3.1171 \(\int \frac{a+b \tan ^{-1}(c x)}{(d+e x^2)^3} \, dx\)

Optimal. Leaf size=893 \[ \frac{3 i b \log \left (\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right ) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b \log \left (-\frac{\sqrt{e} \left (\sqrt{-c^2} x+1\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right ) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b \log \left (-\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (\frac{i \sqrt{e} x}{\sqrt{d}}+1\right ) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}+\frac{3 i b \log \left (\frac{\sqrt{e} \left (\sqrt{-c^2} x+1\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (\frac{i \sqrt{e} x}{\sqrt{d}}+1\right ) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{b \left (5 c^2 d-3 e\right ) \log \left (c^2 x^2+1\right ) c}{16 d^2 \left (c^2 d-e\right )^2}+\frac{b \left (5 c^2 d-3 e\right ) \log \left (e x^2+d\right ) c}{16 d^2 \left (c^2 d-e\right )^2}+\frac{3 i b \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right ) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right ) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}+\frac{3 i b \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (i \sqrt{e} x+\sqrt{d}\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right ) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (i \sqrt{e} x+\sqrt{d}\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right ) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{b c}{8 d \left (c^2 d-e\right ) \left (e x^2+d\right )}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (e x^2+d\right )}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (e x^2+d\right )^2}+\frac{3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \sqrt{e}} \]

[Out]

-(b*c)/(8*d*(c^2*d - e)*(d + e*x^2)) + (x*(a + b*ArcTan[c*x]))/(4*d*(d + e*x^2)^2) + (3*x*(a + b*ArcTan[c*x]))
/(8*d^2*(d + e*x^2)) + (3*(a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]) + (((3*I)/32)*b
*c*Log[(Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c
^2]*d^(5/2)*Sqrt[e]) - (((3*I)/32)*b*c*Log[-((Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*L
og[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) - (((3*I)/32)*b*c*Log[-((Sqrt[e]*(1 - Sqrt[-c^2]*x
))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) + (((3*I)/3
2)*b*c*Log[(Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqr
t[-c^2]*d^(5/2)*Sqrt[e]) - (b*c*(5*c^2*d - 3*e)*Log[1 + c^2*x^2])/(16*d^2*(c^2*d - e)^2) + (b*c*(5*c^2*d - 3*e
)*Log[d + e*x^2])/(16*d^2*(c^2*d - e)^2) + (((3*I)/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sq
rt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) - (((3*I)/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d]
- I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) + (((3*I)/32)*b*c*PolyLog[2,
(Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) - (((3*I)
/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2
)*Sqrt[e])

________________________________________________________________________________________

Rubi [A]  time = 0.948515, antiderivative size = 893, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {199, 205, 4912, 6725, 571, 77, 4908, 2409, 2394, 2393, 2391} \[ \frac{3 i b \log \left (\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right ) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b \log \left (-\frac{\sqrt{e} \left (\sqrt{-c^2} x+1\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right ) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b \log \left (-\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (\frac{i \sqrt{e} x}{\sqrt{d}}+1\right ) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}+\frac{3 i b \log \left (\frac{\sqrt{e} \left (\sqrt{-c^2} x+1\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (\frac{i \sqrt{e} x}{\sqrt{d}}+1\right ) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{b \left (5 c^2 d-3 e\right ) \log \left (c^2 x^2+1\right ) c}{16 d^2 \left (c^2 d-e\right )^2}+\frac{b \left (5 c^2 d-3 e\right ) \log \left (e x^2+d\right ) c}{16 d^2 \left (c^2 d-e\right )^2}+\frac{3 i b \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right ) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right ) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}+\frac{3 i b \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (i \sqrt{e} x+\sqrt{d}\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right ) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (i \sqrt{e} x+\sqrt{d}\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right ) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{b c}{8 d \left (c^2 d-e\right ) \left (e x^2+d\right )}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (e x^2+d\right )}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (e x^2+d\right )^2}+\frac{3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x])/(d + e*x^2)^3,x]

[Out]

