/(-c)^(3/2)-1/4*polylog(2,(-b*x-a+1)*(-c)^(1/2)/((1-a)*(-c)^(1/2)+b*d^(1/2)))*d^(1/2)/(-c)^(3/2)-1/4*pol
ylog(2,(b*x+a+1)*(-c)^(1/2)/((1+a)*(-c)^(1/2)+b*d^(1/2)))*d^(1/2)/(-c)^(3/2)

________________________________________________________________________________________

Rubi [A]  time = 1.53, antiderivative size = 738, normalized size of antiderivative = 1.00, number of steps used = 57, number of rules used = 11, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {6116, 2513, 2409, 2389, 2295, 2394, 2393, 2391, 193, 321, 205} \[ \frac {\sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {-c} (-a-b x+1)}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {-c} (-a-b x+1)}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x+1)}{(a+1) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x+1)}{(a+1) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \left (\log (a+b x-1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )}{2 c^{3/2}}-\frac {\sqrt {d} \left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )}{2 c^{3/2}}+\frac {\sqrt {d} \log (a+b x-1) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (a+b x-1) \log \left (\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (a+b x+1) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(a+1) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x+1) \log \left (-\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{(a+1) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {(-a-b x+1) \log (a+b x-1)}{2 b c}+\frac {x \left (\log (a+b x-1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right )}{2 c}+\frac {(a+b x+1) \log (a+b x+1)}{2 b c}+\frac {x \left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right )}{2 c} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[a + b*x]/(c + d/x^2),x]

[Out]

((1 - a - b*x)*Log[-1 + a + b*x])/(2*b*c) + (x*(Log[-1 + a + b*x] - Log[-((1 - a - b*x)/(a + b*x))] - Log[a +
b*x]))/(2*c) - (Sqrt[d]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*(Log[-1 + a + b*x] - Log[-((1 - a - b*x)/(a + b*x))] - Log
[a + b*x]))/(2*c^(3/2)) + ((1 + a + b*x)*Log[1 + a + b*x])/(2*b*c) + (x*(Log[a + b*x] - Log[1 + a + b*x] + Log
[(1 + a + b*x)/(a + b*x)]))/(2*c) - (Sqrt[d]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*(Log[a + b*x] - Log[1 + a + b*x] + Lo
g[(1 + a + b*x)/(a + b*x)]))/(2*c^(3/2)) + (Sqrt[d]*Log[-1 + a + b*x]*Log[-((b*(Sqrt[d] - Sqrt[-c]*x))/((1 - a
)*Sqrt[-c] - b*Sqrt[d]))])/(4*(-c)^(3/2)) - (Sqrt[d]*Log[1 + a + b*x]*Log[(b*(Sqrt[d] - Sqrt[-c]*x))/((1 + a)*
Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) + (Sqrt[d]*Log[1 + a + b*x]*Log[-((b*(Sqrt[d] + Sqrt[-c]*x))/((1 + a)*S
qrt[-c] - b*Sqrt[d]))])/(4*(-c)^(3/2)) - (Sqrt[d]*Log[-1 + a + b*x]*Log[(b*(Sqrt[d] + Sqrt[-c]*x))/((1 - a)*Sq
rt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 - a - b*x))/((1 - a)*Sqrt[-c] - b*Sqrt
[d])])/(4*(-c)^(3/2)) - (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 - a - b*x))/((1 - a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^
(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 + a + b*x))/((1 + a)*Sqrt[-c] - b*Sqrt[d])])/(4*(-c)^(3/2)) - (Sqrt[
d]*PolyLog[2, (Sqrt[-c]*(1 + a + b*x))/((1 + a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2))

Rule 193

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

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 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2409

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rule 2513

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*(RFx_.), x_Symbol] :> Dist[
p*r, Int[RFx*Log[a + b*x], x], x] + (Dist[q*r, Int[RFx*Log[c + d*x], x], x] - Dist[p*r*Log[a + b*x] + q*r*Log[
c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r], Int[RFx, x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] &&
RationalFunctionQ[RFx, x] && NeQ[b*c - a*d, 0] &&  !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; Integ
ersQ[m, n]]

