3.1.56 \(\int \frac {\text {ArcTan}(a+b x)}{c+\frac {d}{x^2}} \, dx\) [56]

Optimal. Leaf size=668 \[ -\frac {(1+i a+i b x) \log (1+i a+i b x)}{2 b c}-\frac {(1-i a-i b x) \log (-i (i+a+b x))}{2 b c}+\frac {i \sqrt {d} \log (1+i a+i b x) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{i \sqrt {-c}-a \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \log (1-i a-i b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{i \sqrt {-c}+a \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {i \sqrt {d} \log (1-i a-i b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(i+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{i \sqrt {-c}-a \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {i \sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {-c} (i-a-b x)}{i \sqrt {-c}-a \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {-c} (1+i a+i b x)}{(1+i a) \sqrt {-c}-i b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {i \sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {-c} (i+a+b x)}{i \sqrt {-c}+a \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {-c} (i+a+b x)}{i \sqrt {-c}+a \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}} \]

[Out]

-1/2*(1+I*a+I*b*x)*ln(1+I*a+I*b*x)/b/c-1/2*(1-I*a-I*b*x)*ln(-I*(I+a+b*x))/b/c+1/4*I*ln(1+I*a+I*b*x)*ln(-b*(-x*
(-c)^(1/2)+d^(1/2))/(I*(-c)^(1/2)-a*(-c)^(1/2)-b*d^(1/2)))*d^(1/2)/(-c)^(3/2)+1/4*I*ln(1-I*a-I*b*x)*ln(-b*(x*(
-c)^(1/2)+d^(1/2))/((I+a)*(-c)^(1/2)-b*d^(1/2)))*d^(1/2)/(-c)^(3/2)-1/4*I*ln(1+I*a+I*b*x)*ln(b*(x*(-c)^(1/2)+d
^(1/2))/(I*(-c)^(1/2)-a*(-c)^(1/2)+b*d^(1/2)))*d^(1/2)/(-c)^(3/2)-1/4*I*ln(1-I*a-I*b*x)*ln(b*(-x*(-c)^(1/2)+d^
(1/2))/(I*(-c)^(1/2)+a*(-c)^(1/2)+b*d^(1/2)))*d^(1/2)/(-c)^(3/2)+1/4*I*polylog(2,(I-a-b*x)*(-c)^(1/2)/(I*(-c)^
(1/2)-a*(-c)^(1/2)-b*d^(1/2)))*d^(1/2)/(-c)^(3/2)+1/4*I*polylog(2,(I+a+b*x)*(-c)^(1/2)/(I*(-c)^(1/2)+a*(-c)^(1
/2)-b*d^(1/2)))*d^(1/2)/(-c)^(3/2)-1/4*I*polylog(2,(1+I*a+I*b*x)*(-c)^(1/2)/((1+I*a)*(-c)^(1/2)-I*b*d^(1/2)))*
d^(1/2)/(-c)^(3/2)-1/4*I*polylog(2,(I+a+b*x)*(-c)^(1/2)/(I*(-c)^(1/2)+a*(-c)^(1/2)+b*d^(1/2)))*d^(1/2)/(-c)^(3
/2)

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Rubi [A]
time = 0.81, antiderivative size = 668, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5159, 2456, 2436, 2332, 2441, 2440, 2438} \begin {gather*} \frac {i \sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (-a-b x+i)}{-\sqrt {-c} a+i \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (i a+i b x+1)}{(i a+1) \sqrt {-c}-i b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {i \sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x+i)}{\sqrt {-c} a+i \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x+i)}{\sqrt {-c} a+i \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {i \sqrt {d} \log (i a+i b x+1) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{a \left (-\sqrt {-c}\right )-b \sqrt {d}+i \sqrt {-c}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \log (i a+i b x+1) \log \left (\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{a \left (-\sqrt {-c}\right )+b \sqrt {d}+i \sqrt {-c}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \log (-i a-i b x+1) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{a \sqrt {-c}+b \sqrt {d}+i \sqrt {-c}}\right )}{4 (-c)^{3/2}}+\frac {i \sqrt {d} \log (-i a-i b x+1) \log \left (-\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{-b \sqrt {d}+(a+i) \sqrt {-c}}\right )}{4 (-c)^{3/2}}-\frac {(i a+i b x+1) \log (i a+i b x+1)}{2 b c}-\frac {(-i a-i b x+1) \log (-i (a+b x+i))}{2 b c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTan[a + b*x]/(c + d/x^2),x]

