3.1.61 \(\int \frac {\text {ArcTan}(d+e x)}{a+b x^2} \, dx\) [61]
Optimal. Leaf size=543 \[ \frac {i \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} (i+d)+\sqrt {-a} e}\right ) \log (1-i d-i e x)}{4 \sqrt {-a} \sqrt {b}}-\frac {i \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} (i+d)-\sqrt {-a} e}\right ) \log (1-i d-i e x)}{4 \sqrt {-a} \sqrt {b}}-\frac {i \log \left (-\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} (i-d)-\sqrt {-a} e}\right ) \log (1+i d+i e x)}{4 \sqrt {-a} \sqrt {b}}+\frac {i \log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} (i-d)+\sqrt {-a} e}\right ) \log (1+i d+i e x)}{4 \sqrt {-a} \sqrt {b}}-\frac {i \text {PolyLog}\left (2,\frac {\sqrt {b} (i-d-e x)}{\sqrt {b} (i-d)-\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {i \text {PolyLog}\left (2,\frac {\sqrt {b} (i-d-e x)}{\sqrt {b} (i-d)+\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {i \text {PolyLog}\left (2,\frac {\sqrt {b} (i+d+e x)}{\sqrt {b} (i+d)-\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {i \text {PolyLog}\left (2,\frac {\sqrt {b} (i+d+e x)}{\sqrt {b} (i+d)+\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}} \]
[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]
time = 0.47, antiderivative size = 543, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5159, 2456,
2441, 2440, 2438} \begin {gather*} -\frac {i \text {Li}_2\left (\frac {\sqrt {b} (-d-e x+i)}{\sqrt {b} (i-d)-\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {i \text {Li}_2\left (\frac {\sqrt {b} (-d-e x+i)}{\sqrt {b} (i-d)+\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {i \text {Li}_2\left (\frac {\sqrt {b} (d+e x+i)}{\sqrt {b} (d+i)-\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {i \text {Li}_2\left (\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}} \end {gather*}
Antiderivative was successfully verified.
[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*} \int \frac {\tan ^{-1}(d+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 \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 \text {Li}_2\left (\frac {\sqrt {b} (i-d-e x)}{\sqrt {b} (i-d)-\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {i \text {Li}_2\left (\frac {\sqrt {b} (i-d-e x)}{\sqrt {b} (i-d)+\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {i \text {Li}_2\left (\frac {\sqrt {b} (i+d+e x)}{\sqrt {b} (i+d)-\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}+\frac {i \text {Li}_2\left (\frac {\sqrt {b} (i+d+e x)}{\sqrt {b} (i+d)+\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 409, normalized size = 0.75 \begin {gather*} \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))+\text {PolyLog}\left (2,\frac {\sqrt {b} (-i+d+e x)}{\sqrt {b} (-i+d)-\sqrt {-a} e}\right )-\text {PolyLog}\left (2,\frac {\sqrt {b} (-i+d+e x)}{\sqrt {b} (-i+d)+\sqrt {-a} e}\right )-\text {PolyLog}\left (2,\frac {\sqrt {b} (i+d+e x)}{\sqrt {b} (i+d)-\sqrt {-a} e}\right )+\text {PolyLog}\left (2,\frac {\sqrt {b} (i+d+e x)}{\sqrt {b} (i+d)+\sqrt {-a} e}\right )\right )}{4 \sqrt {-a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
[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])
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 2183 vs. \(2 (411 ) = 822\).