Rule 6116

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

Rubi steps

\begin {align*} \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx &=-\left (\frac {1}{2} \int \frac {\log \left (\frac {-1+a+b x}{a+b x}\right )}{c+\frac {d}{x^2}} \, dx\right )+\frac {1}{2} \int \frac {\log \left (\frac {1+a+b x}{a+b x}\right )}{c+\frac {d}{x^2}} \, dx\\ &=-\left (\frac {1}{2} \int \frac {\log (-1+a+b x)}{c+\frac {d}{x^2}} \, dx\right )+\frac {1}{2} \int \frac {\log (1+a+b x)}{c+\frac {d}{x^2}} \, dx-\frac {1}{2} \left (-\log (-1+a+b x)+\log \left (\frac {-1+a+b x}{a+b x}\right )+\log (a+b x)\right ) \int \frac {1}{c+\frac {d}{x^2}} \, dx+\frac {1}{2} \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right ) \int \frac {1}{c+\frac {d}{x^2}} \, dx\\ &=-\left (\frac {1}{2} \int \left (\frac {\log (-1+a+b x)}{c}-\frac {d \log (-1+a+b x)}{c \left (d+c x^2\right )}\right ) \, dx\right )+\frac {1}{2} \int \left (\frac {\log (1+a+b x)}{c}-\frac {d \log (1+a+b x)}{c \left (d+c x^2\right )}\right ) \, dx-\frac {1}{2} \left (-\log (-1+a+b x)+\log \left (\frac {-1+a+b x}{a+b x}\right )+\log (a+b x)\right ) \int \frac {x^2}{d+c x^2} \, dx+\frac {1}{2} \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right ) \int \frac {x^2}{d+c x^2} \, dx\\ &=\frac {x \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}+\frac {x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c}-\frac {\int \log (-1+a+b x) \, dx}{2 c}+\frac {\int \log (1+a+b x) \, dx}{2 c}+\frac {d \int \frac {\log (-1+a+b x)}{d+c x^2} \, dx}{2 c}-\frac {d \int \frac {\log (1+a+b x)}{d+c x^2} \, dx}{2 c}-\frac {\left (d \left (\log (-1+a+b x)-\log \left (\frac {-1+a+b x}{a+b x}\right )-\log (a+b x)\right )\right ) \int \frac {1}{d+c x^2} \, dx}{2 c}+\frac {\left (d \left (-\log (a+b x)+\log (1+a+b x)-\log \left (\frac {1+a+b x}{a+b x}\right )\right )\right ) \int \frac {1}{d+c x^2} \, dx}{2 c}\\ &=\frac {x \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c^{3/2}}+\frac {x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}-\frac {\operatorname {Subst}(\int \log (x) \, dx,x,-1+a+b x)}{2 b c}+\frac {\operatorname {Subst}(\int \log (x) \, dx,x,1+a+b x)}{2 b c}+\frac {d \int \left (\frac {\log (-1+a+b x)}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x\right )}+\frac {\log (-1+a+b x)}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x\right )}\right ) \, dx}{2 c}-\frac {d \int \left (\frac {\log (1+a+b x)}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x\right )}+\frac {\log (1+a+b x)}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x\right )}\right ) \, dx}{2 c}\\ &=\frac {(1-a-b x) \log (-1+a+b x)}{2 b c}+\frac {x \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c^{3/2}}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}+\frac {x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}+\frac {\sqrt {d} \int \frac {\log (-1+a+b x)}{\sqrt {d}-\sqrt {-c} x} \, dx}{4 c}+\frac {\sqrt {d} \int \frac {\log (-1+a+b x)}{\sqrt {d}+\sqrt {-c} x} \, dx}{4 c}-\frac {\sqrt {d} \int \frac {\log (1+a+b x)}{\sqrt {d}-\sqrt {-c} x} \, dx}{4 c}-\frac {\sqrt {d} \int \frac {\log (1+a+b x)}{\sqrt {d}+\sqrt {-c} x} \, dx}{4 c}\\ &=\frac {(1-a-b x) \log (-1+a+b x)}{2 b c}+\frac {x \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c^{3/2}}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}+\frac {x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}+\frac {\sqrt {d} \log (-1+a+b x) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (1+a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (1+a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (-1+a+b x) \log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(-1+a) \sqrt {-c}+b \sqrt {d}}\right )}{-1+a+b x} \, dx}{4 (-c)^{3/2}}+\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{1+a+b x} \, dx}{4 (-c)^{3/2}}+\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{-(-1+a) \sqrt {-c}+b \sqrt {d}}\right )}{-1+a+b x} \, dx}{4 (-c)^{3/2}}-\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{-(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{1+a+b x} \, dx}{4 (-c)^{3/2}}\\ &=\frac {(1-a-b x) \log (-1+a+b x)}{2 b c}+\frac {x \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c^{3/2}}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}+\frac {x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}+\frac {\sqrt {d} \log (-1+a+b x) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (1+a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (1+a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (-1+a+b x) \log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-c} x}{-(-1+a) \sqrt {-c}+b \sqrt {d}}\right )}{x} \, dx,x,-1+a+b x\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-c} x}{(-1+a) \sqrt {-c}+b \sqrt {d}}\right )}{x} \, dx,x,-1+a+b x\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-c} x}{-(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{x} \, dx,x,1+a+b x\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-c} x}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{x} \, dx,x,1+a+b x\right )}{4 (-c)^{3/2}}\\ &=\frac {(1-a-b x) \log (-1+a+b x)}{2 b c}+\frac {x \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c^{3/2}}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}+\frac {x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}+\frac {\sqrt {d} \log (-1+a+b x) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (1+a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (1+a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (-1+a+b x) \log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (1-a-b x)}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (1-a-b x)}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (1+a+b x)}{(1+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (1+a+b x)}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}\\ \end {align*}

