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

Optimal. Leaf size=770 \[ -\frac{i d^2 \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{-\sqrt{b} d+\sqrt{-a-i} c}\right )}{c^3}+\frac{i d^2 \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{-\sqrt{b} d+\sqrt{-a+i} c}\right )}{c^3}-\frac{i d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{b} d+\sqrt{-a-i} c}\right )}{c^3}+\frac{i d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{b} d+\sqrt{-a+i} c}\right )}{c^3}-\frac{i d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (-\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right )}{\sqrt{b} d+\sqrt{-a-i} c}\right )}{c^3}+\frac{i d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (-\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right )}{\sqrt{b} d+\sqrt{-a+i} c}\right )}{c^3}-\frac{i d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right )}{-\sqrt{b} d+\sqrt{-a-i} c}\right )}{c^3}+\frac{i d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right )}{-\sqrt{b} d+\sqrt{-a+i} c}\right )}{c^3}+\frac{i d^2 \log (-i a-i b x+1) \log \left (c \sqrt{x}+d\right )}{c^3}-\frac{i d^2 \log (i a+i b x+1) \log \left (c \sqrt{x}+d\right )}{c^3}-\frac{i d \sqrt{x} \log (-i a-i b x+1)}{c^2}+\frac{i d \sqrt{x} \log (i a+i b x+1)}{c^2}-\frac{2 i \sqrt{a+i} d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right )}{\sqrt{b} c^2}+\frac{2 i \sqrt{-a+i} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{-a+i}}\right )}{\sqrt{b} c^2}-\frac{(i a+i b x+1) \log (i a+i b x+1)}{2 b c}-\frac{(-i a-i b x+1) \log (-i (a+b x+i))}{2 b c} \]

[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[(Sqr
t[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 +
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)]*Lo
g[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*Sqrt[x]*Log[1 - I*a - I*b*x])/c^2 + (I*d^2*Log[d + c*Sqrt[x]]*Log[1 - I*a - I*b*x])/c^3 +
(I*d*Sqrt[x]*Log[1 + I*a + I*b*x])/c^2 - ((1 + I*a + I*b*x)*Log[1 + I*a + I*b*x])/(2*b*c) - (I*d^2*Log[d + c*
Sqrt[x]]*Log[1 + I*a + I*b*x])/c^3 - ((1 - I*a - I*b*x)*Log[(-I)*(I + a + b*x)])/(2*b*c) - (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 = 0.982378, antiderivative size = 770, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 16, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.889, Rules used = {5051, 2408, 2476, 2448, 321, 205, 2454, 2389, 2295, 2462, 260, 2416, 2394, 2393, 2391, 208} \[ -\frac{i d^2 \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{-\sqrt{b} d+\sqrt{-a-i} c}\right )}{c^3}+\frac{i d^2 \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{-\sqrt{b} d+\sqrt{-a+i} c}\right )}{c^3}-\frac{i d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{b} d+\sqrt{-a-i} c}\right )}{c^3}+\frac{i d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{b} d+\sqrt{-a+i} c}\right )}{c^3}-\frac{i d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (-\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right )}{\sqrt{b} d+\sqrt{-a-i} c}\right )}{c^3}+\frac{i d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (-\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right )}{\sqrt{b} d+\sqrt{-a+i} c}\right )}{c^3}-\frac{i d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right )}{-\sqrt{b} d+\sqrt{-a-i} c}\right )}{c^3}+\frac{i d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right )}{-\sqrt{b} d+\sqrt{-a+i} c}\right )}{c^3}+\frac{i d^2 \log (-i a-i b x+1) \log \left (c \sqrt{x}+d\right )}{c^3}-\frac{i d^2 \log (i a+i b x+1) \log \left (c \sqrt{x}+d\right )}{c^3}-\frac{i d \sqrt{x} \log (-i a-i b x+1)}{c^2}+\frac{i d \sqrt{x} \log (i a+i b x+1)}{c^2}-\frac{2 i \sqrt{a+i} d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right )}{\sqrt{b} c^2}+\frac{2 i \sqrt{-a+i} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{-a+i}}\right )}{\sqrt{b} c^2}-\frac{(i a+i b x+1) \log (i a+i b x+1)}{2 b c}-\frac{(-i a-i b x+1) \log (-i (a+b x+i))}{2 b c} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[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[(Sqr
t[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 +
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)]*Lo
g[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*Sqrt[x]*Log[1 - I*a - I*b*x])/c^2 + (I*d^2*Log[d + c*Sqrt[x]]*Log[1 - I*a - I*b*x])/c^3 +
(I*d*Sqrt[x]*Log[1 + I*a + I*b*x])/c^2 - ((1 + I*a + I*b*x)*Log[1 + I*a + I*b*x])/(2*b*c) - (I*d^2*Log[d + c*
Sqrt[x]]*Log[1 + I*a + I*b*x])/c^3 - ((1 - I*a - I*b*x)*Log[(-I)*(I + a + b*x)])/(2*b*c) - (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 5051

