\(\int \frac {\arctan (d+e x)}{a+b x^2} \, dx\) [61]

Optimal result

Integrand size = 16, antiderivative size = 543 \[ \int \frac {\arctan (d+e x)}{a+b x^2} \, dx=\frac {i \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} (i+d)+\sqrt {-a} e}\right ) \log (1-i d-i e x)}{4 \sqrt {-a} \sqrt {b}}-\frac {i \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} (i+d)-\sqrt {-a} e}\right ) \log (1-i d-i e x)}{4 \sqrt {-a} \sqrt {b}}-\frac {i \log \left (-\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} (i-d)-\sqrt {-a} e}\right ) \log (1+i d+i e x)}{4 \sqrt {-a} \sqrt {b}}+\frac {i \log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} (i-d)+\sqrt {-a} e}\right ) \log (1+i d+i e x)}{4 \sqrt {-a} \sqrt {b}}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (i-d-e x)}{\sqrt {b} (i-d)-\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (i-d-e x)}{\sqrt {b} (i-d)+\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (i+d+e x)}{\sqrt {b} (i+d)-\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (i+d+e x)}{\sqrt {b} (i+d)+\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}} \]

[Out]

-1/4*I*ln(1+I*d+I*e*x)*ln(-e*((-a)^(1/2)-x*b^(1/2))/(-e*(-a)^(1/2)+(I-d)*b^(1/2)))/(-a)^(1/2)/b^(1/2)+1/4*I*ln
(1-I*d-I*e*x)*ln(e*((-a)^(1/2)-x*b^(1/2))/(e*(-a)^(1/2)+(I+d)*b^(1/2)))/(-a)^(1/2)/b^(1/2)+1/4*I*ln(1+I*d+I*e*
x)*ln(e*((-a)^(1/2)+x*b^(1/2))/(e*(-a)^(1/2)+(I-d)*b^(1/2)))/(-a)^(1/2)/b^(1/2)-1/4*I*ln(1-I*d-I*e*x)*ln(-e*((
-a)^(1/2)+x*b^(1/2))/(-e*(-a)^(1/2)+(I+d)*b^(1/2)))/(-a)^(1/2)/b^(1/2)-1/4*I*polylog(2,(I-d-e*x)*b^(1/2)/(-e*(
-a)^(1/2)+(I-d)*b^(1/2)))/(-a)^(1/2)/b^(1/2)+1/4*I*polylog(2,(I-d-e*x)*b^(1/2)/(e*(-a)^(1/2)+(I-d)*b^(1/2)))/(
-a)^(1/2)/b^(1/2)-1/4*I*polylog(2,(I+d+e*x)*b^(1/2)/(-e*(-a)^(1/2)+(I+d)*b^(1/2)))/(-a)^(1/2)/b^(1/2)+1/4*I*po
lylog(2,(I+d+e*x)*b^(1/2)/(e*(-a)^(1/2)+(I+d)*b^(1/2)))/(-a)^(1/2)/b^(1/2)

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5159, 2456, 2441, 2440, 2438} \[ \int \frac {\arctan (d+e x)}{a+b x^2} \, dx=-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (-d-e x+i)}{\sqrt {b} (i-d)-\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (-d-e x+i)}{\sqrt {b} (i-d)+\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x+i)}{\sqrt {b} (d+i)-\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x+i)}{\sqrt {b} (d+i)+\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {i \log (-i d-i e x+1) \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} (d+i)}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {i \log (-i d-i e x+1) \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {-a} e+\sqrt {b} (d+i)}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {i \log (i d+i e x+1) \log \left (-\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{-\sqrt {-a} e+\sqrt {b} (-d+i)}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {i \log (i d+i e x+1) \log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} (-d+i)}\right )}{4 \sqrt {-a} \sqrt {b}} \]

[In]

Int[ArcTan[d + e*x]/(a + b*x^2),x]

[Out]

((I/4)*Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*(I + d) + Sqrt[-a]*e)]*Log[1 - I*d - I*e*x])/(Sqrt[-a]*Sqrt[b])
- ((I/4)*Log[-((e*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*(I + d) - Sqrt[-a]*e))]*Log[1 - I*d - I*e*x])/(Sqrt[-a]*Sq
rt[b]) - ((I/4)*Log[-((e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*(I - d) - Sqrt[-a]*e))]*Log[1 + I*d + I*e*x])/(Sqrt[
-a]*Sqrt[b]) + ((I/4)*Log[(e*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*(I - d) + Sqrt[-a]*e)]*Log[1 + I*d + I*e*x])/(Sq
rt[-a]*Sqrt[b]) - ((I/4)*PolyLog[2, (Sqrt[b]*(I - d - e*x))/(Sqrt[b]*(I - d) - Sqrt[-a]*e)])/(Sqrt[-a]*Sqrt[b]
) + ((I/4)*PolyLog[2, (Sqrt[b]*(I - d - e*x))/(Sqrt[b]*(I - d) + Sqrt[-a]*e)])/(Sqrt[-a]*Sqrt[b]) - ((I/4)*Pol
yLog[2, (Sqrt[b]*(I + d + e*x))/(Sqrt[b]*(I + d) - Sqrt[-a]*e)])/(Sqrt[-a]*Sqrt[b]) + ((I/4)*PolyLog[2, (Sqrt[
b]*(I + d + e*x))/(Sqrt[b]*(I + d) + Sqrt[-a]*e)])/(Sqrt[-a]*Sqrt[b])

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 2456

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

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 d-i e x)}{a+b x^2} \, dx-\frac {1}{2} i \int \frac {\log (1+i d+i e x)}{a+b x^2} \, dx \\ & = \frac {1}{2} i \int \left (\frac {\sqrt {-a} \log (1-i d-i e x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \log (1-i d-i e x)}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx-\frac {1}{2} i \int \left (\frac {\sqrt {-a} \log (1+i d+i e x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \log (1+i d+i e x)}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx \\ & = -\frac {i \int \frac {\log (1-i d-i e x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 \sqrt {-a}}-\frac {i \int \frac {\log (1-i d-i e x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 \sqrt {-a}}+\frac {i \int \frac {\log (1+i d+i e x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 \sqrt {-a}}+\frac {i \int \frac {\log (1+i d+i e x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 \sqrt {-a}} \\ & = \frac {i \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} (i+d)+\sqrt {-a} e}\right ) \log (1-i d-i e x)}{4 \sqrt {-a} \sqrt {b}}-\frac {i \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} (i+d)-\sqrt {-a} e}\right ) \log (1-i d-i e x)}{4 \sqrt {-a} \sqrt {b}}-\frac {i \log \left (-\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} (i-d)-\sqrt {-a} e}\right ) \log (1+i d+i e x)}{4 \sqrt {-a} \sqrt {b}}+\frac {i \log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} (i-d)+\sqrt {-a} e}\right ) \log (1+i d+i e x)}{4 \sqrt {-a} \sqrt {b}}-\frac {e \int \frac {\log \left (-\frac {i e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} (1-i d)-i \sqrt {-a} e}\right )}{1-i d-i e x} \, dx}{4 \sqrt {-a} \sqrt {b}}-\frac {e \int \frac {\log \left (\frac {i e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} (1+i d)+i \sqrt {-a} e}\right )}{1+i d+i e x} \, dx}{4 \sqrt {-a} \sqrt {b}}+\frac {e \int \frac {\log \left (-\frac {i e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} (1-i d)-i \sqrt {-a} e}\right )}{1-i d-i e x} \, dx}{4 \sqrt {-a} \sqrt {b}}+\frac {e \int \frac {\log \left (\frac {i e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} (1+i d)+i \sqrt {-a} e}\right )}{1+i d+i e x} \, dx}{4 \sqrt {-a} \sqrt {b}} \\ & = \frac {i \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} (i+d)+\sqrt {-a} e}\right ) \log (1-i d-i e x)}{4 \sqrt {-a} \sqrt {b}}-\frac {i \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} (i+d)-\sqrt {-a} e}\right ) \log (1-i d-i e x)}{4 \sqrt {-a} \sqrt {b}}-\frac {i \log \left (-\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} (i-d)-\sqrt {-a} e}\right ) \log (1+i d+i e x)}{4 \sqrt {-a} \sqrt {b}}+\frac {i \log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} (i-d)+\sqrt {-a} e}\right ) \log (1+i d+i e x)}{4 \sqrt {-a} \sqrt {b}}+\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {b} x}{-\sqrt {b} (1-i d)-i \sqrt {-a} e}\right )}{x} \, dx,x,1-i d-i e x\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {i \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {b} x}{\sqrt {b} (1-i d)-i \sqrt {-a} e}\right )}{x} \, dx,x,1-i d-i e x\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {b} x}{-\sqrt {b} (1+i d)+i \sqrt {-a} e}\right )}{x} \, dx,x,1+i d+i e x\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {i \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {b} x}{\sqrt {b} (1+i d)+i \sqrt {-a} e}\right )}{x} \, dx,x,1+i d+i e x\right )}{4 \sqrt {-a} \sqrt {b}} \\ & = \frac {i \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} (i+d)+\sqrt {-a} e}\right ) \log (1-i d-i e x)}{4 \sqrt {-a} \sqrt {b}}-\frac {i \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} (i+d)-\sqrt {-a} e}\right ) \log (1-i d-i e x)}{4 \sqrt {-a} \sqrt {b}}-\frac {i \log \left (-\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} (i-d)-\sqrt {-a} e}\right ) \log (1+i d+i e x)}{4 \sqrt {-a} \sqrt {b}}+\frac {i \log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} (i-d)+\sqrt {-a} e}\right ) \log (1+i d+i e x)}{4 \sqrt {-a} \sqrt {b}}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (i-d-e x)}{\sqrt {b} (i-d)-\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (i-d-e x)}{\sqrt {b} (i-d)+\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (i+d+e x)}{\sqrt {b} (i+d)-\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (i+d+e x)}{\sqrt {b} (i+d)+\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.75 \[ \int \frac {\arctan (d+e x)}{a+b x^2} \, dx=\frac {i \left (-\log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} (-i+d)+\sqrt {-a} e}\right ) \log (1+i d+i e x)+\log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} (-i+d)+\sqrt {-a} e}\right ) \log (1+i d+i e x)+\log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} (i+d)+\sqrt {-a} e}\right ) \log (-i (i+d+e x))-\log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} (i+d)+\sqrt {-a} e}\right ) \log (-i (i+d+e x))+\operatorname {PolyLog}\left (2,\frac {\sqrt {b} (-i+d+e x)}{\sqrt {b} (-i+d)-\sqrt {-a} e}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {b} (-i+d+e x)}{\sqrt {b} (-i+d)+\sqrt {-a} e}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {b} (i+d+e x)}{\sqrt {b} (i+d)-\sqrt {-a} e}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {b} (i+d+e x)}{\sqrt {b} (i+d)+\sqrt {-a} e}\right )\right )}{4 \sqrt {-a} \sqrt {b}} \]