[Out]

-1/2*((1 + I*a + I*b*x)*Log[1 + I*a + I*b*x])/(b*c) - ((1 - I*a - I*b*x)*Log[(-I)*(I + a + b*x)])/(2*b*c) + ((
I/4)*Sqrt[d]*Log[1 + I*a + I*b*x]*Log[-((b*(Sqrt[d] - Sqrt[-c]*x))/(I*Sqrt[-c] - a*Sqrt[-c] - b*Sqrt[d]))])/(-
c)^(3/2) - ((I/4)*Sqrt[d]*Log[1 - I*a - I*b*x]*Log[(b*(Sqrt[d] - Sqrt[-c]*x))/(I*Sqrt[-c] + a*Sqrt[-c] + b*Sqr
t[d])])/(-c)^(3/2) + ((I/4)*Sqrt[d]*Log[1 - I*a - I*b*x]*Log[-((b*(Sqrt[d] + Sqrt[-c]*x))/((I + a)*Sqrt[-c] -
b*Sqrt[d]))])/(-c)^(3/2) - ((I/4)*Sqrt[d]*Log[1 + I*a + I*b*x]*Log[(b*(Sqrt[d] + Sqrt[-c]*x))/(I*Sqrt[-c] - a*
Sqrt[-c] + b*Sqrt[d])])/(-c)^(3/2) + ((I/4)*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I - a - b*x))/(I*Sqrt[-c] - a*Sqrt[-
c] - b*Sqrt[d])])/(-c)^(3/2) - ((I/4)*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 + I*a + I*b*x))/((1 + I*a)*Sqrt[-c] - I*
b*Sqrt[d])])/(-c)^(3/2) + ((I/4)*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I + a + b*x))/(I*Sqrt[-c] + a*Sqrt[-c] - b*Sqrt
[d])])/(-c)^(3/2) - ((I/4)*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I + a + b*x))/(I*Sqrt[-c] + a*Sqrt[-c] + b*Sqrt[d])])
/(-c)^(3/2)