time = 0.47, size = 2184, normalized size = 4.02
| | |
| method |
result |
size |
| | |
| risch |
\(\frac {\ln \left (-i e x -i d +1\right ) \ln \left (\frac {i b d -e \sqrt {a b}+b \left (-i e x -i d +1\right )-b}{i b d -e \sqrt {a b}-b}\right )}{4 \sqrt {a b}}-\frac {\ln \left (-i e x -i d +1\right ) \ln \left (\frac {i b d +e \sqrt {a b}+b \left (-i e x -i d +1\right )-b}{i b d +e \sqrt {a b}-b}\right )}{4 \sqrt {a b}}+\frac {\dilog \left (\frac {i b d -e \sqrt {a b}+b \left (-i e x -i d +1\right )-b}{i b d -e \sqrt {a b}-b}\right )}{4 \sqrt {a b}}-\frac {\dilog \left (\frac {i b d +e \sqrt {a b}+b \left (-i e x -i d +1\right )-b}{i b d +e \sqrt {a b}-b}\right )}{4 \sqrt {a b}}+\frac {\ln \left (i e x +i d +1\right ) \ln \left (\frac {i b d +e \sqrt {a b}-b \left (i e x +i d +1\right )+b}{i b d +e \sqrt {a b}+b}\right )}{4 \sqrt {a b}}-\frac {\ln \left (i e x +i d +1\right ) \ln \left (\frac {i b d -e \sqrt {a b}-b \left (i e x +i d +1\right )+b}{i b d -e \sqrt {a b}+b}\right )}{4 \sqrt {a b}}+\frac {\dilog \left (\frac {i b d +e \sqrt {a b}-b \left (i e x +i d +1\right )+b}{i b d +e \sqrt {a b}+b}\right )}{4 \sqrt {a b}}-\frac {\dilog \left (\frac {i b d -e \sqrt {a b}-b \left (i e x +i d +1\right )+b}{i b d -e \sqrt {a b}+b}\right )}{4 \sqrt {a b}}\) |
\(494\) |
| derivativedivides |
\(\text {Expression too large to display}\) |
\(2184\) |
| default |
\(\text {Expression too large to display}\) |
\(2184\) |
| | |
|
|
|
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(arctan(e*x+d)/(b*x^2+a),x,method=_RETURNVERBOSE)
[Out]
1/e*(I*e^2/(a*e^2+d^2*b-2*(a*b*e^2)^(1/2)+b)*ln(1-(2*I*b*d+a*e^2+d^2*b-b)*(1+I*(e*x+d))^2/((e*x+d)^2+1)/(-a*e^
2-d^2*b+2*(a*b*e^2)^(1/2)-b))*arctan(e*x+d)+1/2*I/b*e^2*(a*b*e^2)^(1/2)/(a*e^2+d^2*b+2*(a*b*e^2)^(1/2)+b)*ln(1
-(2*I*b*d+a*e^2+d^2*b-b)*(1+I*(e*x+d))^2/((e*x+d)^2+1)/(-a*e^2-d^2*b-2*(a*b*e^2)^(1/2)-b))*arctan(e*x+d)+1/2*I
*(a*b*e^2)^(1/2)/a/(a*e^2+d^2*b+2*(a*b*e^2)^(1/2)+b)*ln(1-(2*I*b*d+a*e^2+d^2*b-b)*(1+I*(e*x+d))^2/((e*x+d)^2+1
)/(-a*e^2-d^2*b-2*(a*b*e^2)^(1/2)-b))*d^2*arctan(e*x+d)+I*e^2/(a*e^2+d^2*b+2*(a*b*e^2)^(1/2)+b)*ln(1-(2*I*b*d+
a*e^2+d^2*b-b)*(1+I*(e*x+d))^2/((e*x+d)^2+1)/(-a*e^2-d^2*b-2*(a*b*e^2)^(1/2)-b))*arctan(e*x+d)-1/2/b*e^2*(a*b*
e^2)^(1/2)/(a*e^2+d^2*b-2*(a*b*e^2)^(1/2)+b)*arctan(e*x+d)^2+e^2/(a*e^2+d^2*b-2*(a*b*e^2)^(1/2)+b)*arctan(e*x+
d)^2-1/2*(a*b*e^2)^(1/2)/a/(a*e^2+d^2*b-2*(a*b*e^2)^(1/2)+b)*arctan(e*x+d)^2-1/2*(a*b*e^2)^(1/2)/a/(a*e^2+d^2*