[In]

Integrate[ArcTan[d + e*x]/(a + b*x^2),x]

[Out]

((I/4)*(-(Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*(-I + d) + Sqrt[-a]*e)]*Log[1 + I*d + I*e*x]) + Log[(e*(Sqrt
[-a] + Sqrt[b]*x))/(-(Sqrt[b]*(-I + d)) + Sqrt[-a]*e)]*Log[1 + I*d + I*e*x] + Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(
Sqrt[b]*(I + d) + Sqrt[-a]*e)]*Log[(-I)*(I + d + e*x)] - Log[(e*(Sqrt[-a] + Sqrt[b]*x))/(-(Sqrt[b]*(I + d)) +
Sqrt[-a]*e)]*Log[(-I)*(I + d + e*x)] + PolyLog[2, (Sqrt[b]*(-I + d + e*x))/(Sqrt[b]*(-I + d) - Sqrt[-a]*e)] -
PolyLog[2, (Sqrt[b]*(-I + d + e*x))/(Sqrt[b]*(-I + d) + Sqrt[-a]*e)] - PolyLog[2, (Sqrt[b]*(I + d + e*x))/(Sqr
t[b]*(I + d) - Sqrt[-a]*e)] + PolyLog[2, (Sqrt[b]*(I + d + e*x))/(Sqrt[b]*(I + d) + Sqrt[-a]*e)]))/(Sqrt[-a]*S
qrt[b])

Maple [A] (verified)

Time = 1.01 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.00

method	result	size
risch	\(\frac {\ln \left (-i e x -i d +1\right ) \ln \left (\frac {i b d -e \sqrt {a b}+b \left (-i e x -i d +1\right )-b}{i b d -e \sqrt {a b}-b}\right ) \sqrt {a b}}{4 a b}-\frac {\ln \left (-i e x -i d +1\right ) \ln \left (\frac {i b d +e \sqrt {a b}+b \left (-i e x -i d +1\right )-b}{i b d +e \sqrt {a b}-b}\right ) \sqrt {a b}}{4 a b}+\frac {\operatorname {dilog}\left (\frac {i b d -e \sqrt {a b}+b \left (-i e x -i d +1\right )-b}{i b d -e \sqrt {a b}-b}\right ) \sqrt {a b}}{4 a b}-\frac {\operatorname {dilog}\left (\frac {i b d +e \sqrt {a b}+b \left (-i e x -i d +1\right )-b}{i b d +e \sqrt {a b}-b}\right ) \sqrt {a b}}{4 a b}+\frac {\ln \left (i e x +i d +1\right ) \ln \left (\frac {i b d +e \sqrt {a b}-b \left (i e x +i d +1\right )+b}{i b d +e \sqrt {a b}+b}\right ) \sqrt {a b}}{4 a b}-\frac {\ln \left (i e x +i d +1\right ) \ln \left (\frac {i b d -e \sqrt {a b}-b \left (i e x +i d +1\right )+b}{i b d -e \sqrt {a b}+b}\right ) \sqrt {a b}}{4 a b}+\frac {\operatorname {dilog}\left (\frac {i b d +e \sqrt {a b}-b \left (i e x +i d +1\right )+b}{i b d +e \sqrt {a b}+b}\right ) \sqrt {a b}}{4 a b}-\frac {\operatorname {dilog}\left (\frac {i b d -e \sqrt {a b}-b \left (i e x +i d +1\right )+b}{i b d -e \sqrt {a b}+b}\right ) \sqrt {a b}}{4 a b}\)	\(542\)
derivativedivides	\(\text {Expression too large to display}\)	\(2184\)
default	\(\text {Expression too large to display}\)	\(2184\)

[In]

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

[Out]

1/4*ln(1-I*d-I*e*x)/a/b*ln((I*b*d-e*(a*b)^(1/2)+b*(1-I*d-I*e*x)-b)/(I*b*d-e*(a*b)^(1/2)-b))*(a*b)^(1/2)-1/4*ln
(1-I*d-I*e*x)/a/b*ln((I*b*d+e*(a*b)^(1/2)+b*(1-I*d-I*e*x)-b)/(I*b*d+e*(a*b)^(1/2)-b))*(a*b)^(1/2)+1/4/a/b*dilo
g((I*b*d-e*(a*b)^(1/2)+b*(1-I*d-I*e*x)-b)/(I*b*d-e*(a*b)^(1/2)-b))*(a*b)^(1/2)-1/4/a/b*dilog((I*b*d+e*(a*b)^(1
/2)+b*(1-I*d-I*e*x)-b)/(I*b*d+e*(a*b)^(1/2)-b))*(a*b)^(1/2)+1/4*ln(1+I*d+I*e*x)/a/b*ln((I*b*d+e*(a*b)^(1/2)-b*
(1+I*d+I*e*x)+b)/(I*b*d+e*(a*b)^(1/2)+b))*(a*b)^(1/2)-1/4*ln(1+I*d+I*e*x)/a/b*ln((I*b*d-e*(a*b)^(1/2)-b*(1+I*d
+I*e*x)+b)/(I*b*d-e*(a*b)^(1/2)+b))*(a*b)^(1/2)+1/4/a/b*dilog((I*b*d+e*(a*b)^(1/2)-b*(1+I*d+I*e*x)+b)/(I*b*d+e
*(a*b)^(1/2)+b))*(a*b)^(1/2)-1/4/a/b*dilog((I*b*d-e*(a*b)^(1/2)-b*(1+I*d+I*e*x)+b)/(I*b*d-e*(a*b)^(1/2)+b))*(a
*b)^(1/2)

Fricas [F]

\[ \int \frac {\arctan (d+e x)}{a+b x^2} \, dx=\int { \frac {\arctan \left (e x + d\right )}{b x^{2} + a} \,d x } \]

[In]

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

[Out]

integral(arctan(e*x + d)/(b*x^2 + a), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\arctan (d+e x)}{a+b x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 14300 vs. \(2 (369) = 738\).