Rule 2332

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

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 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+\frac {d}{x^2}} \, dx &=\frac {1}{2} i \int \frac {\log (1-i a-i b x)}{c+\frac {d}{x^2}} \, dx-\frac {1}{2} i \int \frac {\log (1+i a+i b x)}{c+\frac {d}{x^2}} \, dx\\ &=\frac {1}{2} i \int \left (\frac {\log (1-i a-i b x)}{c}-\frac {d \log (1-i a-i b x)}{c \left (d+c x^2\right )}\right ) \, dx-\frac {1}{2} i \int \left (\frac {\log (1+i a+i b x)}{c}-\frac {d \log (1+i a+i b x)}{c \left (d+c x^2\right )}\right ) \, dx\\ &=\frac {i \int \log (1-i a-i b x) \, dx}{2 c}-\frac {i \int \log (1+i a+i b x) \, dx}{2 c}-\frac {(i d) \int \frac {\log (1-i a-i b x)}{d+c x^2} \, dx}{2 c}+\frac {(i d) \int \frac {\log (1+i a+i b x)}{d+c x^2} \, dx}{2 c}\\ &=-\frac {\text {Subst}(\int \log (x) \, dx,x,1-i a-i b x)}{2 b c}-\frac {\text {Subst}(\int \log (x) \, dx,x,1+i a+i b x)}{2 b c}-\frac {(i d) \int \left (\frac {\log (1-i a-i b x)}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x\right )}+\frac {\log (1-i a-i b x)}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x\right )}\right ) \, dx}{2 c}+\frac {(i d) \int \left (\frac {\log (1+i a+i b x)}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x\right )}+\frac {\log (1+i a+i b x)}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x\right )}\right ) \, dx}{2 c}\\ &=-\frac {(1+i a+i b x) \log (1+i a+i b x)}{2 b c}-\frac {(1-i a-i b x) \log (-i (i+a+b x))}{2 b c}-\frac {\left (i \sqrt {d}\right ) \int \frac {\log (1-i a-i b x)}{\sqrt {d}-\sqrt {-c} x} \, dx}{4 c}-\frac {\left (i \sqrt {d}\right ) \int \frac {\log (1-i a-i b x)}{\sqrt {d}+\sqrt {-c} x} \, dx}{4 c}+\frac {\left (i \sqrt {d}\right ) \int \frac {\log (1+i a+i b x)}{\sqrt {d}-\sqrt {-c} x} \, dx}{4 c}+\frac {\left (i \sqrt {d}\right ) \int \frac {\log (1+i a+i b x)}{\sqrt {d}+\sqrt {-c} x} \, dx}{4 c}\\ &=-\frac {(1+i a+i b x) \log (1+i a+i b x)}{2 b c}-\frac {(1-i a-i b x) \log (-i (i+a+b x))}{2 b c}+\frac {i \sqrt {d} \log (1+i a+i b x) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{i \sqrt {-c}-a \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \log (1-i a-i b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{i \sqrt {-c}+a \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {i \sqrt {d} \log (1-i a-i b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(i+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{i \sqrt {-c}-a \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (-\frac {i b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-i a) \sqrt {-c}-i b \sqrt {d}}\right )}{1-i a-i b x} \, dx}{4 (-c)^{3/2}}+\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (\frac {i b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1+i a) \sqrt {-c}+i b \sqrt {d}}\right )}{1+i a+i b x} \, dx}{4 (-c)^{3/2}}-\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (-\frac {i b \left (\sqrt {d}+\sqrt {-c} x\right )}{-(1-i a) \sqrt {-c}-i b \sqrt {d}}\right )}{1-i a-i b x} \, dx}{4 (-c)^{3/2}}-\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (\frac {i b \left (\sqrt {d}+\sqrt {-c} x\right )}{-(1+i a) \sqrt {-c}+i b \sqrt {d}}\right )}{1+i a+i b x} \, dx}{4 (-c)^{3/2}}\\ &=-\frac {(1+i a+i b x) \log (1+i a+i b x)}{2 b c}-\frac {(1-i a-i b x) \log (-i (i+a+b x))}{2 b c}+\frac {i \sqrt {d} \log (1+i a+i b x) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{i \sqrt {-c}-a \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \log (1-i a-i b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{i \sqrt {-c}+a \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {i \sqrt {d} \log (1-i a-i b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(i+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{i \sqrt {-c}-a \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\left (i \sqrt {d}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-c} x}{-(1-i a) \sqrt {-c}-i b \sqrt {d}}\right )}{x} \, dx,x,1-i a-i b x\right )}{4 (-c)^{3/2}}+\frac {\left (i \sqrt {d}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-c} x}{(1-i a) \sqrt {-c}-i b \sqrt {d}}\right )}{x} \, dx,x,1-i a-i b x\right )}{4 (-c)^{3/2}}+\frac {\left (i \sqrt {d}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-c} x}{-(1+i a) \sqrt {-c}+i b \sqrt {d}}\right )}{x} \, dx,x,1+i a+i b x\right )}{4 (-c)^{3/2}}-\frac {\left (i \sqrt {d}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-c} x}{(1+i