b-2*(a*b*e^2)^(1/2)+b)*d^2*arctan(e*x+d)^2-1/4/b*e^2*(a*b*e^2)^(1/2)/(a*e^2+d^2*b-2*(a*b*e^2)^(1/2)+b)*polylog
(2,(2*I*b*d+a*e^2+d^2*b-b)*(1+I*(e*x+d))^2/((e*x+d)^2+1)/(-a*e^2-d^2*b+2*(a*b*e^2)^(1/2)-b))+1/2*e^2/(a*e^2+d^
2*b-2*(a*b*e^2)^(1/2)+b)*polylog(2,(2*I*b*d+a*e^2+d^2*b-b)*(1+I*(e*x+d))^2/((e*x+d)^2+1)/(-a*e^2-d^2*b+2*(a*b*
e^2)^(1/2)-b))-1/4*(a*b*e^2)^(1/2)/a/(a*e^2+d^2*b-2*(a*b*e^2)^(1/2)+b)*polylog(2,(2*I*b*d+a*e^2+d^2*b-b)*(1+I*
(e*x+d))^2/((e*x+d)^2+1)/(-a*e^2-d^2*b+2*(a*b*e^2)^(1/2)-b))-1/4*(a*b*e^2)^(1/2)/a/(a*e^2+d^2*b-2*(a*b*e^2)^(1
/2)+b)*polylog(2,(2*I*b*d+a*e^2+d^2*b-b)*(1+I*(e*x+d))^2/((e*x+d)^2+1)/(-a*e^2-d^2*b+2*(a*b*e^2)^(1/2)-b))*d^2
+1/2*I*(a*b*e^2)^(1/2)/a/(a*e^2+d^2*b+2*(a*b*e^2)^(1/2)+b)*ln(1-(2*I*b*d+a*e^2+d^2*b-b)*(1+I*(e*x+d))^2/((e*x+
d)^2+1)/(-a*e^2-d^2*b-2*(a*b*e^2)^(1/2)-b))*arctan(e*x+d)-1/2*I*(a*b*e^2)^(1/2)/a/(a*e^2+d^2*b-2*(a*b*e^2)^(1/
2)+b)*ln(1-(2*I*b*d+a*e^2+d^2*b-b)*(1+I*(e*x+d))^2/((e*x+d)^2+1)/(-a*e^2-d^2*b+2*(a*b*e^2)^(1/2)-b))*d^2*arcta
n(e*x+d)-1/2*I/b*e^2*(a*b*e^2)^(1/2)/(a*e^2+d^2*b-2*(a*b*e^2)^(1/2)+b)*ln(1-(2*I*b*d+a*e^2+d^2*b-b)*(1+I*(e*x+
d))^2/((e*x+d)^2+1)/(-a*e^2-d^2*b+2*(a*b*e^2)^(1/2)-b))*arctan(e*x+d)-1/2*I*(a*b*e^2)^(1/2)/a/(a*e^2+d^2*b-2*(
a*b*e^2)^(1/2)+b)*ln(1-(2*I*b*d+a*e^2+d^2*b-b)*(1+I*(e*x+d))^2/((e*x+d)^2+1)/(-a*e^2-d^2*b+2*(a*b*e^2)^(1/2)-b
))*arctan(e*x+d)+1/2/b*e^2*(a*b*e^2)^(1/2)/(a*e^2+d^2*b+2*(a*b*e^2)^(1/2)+b)*arctan(e*x+d)^2+e^2/(a*e^2+d^2*b+
2*(a*b*e^2)^(1/2)+b)*arctan(e*x+d)^2+1/2*(a*b*e^2)^(1/2)/a/(a*e^2+d^2*b+2*(a*b*e^2)^(1/2)+b)*arctan(e*x+d)^2+1
/2*(a*b*e^2)^(1/2)/a/(a*e^2+d^2*b+2*(a*b*e^2)^(1/2)+b)*d^2*arctan(e*x+d)^2+1/4/b*e^2*(a*b*e^2)^(1/2)/(a*e^2+d^
2*b+2*(a*b*e^2)^(1/2)+b)*polylog(2,(2*I*b*d+a*e^2+d^2*b-b)*(1+I*(e*x+d))^2/((e*x+d)^2+1)/(-a*e^2-d^2*b-2*(a*b*
e^2)^(1/2)-b))+1/2*e^2/(a*e^2+d^2*b+2*(a*b*e^2)^(1/2)+b)*polylog(2,(2*I*b*d+a*e^2+d^2*b-b)*(1+I*(e*x+d))^2/((e
*x+d)^2+1)/(-a*e^2-d^2*b-2*(a*b*e^2)^(1/2)-b))+1/4*(a*b*e^2)^(1/2)/a/(a*e^2+d^2*b+2*(a*b*e^2)^(1/2)+b)*polylog
(2,(2*I*b*d+a*e^2+d^2*b-b)*(1+I*(e*x+d))^2/((e*x+d)^2+1)/(-a*e^2-d^2*b-2*(a*b*e^2)^(1/2)-b))+1/4*(a*b*e^2)^(1/
2)/a/(a*e^2+d^2*b+2*(a*b*e^2)^(1/2)+b)*polylog(2,(2*I*b*d+a*e^2+d^2*b-b)*(1+I*(e*x+d))^2/((e*x+d)^2+1)/(-a*e^2
-d^2*b-2*(a*b*e^2)^(1/2)-b))*d^2)
________________________________________________________________________________________
Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 13320 vs. \(2 (389) = 778\).