Time = 2.00 (sec) , antiderivative size = 14300, normalized size of antiderivative = 26.34 \[ \int \frac {\arctan (d+e x)}{a+b x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/8*e*(8*arctan(b*x/sqrt(a*b))*arctan((e^2*x + d*e)/e)/e - (4*arctan(sqrt(b)*x/sqrt(a))*arctan2((2*a*b*d*e^2 +
(a*d*e^3 + (b*d^3 + b*d)*e + (a*e^4 + (b*d^2 + 3*b)*e^2)*x)*sqrt(a)*sqrt(b) + (3*a*b*e^3 + (b^2*d^2 + b^2)*e)
*x)/(b^2*d^4 + a^2*e^4 + 2*b^2*d^2 + 2*(a*b*d^2 + 3*a*b)*e^2 + 4*(a*e^3 + (b*d^2 + b)*e)*sqrt(a)*sqrt(b) + b^2
), (b^2*d^4 + 2*b^2*d^2 + (a*b*d^2 + 3*a*b)*e^2 + (2*b*d*e^2*x + a*e^3 + 3*(b*d^2 + b)*e)*sqrt(a)*sqrt(b) + b^
2 + (a*b*d*e^3 + (b^2*d^3 + b^2*d)*e)*x)/(b^2*d^4 + a^2*e^4 + 2*b^2*d^2 + 2*(a*b*d^2 + 3*a*b)*e^2 + 4*(a*e^3 +
(b*d^2 + b)*e)*sqrt(a)*sqrt(b) + b^2)) + 4*arctan(sqrt(b)*x/sqrt(a))*arctan2((2*a*b*d*e^2 - (a*d*e^3 + (b*d^3
+ b*d)*e + (a*e^4 + (b*d^2 + 3*b)*e^2)*x)*sqrt(a)*sqrt(b) + (3*a*b*e^3 + (b^2*d^2 + b^2)*e)*x)/(b^2*d^4 + a^2
*e^4 + 2*b^2*d^2 + 2*(a*b*d^2 + 3*a*b)*e^2 - 4*(a*e^3 + (b*d^2 + b)*e)*sqrt(a)*sqrt(b) + b^2), (b^2*d^4 + 2*b^
2*d^2 + (a*b*d^2 + 3*a*b)*e^2 - (2*b*d*e^2*x + a*e^3 + 3*(b*d^2 + b)*e)*sqrt(a)*sqrt(b) + b^2 + (a*b*d*e^3 + (
b^2*d^3 + b^2*d)*e)*x)/(b^2*d^4 + a^2*e^4 + 2*b^2*d^2 + 2*(a*b*d^2 + 3*a*b)*e^2 - 4*(a*e^3 + (b*d^2 + b)*e)*sq
rt(a)*sqrt(b) + b^2)) + log(b*x^2 + a)*log((b^12*d^24 + 12*b^12*d^22 + 66*b^12*d^20 + 220*b^12*d^18 + 495*b^12
*d^16 + 792*b^12*d^14 + 924*b^12*d^12 + (a^11*b*d^2 + a^11*b)*e^22 + 792*b^12*d^10 + 11*(a^10*b^2*d^4 + 22*a^1
0*b^2*d^2 + 21*a^10*b^2)*e^20 + 495*b^12*d^8 + 55*(a^9*b^3*d^6 + 39*a^9*b^3*d^4 + 171*a^9*b^3*d^2 + 133*a^9*b^
3)*e^18 + 220*b^12*d^6 + 33*(5*a^8*b^4*d^8 + 260*a^8*b^4*d^6 + 1870*a^8*b^4*d^4 + 3876*a^8*b^4*d^2 + 2261*a^8*
b^4)*e^16 + 66*b^12*d^4 + 330*(a^7*b^5*d^10 + 61*a^7*b^5*d^8 + 570*a^7*b^5*d^6 + 1802*a^7*b^5*d^4 + 2261*a^7*b
^5*d^2 + 969*a^7*b^5)*e^14 + 12*b^12*d^2 + 22*(21*a^6*b^6*d^12 + 1386*a^6*b^6*d^10 + 15015*a^6*b^6*d^8 + 60060
*a^6*b^6*d^6 + 109395*a^6*b^6*d^4 + 92378*a^6*b^6*d^2 + 29393*a^6*b^6)*e^12 + b^12 + 22*(21*a^5*b^7*d^14 + 140
7*a^5*b^7*d^12 + 16401*a^5*b^7*d^10 + 75075*a^5*b^7*d^8 + 169455*a^5*b^7*d^6 + 201773*a^5*b^7*d^4 + 121771*a^5
*b^7*d^2 + 29393*a^5*b^7)*e^10 + 330*(a^4*b^8*d^16 + 64*a^4*b^8*d^14 + 756*a^4*b^8*d^12 + 3696*a^4*b^8*d^10 +
9438*a^4*b^8*d^8 + 13728*a^4*b^8*d^6 + 11492*a^4*b^8*d^4 + 5168*a^4*b^8*d^2 + 969*a^4*b^8)*e^8 + 33*(5*a^3*b^9
*d^18 + 285*a^3*b^9*d^16 + 3220*a^3*b^9*d^14 + 15876*a^3*b^9*d^12 + 42966*a^3*b^9*d^10 + 70070*a^3*b^9*d^8 + 7
0980*a^3*b^9*d^6 + 43860*a^3*b^9*d^4 + 15181*a^3*b^9*d^2 + 2261*a^3*b^9)*e^6 + 55*(a^2*b^10*d^20 + 46*a^2*b^10
*d^18 + 465*a^2*b^10*d^16 + 2184*a^2*b^10*d^14 + 5922*a^2*b^10*d^12 + 10164*a^2*b^10*d^10 + 11466*a^2*b^10*d^8
+ 8520*a^2*b^10*d^6 + 4029*a^2*b^10*d^4 + 1102*a^2*b^10*d^2 + 133*a^2*b^10)*e^4 + 11*(a*b^11*d^22 + 31*a*b^11
*d^20 + 255*a*b^11*d^18 + 1065*a*b^11*d^16 + 2730*a*b^11*d^14 + 4662*a*b^11*d^12 + 5502*a*b^11*d^10 + 4530*a*b
^11*d^8 + 2565*a*b^11*d^6 + 955*a*b^11*d^4 + 211*a*b^11*d^2 + 21*a*b^11)*e^2 + (a^11*b*e^24 + 11*(a^10*b^2*d^2
+ 21*a^10*b^2)*e^22 + 55*(a^9*b^3*d^4 + 38*a^9*b^3*d^2 + 133*a^9*b^3)*e^20 + 33*(5*a^8*b^4*d^6 + 255*a^8*b^4*
d^4 + 1615*a^8*b^4*d^2 + 2261*a^8*b^4)*e^18 + 330*(a^7*b^5*d^8 + 60*a^7*b^5*d^6 + 510*a^7*b^5*d^4 + 1292*a^7*b
^5*d^2 + 969*a^7*b^5)*e^16 + 22*(21*a^6*b^6*d^10 + 1365*a^6*b^6*d^8 + 13650*a^6*b^6*d^6 + 46410*a^6*b^6*d^4 +
62985*a^6*b^6*d^2 + 29393*a^6*b^6)*e^14 + 22*(21*a^5*b^7*d^12 + 1386*a^5*b^7*d^10 + 15015*a^5*b^7*d^8 + 60060*
a^5*b^7*d^6 + 109395*a^5*b^7*d^4 + 92378*a^5*b^7*d^2 + 29393*a^5*b^7)*e^12 + 330*(a^4*b^8*d^14 + 63*a^4*b^8*d^
12 + 693*a^4*b^8*d^10 + 3003*a^4*b^8*d^8 + 6435*a^4*b^8*d^6 + 7293*a^4*b^8*d^4 + 4199*a^4*b^8*d^2 + 969*a^4*b^
8)*e^10 + 33*(5*a^3*b^9*d^16 + 280*a^3*b^9*d^14 + 2940*a^3*b^9*d^12 + 12936*a^3*b^9*d^10 + 30030*a^3*b^9*d^8 +
40040*a^3*b^9*d^6 + 30940*a^3*b^9*d^4 + 12920*a^3*b^9*d^2 + 2261*a^3*b^9)*e^8 + 55*(a^2*b^10*d^18 + 45*a^2*b^
10*d^16 + 420*a^2*b^10*d^14 + 1764*a^2*b^10*d^12 + 4158*a^2*b^10*d^10 + 6006*a^2*b^10*d^8 + 5460*a^2*b^10*d^6
+ 3060*a^2*b^10*d^4 + 969*a^2*b^10*d^2 + 133*a^2*b^10)*e^6 + 11*(a*b^11*d^20 + 30*a*b^11*d^18 + 225*a*b^11*d^1
6 + 840*a*b^11*d^14 + 1890*a*b^11*d^12 + 2772*a*b^11*d^10 + 2730*a*b^11*d^8 + 1800*a*b^11*d^6 + 765*a*b^11*d^4
+ 190*a*b^11*d^2 + 21*a*b^11)*e^4 + (b^12*d^22 + 11*b^12*d^20 + 55*b^12*d^18 + 165*b^12*d^16 + 330*b^12*d^14