-(b*c)/(8*d*(c^2*d - e)*(d + e*x^2)) + (x*(a + b*ArcTan[c*x]))/(4*d*(d + e*x^2)^2) + (3*x*(a + b*ArcTan[c*x]))
/(8*d^2*(d + e*x^2)) + (3*(a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]) + (((3*I)/32)*b
*c*Log[(Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c
^2]*d^(5/2)*Sqrt[e]) - (((3*I)/32)*b*c*Log[-((Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*L
og[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) - (((3*I)/32)*b*c*Log[-((Sqrt[e]*(1 - Sqrt[-c^2]*x
))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) + (((3*I)/3
2)*b*c*Log[(Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqr
t[-c^2]*d^(5/2)*Sqrt[e]) - (b*c*(5*c^2*d - 3*e)*Log[1 + c^2*x^2])/(16*d^2*(c^2*d - e)^2) + (b*c*(5*c^2*d - 3*e
)*Log[d + e*x^2])/(16*d^2*(c^2*d - e)^2) + (((3*I)/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sq
rt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) - (((3*I)/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d]
- I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) + (((3*I)/32)*b*c*PolyLog[2,
(Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) - (((3*I)
/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2
)*Sqrt[e])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 4912

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[b*c, Int[u/(1 + c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x]
&& (IntegerQ[q] || ILtQ[q + 1/2, 0])

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 571

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x
_Symbol] :> Dist[1/n, Subst[Int[(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n], x] /; FreeQ[{a, b, c, d, e,
f, m, n, p, q, r}, x] && EqQ[m - n + 1, 0]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 4908

Int[ArcTan[(c_.)*(x_)]/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Dist[I/2, Int[Log[1 - I*c*x]/(d + e*x^2), x], x] -
Dist[I/2, Int[Log[1 + I*c*x]/(d + e*x^2), x], x] /; FreeQ[{c, d, e}, x]

Rule 2409

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 2394

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 2393

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 2391

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]

