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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 2.37, antiderivative size = 738, normalized size of antiderivative = 1.00, number of steps used = 65, number of rules used = 19, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.056, Rules used = {6116, 190, 44, 2528, 2523, 12, 481, 205, 2525, 446, 72, 2524, 2418, 260, 2416, 2394, 2393, 2391, 208} \[ -\frac {d^2 \text {PolyLog}\left (2,-\frac {\sqrt {b} \left (c \sqrt {x}+d\right )}{\sqrt {-a-1} c-\sqrt {b} d}\right )}{c^3}+\frac {d^2 \text {PolyLog}\left (2,-\frac {\sqrt {b} \left (c \sqrt {x}+d\right )}{\sqrt {1-a} c-\sqrt {b} d}\right )}{c^3}-\frac {d^2 \text {PolyLog}\left (2,\frac {\sqrt {b} \left (c \sqrt {x}+d\right )}{\sqrt {-a-1} c+\sqrt {b} d}\right )}{c^3}+\frac {d^2 \text {PolyLog}\left (2,\frac {\sqrt {b} \left (c \sqrt {x}+d\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )}{c^3}-\frac {d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {-a-1}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-1} c+\sqrt {b} d}\right )}{c^3}+\frac {d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )}{c^3}-\frac {d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {-a-1}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-1} c-\sqrt {b} d}\right )}{c^3}+\frac {d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c-\sqrt {b} d}\right )}{c^3}-\frac {d^2 \log \left (-\frac {-a-b x+1}{a+b x}\right ) \log \left (c \sqrt {x}+d\right )}{c^3}+\frac {d^2 \log \left (\frac {a+b x+1}{a+b x}\right ) \log \left (c \sqrt {x}+d\right )}{c^3}+\frac {d \sqrt {x} \log \left (-\frac {-a-b x+1}{a+b x}\right )}{c^2}-\frac {d \sqrt {x} \log \left (\frac {a+b x+1}{a+b x}\right )}{c^2}-\frac {2 \sqrt {a+1} d \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+1}}\right )}{\sqrt {b} c^2}+\frac {2 \sqrt {1-a} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} c^2}+\frac {(1-a) \log (-a-b x+1)}{2 b c}-\frac {x \log \left (-\frac {-a-b x+1}{a+b x}\right )}{2 c}+\frac {(a+1) \log (a+b x+1)}{2 b c}+\frac {x \log \left (\frac {a+b x+1}{a+b x}\right )}{2 c} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[a + b*x]/(c + d/Sqrt[x]),x]

[Out]

(-2*Sqrt[1 + a]*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[1 + a]])/(Sqrt[b]*c^2) + (2*Sqrt[1 - a]*d*ArcTanh[(Sqrt[b]*Sqr
t[x])/Sqrt[1 - a]])/(Sqrt[b]*c^2) - (d^2*Log[(c*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)
]*Log[d + c*Sqrt[x]])/c^3 + (d^2*Log[(c*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*c + Sqrt[b]*d)]*Log[d +
c*Sqrt[x]])/c^3 - (d^2*Log[(c*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x
]])/c^3 + (d^2*Log[(c*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 +
((1 - a)*Log[1 - a - b*x])/(2*b*c) + (d*Sqrt[x]*Log[-((1 - a - b*x)/(a + b*x))])/c^2 - (x*Log[-((1 - a - b*x)/
(a + b*x))])/(2*c) - (d^2*Log[d + c*Sqrt[x]]*Log[-((1 - a - b*x)/(a + b*x))])/c^3 + ((1 + a)*Log[1 + a + b*x])
/(2*b*c) - (d*Sqrt[x]*Log[(1 + a + b*x)/(a + b*x)])/c^2 + (x*Log[(1 + a + b*x)/(a + b*x)])/(2*c) + (d^2*Log[d
+ c*Sqrt[x]]*Log[(1 + a + b*x)/(a + b*x)])/c^3 - (d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-1 - a]*c -
Sqrt[b]*d))])/c^3 + (d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[1 - a]*c - Sqrt[b]*d))])/c^3 - (d^2*Pol
yLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)])/c^3 + (d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]
))/(Sqrt[1 - a]*c + Sqrt[b]*d)])/c^3

Rule 12

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

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 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[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 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 481

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> -Dist[(a*e^n)/(b*c -
a*d), Int[(e*x)^(m - n)/(a + b*x^n), x], x] + Dist[(c*e^n)/(b*c - a*d), Int[(e*x)^(m - n)/(c + d*x^n), x], x]
/; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1]

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 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