+ 462*b^12*d^12 + 462*b^12*d^10 + 330*b^12*d^8 + 165*b^12*d^6 + 55*b^12*d^4 + 11*b^12*d^2 + b^12)*e^2)*x^2 + 2
*(11*(a^10*b*d^2 + a^10*b)*e^21 + 110*(a^9*b^2*d^4 + 8*a^9*b^2*d^2 + 7*a^9*b^2)*e^19 + 33*(15*a^8*b^3*d^6 + 20
5*a^8*b^3*d^4 + 589*a^8*b^3*d^2 + 399*a^8*b^3)*e^17 + 264*(5*a^7*b^4*d^8 + 90*a^7*b^4*d^6 + 408*a^7*b^4*d^4 +
646*a^7*b^4*d^2 + 323*a^7*b^4)*e^15 + 110*(21*a^6*b^5*d^10 + 441*a^6*b^5*d^8 + 2562*a^6*b^5*d^6 + 6018*a^6*b^5
*d^4 + 6137*a^6*b^5*d^2 + 2261*a^6*b^5)*e^13 + 4*(693*a^5*b^6*d^12 + 15708*a^5*b^6*d^10 + 105105*a^5*b^6*d^8 +
308880*a^5*b^6*d^6 + 449735*a^5*b^6*d^4 + 319124*a^5*b^6*d^2 + 88179*a^5*b^6)*e^11 + 110*(21*a^4*b^7*d^14 + 4
83*a^4*b^7*d^12 + 3465*a^4*b^7*d^10 + 11583*a^4*b^7*d^8 + 20735*a^4*b^7*d^6 + 20553*a^4*b^7*d^4 + 10659*a^4*b^
7*d^2 + 2261*a^4*b^7)*e^9 + 264*(5*a^3*b^8*d^16 + 110*a^3*b^8*d^14 + 798*a^3*b^8*d^12 + 2838*a^3*b^8*d^10 + 57
20*a^3*b^8*d^8 + 6890*a^3*b^8*d^6 + 4930*a^3*b^8*d^4 + 1938*a^3*b^8*d^2 + 323*a^3*b^8)*e^7 + 33*(15*a^2*b^9*d^
18 + 295*a^2*b^9*d^16 + 2044*a^2*b^9*d^14 + 7308*a^2*b^9*d^12 + 15554*a^2*b^9*d^10 + 20930*a^2*b^9*d^8 + 18060
*a^2*b^9*d^6 + 9724*a^2*b^9*d^4 + 2983*a^2*b^9*d^2 + 399*a^2*b^9)*e^5 + 110*(a*b^10*d^20 + 16*a*b^10*d^18 + 99
*a*b^10*d^16 + 336*a*b^10*d^14 + 714*a*b^10*d^12 + 1008*a*b^10*d^10 + 966*a*b^10*d^8 + 624*a*b^10*d^6 + 261*a*
b^10*d^4 + 64*a*b^10*d^2 + 7*a*b^10)*e^3 + (11*a^10*b*e^23 + 110*(a^9*b^2*d^2 + 7*a^9*b^2)*e^21 + 33*(15*a^8*b
^3*d^4 + 190*a^8*b^3*d^2 + 399*a^8*b^3)*e^19 + 264*(5*a^7*b^4*d^6 + 85*a^7*b^4*d^4 + 323*a^7*b^4*d^2 + 323*a^7
*b^4)*e^17 + 110*(21*a^6*b^5*d^8 + 420*a^6*b^5*d^6 + 2142*a^6*b^5*d^4 + 3876*a^6*b^5*d^2 + 2261*a^6*b^5)*e^15
+ 4*(693*a^5*b^6*d^10 + 15015*a^5*b^6*d^8 + 90090*a^5*b^6*d^6 + 218790*a^5*b^6*d^4 + 230945*a^5*b^6*d^2 + 8817
9*a^5*b^6)*e^13 + 110*(21*a^4*b^7*d^12 + 462*a^4*b^7*d^10 + 3003*a^4*b^7*d^8 + 8580*a^4*b^7*d^6 + 12155*a^4*b^
7*d^4 + 8398*a^4*b^7*d^2 + 2261*a^4*b^7)*e^11 + 264*(5*a^3*b^8*d^14 + 105*a^3*b^8*d^12 + 693*a^3*b^8*d^10 + 21
45*a^3*b^8*d^8 + 3575*a^3*b^8*d^6 + 3315*a^3*b^8*d^4 + 1615*a^3*b^8*d^2 + 323*a^3*b^8)*e^9 + 33*(15*a^2*b^9*d^
16 + 280*a^2*b^9*d^14 + 1764*a^2*b^9*d^12 + 5544*a^2*b^9*d^10 + 10010*a^2*b^9*d^8 + 10920*a^2*b^9*d^6 + 7140*a
^2*b^9*d^4 + 2584*a^2*b^9*d^2 + 399*a^2*b^9)*e^7 + 110*(a*b^10*d^18 + 15*a*b^10*d^16 + 84*a*b^10*d^14 + 252*a*
b^10*d^12 + 462*a*b^10*d^10 + 546*a*b^10*d^8 + 420*a*b^10*d^6 + 204*a*b^10*d^4 + 57*a*b^10*d^2 + 7*a*b^10)*e^5
+ 11*(b^11*d^20 + 10*b^11*d^18 + 45*b^11*d^16 + 120*b^11*d^14 + 210*b^11*d^12 + 252*b^11*d^10 + 210*b^11*d^8
+ 120*b^11*d^6 + 45*b^11*d^4 + 10*b^11*d^2 + b^11)*e^3)*x^2 + 11*(b^11*d^22 + 11*b^11*d^20 + 55*b^11*d^18 + 16
5*b^11*d^16 + 330*b^11*d^14 + 462*b^11*d^12 + 462*b^11*d^10 + 330*b^11*d^8 + 165*b^11*d^6 + 55*b^11*d^4 + 11*b
^11*d^2 + b^11)*e + 2*(11*a^10*b*d*e^22 + 110*(a^9*b^2*d^3 + 7*a^9*b^2*d)*e^20 + 33*(15*a^8*b^3*d^5 + 190*a^8*
b^3*d^3 + 399*a^8*b^3*d)*e^18 + 264*(5*a^7*b^4*d^7 + 85*a^7*b^4*d^5 + 323*a^7*b^4*d^3 + 323*a^7*b^4*d)*e^16 +
110*(21*a^6*b^5*d^9 + 420*a^6*b^5*d^7 + 2142*a^6*b^5*d^5 + 3876*a^6*b^5*d^3 + 2261*a^6*b^5*d)*e^14 + 4*(693*a^
5*b^6*d^11 + 15015*a^5*b^6*d^9 + 90090*a^5*b^6*d^7 + 218790*a^5*b^6*d^5 + 230945*a^5*b^6*d^3 + 88179*a^5*b^6*d
)*e^12 + 110*(21*a^4*b^7*d^13 + 462*a^4*b^7*d^11 + 3003*a^4*b^7*d^9 + 8580*a^4*b^7*d^7 + 12155*a^4*b^7*d^5 + 8
398*a^4*b^7*d^3 + 2261*a^4*b^7*d)*e^10 + 264*(5*a^3*b^8*d^15 + 105*a^3*b^8*d^13 + 693*a^3*b^8*d^11 + 2145*a^3*
b^8*d^9 + 3575*a^3*b^8*d^7 + 3315*a^3*b^8*d^5 + 1615*a^3*b^8*d^3 + 323*a^3*b^8*d)*e^8 + 33*(15*a^2*b^9*d^17 +
280*a^2*b^9*d^15 + 1764*a^2*b^9*d^13 + 5544*a^2*b^9*d^11 + 10010*a^2*b^9*d^9 + 10920*a^2*b^9*d^7 + 7140*a^2*b^
9*d^5 + 2584*a^2*b^9*d^3 + 399*a^2*b^9*d)*e^6 + 110*(a*b^10*d^19 + 15*a*b^10*d^17 + 84*a*b^10*d^15 + 252*a*b^1
0*d^13 + 462*a*b^10*d^11 + 546*a*b^10*d^9 + 420*a*b^10*d^7 + 204*a*b^10*d^5 + 57*a*b^10*d^3 + 7*a*b^10*d)*e^4
+ 11*(b^11*d^21 + 10*b^11*d^19 + 45*b^11*d^17 + 120*b^11*d^15 + 210*b^11*d^13 + 252*b^11*d^11 + 210*b^11*d^9 +
120*b^11*d^7 + 45*b^11*d^5 + 10*b^11*d^3 + b^11*d)*e^2)*x)*sqrt(a)*sqrt(b) + 2*(a^11*b*d*e^23 + 11*(a^10*b^2*
d^3 + 21*a^10*b^2*d)*e^21 + 55*(a^9*b^3*d^5 + 38*a^9*b^3*d^3 + 133*a^9*b^3*d)*e^19 + 33*(5*a^8*b^4*d^7 + 255*a
^8*b^4*d^5 + 1615*a^8*b^4*d^3 + 2261*a^8*b^4*d)*e^17 + 330*(a^7*b^5*d^9 + 60*a^7*b^5*d^7 + 510*a^7*b^5*d^5 + 1
