3.1.53 \(\int \frac {\text {ArcTan}(a+b x)}{c+d x^2} \, dx\) [53]

Optimal. Leaf size=543 \[ -\frac {i \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}-(i-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {i \log (1-i a-i b x) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(i+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {i \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}+(i-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {i \log (1-i a-i b x) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(i+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {i \text {PolyLog}\left (2,-\frac {\sqrt {d} (i-a-b x)}{b \sqrt {-c}-(i-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {i \text {PolyLog}\left (2,\frac {\sqrt {d} (i-a-b x)}{b \sqrt {-c}+(i-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {i \text {PolyLog}\left (2,-\frac {\sqrt {d} (i+a+b x)}{b \sqrt {-c}-(i+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {i \text {PolyLog}\left (2,\frac {\sqrt {d} (i+a+b x)}{b \sqrt {-c}+(i+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}} \]

[Out]

-1/4*I*ln(1+I*a+I*b*x)*ln(b*((-c)^(1/2)-x*d^(1/2))/(b*(-c)^(1/2)-(I-a)*d^(1/2)))/(-c)^(1/2)/d^(1/2)+1/4*I*ln(1
-I*a-I*b*x)*ln(b*((-c)^(1/2)-x*d^(1/2))/(b*(-c)^(1/2)+(I+a)*d^(1/2)))/(-c)^(1/2)/d^(1/2)+1/4*I*ln(1+I*a+I*b*x)
*ln(b*((-c)^(1/2)+x*d^(1/2))/(b*(-c)^(1/2)+(I-a)*d^(1/2)))/(-c)^(1/2)/d^(1/2)-1/4*I*ln(1-I*a-I*b*x)*ln(b*((-c)
^(1/2)+x*d^(1/2))/(b*(-c)^(1/2)-(I+a)*d^(1/2)))/(-c)^(1/2)/d^(1/2)-1/4*I*polylog(2,-(I-a-b*x)*d^(1/2)/(b*(-c)^
(1/2)-(I-a)*d^(1/2)))/(-c)^(1/2)/d^(1/2)+1/4*I*polylog(2,(I-a-b*x)*d^(1/2)/(b*(-c)^(1/2)+(I-a)*d^(1/2)))/(-c)^
(1/2)/d^(1/2)-1/4*I*polylog(2,-(I+a+b*x)*d^(1/2)/(b*(-c)^(1/2)-(I+a)*d^(1/2)))/(-c)^(1/2)/d^(1/2)+1/4*I*polylo
g(2,(I+a+b*x)*d^(1/2)/(b*(-c)^(1/2)+(I+a)*d^(1/2)))/(-c)^(1/2)/d^(1/2)

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Rubi [A]
time = 0.49, 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 {d} (-a-b x+i)}{b \sqrt {-c}-(i-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {i \text {Li}_2\left (\frac {\sqrt {d} (-a-b x+i)}{\sqrt {d} (i-a)+b \sqrt {-c}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {i \text {Li}_2\left (-\frac {\sqrt {d} (a+b x+i)}{b \sqrt {-c}-(a+i) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {i \text {Li}_2\left (\frac {\sqrt {d} (a+b x+i)}{\sqrt {d} (a+i)+b \sqrt {-c}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {i \log (i a+i b x+1) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}-(-a+i) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {i \log (-i a-i b x+1) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(a+i) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {i \log (i a+i b x+1) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}+(-a+i) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {i \log (-i a-i b x+1) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(a+i) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTan[a + b*x]/(c + d*x^2),x]

[Out]

