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

Optimal. Leaf size=693 \[ -\frac{i c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-i} d}\right )}{d^2}-\frac{i c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c+\sqrt{-a-i} d}\right )}{d^2}+\frac{i c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a+i} d}\right )}{d^2}+\frac{i c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c+\sqrt{-a+i} d}\right )}{d^2}-\frac{i c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (-\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right )}{\sqrt{b} c+\sqrt{-a-i} d}\right )}{d^2}+\frac{i c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (-\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right )}{\sqrt{b} c+\sqrt{-a+i} d}\right )}{d^2}-\frac{i c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right )}{\sqrt{b} c-\sqrt{-a-i} d}\right )}{d^2}+\frac{i c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right )}{\sqrt{b} c-\sqrt{-a+i} d}\right )}{d^2}-\frac{i c \log \left (-\frac{-a-b x+i}{a+b x}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}+\frac{i c \log \left (\frac{a+b x+i}{a+b x}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}+\frac{i \sqrt{x} \log \left (-\frac{-a-b x+i}{a+b x}\right )}{d}-\frac{i \sqrt{x} \log \left (\frac{a+b x+i}{a+b x}\right )}{d}-\frac{2 i \sqrt{a+i} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right )}{\sqrt{b} d}+\frac{2 i \sqrt{-a+i} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{-a+i}}\right )}{\sqrt{b} d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 2.06471, antiderivative size = 693, 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 = {5052, 190, 43, 2528, 2523, 12, 481, 205, 208, 2524, 2418, 260, 2416, 2394, 2393, 2391} \[ -\frac{i c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-i} d}\right )}{d^2}-\frac{i c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c+\sqrt{-a-i} d}\right )}{d^2}+\frac{i c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a+i} d}\right )}{d^2}+\frac{i c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c+\sqrt{-a+i} d}\right )}{d^2}-\frac{i c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (-\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right )}{\sqrt{b} c+\sqrt{-a-i} d}\right )}{d^2}+\frac{i c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (-\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right )}{\sqrt{b} c+\sqrt{-a+i} d}\right )}{d^2}-\frac{i c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right )}{\sqrt{b} c-\sqrt{-a-i} d}\right )}{d^2}+\frac{i c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right )}{\sqrt{b} c-\sqrt{-a+i} d}\right )}{d^2}-\frac{i c \log \left (-\frac{-a-b x+i}{a+b x}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}+\frac{i c \log \left (\frac{a+b x+i}{a+b x}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}+\frac{i \sqrt{x} \log \left (-\frac{-a-b x+i}{a+b x}\right )}{d}-\frac{i \sqrt{x} \log \left (\frac{a+b x+i}{a+b x}\right )}{d}-\frac{2 i \sqrt{a+i} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right )}{\sqrt{b} d}+\frac{2 i \sqrt{-a+i} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{-a+i}}\right )}{\sqrt{b} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5052