292*a^7*b^5*d^3 + 969*a^7*b^5*d)*e^15 + 22*(21*a^6*b^6*d^11 + 1365*a^6*b^6*d^9 + 13650*a^6*b^6*d^7 + 46410*a^6
*b^6*d^5 + 62985*a^6*b^6*d^3 + 29393*a^6*b^6*d)*e^13 + 22*(21*a^5*b^7*d^13 + 1386*a^5*b^7*d^11 + 15015*a^5*b^7
*d^9 + 60060*a^5*b^7*d^7 + 109395*a^5*b^7*d^5 + 92378*a^5*b^7*d^3 + 29393*a^5*b^7*d)*e^11 + 330*(a^4*b^8*d^15
+ 63*a^4*b^8*d^13 + 693*a^4*b^8*d^11 + 3003*a^4*b^8*d^9 + 6435*a^4*b^8*d^7 + 7293*a^4*b^8*d^5 + 4199*a^4*b^8*d
^3 + 969*a^4*b^8*d)*e^9 + 33*(5*a^3*b^9*d^17 + 280*a^3*b^9*d^15 + 2940*a^3*b^9*d^13 + 12936*a^3*b^9*d^11 + 300
30*a^3*b^9*d^9 + 40040*a^3*b^9*d^7 + 30940*a^3*b^9*d^5 + 12920*a^3*b^9*d^3 + 2261*a^3*b^9*d)*e^7 + 55*(a^2*b^1
0*d^19 + 45*a^2*b^10*d^17 + 420*a^2*b^10*d^15 + 1764*a^2*b^10*d^13 + 4158*a^2*b^10*d^11 + 6006*a^2*b^10*d^9 +
5460*a^2*b^10*d^7 + 3060*a^2*b^10*d^5 + 969*a^2*b^10*d^3 + 133*a^2*b^10*d)*e^5 + 11*(a*b^11*d^21 + 30*a*b^11*d
^19 + 225*a*b^11*d^17 + 840*a*b^11*d^15 + 1890*a*b^11*d^13 + 2772*a*b^11*d^11 + 2730*a*b^11*d^9 + 1800*a*b^11*
d^7 + 765*a*b^11*d^5 + 190*a*b^11*d^3 + 21*a*b^11*d)*e^3 + (b^12*d^23 + 11*b^12*d^21 + 55*b^12*d^19 + 165*b^12
*d^17 + 330*b^12*d^15 + 462*b^12*d^13 + 462*b^12*d^11 + 330*b^12*d^9 + 165*b^12*d^7 + 55*b^12*d^5 + 11*b^12*d^
3 + b^12*d)*e)*x)/(b^12*d^24 + a^12*e^24 + 12*b^12*d^22 + 66*b^12*d^20 + 220*b^12*d^18 + 495*b^12*d^16 + 792*b
^12*d^14 + 924*b^12*d^12 + 12*(a^11*b*d^2 + 23*a^11*b)*e^22 + 792*b^12*d^10 + 66*(a^10*b^2*d^4 + 42*a^10*b^2*d
^2 + 161*a^10*b^2)*e^20 + 495*b^12*d^8 + 44*(5*a^9*b^3*d^6 + 285*a^9*b^3*d^4 + 1995*a^9*b^3*d^2 + 3059*a^9*b^3
)*e^18 + 220*b^12*d^6 + 99*(5*a^8*b^4*d^8 + 340*a^8*b^4*d^6 + 3230*a^8*b^4*d^4 + 9044*a^8*b^4*d^2 + 7429*a^8*b
^4)*e^16 + 66*b^12*d^4 + 264*(3*a^7*b^5*d^10 + 225*a^7*b^5*d^8 + 2550*a^7*b^5*d^6 + 9690*a^7*b^5*d^4 + 14535*a
^7*b^5*d^2 + 7429*a^7*b^5)*e^14 + 12*b^12*d^2 + 4*(231*a^6*b^6*d^12 + 18018*a^6*b^6*d^10 + 225225*a^6*b^6*d^8
+ 1021020*a^6*b^6*d^6 + 2078505*a^6*b^6*d^4 + 1939938*a^6*b^6*d^2 + 676039*a^6*b^6)*e^12 + b^12 + 264*(3*a^5*b
^7*d^14 + 231*a^5*b^7*d^12 + 3003*a^5*b^7*d^10 + 15015*a^5*b^7*d^8 + 36465*a^5*b^7*d^6 + 46189*a^5*b^7*d^4 + 2
9393*a^5*b^7*d^2 + 7429*a^5*b^7)*e^10 + 99*(5*a^4*b^8*d^16 + 360*a^4*b^8*d^14 + 4620*a^4*b^8*d^12 + 24024*a^4*
b^8*d^10 + 64350*a^4*b^8*d^8 + 97240*a^4*b^8*d^6 + 83980*a^4*b^8*d^4 + 38760*a^4*b^8*d^2 + 7429*a^4*b^8)*e^8 +
44*(5*a^3*b^9*d^18 + 315*a^3*b^9*d^16 + 3780*a^3*b^9*d^14 + 19404*a^3*b^9*d^12 + 54054*a^3*b^9*d^10 + 90090*a
^3*b^9*d^8 + 92820*a^3*b^9*d^6 + 58140*a^3*b^9*d^4 + 20349*a^3*b^9*d^2 + 3059*a^3*b^9)*e^6 + 66*(a^2*b^10*d^20
+ 50*a^2*b^10*d^18 + 525*a^2*b^10*d^16 + 2520*a^2*b^10*d^14 + 6930*a^2*b^10*d^12 + 12012*a^2*b^10*d^10 + 1365
0*a^2*b^10*d^8 + 10200*a^2*b^10*d^6 + 4845*a^2*b^10*d^4 + 1330*a^2*b^10*d^2 + 161*a^2*b^10)*e^4 + 12*(a*b^11*d
^22 + 33*a*b^11*d^20 + 275*a*b^11*d^18 + 1155*a*b^11*d^16 + 2970*a*b^11*d^14 + 5082*a*b^11*d^12 + 6006*a*b^11*
d^10 + 4950*a*b^11*d^8 + 2805*a*b^11*d^6 + 1045*a*b^11*d^4 + 231*a*b^11*d^2 + 23*a*b^11)*e^2 + 8*(3*a^11*e^23
+ 11*(3*a^10*b*d^2 + 23*a^10*b)*e^21 + 33*(5*a^9*b^2*d^4 + 70*a^9*b^2*d^2 + 161*a^9*b^2)*e^19 + 99*(5*a^8*b^3*
d^6 + 95*a^8*b^3*d^4 + 399*a^8*b^3*d^2 + 437*a^8*b^3)*e^17 + 22*(45*a^7*b^4*d^8 + 1020*a^7*b^4*d^6 + 5814*a^7*
b^4*d^4 + 11628*a^7*b^4*d^2 + 7429*a^7*b^4)*e^15 + 6*(231*a^6*b^5*d^10 + 5775*a^6*b^5*d^8 + 39270*a^6*b^5*d^6
+ 106590*a^6*b^5*d^4 + 124355*a^6*b^5*d^2 + 52003*a^6*b^5)*e^13 + 6*(231*a^5*b^6*d^12 + 6006*a^5*b^6*d^10 + 45
045*a^5*b^6*d^8 + 145860*a^5*b^6*d^6 + 230945*a^5*b^6*d^4 + 176358*a^5*b^6*d^2 + 52003*a^5*b^6)*e^11 + 22*(45*
a^4*b^7*d^14 + 1155*a^4*b^7*d^12 + 9009*a^4*b^7*d^10 + 32175*a^4*b^7*d^8 + 60775*a^4*b^7*d^6 + 62985*a^4*b^7*d
^4 + 33915*a^4*b^7*d^2 + 7429*a^4*b^7)*e^9 + 99*(5*a^3*b^8*d^16 + 120*a^3*b^8*d^14 + 924*a^3*b^8*d^12 + 3432*a
^3*b^8*d^10 + 7150*a^3*b^8*d^8 + 8840*a^3*b^8*d^6 + 6460*a^3*b^8*d^4 + 2584*a^3*b^8*d^2 + 437*a^3*b^8)*e^7 + 3
3*(5*a^2*b^9*d^18 + 105*a^2*b^9*d^16 + 756*a^2*b^9*d^14 + 2772*a^2*b^9*d^12 + 6006*a^2*b^9*d^10 + 8190*a^2*b^9
*d^8 + 7140*a^2*b^9*d^6 + 3876*a^2*b^9*d^4 + 1197*a^2*b^9*d^2 + 161*a^2*b^9)*e^5 + 11*(3*a*b^10*d^20 + 50*a*b^
10*d^18 + 315*a*b^10*d^16 + 1080*a*b^10*d^14 + 2310*a*b^10*d^12 + 3276*a*b^10*d^10 + 3150*a*b^10*d^8 + 2040*a*
b^10*d^6 + 855*a*b^10*d^4 + 210*a*b^10*d^2 + 23*a*b^10)*e^3 + 3*(b^11*d^22 + 11*b^11*d^20 + 55*b^11*d^18 + 165