Rubi steps

\begin{align*} \int \frac{a+b \tan ^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (d+e x^2\right )}+\frac{3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \sqrt{e}}-(b c) \int \frac{\frac{x}{4 d \left (d+e x^2\right )^2}+\frac{3 x}{8 d^2 \left (d+e x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \sqrt{e}}}{1+c^2 x^2} \, dx\\ &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (d+e x^2\right )}+\frac{3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \sqrt{e}}-(b c) \int \left (\frac{x \left (5 d+3 e x^2\right )}{8 d^2 \left (1+c^2 x^2\right ) \left (d+e x^2\right )^2}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \sqrt{e} \left (1+c^2 x^2\right )}\right ) \, dx\\ &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (d+e x^2\right )}+\frac{3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \sqrt{e}}-\frac{(b c) \int \frac{x \left (5 d+3 e x^2\right )}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )^2} \, dx}{8 d^2}-\frac{(3 b c) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{1+c^2 x^2} \, dx}{8 d^{5/2} \sqrt{e}}\\ &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (d+e x^2\right )}+\frac{3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \sqrt{e}}-\frac{(b c) \operatorname{Subst}\left (\int \frac{5 d+3 e x}{\left (1+c^2 x\right ) (d+e x)^2} \, dx,x,x^2\right )}{16 d^2}-\frac{(3 i b c) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1+c^2 x^2} \, dx}{16 d^{5/2} \sqrt{e}}+\frac{(3 i b c) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1+c^2 x^2} \, dx}{16 d^{5/2} \sqrt{e}}\\ &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (d+e x^2\right )}+\frac{3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \sqrt{e}}-\frac{(b c) \operatorname{Subst}\left (\int \left (\frac{5 c^4 d-3 c^2 e}{\left (c^2 d-e\right )^2 \left (1+c^2 x\right )}-\frac{2 d e}{\left (c^2 d-e\right ) (d+e x)^2}+\frac{e \left (-5 c^2 d+3 e\right )}{\left (-c^2 d+e\right )^2 (d+e x)}\right ) \, dx,x,x^2\right )}{16 d^2}-\frac{(3 i b c) \int \left (\frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 \left (1-\sqrt{-c^2} x\right )}+\frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 \left (1+\sqrt{-c^2} x\right )}\right ) \, dx}{16 d^{5/2} \sqrt{e}}+\frac{(3 i b c) \int \left (\frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 \left (1-\sqrt{-c^2} x\right )}+\frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 \left (1+\sqrt{-c^2} x\right )}\right ) \, dx}{16 d^{5/2} \sqrt{e}}\\ &=-\frac{b c}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (d+e x^2\right )}+\frac{3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \sqrt{e}}-\frac{b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac{b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}-\frac{(3 i b c) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1-\sqrt{-c^2} x} \, dx}{32 d^{5/2} \sqrt{e}}-\frac{(3 i b c) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1+\sqrt{-c^2} x} \, dx}{32 d^{5/2} \sqrt{e}}+\frac{(3 i b c) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1-\sqrt{-c^2} x} \, dx}{32 d^{5/2} \sqrt{e}}+\frac{(3 i b c) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1+\sqrt{-c^2} x} \, dx}{32 d^{5/2} \sqrt{e}}\\ &=-\frac{b c}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (d+e x^2\right )}+\frac{3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \sqrt{e}}+\frac{3 i b c \log \left (\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b c \log \left (-\frac{\sqrt{e} \left (1+\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b c \log \left (-\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}+\frac{3 i b c \log \left (\frac{\sqrt{e} \left (1+\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac{b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}-\frac{(3 b c) \int \frac{\log \left (-\frac{i \sqrt{e} \left (1-\sqrt{-c^2} x\right )}{\sqrt{d} \left (\sqrt{-c^2}-\frac{i \sqrt{e}}{\sqrt{d}}\right )}\right )}{1-\frac{i \sqrt{e} x}{\sqrt{d}}} \, dx}{32 \sqrt{-c^2} d^3}-\frac{(3 b c) \int \frac{\log \left (\frac{i \sqrt{e} \left (1-\sqrt{-c^2} x\right )}{\sqrt{d} \left (\sqrt{-c^2}+\frac{i \sqrt{e}}{\sqrt{d}}\right )}\right )}{1+\frac{i \sqrt{e} x}{\sqrt{d}}} \, dx}{32 \sqrt{-c^2} d^3}+\frac{(3 b c) \int \frac{\log \left (-\frac{i \sqrt{e} \left (1+\sqrt{-c^2} x\right )}{\sqrt{d} \left (-\sqrt{-c^2}-\frac{i \sqrt{e}}{\sqrt{d}}\right )}\right )}{1-\frac{i \sqrt{e} x}{\sqrt{d}}} \, dx}{32 \sqrt{-c^2} d^3}+\frac{(3 b c) \int \frac{\log \left (\frac{i \sqrt{e} \left (1+\sqrt{-c^2} x\right )}{\sqrt{d} \left (-\sqrt{-c^2}+\frac{i \sqrt{e}}{\sqrt{d}}\right )}\right )}{1+\frac{i \sqrt{e} x}{\sqrt{d}}} \, dx}{32 \sqrt{-c^2} d^3}\\ &=-\frac{b c}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (d+e x^2\right )}+\frac{3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \sqrt{e}}+\frac{3 i b c \log \left (\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b c \log \left (-\frac{\sqrt{e} \left (1+\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b c \log \left (-\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}+\frac{3 i b c \log \left (\frac{\sqrt{e} \left (1+\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac{b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac{(3 i b c) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-c^2} x}{-\sqrt{-c^2}-\frac{i \sqrt{e}}{\sqrt{d}}}\right )}{x} \, dx,x,1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{(3 i b c) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-c^2} x}{\sqrt{-c^2}-\frac{i \sqrt{e}}{\sqrt{d}}}\right )}{x} \, dx,x,1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{(3 i b c) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-c^2} x}{-\sqrt{-c^2}+\frac{i \sqrt{e}}{\sqrt{d}}}\right )}{x} \, dx,x,1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}+\frac{(3 i b c) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-c^2} x}{\sqrt{-c^2}+\frac{i \sqrt{e}}{\sqrt{d}}}\right )}{x} \, dx,x,1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}\\ &=-\frac{b c}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (d+e x^2\right )}+\frac{3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \sqrt{e}}+\frac{3 i b c \log \left (\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b c \log \left (-\frac{\sqrt{e} \left (1+\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b c \log \left (-\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}+\frac{3 i b c \log \left (\frac{\sqrt{e} \left (1+\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac{b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac{3 i b c \text{Li}_2\left (\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b c \text{Li}_2\left (\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}+\frac{3 i b c \text{Li}_2\left (\frac{\sqrt{-c^2} \left (\sqrt{d}+i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b c \text{Li}_2\left (\frac{\sqrt{-c^2} \left (\sqrt{d}+i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right )}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}\\ \end{align*}

