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

Optimal. Leaf size=830 \[ \frac{i \log \left (\frac{c \left (\sqrt{-a-i}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-i} c+\sqrt{b} d}\right ) \log \left (\sqrt{x} c+d\right ) d^2}{c^3}-\frac{i \log \left (\frac{c \left (\sqrt{i-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{i-a} c+\sqrt{b} d}\right ) \log \left (\sqrt{x} c+d\right ) d^2}{c^3}+\frac{i \log \left (\frac{c \left (\sqrt{-a-i}+\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-i} c-\sqrt{b} d}\right ) \log \left (\sqrt{x} c+d\right ) d^2}{c^3}-\frac{i \log \left (\frac{c \left (\sqrt{i-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{i-a} c-\sqrt{b} d}\right ) \log \left (\sqrt{x} c+d\right ) d^2}{c^3}+\frac{i \log \left (\sqrt{x} c+d\right ) \log \left (-\frac{-a-b x+i}{a+b x}\right ) d^2}{c^3}-\frac{i \log \left (\sqrt{x} c+d\right ) \log \left (\frac{a+b x+i}{a+b x}\right ) d^2}{c^3}+\frac{i \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (\sqrt{x} c+d\right )}{\sqrt{-a-i} c-\sqrt{b} d}\right ) d^2}{c^3}-\frac{i \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (\sqrt{x} c+d\right )}{\sqrt{i-a} c-\sqrt{b} d}\right ) d^2}{c^3}+\frac{i \text{PolyLog}\left (2,\frac{\sqrt{b} \left (\sqrt{x} c+d\right )}{\sqrt{-a-i} c+\sqrt{b} d}\right ) d^2}{c^3}-\frac{i \text{PolyLog}\left (2,\frac{\sqrt{b} \left (\sqrt{x} c+d\right )}{\sqrt{i-a} c+\sqrt{b} d}\right ) d^2}{c^3}+\frac{2 i \sqrt{a+i} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right ) d}{\sqrt{b} c^2}-\frac{2 i \sqrt{i-a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{i-a}}\right ) d}{\sqrt{b} c^2}-\frac{i \sqrt{x} \log \left (-\frac{-a-b x+i}{a+b x}\right ) d}{c^2}+\frac{i \sqrt{x} \log \left (\frac{a+b x+i}{a+b x}\right ) d}{c^2}+\frac{(i a+1) \log (-a-b x+i)}{2 b c}+\frac{i x \log \left (-\frac{-a-b x+i}{a+b x}\right )}{2 c}+\frac{(1-i a) \log (a+b x+i)}{2 b c}-\frac{i x \log \left (\frac{a+b x+i}{a+b x}\right )}{2 c} \]

[Out]

((2*I)*Sqrt[I + a]*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[I + a]])/(Sqrt[b]*c^2) - ((2*I)*Sqrt[I - a]*d*ArcTanh[(Sqrt
[b]*Sqrt[x])/Sqrt[I - a]])/(Sqrt[b]*c^2) + (I*d^2*Log[(c*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c + S
qrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 - (I*d^2*Log[(c*(Sqrt[I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c + Sqrt[b]*d
)]*Log[d + c*Sqrt[x]])/c^3 + (I*d^2*Log[(c*(Sqrt[-I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c - Sqrt[b]*d)]*Log
[d + c*Sqrt[x]])/c^3 - (I*d^2*Log[(c*(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c - Sqrt[b]*d)]*Log[d + c*S
qrt[x]])/c^3 + ((1 + I*a)*Log[I - a - b*x])/(2*b*c) - (I*d*Sqrt[x]*Log[-((I - a - b*x)/(a + b*x))])/c^2 + ((I/
2)*x*Log[-((I - a - b*x)/(a + b*x))])/c + (I*d^2*Log[d + c*Sqrt[x]]*Log[-((I - a - b*x)/(a + b*x))])/c^3 + ((1
- I*a)*Log[I + a + b*x])/(2*b*c) + (I*d*Sqrt[x]*Log[(I + a + b*x)/(a + b*x)])/c^2 - ((I/2)*x*Log[(I + a + b*x
)/(a + b*x)])/c - (I*d^2*Log[d + c*Sqrt[x]]*Log[(I + a + b*x)/(a + b*x)])/c^3 + (I*d^2*PolyLog[2, -((Sqrt[b]*(
d + c*Sqrt[x]))/(Sqrt[-I - a]*c - Sqrt[b]*d))])/c^3 - (I*d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[I -
a]*c - Sqrt[b]*d))])/c^3 + (I*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-I - a]*c + Sqrt[b]*d)])/c^3 - (I
*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[I - a]*c + Sqrt[b]*d)])/c^3