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

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}(a+b x)}{c+d x^2} \, dx &=\frac {1}{2} i \int \frac {\log (1-i a-i b x)}{c+d x^2} \, dx-\frac {1}{2} i \int \frac {\log (1+i a+i b x)}{c+d x^2} \, dx\\ &=\frac {1}{2} i \int \left (\frac {\sqrt {-c} \log (1-i a-i b x)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \log (1-i a-i b x)}{2 c \left (\sqrt {-c}+\sqrt {d} x\right )}\right ) \, dx-\frac {1}{2} i \int \left (\frac {\sqrt {-c} \log (1+i a+i b x)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \log (1+i a+i b x)}{2 c \left (\sqrt {-c}+\sqrt {d} x\right )}\right ) \, dx\\ &=-\frac {i \int \frac {\log (1-i a-i b x)}{\sqrt {-c}-\sqrt {d} x} \, dx}{4 \sqrt {-c}}-\frac {i \int \frac {\log (1-i a-i b x)}{\sqrt {-c}+\sqrt {d} x} \, dx}{4 \sqrt {-c}}+\frac {i \int \frac {\log (1+i a+i b x)}{\sqrt {-c}-\sqrt {d} x} \, dx}{4 \sqrt {-c}}+\frac {i \int \frac {\log (1+i a+i b x)}{\sqrt {-c}+\sqrt {d} x} \, dx}{4 \sqrt {-c}}\\ &=-\frac {i \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}-(i-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {i \log (1-i a-i b x) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(i+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {i \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}+(i-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {i \log (1-i a-i b x) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(i+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {b \int \frac {\log \left (-\frac {i b \left (\sqrt {-c}-\sqrt {d} x\right )}{-i b \sqrt {-c}+(1-i a) \sqrt {d}}\right )}{1-i a-i b x} \, dx}{4 \sqrt {-c} \sqrt {d}}-\frac {b \int \frac {\log \left (\frac {i b \left (\sqrt {-c}-\sqrt {d} x\right )}{i b \sqrt {-c}+(1+i a) \sqrt {d}}\right )}{1+i a+i b x} \, dx}{4 \sqrt {-c} \sqrt {d}}+\frac {b \int \frac {\log \left (-\frac {i b \left (\sqrt {-c}+\sqrt {d} x\right )}{-i b \sqrt {-c}-(1-i a) \sqrt {d}}\right )}{1-i a-i b x} \, dx}{4 \sqrt {-c} \sqrt {d}}+\frac {b \int \frac {\log \left (\frac {i b \left (\sqrt {-c}+\sqrt {d} x\right )}{i b \sqrt {-c}-(1+i a) \sqrt {d}}\right )}{1+i a+i b x} \, dx}{4 \sqrt {-c} \sqrt {d}}\\ &=-\frac {i \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}-(i-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {i \log (1-i a-i b x) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(i+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {i \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}+(i-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {i \log (1-i a-i b x) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(i+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {d} x}{-i b \sqrt {-c}-(1-i a) \sqrt {d}}\right )}{x} \, dx,x,1-i a-i b x\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {i \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {d} x}{-i b \sqrt {-c}+(1-i a) \sqrt {d}}\right )}{x} \, dx,x,1-i a-i b x\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {d} x}{i b \sqrt {-c}-(1+i a) \sqrt {d}}\right )}{x} \, dx,x,1+i a+i b x\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {i \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {d} x}{i b \sqrt {-c}+(1+i a) \sqrt {d}}\right )}{x} \, dx,x,1+i a+i b x\right )}{4 \sqrt {-c} \sqrt {d}}\\ &=-\frac {i \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}-(i-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {i \log (1-i a-i b x) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(i+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {i \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}+(i-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {i \log (1-i a-i b x) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(i+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {i \text {Li}_2\left (-\frac {\sqrt {d} (i-a-b x)}{b \sqrt {-c}-(i-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {i \text {Li}_2\left (\frac {\sqrt {d} (i-a-b x)}{b \sqrt {-c}+(i-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {i \text {Li}_2\left (-\frac {\sqrt {d} (i+a+b x)}{b \sqrt {-c}-(i+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {i \text {Li}_2\left (\frac {\sqrt {d} (i+a+b x)}{b \sqrt {-c}+(i+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 409, normalized size = 0.75 \begin {gather*} -\frac {i \left (\log (1+i a+i b x) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(-i+a) \sqrt {d}}\right )-\log (-i (i+a+b x)) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(i+a) \sqrt {d}}\right )-\log (1+i a+i b x) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(-i+a) \sqrt {d}}\right )+\log (-i (i+a+b x)) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(i+a) \sqrt {d}}\right )-\text {PolyLog}\left (2,\frac {\sqrt {d} (-i+a+b x)}{-b \sqrt {-c}+(-i+a) \sqrt {d}}\right )+\text {PolyLog}\left (2,\frac {\sqrt {d} (-i+a+b x)}{b \sqrt {-c}+(-i+a) \sqrt {d}}\right )+\text {PolyLog}\left (2,\frac {\sqrt {d} (i+a+b x)}{-b \sqrt {-c}+(i+a) \sqrt {d}}\right )-\text {PolyLog}\left (2,\frac {\sqrt {d} (i+a+b x)}{b \sqrt {-c}+(i+a) \sqrt {d}}\right )\right )}{4 \sqrt {-c} \sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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 = 0.44, size = 2184, normalized size = 4.02