Mathematica [A]  time = 12.1938, size = 1745, normalized size = 1.95 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcTan[c*x])/(d + e*x^2)^3,x]

[Out]

(a*x)/(4*d*(d + e*x^2)^2) + (3*a*x)/(8*d^2*(d + e*x^2)) + (3*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]
) + (b*c*(10*c^2*d*Log[1 + ((c^2*d - e)*Cos[2*ArcTan[c*x]])/(c^2*d + e)] - 6*e*Log[1 + ((c^2*d - e)*Cos[2*ArcT
an[c*x]])/(c^2*d + e)] + (3*c^2*d*(c^2*d - e)*(-4*ArcTan[c*x]*ArcTanh[Sqrt[-(c^2*d*e)]/(c*e*x)] + 2*ArcCos[-((
c^2*d + e)/(c^2*d - e))]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] - (ArcCos[-((c^2*d + e)/(c^2*d - e))] + (2*I)*ArcTa
nh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^2*d*((-I)*e + Sqrt[-(c^2*d*e)])*(-I + c*x))/((c^2*d - e)*(c^2*d + c*Sqr
t[-(c^2*d*e)]*x))] - (ArcCos[-((c^2*d + e)/(c^2*d - e))] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^2
*d*(I*e + Sqrt[-(c^2*d*e)])*(I + c*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] + (ArcCos[-((c^2*d + e)/(
c^2*d - e))] - (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*S
qrt[-(c^2*d*e)])/(Sqrt[c^2*d - e]*E^(I*ArcTan[c*x])*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + (ArcC
os[-((c^2*d + e)/(c^2*d - e))] + (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]
]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTan[c*x]))/(Sqrt[c^2*d - e]*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTa
n[c*x]]])] + I*(PolyLog[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*
(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(c^2*d - c*Sqrt[-(c^2*d*e)
]*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))])))/Sqrt[-(c^2*d*e)] - (3*(c^2*d - e)*e*(-4*ArcTan[c*x]*Arc
Tanh[Sqrt[-(c^2*d*e)]/(c*e*x)] + 2*ArcCos[-((c^2*d + e)/(c^2*d - e))]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] - (Arc
Cos[-((c^2*d + e)/(c^2*d - e))] + (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^2*d*((-I)*e + Sqrt[-(c^2*d
*e)])*(-I + c*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] - (ArcCos[-((c^2*d + e)/(c^2*d - e))] - (2*I)*
ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^2*d*(I*e + Sqrt[-(c^2*d*e)])*(I + c*x))/((c^2*d - e)*(c^2*d + c*Sq
rt[-(c^2*d*e)]*x))] + (ArcCos[-((c^2*d + e)/(c^2*d - e))] - (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTa
nh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)])/(Sqrt[c^2*d - e]*E^(I*ArcTan[c*x])*Sqrt[c^2*d +
e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + (ArcCos[-((c^2*d + e)/(c^2*d - e))] + (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2
*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTan[c*x]))/(Sqrt[c^2*d
- e]*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + I*(PolyLog[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*
(c^2*d - c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)
*Sqrt[-(c^2*d*e)])*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))])))/Sqrt[-(c^2
*d*e)] - (16*c^2*d*(c^2*d - e)*e*ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]])^
2 + (8*c^2*d*e + 4*(5*c^4*d^2 - 8*c^2*d*e + 3*e^2)*ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(c^2*d + e + (c^2*d - e)*Co
s[2*ArcTan[c*x]])))/(32*d^2*(-(c^2*d) + e)^2)