________________________________________________________________________________________

Rubi [A]  time = 2.32187, antiderivative size = 830, normalized size of antiderivative = 1., 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 = {5052, 190, 44, 2528, 2523, 12, 481, 205, 208, 2525, 446, 72, 2524, 2418, 260, 2416, 2394, 2393, 2391} \[ \frac{i \log \left (\frac{c \left (\sqrt{-a-i}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-i} c+\sqrt{b} d}\right ) \log \left (\sqrt{x} c+d\right ) d^2}{c^3}-\frac{i \log \left (\frac{c \left (\sqrt{i-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{i-a} c+\sqrt{b} d}\right ) \log \left (\sqrt{x} c+d\right ) d^2}{c^3}+\frac{i \log \left (\frac{c \left (\sqrt{-a-i}+\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-i} c-\sqrt{b} d}\right ) \log \left (\sqrt{x} c+d\right ) d^2}{c^3}-\frac{i \log \left (\frac{c \left (\sqrt{i-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{i-a} c-\sqrt{b} d}\right ) \log \left (\sqrt{x} c+d\right ) d^2}{c^3}+\frac{i \log \left (\sqrt{x} c+d\right ) \log \left (-\frac{-a-b x+i}{a+b x}\right ) d^2}{c^3}-\frac{i \log \left (\sqrt{x} c+d\right ) \log \left (\frac{a+b x+i}{a+b x}\right ) d^2}{c^3}+\frac{i \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (\sqrt{x} c+d\right )}{\sqrt{-a-i} c-\sqrt{b} d}\right ) d^2}{c^3}-\frac{i \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (\sqrt{x} c+d\right )}{\sqrt{i-a} c-\sqrt{b} d}\right ) d^2}{c^3}+\frac{i \text{PolyLog}\left (2,\frac{\sqrt{b} \left (\sqrt{x} c+d\right )}{\sqrt{-a-i} c+\sqrt{b} d}\right ) d^2}{c^3}-\frac{i \text{PolyLog}\left (2,\frac{\sqrt{b} \left (\sqrt{x} c+d\right )}{\sqrt{i-a} c+\sqrt{b} d}\right ) d^2}{c^3}+\frac{2 i \sqrt{a+i} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right ) d}{\sqrt{b} c^2}-\frac{2 i \sqrt{i-a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{i-a}}\right ) d}{\sqrt{b} c^2}-\frac{i \sqrt{x} \log \left (-\frac{-a-b x+i}{a+b x}\right ) d}{c^2}+\frac{i \sqrt{x} \log \left (\frac{a+b x+i}{a+b x}\right ) d}{c^2}+\frac{(i a+1) \log (-a-b x+i)}{2 b c}+\frac{i x \log \left (-\frac{-a-b x+i}{a+b x}\right )}{2 c}+\frac{(1-i a) \log (a+b x+i)}{2 b c}-\frac{i x \log \left (\frac{a+b x+i}{a+b x}\right )}{2 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*I)*Sqrt[I + a]*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[I + a]])/(Sqrt[b]*c^2) - ((2*I)*Sqrt[I - a]*d*ArcTanh[(Sqrt
[b]*Sqrt[x])/Sqrt[I - a]])/(Sqrt[b]*c^2) + (I*d^2*Log[(c*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c + S
qrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 - (I*d^2*Log[(c*(Sqrt[I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c + Sqrt[b]*d
)]*Log[d + c*Sqrt[x]])/c^3 + (I*d^2*Log[(c*(Sqrt[-I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c - Sqrt[b]*d)]*Log
[d + c*Sqrt[x]])/c^3 - (I*d^2*Log[(c*(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c - Sqrt[b]*d)]*Log[d + c*S
qrt[x]])/c^3 + ((1 + I*a)*Log[I - a - b*x])/(2*b*c) - (I*d*Sqrt[x]*Log[-((I - a - b*x)/(a + b*x))])/c^2 + ((I/
2)*x*Log[-((I - a - b*x)/(a + b*x))])/c + (I*d^2*Log[d + c*Sqrt[x]]*Log[-((I - a - b*x)/(a + b*x))])/c^3 + ((1
- I*a)*Log[I + a + b*x])/(2*b*c) + (I*d*Sqrt[x]*Log[(I + a + b*x)/(a + b*x)])/c^2 - ((I/2)*x*Log[(I + a + b*x
)/(a + b*x)])/c - (I*d^2*Log[d + c*Sqrt[x]]*Log[(I + a + b*x)/(a + b*x)])/c^3 + (I*d^2*PolyLog[2, -((Sqrt[b]*(
d + c*Sqrt[x]))/(Sqrt[-I - a]*c - Sqrt[b]*d))])/c^3 - (I*d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[I -
a]*c - Sqrt[b]*d))])/c^3 + (I*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-I - a]*c + Sqrt[b]*d)])/c^3 - (I
*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[I - a]*c + Sqrt[b]*d)])/c^3