a) \sqrt {-c}+i b \sqrt {d}}\right )}{x} \, dx,x,1+i a+i b x\right )}{4 (-c)^{3/2}}\\ &=-\frac {(1+i a+i b x) \log (1+i a+i b x)}{2 b c}-\frac {(1-i a-i b x) \log (-i (i+a+b x))}{2 b c}+\frac {i \sqrt {d} \log (1+i a+i b x) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{i \sqrt {-c}-a \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \log (1-i a-i b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{i \sqrt {-c}+a \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {i \sqrt {d} \log (1-i a-i b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(i+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{i \sqrt {-c}-a \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {i \sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (i-a-b x)}{i \sqrt {-c}-a \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (1+i a+i b x)}{(1+i a) \sqrt {-c}-i b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {i \sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (i+a+b x)}{i \sqrt {-c}+a \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (i+a+b x)}{i \sqrt {-c}+a \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1536\) vs. \(2(668)=1336\).
time = 16.76, size = 1536, normalized size = 2.30 \begin {gather*} \frac {(a+b x) \text {ArcTan}(a+b x)+\log \left (\frac {1}{\sqrt {1+(a+b x)^2}}\right )}{b c}-\frac {\sqrt {d} \left (-2 \sqrt {c} \text {ArcTan}\left (\frac {(-i+a) \sqrt {c}}{b \sqrt {d}}\right ) \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )-2 a^2 \sqrt {c} \text {ArcTan}\left (\frac {(-i+a) \sqrt {c}}{b \sqrt {d}}\right ) \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )+2 \sqrt {c} \text {ArcTan}\left (\frac {(i+a) \sqrt {c}}{b \sqrt {d}}\right ) \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )+2 a^2 \sqrt {c} \text {ArcTan}\left (\frac {(i+a) \sqrt {c}}{b \sqrt {d}}\right ) \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )-2 b \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )^2+b \sqrt {d} \sqrt {\frac {(-i+a)^2 c+b^2 d}{b^2 d}} e^{-i \text {ArcTan}\left (\frac {(-i+a) \sqrt {c}}{b \sqrt {d}}\right )} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )^2-i a b \sqrt {d} \sqrt {\frac {(-i+a)^2 c+b^2 d}{b^2 d}} e^{-i \text {ArcTan}\left (\frac {(-i+a) \sqrt {c}}{b \sqrt {d}}\right )} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )^2+b \sqrt {d} \sqrt {\frac {(i+a)^2 c+b^2 d}{b^2 d}} e^{-i \text {ArcTan}\left (\frac {(i+a) \sqrt {c}}{b \sqrt {d}}\right )} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )^2+i a b \sqrt {d} \sqrt {\frac {(i+a)^2 c+b^2 d}{b^2 d}} e^{-i \text {ArcTan}\left (\frac {(i+a) \sqrt {c}}{b \sqrt {d}}\right )} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )^2+4 \left (1+a^2\right ) \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \text {ArcTan}(a+b x)+2 i \sqrt {c} \text {ArcTan}\left (\frac {(-i+a) \sqrt {c}}{b \sqrt {d}}\right ) \log \left (1-e^{-2 i \left (\text {ArcTan}\left (\frac {(-i+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )+2 i a^2 \sqrt {c} \text {ArcTan}\left (\frac {(-i+a) \sqrt {c}}{b \sqrt {d}}\right ) \log \left (1-e^{-2 i \left (\text {ArcTan}\left (\frac {(-i+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )+2 i \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (1-e^{-2 i \left (\text {ArcTan}\left (\frac {(-i+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )+2 i a^2 \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (1-e^{-2 i \left (\text {ArcTan}\left (\frac {(-i+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )-2 i \sqrt {c} \text {ArcTan}\left (\frac {(i+a) \sqrt {c}}{b \sqrt {d}}\right ) \log \left (1-e^{-2 i \left (\text {ArcTan}\left (\frac {(i+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )-2 i a^2 \sqrt {c} \text {ArcTan}\left (\frac {(i+a) \sqrt {c}}{b \sqrt {d}}\right ) \log \left (1-e^{-2 i \left (\text {ArcTan}\left (\frac {(i+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )-2 i \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (1-e^{-2 i \left (\text {ArcTan}\left (\frac {(i+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )-2 i a^2 \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (1-e^{-2 i \left (\text {ArcTan}\left (\frac {(i+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )-2 i \sqrt {c} \text {ArcTan}\left (\frac {(-i+a) \sqrt {c}}{b \sqrt {d}}\right ) \log \left (-\sin \left (\text {ArcTan}\left (\frac {(-i+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )\right )-2 i a^2 \sqrt {c} \text {ArcTan}\left (\frac {(-i+a) \sqrt {c}}{b \sqrt {d}}\right ) \log \left (-\sin \left (\text {ArcTan}\left (\frac {(-i+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )\right )+2 i \sqrt {c} \text {ArcTan}\left (\frac {(i+a) \sqrt {c}}{b \sqrt {d}}\right ) \log \left (-\sin \left (\text {ArcTan}\left (\frac {(i+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )\right )+2 i a^2 \sqrt {c} \text {ArcTan}\left (\frac {(i+a) \sqrt {c}}{b \sqrt {d}}\right ) \log \left (-\sin \left (\text {ArcTan}\left (\frac {(i+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )\right )-\left (1+a^2\right ) \sqrt {c} \text {PolyLog}\left (2,e^{-2 i \left (\text {ArcTan}\left (\frac {(-i+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )+\left (1+a^2\right ) \sqrt {c} \text {PolyLog}\left (2,e^{-2 i \left (\text {ArcTan}\left (\frac {(i+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )\right )}{4 \left (1+a^2\right ) c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)*ArcTan[a + b*x] + Log[1/Sqrt[1 + (a + b*x)^2]])/(b*c) - (Sqrt[d]*(-2*Sqrt[c]*ArcTan[((-I + a)*Sqrt[
c])/(b*Sqrt[d])]*ArcTan[(Sqrt[c]*x)/Sqrt[d]] - 2*a^2*Sqrt[c]*ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])]*ArcTan[(Sq
rt[c]*x)/Sqrt[d]] + 2*Sqrt[c]*ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])]*ArcTan[(Sqrt[c]*x)/Sqrt[d]] + 2*a^2*Sqrt[c
]*ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])]*ArcTan[(Sqrt[c]*x)/Sqrt[d]] - 2*b*Sqrt[d]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]^
2 + (b*Sqrt[d]*Sqrt[((-I + a)^2*c + b^2*d)/(b^2*d)]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]^2)/E^(I*ArcTan[((-I + a)*Sqrt[
c])/(b*Sqrt[d])]) - (I*a*b*Sqrt[d]*Sqrt[((-I + a)^2*c + b^2*d)/(b^2*d)]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]^2)/E^(I*Ar
cTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])]) + (b*Sqrt[d]*Sqrt[((I + a)^2*c + b^2*d)/(b^2*d)]*ArcTan[(Sqrt[c]*x)/Sqrt
[d]]^2)/E^(I*ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])]) + (I*a*b*Sqrt[d]*Sqrt[((I + a)^2*c + b^2*d)/(b^2*d)]*ArcTa
n[(Sqrt[c]*x)/Sqrt[d]]^2)/E^(I*ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])]) + 4*(1 + a^2)*Sqrt[c]*ArcTan[(Sqrt[c]*x)
/Sqrt[d]]*ArcTan[a + b*x] + (2*I)*Sqrt[c]*ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])]*Log[1 - E^((-2*I)*(ArcTan[((-
I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] + (2*I)*a^2*Sqrt[c]*ArcTan[((-I + a)*Sqrt[c])/(b
*Sqrt[d])]*Log[1 - E^((-2*I)*(ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] + (2*I)*
Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*Log[1 - E^((-2*I)*(ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c
]*x)/Sqrt[d]]))] + (2*I)*a^2*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*Log[1 - E^((-2*I)*(ArcTan[((-I + a)*Sqrt[c])/
(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] - (2*I)*Sqrt[c]*ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])]*Log[1 - E^
((-2*I)*(ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] - (2*I)*a^2*Sqrt[c]*ArcTan[((I
+ a)*Sqrt[c])/(b*Sqrt[d])]*Log[1 - E^((-2*I)*(ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt
[d]]))] - (2*I)*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*Log[1 - E^((-2*I)*(ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])] +
ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] - (2*I)*a^2*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*Log[1 - E^((-2*I)*(ArcTan[((I
+ a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] - (2*I)*Sqrt[c]*ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[
d])]*Log[-Sin[ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]]] - (2*I)*a^2*Sqrt[c]*ArcTa
n[((-I + a)*Sqrt[c])/(b*Sqrt[d])]*Log[-Sin[ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]
]]] + (2*I)*Sqrt[c]*ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])]*Log[-Sin[ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])] + Arc
Tan[(Sqrt[c]*x)/Sqrt[d]]]] + (2*I)*a^2*Sqrt[c]*ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])]*Log[-Sin[ArcTan[((I + a)*
Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]]] - (1 + a^2)*Sqrt[c]*PolyLog[2, E^((-2*I)*(ArcTan[((-I +
a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] + (1 + a^2)*Sqrt[c]*PolyLog[2, E^((-2*I)*(ArcTan[((I
+ a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))]))/(4*(1 + a^2)*c^2)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.21, size = 14058, normalized size = 21.04

