3.2.12 \(\int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx\) [112]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.76, antiderivative size = 830, 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 = {5160, 196, 46, 2608, 2603, 12, 492, 211, 214, 2605, 457, 78, 2604, 2465, 266, 2463, 2441, 2440, 2438} \begin {gather*} \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 {Li}_2\left (-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-i} c-\sqrt {b} d}\right ) d^2}{c^3}-\frac {i \text {Li}_2\left (-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {i-a} c-\sqrt {b} d}\right ) d^2}{c^3}+\frac {i \text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-i} c+\sqrt {b} d}\right ) d^2}{c^3}-\frac {i \text {Li}_2\left (\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} \text {ArcTan}\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} \end {gather*}

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 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 5160

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]

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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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) \text {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) \text {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 ) \text {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 ) \text {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 \text {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 \text {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) \text {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) \text {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 ) \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 (-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 ) \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 (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 \text {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 \text {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) \text {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) \text {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 ) \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)}{-i+a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{c^3}+\frac {\left (i d^2\right ) \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)}{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 \text {Subst}\left (\int \frac {x}{(a+b x) (-i+a+b x)} \, dx,x,x\right )}{2 c}+\frac {b \text {Subst}\left (\int \frac {x}{(a+b x) (i+a+b x)} \, dx,x,x\right )}{2 c}-\frac {(2 (1-i a) d) \text {Subst}\left (\int \frac {1}{i+a+b x^2} \, dx,x,\sqrt {x}\right )}{c^2}-\frac {(2 (1+i a) d) \text {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 ) \text {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 ) \text {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 \text {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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.50, size = 809, normalized size = 0.97 \begin {gather*} \frac {4 i \sqrt {i+a} \sqrt {b} c d \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )-4 i \sqrt {i-a} \sqrt {b} c d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\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 (d+c \sqrt {x}\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 (d+c \sqrt {x}\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 (d+c \sqrt {x}\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 (d+c \sqrt {x}\right )+c^2 \log (i-a-b x)+i a c^2 \log (i-a-b x)-2 i b c d \sqrt {x} \log \left (\frac {-i+a+b x}{a+b x}\right )+i b c^2 x \log \left (\frac {-i+a+b x}{a+b x}\right )+2 i b d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {-i+a+b x}{a+b x}\right )+c^2 \log (i+a+b x)-i a c^2 \log (i+a+b x)+2 i b c d \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )-i b c^2 x \log \left (\frac {i+a+b x}{a+b x}\right )-2 i b d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )+2 i b d^2 \text {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{-\sqrt {-i-a} c+\sqrt {b} d}\right )+2 i b d^2 \text {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-i-a} c+\sqrt {b} d}\right )-2 i b d^2 \text {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{-\sqrt {i-a} c+\sqrt {b} d}\right )-2 i b d^2 \text {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right )}{2 b c^3} \end {gather*}

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)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.07, size = 388, normalized size = 0.47