Int[ArcTan[(a_) + (b_.)*(x_)]/((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Dist[I/2, Int[Log[1 - I*a - I*b*x]/(c +
d*x^n), x], x] - Dist[I/2, Int[Log[1 + I*a + I*b*x]/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ
[n]

Rule 2408

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> W
ith[{k = Denominator[r]}, Dist[k, Subst[Int[x^(k - 1)*(f + g*x^(k*r))^q*(a + b*Log[c*(d + e*x^k)^n])^p, x], x,
x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x] && FractionQ[r] && IGtQ[p, 0]

Rule 2476

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2462

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[f +
g*x]*(a + b*Log[c*(d + e*x^n)^p]))/g, x] - Dist[(b*e*n*p)/g, Int[(x^(n - 1)*Log[f + g*x])/(d + e*x^n), x], x]
/; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && RationalQ[n]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a+b x)}{c+\frac{d}{\sqrt{x}}} \, dx &=\frac{1}{2} i \int \frac{\log (1-i a-i b x)}{c+\frac{d}{\sqrt{x}}} \, dx-\frac{1}{2} i \int \frac{\log (1+i a+i b x)}{c+\frac{d}{\sqrt{x}}} \, dx\\ &=i \operatorname{Subst}\left (\int \frac{x \log \left (1-i a-i b x^2\right )}{c+\frac{d}{x}} \, dx,x,\sqrt{x}\right )-i \operatorname{Subst}\left (\int \frac{x \log \left (1+i a+i b x^2\right )}{c+\frac{d}{x}} \, dx,x,\sqrt{x}\right )\\ &=i \operatorname{Subst}\left (\int \left (-\frac{d \log \left (1-i a-i b x^2\right )}{c^2}+\frac{x \log \left (1-i a-i b x^2\right )}{c}+\frac{d^2 \log \left (1-i a-i 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 (1+i a+i b x^2\right )}{c^2}+\frac{x \log \left (1+i a+i b x^2\right )}{c}+\frac{d^2 \log \left (1+i a+i b x^2\right )}{c^2 (d+c x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{i \operatorname{Subst}\left (\int x \log \left (1-i a-i b x^2\right ) \, dx,x,\sqrt{x}\right )}{c}-\frac{i \operatorname{Subst}\left (\int x \log \left (1+i a+i b x^2\right ) \, dx,x,\sqrt{x}\right )}{c}-\frac{(i d) \operatorname{Subst}\left (\int \log \left (1-i a-i b x^2\right ) \, dx,x,\sqrt{x}\right )}{c^2}+\frac{(i d) \operatorname{Subst}\left (\int \log \left (1+i a+i b x^2\right ) \, dx,x,\sqrt{x}\right )}{c^2}+\frac{\left (i d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-i a-i 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 (1+i a+i b x^2\right )}{d+c x} \, dx,x,\sqrt{x}\right )}{c^2}\\ &=-\frac{i d \sqrt{x} \log (1-i a-i b x)}{c^2}+\frac{i d^2 \log \left (d+c \sqrt{x}\right ) \log (1-i a-i b x)}{c^3}+\frac{i d \sqrt{x} \log (1+i a+i b x)}{c^2}-\frac{i d^2 \log \left (d+c \sqrt{x}\right ) \log (1+i a+i b x)}{c^3}+\frac{i \operatorname{Subst}(\int \log (1-i a-i b x) \, dx,x,x)}{2 c}-\frac{i \operatorname{Subst}(\int \log (1+i a+i b x) \, dx,x,x)}{2 c}+\frac{(2 b d) \operatorname{Subst}\left (\int \frac{x^2}{1-i a-i b x^2} \, dx,x,\sqrt{x}\right )}{c^2}+\frac{(2 b d) \operatorname{Subst}\left (\int \frac{x^2}{1+i a+i b x^2} \, dx,x,\sqrt{x}\right )}{c^2}-\frac{\left (2 b d^2\right ) \operatorname{Subst}\left (\int \frac{x \log (d+c x)}{1-i a-i b x^2} \, dx,x,\sqrt{x}\right )}{c^3}-\frac{\left (2 b d^2\right ) \operatorname{Subst}\left (\int \frac{x \log (d+c x)}{1+i a+i b x^2} \, dx,x,\sqrt{x}\right )}{c^3}\\ &=-\frac{i d \sqrt{x} \log (1-i a-i b x)}{c^2}+\frac{i d^2 \log \left (d+c \sqrt{x}\right ) \log (1-i a-i b x)}{c^3}+\frac{i d \sqrt{x} \log (1+i a+i b x)}{c^2}-\frac{i d^2 \log \left (d+c \sqrt{x}\right ) \log (1+i a+i b x)}{c^3}-\frac{\operatorname{Subst}(\int \log (x) \, dx,x,1-i a-i b x)}{2 b c}-\frac{\operatorname{Subst}(\int \log (x) \, dx,x,1+i a+i b x)}{2 b c}+\frac{(2 (i-a) d) \operatorname{Subst}\left (\int \frac{1}{1+i a+i b x^2} \, dx,x,\sqrt{x}\right )}{c^2}-\frac{(2 (i+a) d) \operatorname{Subst}\left (\int \frac{1}{1-i a-i b x^2} \, dx,x,\sqrt{x}\right )}{c^2}-\frac{\left (2 b d^2\right ) \operatorname{Subst}\left (\int \left (-\frac{i \log (d+c x)}{2 \sqrt{b} \left (\sqrt{-i-a}-\sqrt{b} x\right )}+\frac{i \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 b d^2\right ) \operatorname{Subst}\left (\int \left (\frac{i \log (d+c x)}{2 \sqrt{b} \left (\sqrt{i-a}-\sqrt{b} x\right )}-\frac{i \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{i d \sqrt{x} \log (1-i a-i b x)}{c^2}+\frac{i d^2 \log \left (d+c \sqrt{x}\right ) \log (1-i a-i b x)}{c^3}+\frac{i d \sqrt{x} \log (1+i a+i b x)}{c^2}-\frac{(1+i a+i b x) \log (1+i a+i b x)}{2 b c}-\frac{i d^2 \log \left (d+c \sqrt{x}\right ) \log (1+i a+i b x)}{c^3}-\frac{(1-i a-i b x) \log (-i (i+a+b x))}{2 b c}+\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{i d \sqrt{x} \log (1-i a-i b x)}{c^2}+\frac{i d^2 \log \left (d+c \sqrt{x}\right ) \log (1-i a-i b x)}{c^3}+\frac{i d \sqrt{x} \log (1+i a+i b x)}{c^2}-\frac{(1+i a+i b x) \log (1+i a+i b x)}{2 b c}-\frac{i d^2 \log \left (d+c \sqrt{x}\right ) \log (1+i a+i b x)}{c^3}-\frac{(1-i a-i b x) \log (-i (i+a+b x))}{2 b c}+\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{i d \sqrt{x} \log (1-i a-i b x)}{c^2}+\frac{i d^2 \log \left (d+c \sqrt{x}\right ) \log (1-i a-i b x)}{c^3}+\frac{i d \sqrt{x} \log (1+i a+i b x)}{c^2}-\frac{(1+i a+i b x) \log (1+i a+i b x)}{2 b c}-\frac{i d^2 \log \left (d+c \sqrt{x}\right ) \log (1+i a+i b x)}{c^3}-\frac{(1-i a-i b x) \log (-i (i+a+b x))}{2 b c}+\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{i d \sqrt{x} \log (1-i a-i b x)}{c^2}+\frac{i d^2 \log \left (d+c \sqrt{x}\right ) \log (1-i a-i b x)}{c^3}+\frac{i d \sqrt{x} \log (1+i a+i b x)}{c^2}-\frac{(1+i a+i b x) \log (1+i a+i b x)}{2 b c}-\frac{i d^2 \log \left (d+c \sqrt{x}\right ) \log (1+i a+i b x)}{c^3}-\frac{(1-i a-i b x) \log (-i (i+a+b x))}{2 b c}-\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 = 0.718889, size = 666, normalized size = 0.86 \[ \frac{i \left (-2 d^2 \left (\text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{b} d-\sqrt{-a-i} c}\right )+\text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{b} d+\sqrt{-a-i} c}\right )+\log \left (c \sqrt{x}+d\right ) \left (\log \left (\frac{c \left (-\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right )}{\sqrt{b} d+\sqrt{-a-i} c}\right )+\log \left (\frac{c \left (\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right )}{-\sqrt{b} d+\sqrt{-a-i} c}\right )\right )\right )+2 