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

Mathematica [A]  time = 0.685635, size = 809, normalized size = 0.97 \[ \frac{i a \log (-a-b x+i) c^2+\log (-a-b x+i) c^2+i b x \log \left (\frac{a+b x-i}{a+b x}\right ) c^2-i a \log (a+b x+i) c^2+\log (a+b x+i) c^2-i b x \log \left (\frac{a+b x+i}{a+b x}\right ) c^2+4 i \sqrt{a+i} \sqrt{b} d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right ) c-4 i \sqrt{i-a} \sqrt{b} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{i-a}}\right ) c-2 i b d \sqrt{x} \log \left (\frac{a+b x-i}{a+b x}\right ) c+2 i b d \sqrt{x} \log \left (\frac{a+b x+i}{a+b x}\right ) c+2 i b d^2 \log \left (\frac{c \left (\sqrt{-a-i}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-i} c+\sqrt{b} d}\right ) \log \left (\sqrt{x} c+d\right )-2 i b d^2 \log \left (\frac{c \left (\sqrt{i-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{i-a} c+\sqrt{b} d}\right ) \log \left (\sqrt{x} c+d\right )+2 i b d^2 \log \left (\frac{c \left (\sqrt{-a-i}+\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-i} c-\sqrt{b} d}\right ) \log \left (\sqrt{x} c+d\right )-2 i b d^2 \log \left (\frac{c \left (\sqrt{i-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{i-a} c-\sqrt{b} d}\right ) \log \left (\sqrt{x} c+d\right )+2 i b d^2 \log \left (\sqrt{x} c+d\right ) \log \left (\frac{a+b x-i}{a+b x}\right )-2 i b d^2 \log \left (\sqrt{x} c+d\right ) \log \left (\frac{a+b x+i}{a+b x}\right )+2 i b d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (\sqrt{x} c+d\right )}{\sqrt{b} d-\sqrt{-a-i} c}\right )+2 i b d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (\sqrt{x} c+d\right )}{\sqrt{-a-i} c+\sqrt{b} d}\right )-2 i b d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (\sqrt{x} c+d\right )}{\sqrt{b} d-\sqrt{i-a} c}\right )-2 i b d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (\sqrt{x} c+d\right )}{\sqrt{i-a} c+\sqrt{b} d}\right )}{2 b c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*I)*Sqrt[I + a]*Sqrt[b]*c*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[I + a]] - (4*I)*Sqrt[I - a]*Sqrt[b]*c*d*ArcTanh[(
Sqrt[b]*Sqrt[x])/Sqrt[I - a]] + (2*I)*b*d^2*Log[(c*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c + Sqrt[b]
*d)]*Log[d + c*Sqrt[x]] - (2*I)*b*d^2*Log[(c*(Sqrt[I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c + Sqrt[b]*d)]*Log
[d + c*Sqrt[x]] + (2*I)*b*d^2*Log[(c*(Sqrt[-I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c - Sqrt[b]*d)]*Log[d + c
*Sqrt[x]] - (2*I)*b*d^2*Log[(c*(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]
] + c^2*Log[I - a - b*x] + I*a*c^2*Log[I - a - b*x] - (2*I)*b*c*d*Sqrt[x]*Log[(-I + a + b*x)/(a + b*x)] + I*b*
c^2*x*Log[(-I + a + b*x)/(a + b*x)] + (2*I)*b*d^2*Log[d + c*Sqrt[x]]*Log[(-I + a + b*x)/(a + b*x)] + c^2*Log[I
+ a + b*x] - I*a*c^2*Log[I + a + b*x] + (2*I)*b*c*d*Sqrt[x]*Log[(I + a + b*x)/(a + b*x)] - I*b*c^2*x*Log[(I +
a + b*x)/(a + b*x)] - (2*I)*b*d^2*Log[d + c*Sqrt[x]]*Log[(I + a + b*x)/(a + b*x)] + (2*I)*b*d^2*PolyLog[2, (S
qrt[b]*(d + c*Sqrt[x]))/(-(Sqrt[-I - a]*c) + Sqrt[b]*d)] + (2*I)*b*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(S
qrt[-I - a]*c + Sqrt[b]*d)] - (2*I)*b*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(-(Sqrt[I - a]*c) + Sqrt[b]*d)]
- (2*I)*b*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[I - a]*c + Sqrt[b]*d)])/(2*b*c^3)