method	result	size
risch	\(\frac {\ln \left (-i b x -i a +1\right ) \ln \left (\frac {i a d -b \sqrt {c d}+\left (-i b x -i a +1\right ) d -d}{i a d -b \sqrt {c d}-d}\right )}{4 \sqrt {c d}}-\frac {\ln \left (-i b x -i a +1\right ) \ln \left (\frac {i a d +b \sqrt {c d}+\left (-i b x -i a +1\right ) d -d}{i a d +b \sqrt {c d}-d}\right )}{4 \sqrt {c d}}+\frac {\dilog \left (\frac {i a d -b \sqrt {c d}+\left (-i b x -i a +1\right ) d -d}{i a d -b \sqrt {c d}-d}\right )}{4 \sqrt {c d}}-\frac {\dilog \left (\frac {i a d +b \sqrt {c d}+\left (-i b x -i a +1\right ) d -d}{i a d +b \sqrt {c d}-d}\right )}{4 \sqrt {c d}}+\frac {\ln \left (i b x +i a +1\right ) \ln \left (\frac {i a d +b \sqrt {c d}-\left (i b x +i a +1\right ) d +d}{i a d +b \sqrt {c d}+d}\right )}{4 \sqrt {c d}}-\frac {\ln \left (i b x +i a +1\right ) \ln \left (\frac {i a d -b \sqrt {c d}-\left (i b x +i a +1\right ) d +d}{i a d -b \sqrt {c d}+d}\right )}{4 \sqrt {c d}}+\frac {\dilog \left (\frac {i a d +b \sqrt {c d}-\left (i b x +i a +1\right ) d +d}{i a d +b \sqrt {c d}+d}\right )}{4 \sqrt {c d}}-\frac {\dilog \left (\frac {i a d -b \sqrt {c d}-\left (i b x +i a +1\right ) d +d}{i a d -b \sqrt {c d}+d}\right )}{4 \sqrt {c d}}\)	\(494\)
derivativedivides	\(\text {Expression too large to display}\)	\(2184\)
default	\(\text {Expression too large to display}\)	\(2184\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 8520 vs. \(2 (369) = 738\).
time = 5.42, size = 8520, normalized size = 15.69 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b*(8*arctan(d*x/sqrt(c*d))*arctan((b^2*x + a*b)/b)/b - (4*arctan(sqrt(d)*x/sqrt(c))*arctan2((2*a*b^2*c*d +
(a*b^3*c + (a^3 + a)*b*d + (b^4*c + (a^2 + 3)*b^2*d)*x)*sqrt(c)*sqrt(d) + (3*b^3*c*d + (a^2 + 1)*b*d^2)*x)/(b
^4*c^2 + 2*(a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*d^2 + 4*(b^3*c + (a^2 + 1)*b*d)*sqrt(c)*sqrt(d)), ((a^2 + 3)*
b^2*c*d + (a^4 + 2*a^2 + 1)*d^2 + (2*a*b^2*d*x + b^3*c + 3*(a^2 + 1)*b*d)*sqrt(c)*sqrt(d) + (a*b^3*c*d + (a^3
+ a)*b*d^2)*x)/(b^4*c^2 + 2*(a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*d^2 + 4*(b^3*c + (a^2 + 1)*b*d)*sqrt(c)*sqrt
(d))) + 4*arctan(sqrt(d)*x/sqrt(c))*arctan2((2*a*b^2*c*d - (a*b^3*c + (a^3 + a)*b*d + (b^4*c + (a^2 + 3)*b^2*d
)*x)*sqrt(c)*sqrt(d) + (3*b^3*c*d + (a^2 + 1)*b*d^2)*x)/(b^4*c^2 + 2*(a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*d^2