Int[ArcCot[(a_) + (b_.)*(x_)]/((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Dist[I/2, Int[Log[(-I + a + b*x)/(a + b*
x)]/(c + d*x^n), x], x] - Dist[I/2, Int[Log[(I + a + b*x)/(a + b*x)]/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d}
, 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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.73575, size = 618, normalized size = 0.89 \[ -\frac{i \left (c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-i} d}\right )+c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c+\sqrt{-a-i} d}\right )-c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a+i} d}\right )-c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c+\sqrt{-a+i} d}\right )+c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (-\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right )}{\sqrt{b} c+\sqrt{-a-i} d}\right )-c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (-\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right )}{\sqrt{b} c+\sqrt{-a+i} d}\right )+c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right )}{-\sqrt{b} c+\sqrt{-a-i} d}\right )-c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right )}{-\sqrt{b} c+\sqrt{-a+i} d}\right )+c \log \left (\frac{a+b x-i}{a+b x}\right ) \log \left (c+d \sqrt{x}\right )-c \log \left (\frac{a+b x+i}{a+b x}\right ) \log \left (c+d \sqrt{x}\right )-d \sqrt{x} \log \left (\frac{a+b x-i}{a+b x}\right )+d \sqrt{x} \log \left (\frac{a+b x+i}{a+b x}\right )+\frac{2 \sqrt{a+i} d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right )}{\sqrt{b}}-\frac{2 \sqrt{-a+i} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{-a+i}}\right )}{\sqrt{b}}\right )}{d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.231, size = 343, normalized size = 0.5 \begin{align*} 2\,{\frac{{\rm arccot} \left (bx+a\right )\sqrt{x}}{d}}-2\,{\frac{{\rm arccot} \left (bx+a\right )c\ln \left ( c+d\sqrt{x} \right ) }{{d}^{2}}}-c\sum _{{\it \_R1}={\it RootOf} \left ({b}^{2}{{\it \_Z}}^{4}-4\,c{b}^{2}{{\it \_Z}}^{3}+ \left ( 2\,ab{d}^{2}+6\,{b}^{2}{c}^{2} \right ){{\it \_Z}}^{2}+ \left ( -4\,abc{d}^{2}-4\,{b}^{2}{c}^{3} \right ){\it \_Z}+{a}^{2}{d}^{4}+2\,ab{c}^{2}{d}^{2}+{b}^{2}{c}^{4}+{d}^{4} \right ) }{\frac{1}{{{\it \_R1}}^{2}b-2\,{\it \_R1}\,bc+a{d}^{2}+{c}^{2}b} \left ( \ln \left ( c+d\sqrt{x} \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ( -d\sqrt{x}+{\it \_R1}-c \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ( -d\sqrt{x}+{\it \_R1}-c \right ) } \right ) \right ) }+\sum _{{\it \_R}={\it RootOf} \left ({b}^{2}{{\it \_Z}}^{4}-4\,c{b}^{2}{{\it \_Z}}^{3}+ \left ( 2\,ab{d}^{2}+6\,{b}^{2}{c}^{2} \right ){{\it \_Z}}^{2}+ \left ( -4\,abc{d}^{2}-4\,{b}^{2}{c}^{3} \right ){\it \_Z}+{a}^{2}{d}^{4}+2\,ab{c}^{2}{d}^{2}+{b}^{2}{c}^{4}+{d}^{4} \right ) }{\frac{{{\it \_R}}^{2}-2\,{\it \_R}\,c+{c}^{2}}{b{{\it \_R}}^{3}-3\,cb{{\it \_R}}^{2}+a{d}^{2}{\it \_R}+3\,{c}^{2}b{\it \_R}-ac{d}^{2}-b{c}^{3}}\ln \left ( d\sqrt{x}-{\it \_R}+c \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arccot(b*x+a)*x^(1/2)/d-2*arccot(b*x+a)*c/d^2*ln(c+d*x^(1/2))-c*sum(1/(_R1^2*b-2*_R1*b*c+a*d^2+b*c^2)*(ln(c+
d*x^(1/2))*ln((-d*x^(1/2)+_R1-c)/_R1)+dilog((-d*x^(1/2)+_R1-c)/_R1)),_R1=RootOf(b^2*_Z^4-4*c*b^2*_Z^3+(2*a*b*d
^2+6*b^2*c^2)*_Z^2+(-4*a*b*c*d^2-4*b^2*c^3)*_Z+a^2*d^4+2*a*b*c^2*d^2+b^2*c^4+d^4))+sum((_R^2-2*_R*c+c^2)/(_R^3
*b-3*_R^2*b*c+_R*a*d^2+3*_R*b*c^2-a*c*d^2-b*c^3)*ln(d*x^(1/2)-_R+c),_R=RootOf(b^2*_Z^4-4*c*b^2*_Z^3+(2*a*b*d^2
+6*b^2*c^2)*_Z^2+(-4*a*b*c*d^2-4*b^2*c^3)*_Z+a^2*d^4+2*a*b*c^2*d^2+b^2*c^4+d^4))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\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(arccot(b*x+a)/(c+d*x^(1/2)),x, algorithm="maxima")

[Out]

integrate(arccot(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{arccot}\left (b x + a\right ) - c \operatorname{arccot}\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(arccot(b*x+a)/(c+d*x^(1/2)),x, algorithm="fricas")

[Out]

integral((d*sqrt(x)*arccot(b*x + a) - c*arccot(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(acot(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{arccot}\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(arccot(b*x+a)/(c+d*x^(1/2)),x, algorithm="giac")

[Out]

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