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Mathematica [C]  time = 36.35, size = 5552, normalized size = 7.52 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2} \operatorname {arcoth}\left (b x + a\right )}{c x^{2} + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 1.88, size = 19686, normalized size = 26.67 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [C]  time = 0.55, size = 647, normalized size = 0.88 \[ -{\left (\frac {d \arctan \left (\frac {c x}{\sqrt {c d}}\right )}{\sqrt {c d} c} - \frac {x}{c}\right )} \operatorname {arcoth}\left (b x + a\right ) + \frac {2 \, {\left (a + 1\right )} c \log \left (b x + a + 1\right ) - 2 \, {\left (a - 1\right )} c \log \left (b x + a - 1\right ) + {\left (b \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (\frac {b^{2} c x^{2} + 2 \, {\left (a + 1\right )} b c x + {\left (a^{2} + 2 \, a + 1\right )} c}{b^{2} d + {\left (a^{2} + 2 \, a + 1\right )} c}\right ) - b \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (\frac {b^{2} c x^{2} + 2 \, {\left (a - 1\right )} b c x + {\left (a^{2} - 2 \, a + 1\right )} c}{b^{2} d + {\left (a^{2} - 2 \, a + 1\right )} c}\right ) + i \, b {\rm Li}_2\left (\frac {{\left (a + 1\right )} b c x + b^{2} d - {\left (i \, b^{2} x + {\left (-i \, a - i\right )} b\right )} \sqrt {c} \sqrt {d}}{{\left (2 i \, a + 2 i\right )} b \sqrt {c} \sqrt {d} + b^{2} d - {\left (a^{2} + 2 \, a + 1\right )} c}\right ) - i \, b {\rm Li}_2\left (-\frac {{\left (a + 1\right )} b c x + b^{2} d + {\left (i \, b^{2} x + {\left (-i \, a - i\right )} b\right )} \sqrt {c} \sqrt {d}}{{\left (2 i \, a + 2 i\right )} b \sqrt {c} \sqrt {d} - b^{2} d + {\left (a^{2} + 2 \, a + 1\right )} c}\right ) - i \, b {\rm Li}_2\left (\frac {{\left (a - 1\right )} b c x + b^{2} d - {\left (i \, b^{2} x + {\left (-i \, a + i\right )} b\right )} \sqrt {c} \sqrt {d}}{{\left (2 i \, a - 2 i\right )} b \sqrt {c} \sqrt {d} + b^{2} d - {\left (a^{2} - 2 \, a + 1\right )} c}\right ) + i \, b {\rm Li}_2\left (-\frac {{\left (a - 1\right )} b c x + b^{2} d + {\left (i \, b^{2} x + {\left (-i \, a + i\right )} b\right )} \sqrt {c} \sqrt {d}}{{\left (2 i \, a - 2 i\right )} b \sqrt {c} \sqrt {d} - b^{2} d + {\left (a^{2} - 2 \, a + 1\right )} c}\right ) - {\left (b \arctan \left (\frac {{\left (b^{2} x + {\left (a + 1\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} d + {\left (a^{2} + 2 \, a + 1\right )} c}, \frac {{\left (a + 1\right )} b c x + {\left (a^{2} + 2 \, a + 1\right )} c}{b^{2} d + {\left (a^{2} + 2 \, a + 1\right )} c}\right ) - b \arctan \left (\frac {{\left (b^{2} x + {\left (a - 1\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} d + {\left (a^{2} - 2 \, a + 1\right )} c}, \frac {{\left (a - 1\right )} b c x + {\left (a^{2} - 2 \, a + 1\right )} c}{b^{2} d + {\left (a^{2} - 2 \, a + 1\right )} c}\right )\right )} \log \left (c x^{2} + d\right )\right )} \sqrt {c} \sqrt {d}}{4 \, b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acoth(a + b*x)/(c + d/x^2),x)

[Out]

int(acoth(a + b*x)/(c + d/x^2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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