________________________________________________________________________________________

Maple [B]  time = 0.971, size = 4027, normalized size = 4.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctan(c*x))/(e*x^2+d)^3,x)

[Out]

1/8*c^5*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*e+5/16*c^5*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*d-e)*ln((1+I*c
*x)^4/(c^2*x^2+1)^2*c^2*d+2*c^2*d*(1+I*c*x)^2/(c^2*x^2+1)-(1+I*c*x)^4/(c^2*x^2+1)^2*e+c^2*d+2*(1+I*c*x)^2/(c^2
*x^2+1)*e-e)-5/4*c^5*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*d-e)*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))+3/4*c^5*b*arctan(c*x)
^2/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)+3/8*c^5*b*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(
c^2*e*d)^(1/2)-e))/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)-3/4*c^6*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*
d)^2*e^2/d*arctan(c*x)*x^3+3/8*c^4*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2/d^2*arctan(c*x)*x^3*e^3+5/8*c
^4*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2/d*arctan(c*x)*x*e^2+5/16*c^2*b*(d*e)^(1/2)*e/d^2*arctanh(1/4*
(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c^2*d+2*e)/c/(d*e)^(1/2))/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*d-e)-3/8*I*c*b*(
c^2*e*d)^(1/2)/(c^4*d^2-2*c^2*d*e+e^2)/d^2*arctan(c*x)*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*e
*d)^(1/2)-e))+5/4*I*c^7*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*arctan(c*x)*x^2*e+3/8*a/d^2/(d*e)^(1/2)*
arctan(e*x/(d*e)^(1/2))-3/16*I*c^7*b*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*e*d)^(1/2)-e))*arct
an(c*x)/e/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)*d+3/16*I*c^3*b*(c^2*e*d)^(1/2)/(c^4*d^2-2*c^2*d*e+e^2)/e/d
*arctan(c*x)*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*e*d)^(1/2)-e))-9/8*I*c^3*b*ln(1-(c^2*d-e)*(
1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*e*d)^(1/2)-e))*arctan(c*x)*e/d/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)
+3/4*I*c*b*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*e*d)^(1/2)-e))*arctan(c*x)*e^2/d^2/(c^4*d^2-2
*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)+3/16*I/c*b*(c^2*e*d)^(1/2)/d^3*e/(c^4*d^2-2*c^2*d*e+e^2)*arctan(c*x)*ln(1-(c^2
*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*e*d)^(1/2)-e))-3/16*I/c*b*e^3*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2
+1)/(-c^2*d+2*(c^2*e*d)^(1/2)-e))*arctan(c*x)/d^3/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)+5/8*I*c^7*b/(c^4*d
^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2/d*arctan(c*x)*x^4*e^2-3/8*I*c^5*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2
*d)^2/d^2*arctan(c*x)*x^4*e^3-3/4*I*c^5*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2/d*arctan(c*x)*x^2*e^2+3/
8*c^2*a/d^2*x/(c^2*e*x^2+c^2*d)+1/4*c^4*a*x/d/(c^2*e*x^2+c^2*d)^2+1/8*c^7*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2
+c^2*d)^2*e*x^2-3/16*c*b*(c^2*e*d)^(1/2)/d^2/(c^4*d^2-2*c^2*d*e+e^2)*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+
1)/(-c^2*d-2*(c^2*e*d)^(1/2)-e))+1/8*c^2*b*(d*e)^(1/2)/d^2*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*