- 4*(b^3*c + (a^2 + 1)*b*d)*sqrt(c)*sqrt(d)), ((a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*d^2 - (2*a*b^2*d*x + b^3
*c + 3*(a^2 + 1)*b*d)*sqrt(c)*sqrt(d) + (a*b^3*c*d + (a^3 + a)*b*d^2)*x)/(b^4*c^2 + 2*(a^2 + 3)*b^2*c*d + (a^4
+ 2*a^2 + 1)*d^2 - 4*(b^3*c + (a^2 + 1)*b*d)*sqrt(c)*sqrt(d))) + log(d*x^2 + c)*log(((a^2 + 1)*b^22*c^11*d +
11*(a^4 + 22*a^2 + 21)*b^20*c^10*d^2 + 55*(a^6 + 39*a^4 + 171*a^2 + 133)*b^18*c^9*d^3 + 33*(5*a^8 + 260*a^6 +
1870*a^4 + 3876*a^2 + 2261)*b^16*c^8*d^4 + 330*(a^10 + 61*a^8 + 570*a^6 + 1802*a^4 + 2261*a^2 + 969)*b^14*c^7*
d^5 + 22*(21*a^12 + 1386*a^10 + 15015*a^8 + 60060*a^6 + 109395*a^4 + 92378*a^2 + 29393)*b^12*c^6*d^6 + 22*(21*
a^14 + 1407*a^12 + 16401*a^10 + 75075*a^8 + 169455*a^6 + 201773*a^4 + 121771*a^2 + 29393)*b^10*c^5*d^7 + 330*(
a^16 + 64*a^14 + 756*a^12 + 3696*a^10 + 9438*a^8 + 13728*a^6 + 11492*a^4 + 5168*a^2 + 969)*b^8*c^4*d^8 + 33*(5
*a^18 + 285*a^16 + 3220*a^14 + 15876*a^12 + 42966*a^10 + 70070*a^8 + 70980*a^6 + 43860*a^4 + 15181*a^2 + 2261)
*b^6*c^3*d^9 + 55*(a^20 + 46*a^18 + 465*a^16 + 2184*a^14 + 5922*a^12 + 10164*a^10 + 11466*a^8 + 8520*a^6 + 402
9*a^4 + 1102*a^2 + 133)*b^4*c^2*d^10 + 11*(a^22 + 31*a^20 + 255*a^18 + 1065*a^16 + 2730*a^14 + 4662*a^12 + 550
2*a^10 + 4530*a^8 + 2565*a^6 + 955*a^4 + 211*a^2 + 21)*b^2*c*d^11 + (a^24 + 12*a^22 + 66*a^20 + 220*a^18 + 495
*a^16 + 792*a^14 + 924*a^12 + 792*a^10 + 495*a^8 + 220*a^6 + 66*a^4 + 12*a^2 + 1)*d^12 + (b^24*c^11*d + 11*(a^
2 + 21)*b^22*c^10*d^2 + 55*(a^4 + 38*a^2 + 133)*b^20*c^9*d^3 + 33*(5*a^6 + 255*a^4 + 1615*a^2 + 2261)*b^18*c^8
*d^4 + 330*(a^8 + 60*a^6 + 510*a^4 + 1292*a^2 + 969)*b^16*c^7*d^5 + 22*(21*a^10 + 1365*a^8 + 13650*a^6 + 46410
*a^4 + 62985*a^2 + 29393)*b^14*c^6*d^6 + 22*(21*a^12 + 1386*a^10 + 15015*a^8 + 60060*a^6 + 109395*a^4 + 92378*
a^2 + 29393)*b^12*c^5*d^7 + 330*(a^14 + 63*a^12 + 693*a^10 + 3003*a^8 + 6435*a^6 + 7293*a^4 + 4199*a^2 + 969)*
b^10*c^4*d^8 + 33*(5*a^16 + 280*a^14 + 2940*a^12 + 12936*a^10 + 30030*a^8 + 40040*a^6 + 30940*a^4 + 12920*a^2
+ 2261)*b^8*c^3*d^9 + 55*(a^18 + 45*a^16 + 420*a^14 + 1764*a^12 + 4158*a^10 + 6006*a^8 + 5460*a^6 + 3060*a^4 +
969*a^2 + 133)*b^6*c^2*d^10 + 11*(a^20 + 30*a^18 + 225*a^16 + 840*a^14 + 1890*a^12 + 2772*a^10 + 2730*a^8 + 1
800*a^6 + 765*a^4 + 190*a^2 + 21)*b^4*c*d^11 + (a^22 + 11*a^20 + 55*a^18 + 165*a^16 + 330*a^14 + 462*a^12 + 46
2*a^10 + 330*a^8 + 165*a^6 + 55*a^4 + 11*a^2 + 1)*b^2*d^12)*x^2 + 2*(11*(a^2 + 1)*b^21*c^10*d + 110*(a^4 + 8*a