*b^11*d^16 + 330*b^11*d^14 + 462*b^11*d^12 + 462*b^11*d^10 + 330*b^11*d^8 + 165*b^11*d^6 + 55*b^11*d^4 + 11*b^
11*d^2 + b^11)*e)*sqrt(a)*sqrt(b))) - log(b*x^2 + a)*log((b^12*d^24 + 12*b^12*d^22 + 66*b^12*d^20 + 220*b^12*d
^18 + 495*b^12*d^16 + 792*b^12*d^14 + 924*b^12*d^12 + (a^11*b*d^2 + a^11*b)*e^22 + 792*b^12*d^10 + 11*(a^10*b^
2*d^4 + 22*a^10*b^2*d^2 + 21*a^10*b^2)*e^20 + 495*b^12*d^8 + 55*(a^9*b^3*d^6 + 39*a^9*b^3*d^4 + 171*a^9*b^3*d^
2 + 133*a^9*b^3)*e^18 + 220*b^12*d^6 + 33*(5*a^8*b^4*d^8 + 260*a^8*b^4*d^6 + 1870*a^8*b^4*d^4 + 3876*a^8*b^4*d
^2 + 2261*a^8*b^4)*e^16 + 66*b^12*d^4 + 330*(a^7*b^5*d^10 + 61*a^7*b^5*d^8 + 570*a^7*b^5*d^6 + 1802*a^7*b^5*d^
4 + 2261*a^7*b^5*d^2 + 969*a^7*b^5)*e^14 + 12*b^12*d^2 + 22*(21*a^6*b^6*d^12 + 1386*a^6*b^6*d^10 + 15015*a^6*b
^6*d^8 + 60060*a^6*b^6*d^6 + 109395*a^6*b^6*d^4 + 92378*a^6*b^6*d^2 + 29393*a^6*b^6)*e^12 + b^12 + 22*(21*a^5*
b^7*d^14 + 1407*a^5*b^7*d^12 + 16401*a^5*b^7*d^10 + 75075*a^5*b^7*d^8 + 169455*a^5*b^7*d^6 + 201773*a^5*b^7*d^
4 + 121771*a^5*b^7*d^2 + 29393*a^5*b^7)*e^10 + 330*(a^4*b^8*d^16 + 64*a^4*b^8*d^14 + 756*a^4*b^8*d^12 + 3696*a
^4*b^8*d^10 + 9438*a^4*b^8*d^8 + 13728*a^4*b^8*d^6 + 11492*a^4*b^8*d^4 + 5168*a^4*b^8*d^2 + 969*a^4*b^8)*e^8 +
33*(5*a^3*b^9*d^18 + 285*a^3*b^9*d^16 + 3220*a^3*b^9*d^14 + 15876*a^3*b^9*d^12 + 42966*a^3*b^9*d^10 + 70070*a
^3*b^9*d^8 + 70980*a^3*b^9*d^6 + 43860*a^3*b^9*d^4 + 15181*a^3*b^9*d^2 + 2261*a^3*b^9)*e^6 + 55*(a^2*b^10*d^20
+ 46*a^2*b^10*d^18 + 465*a^2*b^10*d^16 + 2184*a^2*b^10*d^14 + 5922*a^2*b^10*d^12 + 10164*a^2*b^10*d^10 + 1146
6*a^2*b^10*d^8 + 8520*a^2*b^10*d^6 + 4029*a^2*b^10*d^4 + 1102*a^2*b^10*d^2 + 133*a^2*b^10)*e^4 + 11*(a*b^11*d^
22 + 31*a*b^11*d^20 + 255*a*b^11*d^18 + 1065*a*b^11*d^16 + 2730*a*b^11*d^14 + 4662*a*b^11*d^12 + 5502*a*b^11*d
^10 + 4530*a*b^11*d^8 + 2565*a*b^11*d^6 + 955*a*b^11*d^4 + 211*a*b^11*d^2 + 21*a*b^11)*e^2 + (a^11*b*e^24 + 11
*(a^10*b^2*d^2 + 21*a^10*b^2)*e^22 + 55*(a^9*b^3*d^4 + 38*a^9*b^3*d^2 + 133*a^9*b^3)*e^20 + 33*(5*a^8*b^4*d^6
+ 255*a^8*b^4*d^4 + 1615*a^8*b^4*d^2 + 2261*a^8*b^4)*e^18 + 330*(a^7*b^5*d^8 + 60*a^7*b^5*d^6 + 510*a^7*b^5*d^
4 + 1292*a^7*b^5*d^2 + 969*a^7*b^5)*e^16 + 22*(21*a^6*b^6*d^10 + 1365*a^6*b^6*d^8 + 13650*a^6*b^6*d^6 + 46410*
a^6*b^6*d^4 + 62985*a^6*b^6*d^2 + 29393*a^6*b^6)*e^14 + 22*(21*a^5*b^7*d^12 + 1386*a^5*b^7*d^10 + 15015*a^5*b^
7*d^8 + 60060*a^5*b^7*d^6 + 109395*a^5*b^7*d^4 + 92378*a^5*b^7*d^2 + 29393*a^5*b^7)*e^12 + 330*(a^4*b^8*d^14 +
63*a^4*b^8*d^12 + 693*a^4*b^8*d^10 + 3003*a^4*b^8*d^8 + 6435*a^4*b^8*d^6 + 7293*a^4*b^8*d^4 + 4199*a^4*b^8*d^
2 + 969*a^4*b^8)*e^10 + 33*(5*a^3*b^9*d^16 + 280*a^3*b^9*d^14 + 2940*a^3*b^9*d^12 + 12936*a^3*b^9*d^10 + 30030
*a^3*b^9*d^8 + 40040*a^3*b^9*d^6 + 30940*a^3*b^9*d^4 + 12920*a^3*b^9*d^2 + 2261*a^3*b^9)*e^8 + 55*(a^2*b^10*d^
18 + 45*a^2*b^10*d^16 + 420*a^2*b^10*d^14 + 1764*a^2*b^10*d^12 + 4158*a^2*b^10*d^10 + 6006*a^2*b^10*d^8 + 5460
*a^2*b^10*d^6 + 3060*a^2*b^10*d^4 + 969*a^2*b^10*d^2 + 133*a^2*b^10)*e^6 + 11*(a*b^11*d^20 + 30*a*b^11*d^18 +
225*a*b^11*d^16 + 840*a*b^11*d^14 + 1890*a*b^11*d^12 + 2772*a*b^11*d^10 + 2730*a*b^11*d^8 + 1800*a*b^11*d^6 +
765*a*b^11*d^4 + 190*a*b^11*d^2 + 21*a*b^11)*e^4 + (b^12*d^22 + 11*b^12*d^20 + 55*b^12*d^18 + 165*b^12*d^16 +
330*b^12*d^14 + 462*b^12*d^12 + 462*b^12*d^10 + 330*b^12*d^8 + 165*b^12*d^6 + 55*b^12*d^4 + 11*b^12*d^2 + b^12
)*e^2)*x^2 - 2*(11*(a^10*b*d^2 + a^10*b)*e^21 + 110*(a^9*b^2*d^4 + 8*a^9*b^2*d^2 + 7*a^9*b^2)*e^19 + 33*(15*a^
8*b^3*d^6 + 205*a^8*b^3*d^4 + 589*a^8*b^3*d^2 + 399*a^8*b^3)*e^17 + 264*(5*a^7*b^4*d^8 + 90*a^7*b^4*d^6 + 408*
a^7*b^4*d^4 + 646*a^7*b^4*d^2 + 323*a^7*b^4)*e^15 + 110*(21*a^6*b^5*d^10 + 441*a^6*b^5*d^8 + 2562*a^6*b^5*d^6
+ 6018*a^6*b^5*d^4 + 6137*a^6*b^5*d^2 + 2261*a^6*b^5)*e^13 + 4*(693*a^5*b^6*d^12 + 15708*a^5*b^6*d^10 + 105105
*a^5*b^6*d^8 + 308880*a^5*b^6*d^6 + 449735*a^5*b^6*d^4 + 319124*a^5*b^6*d^2 + 88179*a^5*b^6)*e^11 + 110*(21*a^
4*b^7*d^14 + 483*a^4*b^7*d^12 + 3465*a^4*b^7*d^10 + 11583*a^4*b^7*d^8 + 20735*a^4*b^7*d^6 + 20553*a^4*b^7*d^4
+ 10659*a^4*b^7*d^2 + 2261*a^4*b^7)*e^9 + 264*(5*a^3*b^8*d^16 + 110*a^3*b^8*d^14 + 798*a^3*b^8*d^12 + 2838*a^3
*b^8*d^10 + 5720*a^3*b^8*d^8 + 6890*a^3*b^8*d^6 + 4930*a^3*b^8*d^4 + 1938*a^3*b^8*d^2 + 323*a^3*b^8)*e^7 + 33*
(15*a^2*b^9*d^18 + 295*a^2*b^9*d^16 + 2044*a^2*b^9*d^14 + 7308*a^2*b^9*d^12 + 15554*a^2*b^9*d^10 + 20930*a^2*b
