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

Optimal. Leaf size=673 \[ \frac{i c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-i} d}\right )}{d^2}+\frac{i c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c+\sqrt{-a-i} d}\right )}{d^2}-\frac{i c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a+i} d}\right )}{d^2}-\frac{i c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c+\sqrt{-a+i} d}\right )}{d^2}+\frac{i c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (-\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right )}{\sqrt{b} c+\sqrt{-a-i} d}\right )}{d^2}-\frac{i c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (-\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right )}{\sqrt{b} c+\sqrt{-a+i} d}\right )}{d^2}+\frac{i c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right )}{\sqrt{b} c-\sqrt{-a-i} d}\right )}{d^2}-\frac{i c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right )}{\sqrt{b} c-\sqrt{-a+i} d}\right )}{d^2}-\frac{i c \log (-i a-i b x+1) \log \left (c+d \sqrt{x}\right )}{d^2}+\frac{i c \log (i a+i b x+1) \log \left (c+d \sqrt{x}\right )}{d^2}+\frac{i \sqrt{x} \log (-i a-i b x+1)}{d}-\frac{i \sqrt{x} \log (i a+i b x+1)}{d}+\frac{2 i \sqrt{a+i} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right )}{\sqrt{b} d}-\frac{2 i \sqrt{-a+i} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{-a+i}}\right )}{\sqrt{b} d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.898424, antiderivative size = 673, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 13, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.722, Rules used = {5051, 2408, 2466, 2448, 321, 205, 2462, 260, 2416, 2394, 2393, 2391, 208} \[ \frac{i c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-i} d}\right )}{d^2}+\frac{i c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c+\sqrt{-a-i} d}\right )}{d^2}-\frac{i c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a+i} d}\right )}{d^2}-\frac{i c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c+\sqrt{-a+i} d}\right )}{d^2}+\frac{i c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (-\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right )}{\sqrt{b} c+\sqrt{-a-i} d}\right )}{d^2}-\frac{i c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (-\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right )}{\sqrt{b} c+\sqrt{-a+i} d}\right )}{d^2}+\frac{i c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right )}{\sqrt{b} c-\sqrt{-a-i} d}\right )}{d^2}-\frac{i c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right )}{\sqrt{b} c-\sqrt{-a+i} d}\right )}{d^2}-\frac{i c \log (-i a-i b x+1) \log \left (c+d \sqrt{x}\right )}{d^2}+\frac{i c \log (i a+i b x+1) \log \left (c+d \sqrt{x}\right )}{d^2}+\frac{i \sqrt{x} \log (-i a-i b x+1)}{d}-\frac{i \sqrt{x} \log (i a+i b x+1)}{d}+\frac{2 i \sqrt{a+i} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right )}{\sqrt{b} d}-\frac{2 i \sqrt{-a+i} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{-a+i}}\right )}{\sqrt{b} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 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 2466

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

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

Mathematica [A]  time = 0.517766, size = 604, normalized size = 0.9 \[ \frac{i \left (c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-i} d}\right )+c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c+\sqrt{-a-i} d}\right )-c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a+i} d}\right )-c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c+\sqrt{-a+i} d}\right )+c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (-\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right )}{\sqrt{b} c+\sqrt{-a-i} d}\right )-c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (-\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right )}{\sqrt{b} c+\sqrt{-a+i} d}\right )+c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right )}{-\sqrt{b} c+\sqrt{-a-i} d}\right )-c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right )}{-\sqrt{b} c+\sqrt{-a+i} d}\right )+c \log (i a+i b x+1) \log \left (c+d \sqrt{x}\right )-c \log (-i (a+b x+i)) \log \left (c+d \sqrt{x}\right )-d \sqrt{x} \log (i a+i b x+1)+d \sqrt{x} \log (-i (a+b x+i))+\frac{2 \sqrt{a+i} d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right )}{\sqrt{b}}-\frac{2 \sqrt{-a+i} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{-a+i}}\right )}{\sqrt{b}}\right )}{d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.338, size = 344, normalized size = 0.5 \begin{align*} 2\,{\frac{\arctan \left ( bx+a \right ) \sqrt{x}}{d}}-2\,{\frac{\arctan \left ( bx+a \right ) c\ln \left ( c+d\sqrt{x} \right ) }{{d}^{2}}}+c\sum _{{\it \_R1}={\it RootOf} \left ({b}^{2}{{\it \_Z}}^{4}-4\,c{b}^{2}{{\it \_Z}}^{3}+ \left ( 2\,ab{d}^{2}+6\,{b}^{2}{c}^{2} \right ){{\it \_Z}}^{2}+ \left ( -4\,bca{d}^{2}-4\,{c}^{3}{b}^{2} \right ){\it \_Z}+{a}^{2}{d}^{4}+2\,ab{c}^{2}{d}^{2}+{b}^{2}{c}^{4}+{d}^{4} \right ) }{\frac{1}{{{\it \_R1}}^{2}b-2\,{\it \_R1}\,bc+a{d}^{2}+{c}^{2}b} \left ( \ln \left ( c+d\sqrt{x} \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ( -d\sqrt{x}+{\it \_R1}-c \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ( -d\sqrt{x}+{\it \_R1}-c \right ) } \right ) \right ) }-\sum _{{\it \_R}={\it RootOf} \left ({b}^{2}{{\it \_Z}}^{4}-4\,c{b}^{2}{{\it \_Z}}^{3}+ \left ( 2\,ab{d}^{2}+6\,{b}^{2}{c}^{2} \right ){{\it \_Z}}^{2}+ \left ( -4\,bca{d}^{2}-4\,{c}^{3}{b}^{2} \right ){\it \_Z}+{a}^{2}{d}^{4}+2\,ab{c}^{2}{d}^{2}+{b}^{2}{c}^{4}+{d}^{4} \right ) }{\frac{{{\it \_R}}^{2}-2\,{\it \_R}\,c+{c}^{2}}{b{{\it \_R}}^{3}-3\,bc{{\it \_R}}^{2}+a{d}^{2}{\it \_R}+3\,{c}^{2}b{\it \_R}-ac{d}^{2}-b{c}^{3}}\ln \left ( d\sqrt{x}-{\it \_R}+c \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(b*x+a)/(c+d*x^(1/2)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(b*x+a)/(c+d*x^(1/2)),x, algorithm="giac")

[Out]

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