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

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Mathematica [A]  time = 0.74, size = 719, normalized size = 0.97 \[ \frac {-a c^2 \log (-a-b x+1)+c^2 \log (-a-b x+1)-b c^2 x \log \left (\frac {a+b x-1}{a+b x}\right )+a c^2 \log (a+b x+1)+c^2 \log (a+b x+1)+b c^2 x \log \left (\frac {a+b x+1}{a+b x}\right )-2 b d^2 \text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {b} d-\sqrt {-a-1} c}\right )-2 b d^2 \text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-1} c+\sqrt {b} d}\right )+2 b d^2 \text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {b} d-\sqrt {1-a} c}\right )+2 b d^2 \text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )-2 b d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {-a-1}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-1} c+\sqrt {b} d}\right )+2 b d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )-2 b d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {-a-1}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-1} c-\sqrt {b} d}\right )+2 b d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c-\sqrt {b} d}\right )-2 b d^2 \log \left (\frac {a+b x-1}{a+b x}\right ) \log \left (c \sqrt {x}+d\right )+2 b d^2 \log \left (\frac {a+b x+1}{a+b x}\right ) \log \left (c \sqrt {x}+d\right )+2 b c d \sqrt {x} \log \left (\frac {a+b x-1}{a+b x}\right )-2 b c d \sqrt {x} \log \left (\frac {a+b x+1}{a+b x}\right )-4 \sqrt {a+1} \sqrt {b} c d \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+1}}\right )+4 \sqrt {1-a} \sqrt {b} c d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{2 b c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {c x \operatorname {arcoth}\left (b x + a\right ) - d \sqrt {x} \operatorname {arcoth}\left (b x + a\right )}{c^{2} x - d^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(b*x+a)/(c+d/x^(1/2)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(b*x+a)/(c+d/x^(1/2)),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.12, size = 970, normalized size = 1.31 \[ \frac {\mathrm {arccoth}\left (b x +a \right ) x}{c}-\frac {2 \,\mathrm {arccoth}\left (b x +a \right ) d \sqrt {x}}{c^{2}}+\frac {2 \,\mathrm {arccoth}\left (b x +a \right ) d^{2} \ln \left (d +c \sqrt {x}\right )}{c^{3}}-\frac {d^{2} \ln \left (d +c \sqrt {x}\right ) \ln \left (\frac {-b \left (d +c \sqrt {x}\right )+b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}{b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )}{c^{3}}-\frac {d^{2} \ln \left (d +c \sqrt {x}\right ) \ln \left (\frac {b \left (d +c \sqrt {x}\right )-b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}{-b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )}{c^{3}}-\frac {d^{2} \dilog \left (\frac {-b \left (d +c \sqrt {x}\right )+b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}{b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )}{c^{3}}-\frac {d^{2} \dilog \left (\frac {b \left (d +c \sqrt {x}\right )-b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}{-b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )}{c^{3}}+\frac {d^{2} \ln \left (d +c \sqrt {x}\right ) \ln \left (\frac {-b \left (d +c \sqrt {x}\right )+b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}{b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )}{c^{3}}+\frac {d^{2} \ln \left (d +c \sqrt {x}\right ) \ln \left (\frac {b \left (d +c \sqrt {x}\right )-b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}{-b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )}{c^{3}}+\frac {d^{2} \dilog \left (\frac {-b \left (d +c \sqrt {x}\right )+b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}{b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )}{c^{3}}+\frac {d^{2} \dilog \left (\frac {b \left (d +c \sqrt {x}\right )-b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}{-b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )}{c^{3}}+\frac {\ln \left (b \left (d +c \sqrt {x}\right )^{2}-2 \left (d +c \sqrt {x}\right ) b d +a \,c^{2}+b \,d^{2}+c^{2}\right )}{2 b c}-\frac {2 d \arctan \left (\frac {2 b \left (d +c \sqrt {x}\right )-2 b d}{2 \sqrt {a b \,c^{2}+b \,c^{2}}}\right )}{c \sqrt {a b \,c^{2}+b \,c^{2}}}+\frac {a \ln \left (b \left (d +c \sqrt {x}\right )^{2}-2 \left (d +c \sqrt {x}\right ) b d +a \,c^{2}+b \,d^{2}+c^{2}\right )}{2 b c}-\frac {2 a d \arctan \left (\frac {2 b \left (d +c \sqrt {x}\right )-2 b d}{2 \sqrt {a b \,c^{2}+b \,c^{2}}}\right )}{c \sqrt {a b \,c^{2}+b \,c^{2}}}-\frac {a \ln \left (b \left (d +c \sqrt {x}\right )^{2}-2 \left (d +c \sqrt {x}\right ) b d +a \,c^{2}+b \,d^{2}-c^{2}\right )}{2 b c}+\frac {2 a d \arctan \left (\frac {2 b \left (d +c \sqrt {x}\right )-2 b d}{2 \sqrt {a b \,c^{2}-b \,c^{2}}}\right )}{c \sqrt {a b \,c^{2}-b \,c^{2}}}+\frac {\ln \left (b \left (d +c \sqrt {x}\right )^{2}-2 \left (d +c \sqrt {x}\right ) b d +a \,c^{2}+b \,d^{2}-c^{2}\right )}{2 b c}-\frac {2 d \arctan \left (\frac {2 b \left (d +c \sqrt {x}\right )-2 b d}{2 \sqrt {a b \,c^{2}-b \,c^{2}}}\right )}{c \sqrt {a b \,c^{2}-b \,c^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(b*x+a)/(c+d/x^(1/2)),x)