time = 1.16, size = 13320, normalized size = 24.53 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arctan(e*x+d)/(b*x^2+a),x, algorithm="maxima")
[Out]
arctan(x*e + d)*arctan(b*x/sqrt(a*b))/sqrt(a*b) - arctan((x*e^2 + d*e)*e^(-1))*arctan(b*x/sqrt(a*b))/sqrt(a*b)
+ 1/8*(8*arctan((x*e^2 + d*e)*e^(-1))*arctan(b*x/sqrt(a*b))*e^(-1) - (4*arctan(sqrt(b)*x/sqrt(a))*arctan2((2*
a*b*d*e^2 + (b*d^3*e + (a*e^3 + b*e)*d + (b*d^2*e^2 + a*e^4 + 3*b*e^2)*x)*sqrt(a)*sqrt(b) + (b^2*d^2*e + 3*a*b
*e^3 + b^2*e)*x)/(b^2*d^4 + 2*(a*b*e^2 + b^2)*d^2 + a^2*e^4 + 6*a*b*e^2 + 4*(b*d^2*e + a*e^3 + b*e)*sqrt(a)*sq
rt(b) + b^2), (b^2*d^4 + (a*b*e^2 + 2*b^2)*d^2 + 3*a*b*e^2 + (2*b*d*x*e^2 + 3*b*d^2*e + a*e^3 + 3*b*e)*sqrt(a)
*sqrt(b) + b^2 + (b^2*d^3*e + (a*b*e^3 + b^2*e)*d)*x)/(b^2*d^4 + 2*(a*b*e^2 + b^2)*d^2 + a^2*e^4 + 6*a*b*e^2 +
4*(b*d^2*e + a*e^3 + b*e)*sqrt(a)*sqrt(b) + b^2)) + 4*arctan(sqrt(b)*x/sqrt(a))*arctan2((2*a*b*d*e^2 - (b*d^3
*e + (a*e^3 + b*e)*d + (b*d^2*e^2 + a*e^4 + 3*b*e^2)*x)*sqrt(a)*sqrt(b) + (b^2*d^2*e + 3*a*b*e^3 + b^2*e)*x)/(
b^2*d^4 + 2*(a*b*e^2 + b^2)*d^2 + a^2*e^4 + 6*a*b*e^2 - 4*(b*d^2*e + a*e^3 + b*e)*sqrt(a)*sqrt(b) + b^2), (b^2
*d^4 + (a*b*e^2 + 2*b^2)*d^2 + 3*a*b*e^2 - (2*b*d*x*e^2 + 3*b*d^2*e + a*e^3 + 3*b*e)*sqrt(a)*sqrt(b) + b^2 + (
b^2*d^3*e + (a*b*e^3 + b^2*e)*d)*x)/(b^2*d^4 + 2*(a*b*e^2 + b^2)*d^2 + a^2*e^4 + 6*a*b*e^2 - 4*(b*d^2*e + a*e^
3 + b*e)*sqrt(a)*sqrt(b) + b^2)) + log(b*x^2 + a)*log((b^12*d^24 + (11*a*b^11*e^2 + 12*b^12)*d^22 + 11*(5*a^2*
b^10*e^4 + 31*a*b^11*e^2 + 6*b^12)*d^20 + 55*(3*a^3*b^9*e^6 + 46*a^2*b^10*e^4 + 51*a*b^11*e^2 + 4*b^12)*d^18 +
165*(2*a^4*b^8*e^8 + 57*a^3*b^9*e^6 + 155*a^2*b^10*e^4 + 71*a*b^11*e^2 + 3*b^12)*d^16 + 66*(7*a^5*b^7*e^10 +
320*a^4*b^8*e^8 + 1610*a^3*b^9*e^6 + 1820*a^2*b^10*e^4 + 455*a*b^11*e^2 + 12*b^12)*d^14 + 462*(a^6*b^6*e^12 +
67*a^5*b^7*e^10 + 540*a^4*b^8*e^8 + 1134*a^3*b^9*e^6 + 705*a^2*b^10*e^4 + 111*a*b^11*e^2 + 2*b^12)*d^12 + a^11
*b*e^22 + 231*a^10*b^2*e^20 + 7315*a^9*b^3*e^18 + 74613*a^8*b^4*e^16 + 319770*a^7*b^5*e^14 + 646646*a^6*b^6*e^