^9*d^8 + 18060*a^2*b^9*d^6 + 9724*a^2*b^9*d^4 + 2983*a^2*b^9*d^2 + 399*a^2*b^9)*e^5 + 110*(a*b^10*d^20 + 16*a*
b^10*d^18 + 99*a*b^10*d^16 + 336*a*b^10*d^14 + 714*a*b^10*d^12 + 1008*a*b^10*d^10 + 966*a*b^10*d^8 + 624*a*b^1
0*d^6 + 261*a*b^10*d^4 + 64*a*b^10*d^2 + 7*a*b^10)*e^3 + (11*a^10*b*e^23 + 110*(a^9*b^2*d^2 + 7*a^9*b^2)*e^21
+ 33*(15*a^8*b^3*d^4 + 190*a^8*b^3*d^2 + 399*a^8*b^3)*e^19 + 264*(5*a^7*b^4*d^6 + 85*a^7*b^4*d^4 + 323*a^7*b^4
*d^2 + 323*a^7*b^4)*e^17 + 110*(21*a^6*b^5*d^8 + 420*a^6*b^5*d^6 + 2142*a^6*b^5*d^4 + 3876*a^6*b^5*d^2 + 2261*
a^6*b^5)*e^15 + 4*(693*a^5*b^6*d^10 + 15015*a^5*b^6*d^8 + 90090*a^5*b^6*d^6 + 218790*a^5*b^6*d^4 + 230945*a^5*
b^6*d^2 + 88179*a^5*b^6)*e^13 + 110*(21*a^4*b^7*d^12 + 462*a^4*b^7*d^10 + 3003*a^4*b^7*d^8 + 8580*a^4*b^7*d^6
+ 12155*a^4*b^7*d^4 + 8398*a^4*b^7*d^2 + 2261*a^4*b^7)*e^11 + 264*(5*a^3*b^8*d^14 + 105*a^3*b^8*d^12 + 693*a^3
*b^8*d^10 + 2145*a^3*b^8*d^8 + 3575*a^3*b^8*d^6 + 3315*a^3*b^8*d^4 + 1615*a^3*b^8*d^2 + 323*a^3*b^8)*e^9 + 33*
(15*a^2*b^9*d^16 + 280*a^2*b^9*d^14 + 1764*a^2*b^9*d^12 + 5544*a^2*b^9*d^10 + 10010*a^2*b^9*d^8 + 10920*a^2*b^
9*d^6 + 7140*a^2*b^9*d^4 + 2584*a^2*b^9*d^2 + 399*a^2*b^9)*e^7 + 110*(a*b^10*d^18 + 15*a*b^10*d^16 + 84*a*b^10
*d^14 + 252*a*b^10*d^12 + 462*a*b^10*d^10 + 546*a*b^10*d^8 + 420*a*b^10*d^6 + 204*a*b^10*d^4 + 57*a*b^10*d^2 +
7*a*b^10)*e^5 + 11*(b^11*d^20 + 10*b^11*d^18 + 45*b^11*d^16 + 120*b^11*d^14 + 210*b^11*d^12 + 252*b^11*d^10 +
210*b^11*d^8 + 120*b^11*d^6 + 45*b^11*d^4 + 10*b^11*d^2 + b^11)*e^3)*x^2 + 11*(b^11*d^22 + 11*b^11*d^20 + 55*
b^11*d^18 + 165*b^11*d^16 + 330*b^11*d^14 + 462*b^11*d^12 + 462*b^11*d^10 + 330*b^11*d^8 + 165*b^11*d^6 + 55*b
^11*d^4 + 11*b^11*d^2 + b^11)*e + 2*(11*a^10*b*d*e^22 + 110*(a^9*b^2*d^3 + 7*a^9*b^2*d)*e^20 + 33*(15*a^8*b^3*
d^5 + 190*a^8*b^3*d^3 + 399*a^8*b^3*d)*e^18 + 264*(5*a^7*b^4*d^7 + 85*a^7*b^4*d^5 + 323*a^7*b^4*d^3 + 323*a^7*
b^4*d)*e^16 + 110*(21*a^6*b^5*d^9 + 420*a^6*b^5*d^7 + 2142*a^6*b^5*d^5 + 3876*a^6*b^5*d^3 + 2261*a^6*b^5*d)*e^
14 + 4*(693*a^5*b^6*d^11 + 15015*a^5*b^6*d^9 + 90090*a^5*b^6*d^7 + 218790*a^5*b^6*d^5 + 230945*a^5*b^6*d^3 + 8
8179*a^5*b^6*d)*e^12 + 110*(21*a^4*b^7*d^13 + 462*a^4*b^7*d^11 + 3003*a^4*b^7*d^9 + 8580*a^4*b^7*d^7 + 12155*a
^4*b^7*d^5 + 8398*a^4*b^7*d^3 + 2261*a^4*b^7*d)*e^10 + 264*(5*a^3*b^8*d^15 + 105*a^3*b^8*d^13 + 693*a^3*b^8*d^
11 + 2145*a^3*b^8*d^9 + 3575*a^3*b^8*d^7 + 3315*a^3*b^8*d^5 + 1615*a^3*b^8*d^3 + 323*a^3*b^8*d)*e^8 + 33*(15*a
^2*b^9*d^17 + 280*a^2*b^9*d^15 + 1764*a^2*b^9*d^13 + 5544*a^2*b^9*d^11 + 10010*a^2*b^9*d^9 + 10920*a^2*b^9*d^7
+ 7140*a^2*b^9*d^5 + 2584*a^2*b^9*d^3 + 399*a^2*b^9*d)*e^6 + 110*(a*b^10*d^19 + 15*a*b^10*d^17 + 84*a*b^10*d^
15 + 252*a*b^10*d^13 + 462*a*b^10*d^11 + 546*a*b^10*d^9 + 420*a*b^10*d^7 + 204*a*b^10*d^5 + 57*a*b^10*d^3 + 7*
a*b^10*d)*e^4 + 11*(b^11*d^21 + 10*b^11*d^19 + 45*b^11*d^17 + 120*b^11*d^15 + 210*b^11*d^13 + 252*b^11*d^11 +
210*b^11*d^9 + 120*b^11*d^7 + 45*b^11*d^5 + 10*b^11*d^3 + b^11*d)*e^2)*x)*sqrt(a)*sqrt(b) + 2*(a^11*b*d*e^23 +
11*(a^10*b^2*d^3 + 21*a^10*b^2*d)*e^21 + 55*(a^9*b^3*d^5 + 38*a^9*b^3*d^3 + 133*a^9*b^3*d)*e^19 + 33*(5*a^8*b
^4*d^7 + 255*a^8*b^4*d^5 + 1615*a^8*b^4*d^3 + 2261*a^8*b^4*d)*e^17 + 330*(a^7*b^5*d^9 + 60*a^7*b^5*d^7 + 510*a
^7*b^5*d^5 + 1292*a^7*b^5*d^3 + 969*a^7*b^5*d)*e^15 + 22*(21*a^6*b^6*d^11 + 1365*a^6*b^6*d^9 + 13650*a^6*b^6*d
^7 + 46410*a^6*b^6*d^5 + 62985*a^6*b^6*d^3 + 29393*a^6*b^6*d)*e^13 + 22*(21*a^5*b^7*d^13 + 1386*a^5*b^7*d^11 +
15015*a^5*b^7*d^9 + 60060*a^5*b^7*d^7 + 109395*a^5*b^7*d^5 + 92378*a^5*b^7*d^3 + 29393*a^5*b^7*d)*e^11 + 330*
(a^4*b^8*d^15 + 63*a^4*b^8*d^13 + 693*a^4*b^8*d^11 + 3003*a^4*b^8*d^9 + 6435*a^4*b^8*d^7 + 7293*a^4*b^8*d^5 +
4199*a^4*b^8*d^3 + 969*a^4*b^8*d)*e^9 + 33*(5*a^3*b^9*d^17 + 280*a^3*b^9*d^15 + 2940*a^3*b^9*d^13 + 12936*a^3*
b^9*d^11 + 30030*a^3*b^9*d^9 + 40040*a^3*b^9*d^7 + 30940*a^3*b^9*d^5 + 12920*a^3*b^9*d^3 + 2261*a^3*b^9*d)*e^7
+ 55*(a^2*b^10*d^19 + 45*a^2*b^10*d^17 + 420*a^2*b^10*d^15 + 1764*a^2*b^10*d^13 + 4158*a^2*b^10*d^11 + 6006*a
^2*b^10*d^9 + 5460*a^2*b^10*d^7 + 3060*a^2*b^10*d^5 + 969*a^2*b^10*d^3 + 133*a^2*b^10*d)*e^5 + 11*(a*b^11*d^21
+ 30*a*b^11*d^19 + 225*a*b^11*d^17 + 840*a*b^11*d^15 + 1890*a*b^11*d^13 + 2772*a*b^11*d^11 + 2730*a*b^11*d^9
+ 1800*a*b^11*d^7 + 765*a*b^11*d^5 + 190*a*b^11*d^3 + 21*a*b^11*d)*e^3 + (b^12*d^23 + 11*b^12*d^21 + 55*b^12*d
^19 + 165*b^12*d^17 + 330*b^12*d^15 + 462*b^12*d^13 + 462*b^12*d^11 + 330*b^12*d^9 + 165*b^12*d^7 + 55*b^12*d^