[Out]

arccoth(b*x+a)/c*x-2*arccoth(b*x+a)/c^2*d*x^(1/2)+2*arccoth(b*x+a)*d^2/c^3*ln(d+c*x^(1/2))-1/c^3*d^2*ln(d+c*x^
(1/2))*ln((-b*(d+c*x^(1/2))+b*d+(-a*b*c^2-b*c^2)^(1/2))/(b*d+(-a*b*c^2-b*c^2)^(1/2)))-1/c^3*d^2*ln(d+c*x^(1/2)
)*ln((b*(d+c*x^(1/2))-b*d+(-a*b*c^2-b*c^2)^(1/2))/(-b*d+(-a*b*c^2-b*c^2)^(1/2)))-1/c^3*d^2*dilog((-b*(d+c*x^(1
/2))+b*d+(-a*b*c^2-b*c^2)^(1/2))/(b*d+(-a*b*c^2-b*c^2)^(1/2)))-1/c^3*d^2*dilog((b*(d+c*x^(1/2))-b*d+(-a*b*c^2-
b*c^2)^(1/2))/(-b*d+(-a*b*c^2-b*c^2)^(1/2)))+1/c^3*d^2*ln(d+c*x^(1/2))*ln((-b*(d+c*x^(1/2))+b*d+(-a*b*c^2+b*c^
2)^(1/2))/(b*d+(-a*b*c^2+b*c^2)^(1/2)))+1/c^3*d^2*ln(d+c*x^(1/2))*ln((b*(d+c*x^(1/2))-b*d+(-a*b*c^2+b*c^2)^(1/
2))/(-b*d+(-a*b*c^2+b*c^2)^(1/2)))+1/c^3*d^2*dilog((-b*(d+c*x^(1/2))+b*d+(-a*b*c^2+b*c^2)^(1/2))/(b*d+(-a*b*c^
2+b*c^2)^(1/2)))+1/c^3*d^2*dilog((b*(d+c*x^(1/2))-b*d+(-a*b*c^2+b*c^2)^(1/2))/(-b*d+(-a*b*c^2+b*c^2)^(1/2)))+1
/2/b/c*ln(b*(d+c*x^(1/2))^2-2*(d+c*x^(1/2))*b*d+a*c^2+b*d^2+c^2)-2/c*d/(a*b*c^2+b*c^2)^(1/2)*arctan(1/2*(2*b*(
d+c*x^(1/2))-2*b*d)/(a*b*c^2+b*c^2)^(1/2))+1/2/b/c*a*ln(b*(d+c*x^(1/2))^2-2*(d+c*x^(1/2))*b*d+a*c^2+b*d^2+c^2)
-2/c*a*d/(a*b*c^2+b*c^2)^(1/2)*arctan(1/2*(2*b*(d+c*x^(1/2))-2*b*d)/(a*b*c^2+b*c^2)^(1/2))-1/2/b/c*a*ln(b*(d+c
*x^(1/2))^2-2*(d+c*x^(1/2))*b*d+a*c^2+b*d^2-c^2)+2/c*a*d/(a*b*c^2-b*c^2)^(1/2)*arctan(1/2*(2*b*(d+c*x^(1/2))-2
*b*d)/(a*b*c^2-b*c^2)^(1/2))+1/2/b/c*ln(b*(d+c*x^(1/2))^2-2*(d+c*x^(1/2))*b*d+a*c^2+b*d^2-c^2)-2/c*d/(a*b*c^2-
b*c^2)^(1/2)*arctan(1/2*(2*b*(d+c*x^(1/2))-2*b*d)/(a*b*c^2-b*c^2)^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (b x + a + 1\right )} \log \left (b x + a + 1\right ) - {\left (b x + a - 1\right )} \log \left (b x + a - 1\right )}{2 \, b c} - \frac {1}{2} \, \int \frac {d \log \left (b x + a + 1\right ) - d \log \left (b x + a - 1\right )}{c^{2} \sqrt {x} + c d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(b*x+a)/(c+d/x^(1/2)),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acoth(a + b*x)/(c + d/x^(1/2)),x)

[Out]

int(acoth(a + b*x)/(c + d/x^(1/2)), 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**(1/2)),x)

[Out]

Timed out

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