12 + 646646*a^5*b^7*e^10 + 319770*a^4*b^8*e^8 + 74613*a^3*b^9*e^6 + 7315*a^2*b^10*e^4 + 231*a*b^11*e^2 + b^12
+ 66*(5*a^7*b^5*e^14 + 462*a^6*b^6*e^12 + 5467*a^5*b^7*e^10 + 18480*a^4*b^8*e^8 + 21483*a^3*b^9*e^6 + 8470*a^2
*b^10*e^4 + 917*a*b^11*e^2 + 12*b^12)*d^10 + 165*(a^8*b^4*e^16 + 122*a^7*b^5*e^14 + 2002*a^6*b^6*e^12 + 10010*
a^5*b^7*e^10 + 18876*a^4*b^8*e^8 + 14014*a^3*b^9*e^6 + 3822*a^2*b^10*e^4 + 302*a*b^11*e^2 + 3*b^12)*d^8 + 55*(
a^9*b^3*e^18 + 156*a^8*b^4*e^16 + 3420*a^7*b^5*e^14 + 24024*a^6*b^6*e^12 + 67782*a^5*b^7*e^10 + 82368*a^4*b^8*
e^8 + 42588*a^3*b^9*e^6 + 8520*a^2*b^10*e^4 + 513*a*b^11*e^2 + 4*b^12)*d^6 + 11*(a^10*b^2*e^20 + 195*a^9*b^3*e
^18 + 5610*a^8*b^4*e^16 + 54060*a^7*b^5*e^14 + 218790*a^6*b^6*e^12 + 403546*a^5*b^7*e^10 + 344760*a^4*b^8*e^8
+ 131580*a^3*b^9*e^6 + 20145*a^2*b^10*e^4 + 955*a*b^11*e^2 + 6*b^12)*d^4 + (a^11*b*e^22 + 242*a^10*b^2*e^20 +
9405*a^9*b^3*e^18 + 127908*a^8*b^4*e^16 + 746130*a^7*b^5*e^14 + 2032316*a^6*b^6*e^12 + 2678962*a^5*b^7*e^10 +
1705440*a^4*b^8*e^8 + 500973*a^3*b^9*e^6 + 60610*a^2*b^10*e^4 + 2321*a*b^11*e^2 + 12*b^12)*d^2 + (b^12*d^22*e^
2 + 11*(a*b^11*e^4 + b^12*e^2)*d^20 + 55*(a^2*b^10*e^6 + 6*a*b^11*e^4 + b^12*e^2)*d^18 + 165*(a^3*b^9*e^8 + 15
*a^2*b^10*e^6 + 15*a*b^11*e^4 + b^12*e^2)*d^16 + 330*(a^4*b^8*e^10 + 28*a^3*b^9*e^8 + 70*a^2*b^10*e^6 + 28*a*b
^11*e^4 + b^12*e^2)*d^14 + 462*(a^5*b^7*e^12 + 45*a^4*b^8*e^10 + 210*a^3*b^9*e^8 + 210*a^2*b^10*e^6 + 45*a*b^1
1*e^4 + b^12*e^2)*d^12 + a^11*b*e^24 + 231*a^10*b^2*e^22 + 7315*a^9*b^3*e^20 + 74613*a^8*b^4*e^18 + 319770*a^7
*b^5*e^16 + 646646*a^6*b^6*e^14 + 646646*a^5*b^7*e^12 + 319770*a^4*b^8*e^10 + 74613*a^3*b^9*e^8 + 7315*a^2*b^1
0*e^6 + 231*a*b^11*e^4 + b^12*e^2 + 462*(a^6*b^6*e^14 + 66*a^5*b^7*e^12 + 495*a^4*b^8*e^10 + 924*a^3*b^9*e^8 +
495*a^2*b^10*e^6 + 66*a*b^11*e^4 + b^12*e^2)*d^10 + 330*(a^7*b^5*e^16 + 91*a^6*b^6*e^14 + 1001*a^5*b^7*e^12 +
3003*a^4*b^8*e^10 + 3003*a^3*b^9*e^8 + 1001*a^2*b^10*e^6 + 91*a*b^11*e^4 + b^12*e^2)*d^8 + 165*(a^8*b^4*e^18
