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

Optimal. Leaf size=619 \[ \frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-1} d}\right )}{d^2}+\frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{-a-1} d+\sqrt{b} c}\right )}{d^2}-\frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right )}{d^2}-\frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{1-a} d+\sqrt{b} c}\right )}{d^2}+\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} d+\sqrt{b} c}\right )}{d^2}-\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} d+\sqrt{b} c}\right )}{d^2}+\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-1} d}\right )}{d^2}-\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right )}{d^2}+\frac{c \log \left (-\frac{-a-b x+1}{a+b x}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{c \log \left (\frac{a+b x+1}{a+b x}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{\sqrt{x} \log \left (-\frac{-a-b x+1}{a+b x}\right )}{d}+\frac{\sqrt{x} \log \left (\frac{a+b x+1}{a+b x}\right )}{d}+\frac{2 \sqrt{a+1} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right )}{\sqrt{b} d}-\frac{2 \sqrt{1-a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} d} \]

[Out]

(2*Sqrt[1 + a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[1 + a]])/(Sqrt[b]*d) - (2*Sqrt[1 - a]*ArcTanh[(Sqrt[b]*Sqrt[x])/S
qrt[1 - a]])/(Sqrt[b]*d) + (c*Log[(d*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[-1 - a]*d)]*Log[c + d
*Sqrt[x]])/d^2 - (c*Log[(d*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[1 - a]*d)]*Log[c + d*Sqrt[x]])/d
^2 + (c*Log[-((d*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c - Sqrt[-1 - a]*d))]*Log[c + d*Sqrt[x]])/d^2 - (c
*Log[-((d*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c - Sqrt[1 - a]*d))]*Log[c + d*Sqrt[x]])/d^2 - (Sqrt[x]*Lo
g[-((1 - a - b*x)/(a + b*x))])/d + (c*Log[c + d*Sqrt[x]]*Log[-((1 - a - b*x)/(a + b*x))])/d^2 + (Sqrt[x]*Log[(
1 + a + b*x)/(a + b*x)])/d - (c*Log[c + d*Sqrt[x]]*Log[(1 + a + b*x)/(a + b*x)])/d^2 + (c*PolyLog[2, (Sqrt[b]*
(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[-1 - a]*d)])/d^2 + (c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqr
t[-1 - a]*d)])/d^2 - (c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[1 - a]*d)])/d^2 - (c*PolyLog[2,
(Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqrt[1 - a]*d)])/d^2

________________________________________________________________________________________

Rubi [A]  time = 2.19896, antiderivative size = 619, normalized size of antiderivative = 1., number of steps used = 55, number of rules used = 16, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.889, Rules used = {6116, 190, 43, 2528, 2523, 12, 481, 205, 2524, 2418, 260, 2416, 2394, 2393, 2391, 208} \[ \frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-1} d}\right )}{d^2}+\frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{-a-1} d+\sqrt{b} c}\right )}{d^2}-\frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right )}{d^2}-\frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{1-a} d+\sqrt{b} c}\right )}{d^2}+\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} d+\sqrt{b} c}\right )}{d^2}-\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} d+\sqrt{b} c}\right )}{d^2}+\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-1} d}\right )}{d^2}-\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right )}{d^2}+\frac{c \log \left (-\frac{-a-b x+1}{a+b x}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{c \log \left (\frac{a+b x+1}{a+b x}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{\sqrt{x} \log \left (-\frac{-a-b x+1}{a+b x}\right )}{d}+\frac{\sqrt{x} \log \left (\frac{a+b x+1}{a+b x}\right )}{d}+\frac{2 \sqrt{a+1} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right )}{\sqrt{b} d}-\frac{2 \sqrt{1-a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 + a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[1 + a]])/(Sqrt[b]*d) - (2*Sqrt[1 - a]*ArcTanh[(Sqrt[b]*Sqrt[x])/S
qrt[1 - a]])/(Sqrt[b]*d) + (c*Log[(d*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[-1 - a]*d)]*Log[c + d
*Sqrt[x]])/d^2 - (c*Log[(d*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[1 - a]*d)]*Log[c + d*Sqrt[x]])/d
^2 + (c*Log[-((d*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c - Sqrt[-1 - a]*d))]*Log[c + d*Sqrt[x]])/d^2 - (c
*Log[-((d*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c - Sqrt[1 - a]*d))]*Log[c + d*Sqrt[x]])/d^2 - (Sqrt[x]*Lo
g[-((1 - a - b*x)/(a + b*x))])/d + (c*Log[c + d*Sqrt[x]]*Log[-((1 - a - b*x)/(a + b*x))])/d^2 + (Sqrt[x]*Log[(
1 + a + b*x)/(a + b*x)])/d - (c*Log[c + d*Sqrt[x]]*Log[(1 + a + b*x)/(a + b*x)])/d^2 + (c*PolyLog[2, (Sqrt[b]*
(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[-1 - a]*d)])/d^2 + (c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqr
t[-1 - a]*d)])/d^2 - (c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[1 - a]*d)])/d^2 - (c*PolyLog[2,
(Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqrt[1 - a]*d)])/d^2

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]

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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 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 12

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

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 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 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 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 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 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 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 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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.666141, size = 575, normalized size = 0.93 \[ \frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-1} d}\right )+c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{-a-1} d+\sqrt{b} c}\right )-c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right )-c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{1-a} d+\sqrt{b} c}\right )+c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} d+\sqrt{b} c}\right )-c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} d+\sqrt{b} c}\right )+c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} d-\sqrt{b} c}\right )-c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} d-\sqrt{b} c}\right )+c \log \left (\frac{a+b x-1}{a+b x}\right ) \log \left (c+d \sqrt{x}\right )-c \log \left (\frac{a+b x+1}{a+b x}\right ) \log \left (c+d \sqrt{x}\right )-d \sqrt{x} \log \left (\frac{a+b x-1}{a+b x}\right )+d \sqrt{x} \log \left (\frac{a+b x+1}{a+b x}\right )+\frac{2 \sqrt{a+1} d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right )}{\sqrt{b}}-\frac{2 \sqrt{1-a} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b}}}{d^2} \]

Antiderivative was successfully verified.

[In]

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

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Maple [A]  time = 0.191, size = 738, normalized size = 1.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (b x + a\right )}{d \sqrt{x} + c}\,{d x} \end{align*}

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]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{d \sqrt{x} \operatorname{arcoth}\left (b x + a\right ) - c \operatorname{arcoth}\left (b x + a\right )}{d^{2} x - c^{2}}, x\right ) \end{align*}

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (b x + a\right )}{d \sqrt{x} + c}\,{d x} \end{align*}

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)/(d*sqrt(x) + c), x)