method	result	size
derivativedivides	\(\frac {\mathrm {arccot}\left (b x +a \right ) x}{c}-\frac {2 \,\mathrm {arccot}\left (b x +a \right ) d \sqrt {x}}{c^{2}}+\frac {2 \,\mathrm {arccot}\left (b x +a \right ) d^{2} \ln \left (d +c \sqrt {x}\right )}{c^{3}}+\frac {4 b \left (-\frac {c \left (\munderset {\textit {\_R} =\RootOf \left (b^{2} \textit {\_Z}^{4}-4 b^{2} d \,\textit {\_Z}^{3}+\left (2 a b \,c^{2}+6 b^{2} d^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b \,c^{2} d -4 b^{2} d^{3}\right ) \textit {\_Z} +a^{2} c^{4}+2 a b \,c^{2} d^{2}+b^{2} d^{4}+c^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{3}+5 \textit {\_R}^{2} d -7 \textit {\_R} \,d^{2}+3 d^{3}\right ) \ln \left (c \sqrt {x}-\textit {\_R} +d \right )}{\textit {\_R}^{3} b -3 \textit {\_R}^{2} b d +\textit {\_R} a \,c^{2}+3 \textit {\_R} b \,d^{2}-a \,c^{2} d -b \,d^{3}}\right )}{8 b}+\frac {c \,d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (b^{2} \textit {\_Z}^{4}-4 b^{2} d \,\textit {\_Z}^{3}+\left (2 a b \,c^{2}+6 b^{2} d^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b \,c^{2} d -4 b^{2} d^{3}\right ) \textit {\_Z} +a^{2} c^{4}+2 a b \,c^{2} d^{2}+b^{2} d^{4}+c^{4}\right )}{\sum }\frac {\ln \left (d +c \sqrt {x}\right ) \ln \left (\frac {-c \sqrt {x}+\textit {\_R1} -d}{\textit {\_R1}}\right )+\dilog \left (\frac {-c \sqrt {x}+\textit {\_R1} -d}{\textit {\_R1}}\right )}{\textit {\_R1}^{2} b -2 \textit {\_R1} b d +a \,c^{2}+d^{2} b}\right )}{4 b}\right )}{c^{2}}\)	\(388\)
default	\(\frac {\mathrm {arccot}\left (b x +a \right ) x}{c}-\frac {2 \,\mathrm {arccot}\left (b x +a \right ) d \sqrt {x}}{c^{2}}+\frac {2 \,\mathrm {arccot}\left (b x +a \right ) d^{2} \ln \left (d +c \sqrt {x}\right )}{c^{3}}+\frac {4 b \left (-\frac {c \left (\munderset {\textit {\_R} =\RootOf \left (b^{2} \textit {\_Z}^{4}-4 b^{2} d \,\textit {\_Z}^{3}+\left (2 a b \,c^{2}+6 b^{2} d^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b \,c^{2} d -4 b^{2} d^{3}\right ) \textit {\_Z} +a^{2} c^{4}+2 a b \,c^{2} d^{2}+b^{2} d^{4}+c^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{3}+5 \textit {\_R}^{2} d -7 \textit {\_R} \,d^{2}+3 d^{3}\right ) \ln \left (c \sqrt {x}-\textit {\_R} +d \right )}{\textit {\_R}^{3} b -3 \textit {\_R}^{2} b d +\textit {\_R} a \,c^{2}+3 \textit {\_R} b \,d^{2}-a \,c^{2} d -b \,d^{3}}\right )}{8 b}+\frac {c \,d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (b^{2} \textit {\_Z}^{4}-4 b^{2} d \,\textit {\_Z}^{3}+\left (2 a b \,c^{2}+6 b^{2} d^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b \,c^{2} d -4 b^{2} d^{3}\right ) \textit {\_Z} +a^{2} c^{4}+2 a b \,c^{2} d^{2}+b^{2} d^{4}+c^{4}\right )}{\sum }\frac {\ln \left (d +c \sqrt {x}\right ) \ln \left (\frac {-c \sqrt {x}+\textit {\_R1} -d}{\textit {\_R1}}\right )+\dilog \left (\frac {-c \sqrt {x}+\textit {\_R1} -d}{\textit {\_R1}}\right )}{\textit {\_R1}^{2} b -2 \textit {\_R1} b d +a \,c^{2}+d^{2} b}\right )}{4 b}\right )}{c^{2}}\)	\(388\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arccot(b*x+a)/c*x-2*arccot(b*x+a)/c^2*d*x^(1/2)+2*arccot(b*x+a)*d^2/c^3*ln(d+c*x^(1/2))+4*b/c^2*(-1/8*c/b*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=Ro
otOf(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))+1/4*c*d^2/b*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)))

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

[Out]

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

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

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {acot}\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(acot(a + b*x)/(c + d/x^(1/2)),x)

[Out]

int(acot(a + b*x)/(c + d/x^(1/2)), x)

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