+ 120*a^7*b^5*e^16 + 1820*a^6*b^6*e^14 + 8008*a^5*b^7*e^12 + 12870*a^4*b^8*e^10 + 8008*a^3*b^9*e^8 + 1820*a^2*
b^10*e^6 + 120*a*b^11*e^4 + b^12*e^2)*d^6 + 55*(a^9*b^3*e^20 + 153*a^8*b^4*e^18 + 3060*a^7*b^5*e^16 + 18564*a^
6*b^6*e^14 + 43758*a^5*b^7*e^12 + 43758*a^4*b^8*e^10 + 18564*a^3*b^9*e^8 + 3060*a^2*b^10*e^6 + 153*a*b^11*e^4
+ b^12*e^2)*d^4 + 11*(a^10*b^2*e^22 + 190*a^9*b^3*e^20 + 4845*a^8*b^4*e^18 + 38760*a^7*b^5*e^16 + 125970*a^6*b
^6*e^14 + 184756*a^5*b^7*e^12 + 125970*a^4*b^8*e^10 + 38760*a^3*b^9*e^8 + 4845*a^2*b^10*e^6 + 190*a*b^11*e^4 +
b^12*e^2)*d^2)*x^2 + 2*(11*b^11*d^22*e + 11*(10*a*b^10*e^3 + 11*b^11*e)*d^20 + 55*(9*a^2*b^9*e^5 + 32*a*b^10*
e^3 + 11*b^11*e)*d^18 + 165*(8*a^3*b^8*e^7 + 59*a^2*b^9*e^5 + 66*a*b^10*e^3 + 11*b^11*e)*d^16 + 66*(35*a^4*b^7
*e^9 + 440*a^3*b^8*e^7 + 1022*a^2*b^9*e^5 + 560*a*b^10*e^3 + 55*b^11*e)*d^14 + 462*(6*a^5*b^6*e^11 + 115*a^4*b
^7*e^9 + 456*a^3*b^8*e^7 + 522*a^2*b^9*e^5 + 170*a*b^10*e^3 + 11*b^11*e)*d^12 + 11*a^10*b*e^21 + 770*a^9*b^2*e
^19 + 13167*a^8*b^3*e^17 + 85272*a^7*b^4*e^15 + 248710*a^6*b^5*e^13 + 352716*a^5*b^6*e^11 + 248710*a^4*b^7*e^9
+ 85272*a^3*b^8*e^7 + 13167*a^2*b^9*e^5 + 770*...
________________________________________________________________________________________
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(e*x+d)/(b*x^2+a),x, algorithm="fricas")
[Out]
integral(arctan(x*e + d)/(b*x^2 + a), x)
________________________________________________________________________________________
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(e*x+d)/(b*x**2+a),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [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(e*x+d)/(b*x^2+a),x, algorithm="giac")
[Out]
sage0*x
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {atan}\left (d+e\,x\right )}{b\,x^2+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(atan(d + e*x)/(a + b*x^2),x)
[Out]
int(atan(d + e*x)/(a + b*x^2), x)
________________________________________________________________________________________