method	result	size
risch	\(-\frac {i \ln \left (i b x +i a +1\right ) x}{2 c}+\frac {a \arctan \left (b x +a \right )}{b c}+\frac {i \ln \left (-i b x -i a +1\right ) x}{2 c}-\frac {\ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{2 b c}+\frac {1}{b c}-\frac {d \dilog \left (\frac {-i a c +b \sqrt {c d}-\left (-i b x -i a +1\right ) c +c}{-i a c +b \sqrt {c d}+c}\right )}{4 c \sqrt {c d}}+\frac {d \dilog \left (\frac {i a c +b \sqrt {c d}+\left (-i b x -i a +1\right ) c -c}{i a c +b \sqrt {c d}-c}\right )}{4 c \sqrt {c d}}-\frac {d \dilog \left (\frac {i a c +b \sqrt {c d}-\left (i b x +i a +1\right ) c +c}{i a c +b \sqrt {c d}+c}\right )}{4 c \sqrt {c d}}+\frac {d \dilog \left (\frac {-i a c +b \sqrt {c d}+\left (i b x +i a +1\right ) c -c}{-i a c +b \sqrt {c d}-c}\right )}{4 c \sqrt {c d}}-\frac {d \ln \left (-i b x -i a +1\right ) \ln \left (\frac {-i a c +b \sqrt {c d}-\left (-i b x -i a +1\right ) c +c}{-i a c +b \sqrt {c d}+c}\right )}{4 c \sqrt {c d}}+\frac {d \ln \left (-i b x -i a +1\right ) \ln \left (\frac {i a c +b \sqrt {c d}+\left (-i b x -i a +1\right ) c -c}{i a c +b \sqrt {c d}-c}\right )}{4 c \sqrt {c d}}-\frac {d \ln \left (i b x +i a +1\right ) \ln \left (\frac {i a c +b \sqrt {c d}-\left (i b x +i a +1\right ) c +c}{i a c +b \sqrt {c d}+c}\right )}{4 c \sqrt {c d}}+\frac {d \ln \left (i b x +i a +1\right ) \ln \left (\frac {-i a c +b \sqrt {c d}+\left (i b x +i a +1\right ) c -c}{-i a c +b \sqrt {c d}-c}\right )}{4 c \sqrt {c d}}\)	\(603\)
derivativedivides	\(\text {Expression too large to display}\)	\(14058\)
default	\(\text {Expression too large to display}\)	\(14058\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 8518 vs. \(2 (466) = 932\).
time = 1.20, size = 8518, normalized size = 12.75 \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)/(c+d/x^2),x, algorithm="maxima")

