3.1.81 \(\int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx\) [81]

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

[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 = 1.71, 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 = {6251, 196, 46, 2608, 2603, 12, 492, 211, 2605, 457, 78, 2604, 2465, 266, 2463, 2441, 2440, 2438, 214} \begin {gather*} -\frac {2 \sqrt {a+1} d \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+1}}\right )}{\sqrt {b} c^2}-\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 {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} \end {gather*}

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 46

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] && Lt
Q[m + n + 2, 0])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 196

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 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 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[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 457

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 492

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 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 2440

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 2441

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 2463

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 2465

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 2603

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 2604

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 2605

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 2608

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 6251

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\\ &=-\text {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 )+\text {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 )\\ &=-\text {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 )+\text {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 {\text {Subst}\left (\int x \log \left (\frac {-1+a+b x^2}{a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{c}+\frac {\text {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 \text {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 \text {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 \text {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 \text {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 {\text {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 {\text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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) \text {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) \text {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 \text {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 \text {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 \text {Subst}\left (\int \frac {x}{(-1+a+b x) (a+b x)} \, dx,x,x\right )}{2 c}-\frac {b \text {Subst}\left (\int \frac {x}{(-a-b x) (1+a+b x)} \, dx,x,x\right )}{2 c}-\frac {(2 (1-a) d) \text {Subst}\left (\int \frac {1}{-1+a+b x^2} \, dx,x,\sqrt {x}\right )}{c^2}-\frac {(2 a d) \text {Subst}\left (\int \frac {1}{-a-b x^2} \, dx,x,\sqrt {x}\right )}{c^2}-\frac {(2 a d) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{c^2}-\frac {(2 (1+a) d) \text {Subst}\left (\int \frac {1}{1+a+b x^2} \, dx,x,\sqrt {x}\right )}{c^2}+\frac {\left (2 b d^2\right ) \text {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 ) \text {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 \text {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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 = 0.47, size = 719, normalized size = 0.97 \begin {gather*} \frac {-4 \sqrt {1+a} \sqrt {b} c d \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )+4 \sqrt {1-a} \sqrt {b} c d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )-2 b 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 )+2 b 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 )-2 b 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 )+2 b 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^2 \log (1-a-b x)-a c^2 \log (1-a-b x)+2 b c d \sqrt {x} \log \left (\frac {-1+a+b x}{a+b x}\right )-b c^2 x \log \left (\frac {-1+a+b x}{a+b x}\right )-2 b d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {-1+a+b x}{a+b x}\right )+c^2 \log (1+a+b x)+a c^2 \log (1+a+b x)-2 b c d \sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )+b c^2 x \log \left (\frac {1+a+b x}{a+b x}\right )+2 b d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )-2 b d^2 \text {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{-\sqrt {-1-a} c+\sqrt {b} d}\right )-2 b d^2 \text {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-1-a} c+\sqrt {b} d}\right )+2 b d^2 \text {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{-\sqrt {1-a} c+\sqrt {b} d}\right )+2 b d^2 \text {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )}{2 b c^3} \end {gather*}

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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Maple [A]
time = 0.33, size = 1001, normalized size = 1.36 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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))+4*b/c^2*(1/8*c/b^2*
ln(a*c^2+b*d^2-2*b*d*(d+c*x^(1/2))+b*(d+c*x^(1/2))^2+c^2)-1/2*c/b*d/(a*b*c^2+b*c^2)^(1/2)*arctan(1/2*(-2*b*d+2
*b*(d+c*x^(1/2)))/(a*b*c^2+b*c^2)^(1/2))+1/8*c/b^2*a*ln(a*c^2+b*d^2-2*b*d*(d+c*x^(1/2))+b*(d+c*x^(1/2))^2+c^2)
-1/2*c/b*a*d/(a*b*c^2+b*c^2)^(1/2)*arctan(1/2*(-2*b*d+2*b*(d+c*x^(1/2)))/(a*b*c^2+b*c^2)^(1/2))+1/8*c/b^2*ln(a
*c^2+b*d^2-2*b*d*(d+c*x^(1/2))+b*(d+c*x^(1/2))^2-c^2)-1/2*c/b*d/(a*b*c^2-b*c^2)^(1/2)*arctan(1/2*(-2*b*d+2*b*(
d+c*x^(1/2)))/(a*b*c^2-b*c^2)^(1/2))-1/8*c/b^2*a*ln(a*c^2+b*d^2-2*b*d*(d+c*x^(1/2))+b*(d+c*x^(1/2))^2-c^2)+1/2
*c/b*a*d/(a*b*c^2-b*c^2)^(1/2)*arctan(1/2*(-2*b*d+2*b*(d+c*x^(1/2)))/(a*b*c^2-b*c^2)^(1/2))-1/4/c*d^2/b*ln(d+c
*x^(1/2))*ln((b*d-b*(d+c*x^(1/2))+(-a*b*c^2-b*c^2)^(1/2))/(b*d+(-a*b*c^2-b*c^2)^(1/2)))-1/4/c*d^2/b*ln(d+c*x^(
1/2))*ln((-b*d+b*(d+c*x^(1/2))+(-a*b*c^2-b*c^2)^(1/2))/(-b*d+(-a*b*c^2-b*c^2)^(1/2)))-1/4/c*d^2/b*dilog((b*d-b
*(d+c*x^(1/2))+(-a*b*c^2-b*c^2)^(1/2))/(b*d+(-a*b*c^2-b*c^2)^(1/2)))-1/4/c*d^2/b*dilog((-b*d+b*(d+c*x^(1/2))+(
-a*b*c^2-b*c^2)^(1/2))/(-b*d+(-a*b*c^2-b*c^2)^(1/2)))+1/4/c*d^2/b*ln(d+c*x^(1/2))*ln((b*d-b*(d+c*x^(1/2))+(-a*
b*c^2+b*c^2)^(1/2))/(b*d+(-a*b*c^2+b*c^2)^(1/2)))+1/4/c*d^2/b*ln(d+c*x^(1/2))*ln((-b*d+b*(d+c*x^(1/2))+(-a*b*c
^2+b*c^2)^(1/2))/(-b*d+(-a*b*c^2+b*c^2)^(1/2)))+1/4/c*d^2/b*dilog((b*d-b*(d+c*x^(1/2))+(-a*b*c^2+b*c^2)^(1/2))
/(b*d+(-a*b*c^2+b*c^2)^(1/2)))+1/4/c*d^2/b*dilog((-b*d+b*(d+c*x^(1/2))+(-a*b*c^2+b*c^2)^(1/2))/(-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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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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