^2 + 7)*b^19*c^9*d^2 + 33*(15*a^6 + 205*a^4 + 589*a^2 + 399)*b^17*c^8*d^3 + 264*(5*a^8 + 90*a^6 + 408*a^4 + 64
6*a^2 + 323)*b^15*c^7*d^4 + 110*(21*a^10 + 441*a^8 + 2562*a^6 + 6018*a^4 + 6137*a^2 + 2261)*b^13*c^6*d^5 + 4*(
693*a^12 + 15708*a^10 + 105105*a^8 + 308880*a^6 + 449735*a^4 + 319124*a^2 + 88179)*b^11*c^5*d^6 + 110*(21*a^14
+ 483*a^12 + 3465*a^10 + 11583*a^8 + 20735*a^6 + 20553*a^4 + 10659*a^2 + 2261)*b^9*c^4*d^7 + 264*(5*a^16 + 11
0*a^14 + 798*a^12 + 2838*a^10 + 5720*a^8 + 6890*a^6 + 4930*a^4 + 1938*a^2 + 323)*b^7*c^3*d^8 + 33*(15*a^18 + 2
95*a^16 + 2044*a^14 + 7308*a^12 + 15554*a^10 + 20930*a^8 + 18060*a^6 + 9724*a^4 + 2983*a^2 + 399)*b^5*c^2*d^9
+ 110*(a^20 + 16*a^18 + 99*a^16 + 336*a^14 + 714*a^12 + 1008*a^10 + 966*a^8 + 624*a^6 + 261*a^4 + 64*a^2 + 7)*
b^3*c*d^10 + 11*(a^22 + 11*a^20 + 55*a^18 + 165*a^16 + 330*a^14 + 462*a^12 + 462*a^10 + 330*a^8 + 165*a^6 + 55
*a^4 + 11*a^2 + 1)*b*d^11 + (11*b^23*c^10*d + 110*(a^2 + 7)*b^21*c^9*d^2 + 33*(15*a^4 + 190*a^2 + 399)*b^19*c^
8*d^3 + 264*(5*a^6 + 85*a^4 + 323*a^2 + 323)*b^17*c^7*d^4 + 110*(21*a^8 + 420*a^6 + 2142*a^4 + 3876*a^2 + 2261
)*b^15*c^6*d^5 + 4*(693*a^10 + 15015*a^8 + 90090*a^6 + 218790*a^4 + 230945*a^2 + 88179)*b^13*c^5*d^6 + 110*(21
*a^12 + 462*a^10 + 3003*a^8 + 8580*a^6 + 12155*a^4 + 8398*a^2 + 2261)*b^11*c^4*d^7 + 264*(5*a^14 + 105*a^12 +
693*a^10 + 2145*a^8 + 3575*a^6 + 3315*a^4 + 1615*a^2 + 323)*b^9*c^3*d^8 + 33*(15*a^16 + 280*a^14 + 1764*a^12 +
5544*a^10 + 10010*a^8 + 10920*a^6 + 7140*a^4 + 2584*a^2 + 399)*b^7*c^2*d^9 + 110*(a^18 + 15*a^16 + 84*a^14 +
252*a^12 + 462*a^10 + 546*a^8 + 420*a^6 + 204*a^4 + 57*a^2 + 7)*b^5*c*d^10 + 11*(a^20 + 10*a^18 + 45*a^16 + 12
0*a^14 + 210*a^12 + 252*a^10 + 210*a^8 + 120*a^6 + 45*a^4 + 10*a^2 + 1)*b^3*d^11)*x^2 + 2*(11*a*b^22*c^10*d +
110*(a^3 + 7*a)*b^20*c^9*d^2 + 33*(15*a^5 + 190...

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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)/(d*x^2+c),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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

[Out]

sage0*x

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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 )}{d\,x^2+c} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(a + b*x)/(c + d*x^2),x)

[Out]

int(atan(a + b*x)/(c + d*x^2), x)

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