\(\int \frac {\arctan (a+b x)}{c+d \sqrt {x}} \, dx\) [58]

Optimal result

Integrand size = 18, antiderivative size = 673 \[ \int \frac {\arctan (a+b x)}{c+d \sqrt {x}} \, dx=\frac {2 i \sqrt {i+a} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} d}-\frac {2 i \sqrt {i-a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} d}+\frac {i c \log \left (\frac {d \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {i c \log \left (\frac {d \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log \left (-\frac {d \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {i c \log \left (-\frac {d \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i \sqrt {x} \log (1-i a-i b x)}{d}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log (1-i a-i b x)}{d^2}-\frac {i \sqrt {x} \log (1+i a+i b x)}{d}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log (1+i a+i b x)}{d^2}+\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right )}{d^2}+\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right )}{d^2}-\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right )}{d^2}-\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right )}{d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 673, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {5159, 2455, 2516, 2498, 327, 211, 2512, 266, 2463, 2441, 2440, 2438, 214} \[ \int \frac {\arctan (a+b x)}{c+d \sqrt {x}} \, dx=\frac {2 i \sqrt {a+i} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+i}}\right )}{\sqrt {b} d}-\frac {2 i \sqrt {-a+i} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a+i}}\right )}{\sqrt {b} d}+\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-a-i} d}\right )}{d^2}+\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-a-i} d}\right )}{d^2}-\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right )}{d^2}-\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right )}{d^2}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (-\sqrt {b} \sqrt {x}+\sqrt {-a-i}\right )}{\sqrt {b} c+\sqrt {-a-i} d}\right )}{d^2}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (-\sqrt {b} \sqrt {x}+\sqrt {-a+i}\right )}{\sqrt {b} c+\sqrt {-a+i} d}\right )}{d^2}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {d \left (\sqrt {b} \sqrt {x}+\sqrt {-a-i}\right )}{\sqrt {b} c-\sqrt {-a-i} d}\right )}{d^2}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {d \left (\sqrt {b} \sqrt {x}+\sqrt {-a+i}\right )}{\sqrt {b} c-\sqrt {-a+i} d}\right )}{d^2}-\frac {i c \log (-i a-i b x+1) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log (i a+i b x+1) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i \sqrt {x} \log (-i a-i b x+1)}{d}-\frac {i \sqrt {x} \log (i a+i b x+1)}{d} \]

[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 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 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 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 2516

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

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 604, normalized size of antiderivative = 0.90 \[ \int \frac {\arctan (a+b x)}{c+d \sqrt {x}} \, dx=\frac {i \left (\frac {2 \sqrt {i+a} d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b}}-\frac {2 \sqrt {i-a} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b}}+c \log \left (\frac {d \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )-c \log \left (\frac {d \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )+c \log \left (\frac {d \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{-\sqrt {b} c+\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )-c \log \left (\frac {d \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{-\sqrt {b} c+\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )-d \sqrt {x} \log (1+i a+i b x)+c \log \left (c+d \sqrt {x}\right ) \log (1+i a+i b x)+d \sqrt {x} \log (-i (i+a+b x))-c \log \left (c+d \sqrt {x}\right ) \log (-i (i+a+b x))+c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right )+c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right )-c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right )-c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right )\right )}{d^2} \]

[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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.24 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.54

method	result	size
derivativedivides	\(\frac {2 \arctan \left (b x +a \right ) \sqrt {x}}{d}-\frac {2 \arctan \left (b x +a \right ) c \ln \left (c +d \sqrt {x}\right )}{d^{2}}-\frac {4 b \left (\frac {d^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} c \,\textit {\_Z}^{3}+\left (2 a b \,d^{2}+6 b^{2} c^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c \,d^{2}-4 b^{2} c^{3}\right ) \textit {\_Z} +a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}+d^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{2}-2 \textit {\_R} c +c^{2}\right ) \ln \left (d \sqrt {x}-\textit {\_R} +c \right )}{b \,\textit {\_R}^{3}-3 \textit {\_R}^{2} b c +\textit {\_R} a \,d^{2}+3 \textit {\_R} b \,c^{2}-a c \,d^{2}-b \,c^{3}}\right )}{4 b}-\frac {c \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} c \,\textit {\_Z}^{3}+\left (2 a b \,d^{2}+6 b^{2} c^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c \,d^{2}-4 b^{2} c^{3}\right ) \textit {\_Z} +a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}+d^{4}\right )}{\sum }\frac {\ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {-d \sqrt {x}+\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d \sqrt {x}+\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2} b -2 \textit {\_R1} b c +a \,d^{2}+b \,c^{2}}\right )}{4 b}\right )}{d^{2}}\)	\(364\)
default	\(\frac {2 \arctan \left (b x +a \right ) \sqrt {x}}{d}-\frac {2 \arctan \left (b x +a \right ) c \ln \left (c +d \sqrt {x}\right )}{d^{2}}-\frac {4 b \left (\frac {d^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} c \,\textit {\_Z}^{3}+\left (2 a b \,d^{2}+6 b^{2} c^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c \,d^{2}-4 b^{2} c^{3}\right ) \textit {\_Z} +a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}+d^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{2}-2 \textit {\_R} c +c^{2}\right ) \ln \left (d \sqrt {x}-\textit {\_R} +c \right )}{b \,\textit {\_R}^{3}-3 \textit {\_R}^{2} b c +\textit {\_R} a \,d^{2}+3 \textit {\_R} b \,c^{2}-a c \,d^{2}-b \,c^{3}}\right )}{4 b}-\frac {c \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} c \,\textit {\_Z}^{3}+\left (2 a b \,d^{2}+6 b^{2} c^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c \,d^{2}-4 b^{2} c^{3}\right ) \textit {\_Z} +a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}+d^{4}\right )}{\sum }\frac {\ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {-d \sqrt {x}+\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d \sqrt {x}+\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2} b -2 \textit {\_R1} b c +a \,d^{2}+b \,c^{2}}\right )}{4 b}\right )}{d^{2}}\)	\(364\)

[In]

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

[Out]

2*arctan(b*x+a)/d*x^(1/2)-2*arctan(b*x+a)*c/d^2*ln(c+d*x^(1/2))-4*b/d^2*(1/4*d^2/b*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*b^2*c*_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))-1/4*c*d^2/b*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*b^2*c*_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)))

Fricas [F]

\[ \int \frac {\arctan (a+b x)}{c+d \sqrt {x}} \, dx=\int { \frac {\arctan \left (b x + a\right )}{d \sqrt {x} + c} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\arctan (a+b x)}{c+d \sqrt {x}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\arctan (a+b x)}{c+d \sqrt {x}} \, dx=\int { \frac {\arctan \left (b x + a\right )}{d \sqrt {x} + c} \,d x } \]

[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)

Giac [F(-2)]

Exception generated. \[ \int \frac {\arctan (a+b x)}{c+d \sqrt {x}} \, dx=\text {Exception raised: TypeError} \]

[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:The choice was done assuming 0=[0]Warning, replacing 0 by -24, a substitution variable should perhaps be pu
rged.Warnin

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (a+b x)}{c+d \sqrt {x}} \, dx=\int \frac {\mathrm {atan}\left (a+b\,x\right )}{c+d\,\sqrt {x}} \,d x \]

[In]

int(atan(a + b*x)/(c + d*x^(1/2)),x)

[Out]

int(atan(a + b*x)/(c + d*x^(1/2)), x)