________________________________________________________________________________________

Maple [C]  time = 0.214, size = 376, normalized size = 0.5 \begin{align*}{\frac{x{\rm arccot} \left (bx+a\right )}{c}}-2\,{\frac{{\rm arccot} \left (bx+a\right )d\sqrt{x}}{{c}^{2}}}+2\,{\frac{{\rm arccot} \left (bx+a\right ){d}^{2}\ln \left ( d+c\sqrt{x} \right ) }{{c}^{3}}}+{\frac{{d}^{2}}{c}\sum _{{\it \_R1}={\it RootOf} \left ({b}^{2}{{\it \_Z}}^{4}-4\,{b}^{2}d{{\it \_Z}}^{3}+ \left ( 2\,{c}^{2}ba+6\,{b}^{2}{d}^{2} \right ){{\it \_Z}}^{2}+ \left ( -4\,ab{c}^{2}d-4\,{b}^{2}{d}^{3} \right ){\it \_Z}+{a}^{2}{c}^{4}+2\,ab{c}^{2}{d}^{2}+{b}^{2}{d}^{4}+{c}^{4} \right ) }{\frac{1}{{{\it \_R1}}^{2}b-2\,{\it \_R1}\,bd+a{c}^{2}+b{d}^{2}} \left ( \ln \left ( d+c\sqrt{x} \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ( -c\sqrt{x}+{\it \_R1}-d \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ( -c\sqrt{x}+{\it \_R1}-d \right ) } \right ) \right ) }}+{\frac{1}{2\,c}\sum _{{\it \_R}={\it RootOf} \left ({b}^{2}{{\it \_Z}}^{4}-4\,{b}^{2}d{{\it \_Z}}^{3}+ \left ( 2\,{c}^{2}ba+6\,{b}^{2}{d}^{2} \right ){{\it \_Z}}^{2}+ \left ( -4\,ab{c}^{2}d-4\,{b}^{2}{d}^{3} \right ){\it \_Z}+{a}^{2}{c}^{4}+2\,ab{c}^{2}{d}^{2}+{b}^{2}{d}^{4}+{c}^{4} \right ) }{\frac{{{\it \_R}}^{3}-5\,{{\it \_R}}^{2}d+7\,{\it \_R}\,{d}^{2}-3\,{d}^{3}}{b{{\it \_R}}^{3}-3\,bd{{\it \_R}}^{2}+a{c}^{2}{\it \_R}+3\,b{d}^{2}{\it \_R}-a{c}^{2}d-b{d}^{3}}\ln \left ( c\sqrt{x}-{\it \_R}+d \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]

arccot(b*x+a)/c*x-2*arccot(b*x+a)/c^2*d*x^(1/2)+2*arccot(b*x+a)/c^3*d^2*ln(d+c*x^(1/2))+1/c*d^2*sum(1/(_R1^2*b
-2*_R1*b*d+a*c^2+b*d^2)*(ln(d+c*x^(1/2))*ln((-c*x^(1/2)+_R1-d)/_R1)+dilog((-c*x^(1/2)+_R1-d)/_R1)),_R1=RootOf(
b^2*_Z^4-4*b^2*d*_Z^3+(2*a*b*c^2+6*b^2*d^2)*_Z^2+(-4*a*b*c^2*d-4*b^2*d^3)*_Z+a^2*c^4+2*a*b*c^2*d^2+b^2*d^4+c^4
))+1/2/c*sum((_R^3-5*_R^2*d+7*_R*d^2-3*d^3)/(_R^3*b-3*_R^2*b*d+_R*a*c^2+3*_R*b*d^2-a*c^2*d-b*d^3)*ln(c*x^(1/2)
-_R+d),_R=RootOf(b^2*_Z^4-4*b^2*d*_Z^3+(2*a*b*c^2+6*b^2*d^2)*_Z^2+(-4*a*b*c^2*d-4*b^2*d^3)*_Z+a^2*c^4+2*a*b*c^
2*d^2+b^2*d^4+c^4))

________________________________________________________________________________________

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

________________________________________________________________________________________

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

________________________________________________________________________________________

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

________________________________________________________________________________________

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