5 + 11*b^12*d^3 + b^12*d)*e)*x)/(b^12*d^24 + a^12*e^24 + 12*b^12*d^22 + 66*b^12*d^20 + 220*b^12*d^18 + 495*b^1
2*d^16 + 792*b^12*d^14 + 924*b^12*d^12 + 12*(a^11*b*d^2 + 23*a^11*b)*e^22 + 792*b^12*d^10 + 66*(a^10*b^2*d^4 +
42*a^10*b^2*d^2 + 161*a^10*b^2)*e^20 + 495*b^12*d^8 + 44*(5*a^9*b^3*d^6 + 285*a^9*b^3*d^4 + 1995*a^9*b^3*d^2
+ 3059*a^9*b^3)*e^18 + 220*b^12*d^6 + 99*(5*a^8*b^4*d^8 + 340*a^8*b^4*d^6 + 3230*a^8*b^4*d^4 + 9044*a^8*b^4*d^
2 + 7429*a^8*b^4)*e^16 + 66*b^12*d^4 + 264*(3*a^7*b^5*d^10 + 225*a^7*b^5*d^8 + 2550*a^7*b^5*d^6 + 9690*a^7*b^5
*d^4 + 14535*a^7*b^5*d^2 + 7429*a^7*b^5)*e^14 + 12*b^12*d^2 + 4*(231*a^6*b^6*d^12 + 18018*a^6*b^6*d^10 + 22522
5*a^6*b^6*d^8 + 1021020*a^6*b^6*d^6 + 2078505*a^6*b^6*d^4 + 1939938*a^6*b^6*d^2 + 676039*a^6*b^6)*e^12 + b^12
+ 264*(3*a^5*b^7*d^14 + 231*a^5*b^7*d^12 + 3003*a^5*b^7*d^10 + 15015*a^5*b^7*d^8 + 36465*a^5*b^7*d^6 + 46189*a
^5*b^7*d^4 + 29393*a^5*b^7*d^2 + 7429*a^5*b^7)*e^10 + 99*(5*a^4*b^8*d^16 + 360*a^4*b^8*d^14 + 4620*a^4*b^8*d^1
2 + 24024*a^4*b^8*d^10 + 64350*a^4*b^8*d^8 + 97240*a^4*b^8*d^6 + 83980*a^4*b^8*d^4 + 38760*a^4*b^8*d^2 + 7429*
a^4*b^8)*e^8 + 44*(5*a^3*b^9*d^18 + 315*a^3*b^9*d^16 + 3780*a^3*b^9*d^14 + 19404*a^3*b^9*d^12 + 54054*a^3*b^9*
d^10 + 90090*a^3*b^9*d^8 + 92820*a^3*b^9*d^6 + 58140*a^3*b^9*d^4 + 20349*a^3*b^9*d^2 + 3059*a^3*b^9)*e^6 + 66*
(a^2*b^10*d^20 + 50*a^2*b^10*d^18 + 525*a^2*b^10*d^16 + 2520*a^2*b^10*d^14 + 6930*a^2*b^10*d^12 + 12012*a^2*b^
10*d^10 + 13650*a^2*b^10*d^8 + 10200*a^2*b^10*d^6 + 4845*a^2*b^10*d^4 + 1330*a^2*b^10*d^2 + 161*a^2*b^10)*e^4
+ 12*(a*b^11*d^22 + 33*a*b^11*d^20 + 275*a*b^11*d^18 + 1155*a*b^11*d^16 + 2970*a*b^11*d^14 + 5082*a*b^11*d^12
+ 6006*a*b^11*d^10 + 4950*a*b^11*d^8 + 2805*a*b^11*d^6 + 1045*a*b^11*d^4 + 231*a*b^11*d^2 + 23*a*b^11)*e^2 - 8
*(3*a^11*e^23 + 11*(3*a^10*b*d^2 + 23*a^10*b)*e^21 + 33*(5*a^9*b^2*d^4 + 70*a^9*b^2*d^2 + 161*a^9*b^2)*e^19 +
99*(5*a^8*b^3*d^6 + 95*a^8*b^3*d^4 + 399*a^8*b^3*d^2 + 437*a^8*b^3)*e^17 + 22*(45*a^7*b^4*d^8 + 1020*a^7*b^4*d
^6 + 5814*a^7*b^4*d^4 + 11628*a^7*b^4*d^2 + 7429*a^7*b^4)*e^15 + 6*(231*a^6*b^5*d^10 + 5775*a^6*b^5*d^8 + 3927
0*a^6*b^5*d^6 + 106590*a^6*b^5*d^4 + 124355*a^6*b^5*d^2 + 52003*a^6*b^5)*e^13 + 6*(231*a^5*b^6*d^12 + 6006*a^5
*b^6*d^10 + 45045*a^5*b^6*d^8 + 145860*a^5*b^6*d^6 + 230945*a^5*b^6*d^4 + 176358*a^5*b^6*d^2 + 52003*a^5*b^6)*
e^11 + 22*(45*a^4*b^7*d^14 + 1155*a^4*b^7*d^12 + 9009*a^4*b^7*d^10 + 32175*a^4*b^7*d^8 + 60775*a^4*b^7*d^6 + 6
2985*a^4*b^7*d^4 + 33915*a^4*b^7*d^2 + 7429*a^4*b^7)*e^9 + 99*(5*a^3*b^8*d^16 + 120*a^3*b^8*d^14 + 924*a^3*b^8
*d^12 + 3432*a^3*b^8*d^10 + 7150*a^3*b^8*d^8 + 8840*a^3*b^8*d^6 + 6460*a^3*b^8*d^4 + 2584*a^3*b^8*d^2 + 437*a^
3*b^8)*e^7 + 33*(5*a^2*b^9*d^18 + 105*a^2*b^9*d^16 + 756*a^2*b^9*d^14 + 2772*a^2*b^9*d^12 + 6006*a^2*b^9*d^10
+ 8190*a^2*b^9*d^8 + 7140*a^2*b^9*d^6 + 3876*a^2*b^9*d^4 + 1197*a^2*b^9*d^2 + 161*a^2*b^9)*e^5 + 11*(3*a*b^10*
d^20 + 50*a*b^10*d^18 + 315*a*b^10*d^16 + 1080*a*b^10*d^14 + 2310*a*b^10*d^12 + 3276*a*b^10*d^10 + 3150*a*b^10
*d^8 + 2040*a*b^10*d^6 + 855*a*b^10*d^4 + 210*a*b^10*d^2 + 23*a*b^10)*e^3 + 3*(b^11*d^22 + 11*b^11*d^20 + 55*b
^11*d^18 + 165*b^11*d^16 + 330*b^11*d^14 + 462*b^11*d^12 + 462*b^11*d^10 + 330*b^11*d^8 + 165*b^11*d^6 + 55*b^
11*d^4 + 11*b^11*d^2 + b^11)*e)*sqrt(a)*sqrt(b))) + 2*dilog(-(a*e^2 + (b*d + I*b)*e*x + (I*e^2*x + (-I*d + 1)*
e)*sqrt(a)*sqrt(b))/(b*d^2 - 2*sqrt(a)*sqrt(b)*(-I*d + 1)*e - a*e^2 + 2*I*b*d - b)) - 2*dilog(-(a*e^2 + (b*d +
I*b)*e*x - (I*e^2*x + (-I*d + 1)*e)*sqrt(a)*sqrt(b))/(b*d^2 + 2*sqrt(a)*sqrt(b)*(-I*d + 1)*e - a*e^2 + 2*I*b*
d - b)) - 2*dilog(-(a*e^2 + (b*d - I*b)*e*x + (I*e^2*x + (-I*d - 1)*e)*sqrt(a)*sqrt(b))/(b*d^2 - 2*sqrt(a)*sqr
t(b)*(-I*d - 1)*e - a*e^2 - 2*I*b*d - b)) + 2*dilog(-(a*e^2 + (b*d - I*b)*e*x - (I*e^2*x + (-I*d - 1)*e)*sqrt(
a)*sqrt(b))/(b*d^2 + 2*sqrt(a)*sqrt(b)*(-I*d - 1)*e - a*e^2 - 2*I*b*d - b)))/e)/sqrt(a*b) + arctan(e*x + d)*ar
ctan(b*x/sqrt(a*b))/sqrt(a*b) - arctan(b*x/sqrt(a*b))*arctan((e^2*x + d*e)/e)/sqrt(a*b)

Giac [F]

\[ \int \frac {\arctan (d+e x)}{a+b x^2} \, dx=\int { \frac {\arctan \left (e x + d\right )}{b x^{2} + a} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (d+e x)}{a+b x^2} \, dx=\int \frac {\mathrm {atan}\left (d+e\,x\right )}{b\,x^2+a} \,d x \]

[In]

int(atan(d + e*x)/(a + b*x^2),x)

[Out]

int(atan(d + e*x)/(a + b*x^2), x)