[Out]

-(d*arctan(c*x/sqrt(c*d))/(sqrt(c*d)*c) - x/c)*arctan(b*x + a) + 1/8*(8*a*c*arctan(b*x + a) + (4*b*arctan(sqrt
(c)*x/sqrt(d))*arctan2((2*a*b^2*c*d + (a*b^3*d + (a^3 + a)*b*c + (b^4*d + (a^2 + 3)*b^2*c)*x)*sqrt(c)*sqrt(d)
+ (3*b^3*c*d + (a^2 + 1)*b*c^2)*x)/(b^4*d^2 + 2*(a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*c^2 + 4*(b^3*d + (a^2 +
1)*b*c)*sqrt(c)*sqrt(d)), ((a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*c^2 + (2*a*b^2*c*x + b^3*d + 3*(a^2 + 1)*b*c)
*sqrt(c)*sqrt(d) + (a*b^3*c*d + (a^3 + a)*b*c^2)*x)/(b^4*d^2 + 2*(a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*c^2 + 4
*(b^3*d + (a^2 + 1)*b*c)*sqrt(c)*sqrt(d))) + 4*b*arctan(sqrt(c)*x/sqrt(d))*arctan2((2*a*b^2*c*d - (a*b^3*d + (
a^3 + a)*b*c + (b^4*d + (a^2 + 3)*b^2*c)*x)*sqrt(c)*sqrt(d) + (3*b^3*c*d + (a^2 + 1)*b*c^2)*x)/(b^4*d^2 + 2*(a
^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*c^2 - 4*(b^3*d + (a^2 + 1)*b*c)*sqrt(c)*sqrt(d)), ((a^2 + 3)*b^2*c*d + (a^
4 + 2*a^2 + 1)*c^2 - (2*a*b^2*c*x + b^3*d + 3*(a^2 + 1)*b*c)*sqrt(c)*sqrt(d) + (a*b^3*c*d + (a^3 + a)*b*c^2)*x
)/(b^4*d^2 + 2*(a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*c^2 - 4*(b^3*d + (a^2 + 1)*b*c)*sqrt(c)*sqrt(d))) + b*log
(c*x^2 + d)*log(((a^2 + 1)*b^22*c*d^11 + 11*(a^4 + 22*a^2 + 21)*b^20*c^2*d^10 + 55*(a^6 + 39*a^4 + 171*a^2 + 1
33)*b^18*c^3*d^9 + 33*(5*a^8 + 260*a^6 + 1870*a^4 + 3876*a^2 + 2261)*b^16*c^4*d^8 + 330*(a^10 + 61*a^8 + 570*a
^6 + 1802*a^4 + 2261*a^2 + 969)*b^14*c^5*d^7 + 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^7*d^5 + 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^8*d^4 + 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^9*d^3 + 55*(a^20 + 46*a^18 + 465*a^16 + 2184*a^14 + 5922*a^12
+ 10164*a^10 + 11466*a^8 + 8520*a^6 + 4029*a^4 + 1102*a^2 + 133)*b^4*c^10*d^2 + 11*(a^22 + 31*a^20 + 255*a^18
+ 1065*a^16 + 2730*a^14 + 4662*a^12 + 5502*a^10 + 4530*a^8 + 2565*a^6 + 955*a^4 + 211*a^2 + 21)*b^2*c^11*d + (
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)*c^12 + (b^24*c*d^11 + 11*(a^2 + 21)*b^22*c^2*d^10 + 55*(a^4 + 38*a^2 + 133)*b^20*c^3*d^9 + 33*(5*
a^6 + 255*a^4 + 1615*a^2 + 2261)*b^18*c^4*d^8 + 330*(a^8 + 60*a^6 + 510*a^4 + 1292*a^2 + 969)*b^16*c^5*d^7 + 2
2*(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 + 15
015*a^8 + 60060*a^6 + 109395*a^4 + 92378*a^2 + 29393)*b^12*c^7*d^5 + 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^8*d^4 + 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^9*d^3 + 55*(a^18 + 45*a^16 + 420*a^14 + 1764*a^12 + 415
8*a^10 + 6006*a^8 + 5460*a^6 + 3060*a^4 + 969*a^2 + 133)*b^6*c^10*d^2 + 11*(a^20 + 30*a^18 + 225*a^16 + 840*a^
14 + 1890*a^12 + 2772*a^10 + 2730*a^8 + 1800*a^6 + 765*a^4 + 190*a^2 + 21)*b^4*c^11*d + (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^2*c^12)*x^2 + 2*(
11*(a^2 + 1)*b^21*c*d^10 + 110*(a^4 + 8*a^2 + 7)*b^19*c^2*d^9 + 33*(15*a^6 + 205*a^4 + 589*a^2 + 399)*b^17*c^3
*d^8 + 264*(5*a^8 + 90*a^6 + 408*a^4 + 646*a^2 + 323)*b^15*c^4*d^7 + 110*(21*a^10 + 441*a^8 + 2562*a^6 + 6018*
a^4 + 6137*a^2 + 2261)*b^13*c^5*d^6 + 4*(693*a^12 + 15708*a^10 + 105105*a^8 + 308880*a^6 + 449735*a^4 + 319124
*a^2 + 88179)*b^11*c^6*d^5 + 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^7*d^4 + 264*(5*a^16 + 110*a^14 + 798*a^12 + 2838*a^10 + 5720*a^8 + 6890*a^6 + 4930*a^4 + 1938
*a^2 + 323)*b^7*c^8*d^3 + 33*(15*a^18 + 295*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^9*d^2 + 110*(a^20 + 16*a^18 + 99*a^16 + 336*a^14 + 714*a^12 + 1008*a^10 + 9
66*a^8 + 624*a^6 + 261*a^4 + 64*a^2 + 7)*b^3*c^10*d + 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*c^11 + (11*b^23*c*d^10 + 110*(a^2 + 7)*b^21*c^2*
d^9 + 33*(15*a^4 + 190*a^2 + 399)*b^19*c^3*d^8 + 264*(5*a^6 + 85*a^4 + 323*a^2 + 323)*b^17*c^4*d^7 + 110*(21*a
^8 + 420*a^6 + 2142*a^4 + 3876*a^2 + 2261)*b^15*c^5*d^6 + 4*(693*a^10 + 15015*a^8 + 90090*a^6 + 218790*a^4 + 2
30945*a^2 + 88179)*b^13*c^6*d^5 + 110*(21*a^12 + 462*a^10 + 3003*a^8 + 8580*a^6 + 12155*a^4 + 8398*a^2 + 2261)
*b^11*c^7*d^4 + 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^8*d
^3 + 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
^9*d^2 + 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
^10*d + 11*(a^20 + 10*a^18 + 45*a^16 + 120*a^14 + 210*a^12 + 252*a^10 + 210*a^8 + 120*a^6 + 45*a^4 + 10*a^2 +
1)*b^3*c^11)*x^2 + 2*(11*a*b^22*c*d^10 + 110*(a...

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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