d^2 \left (\text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{b} d-\sqrt{-a+i} c}\right )+\text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{b} d+\sqrt{-a+i} c}\right )+\log \left (c \sqrt{x}+d\right ) \left (\log \left (\frac{c \left (-\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right )}{\sqrt{b} d+\sqrt{-a+i} c}\right )+\log \left (\frac{c \left (\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right )}{-\sqrt{b} d+\sqrt{-a+i} c}\right )\right )\right )-\frac{c^2 (a+b x-i) \log (i a+i b x+1)}{b}+\frac{c^2 (a+b x+i) \log (-i (a+b x+i))}{b}-2 d^2 \log (i a+i b x+1) \log \left (c \sqrt{x}+d\right )+2 d^2 \log (-i (a+b x+i)) \log \left (c \sqrt{x}+d\right )+2 c d \sqrt{x} \log (i a+i b x+1)-2 c d \sqrt{x} \log (-i (a+b x+i))+4 c d \left (\sqrt{x}-\frac{\sqrt{a+i} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right )}{\sqrt{b}}\right )-4 c d \left (\sqrt{x}-\frac{\sqrt{-a+i} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{-a+i}}\right )}{\sqrt{b}}\right )\right )}{2 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/2)*(4*c*d*(Sqrt[x] - (Sqrt[I + a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[I + a]])/Sqrt[b]) - 4*c*d*(Sqrt[x] - (Sqrt
[I - a]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[I - a]])/Sqrt[b]) + 2*c*d*Sqrt[x]*Log[1 + I*a + I*b*x] - (c^2*(-I + a +
b*x)*Log[1 + I*a + I*b*x])/b - 2*d^2*Log[d + c*Sqrt[x]]*Log[1 + I*a + I*b*x] - 2*c*d*Sqrt[x]*Log[(-I)*(I + a
+ b*x)] + (c^2*(I + a + b*x)*Log[(-I)*(I + a + b*x)])/b + 2*d^2*Log[d + c*Sqrt[x]]*Log[(-I)*(I + a + b*x)] - 2
*d^2*((Log[(c*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c + Sqrt[b]*d)] + Log[(c*(Sqrt[-I - a] + Sqrt[b]
*Sqrt[x]))/(Sqrt[-I - a]*c - Sqrt[b]*d)])*Log[d + c*Sqrt[x]] + PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(-(Sqrt[-I
- a]*c) + Sqrt[b]*d)] + PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-I - a]*c + Sqrt[b]*d)]) + 2*d^2*((Log[(c*
(Sqrt[I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c + Sqrt[b]*d)] + Log[(c*(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[
I - a]*c - Sqrt[b]*d)])*Log[d + c*Sqrt[x]] + PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(-(Sqrt[I - a]*c) + Sqrt[b]*
d)] + PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[I - a]*c + Sqrt[b]*d)])))/c^3

________________________________________________________________________________________

Maple [C]  time = 0.318, size = 377, normalized size = 0.5 \begin{align*}{\frac{x\arctan \left ( bx+a \right ) }{c}}-2\,{\frac{\arctan \left ( bx+a \right ) d\sqrt{x}}{{c}^{2}}}+2\,{\frac{\arctan \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}ab+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}ab+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(arctan(b*x+a)/(c+d/x^(1/2)),x)

[Out]

arctan(b*x+a)/c*x-2*arctan(b*x+a)/c^2*d*x^(1/2)+2*arctan(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{\arctan \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(arctan(b*x+a)/(c+d/x^(1/2)),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

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