3.1.59 \(\int \frac {\text {ArcTan}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx\) [59]

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

[Out]

-1/2*(1+I*a+I*b*x)*ln(1+I*a+I*b*x)/b/c-1/2*(1-I*a-I*b*x)*ln(-I*(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^2*ln(1+I*a+I*b*x)*ln(d+c*x^(1/2))/c^3+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-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(1-I*a-I*b*x)
*ln(d+c*x^(1/2))/c^3-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+I*d*ln(1+I*a+I*b*x)*x^(1/2)/c^2-I*d*ln(1
-I*a-I*b*x)*x^(1/2)/c^2-2*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)+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 = 0.81, antiderivative size = 770, normalized size of antiderivative = 1.00, 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 = {5159, 2455, 2526, 2498, 327, 211, 2504, 2436, 2332, 2512, 266, 2463, 2441, 2440, 2438, 214} \begin {gather*} -\frac {2 i \sqrt {a+i} d \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+i}}\right )}{\sqrt {b} c^2}-\frac {i d^2 \text {Li}_2\left (-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-i} c-\sqrt {b} d}\right )}{c^3}+\frac {i d^2 \text {Li}_2\left (-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {i-a} c-\sqrt {b} d}\right )}{c^3}-\frac {i d^2 \text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-i} c+\sqrt {b} d}\right )}{c^3}+\frac {i d^2 \text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {i-a} c+\sqrt {b} d}\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 \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} \end {gather*}

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 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 327

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 2332

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

Rule 2436

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 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 2455

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 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 2498

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 2504

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 2512

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 2526

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 5159

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]

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

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Mathematica [A]
time = 0.60, size = 666, normalized size = 0.86 \begin {gather*} \frac {i \left (4 c d \left (\sqrt {x}-\frac {\sqrt {i+a} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b}}\right )-4 c d \left (\sqrt {x}-\frac {\sqrt {i-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b}}\right )+2 c d \sqrt {x} \log (1+i a+i b x)-\frac {c^2 (-i+a+b x) \log (1+i a+i b x)}{b}-2 d^2 \log \left (d+c \sqrt {x}\right ) \log (1+i a+i b x)-2 c d \sqrt {x} \log (-i (i+a+b x))+\frac {c^2 (i+a+b x) \log (-i (i+a+b x))}{b}+2 d^2 \log \left (d+c \sqrt {x}\right ) \log (-i (i+a+b x))-2 d^2 \left (\left (\log \left (\frac {c \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-i-a} c+\sqrt {b} d}\right )+\log \left (\frac {c \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-i-a} c-\sqrt {b} d}\right )\right ) \log \left (d+c \sqrt {x}\right )+\text {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{-\sqrt {-i-a} c+\sqrt {b} d}\right )+\text {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-i-a} c+\sqrt {b} d}\right )\right )+2 d^2 \left (\left (\log \left (\frac {c \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right )+\log \left (\frac {c \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c-\sqrt {b} d}\right )\right ) \log \left (d+c \sqrt {x}\right )+\text {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{-\sqrt {i-a} c+\sqrt {b} d}\right )+\text {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right )\right )\right )}{2 c^3} \end {gather*}

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.06, size = 388, normalized size = 0.50

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(arctan(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(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)

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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(atan(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(arctan(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 {atan}\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(atan(a + b*x)/(c + d/x^(1/2)),x)

[Out]

int(atan(a + b*x)/(c + d/x^(1/2)), x)

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