c^2*d+2*e)/c/(d*e)^(1/2))/(c^4*d^2-2*c^2*d*e+e^2)-3/8*c*b*(c^2*e*d)^(1/2)/d^2/(c^4*d^2-2*c^2*d*e+e^2)*arctan(c
*x)^2-3/16*b*(d*e)^(1/2)/d^3*e*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c^2*d+2*e)/c/(d*e)^(1/2))/(c
^4*d^2-2*c^2*d*e+e^2)-5/4*c^6*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*e*arctan(c*x)*x-3/16/c*b*e^3*arcta
n(c*x)^2/d^3/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)-3/32*c^7*b*d*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1
)/(-c^2*d+2*(c^2*e*d)^(1/2)-e))/e/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)+3/16*c^4*b*(d*e)^(1/2)/d*arctanh(1
/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c^2*d+2*e)/c/(d*e)^(1/2))/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*d-e)+3/4*c*b*
e^2*arctan(c*x)^2/d^2/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)-1/2*c^3*b/(c^4*d^2-2*c^2*d*e+e^2)*e/d/(c^2*d-e
)*ln((1+I*c*x)^4/(c^2*x^2+1)^2*c^2*d+2*c^2*d*(1+I*c*x)^2/(c^2*x^2+1)-(1+I*c*x)^4/(c^2*x^2+1)^2*e+c^2*d+2*(1+I*
c*x)^2/(c^2*x^2+1)*e-e)+5/8*I*c^7*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*d*arctan(c*x)-3/8*I*c^5*b/(c^4
*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*e*arctan(c*x)+3/4*I*c^5*b*arctan(c*x)*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*
x^2+1)/(-c^2*d+2*(c^2*e*d)^(1/2)-e))/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)+1/8*c^5*b/(c^4*d^2-2*c^2*d*e+e^
2)/(c^2*e*x^2+c^2*d)^2/d*x^2*e^2+1/8*c^7*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2/d*x^4*e^2-3/16*b*(d*e)^
(1/2)/d^3*e^2*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c^2*d+2*e)/c/(d*e)^(1/2))/(c^4*d^2-2*c^2*d*e+
e^2)/(c^2*d-e)+5/16*c^4*b*(d*e)^(1/2)/d/e*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c^2*d+2*e)/c/(d*e
)^(1/2))/(c^4*d^2-2*c^2*d*e+e^2)-5/16*c^6*b*(d*e)^(1/2)/e*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c
^2*d+2*e)/c/(d*e)^(1/2))/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*d-e)-9/8*c^3*b*arctan(c*x)^2*e/d/(c^4*d^2-2*c^2*d*e+e^2)
^2*(c^2*e*d)^(1/2)-9/16*c^3*b*e*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*e*d)^(1/2)-e))/d/(c
^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)+3/8*c*b*e^2*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2
*e*d)^(1/2)-e))/d^2/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)+3/16*c*b/(c^4*d^2-2*c^2*d*e+e^2)/d^2*e^2/(c^2*d-
e)*ln((1+I*c*x)^4/(c^2*x^2+1)^2*c^2*d+2*c^2*d*(1+I*c*x)^2/(c^2*x^2+1)-(1+I*c*x)^4/(c^2*x^2+1)^2*e+c^2*d+2*(1+I
*c*x)^2/(c^2*x^2+1)*e-e)-3/4*c*b/(c^4*d^2-2*c^2*d*e+e^2)/d^2*e^2/(c^2*d-e)*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))+3/1
6/c*b*(c^2*e*d)^(1/2)/d^3*e/(c^4*d^2-2*c^2*d*e+e^2)*arctan(c*x)^2+3/32/c*b*(c^2*e*d)^(1/2)/d^3*e/(c^4*d^2-2*c^
2*d*e+e^2)*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*e*d)^(1/2)-e))-3/32/c*b*e^3*polylog(2,(c
^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*e*d)^(1/2)-e))/d^3/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)+3/
32*c^3*b*(c^2*e*d)^(1/2)/d/e/(c^4*d^2-2*c^2*d*e+e^2)*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^
2*e*d)^(1/2)-e))-3/16*c^7*b*d*arctan(c*x)^2/e/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)+3/16*c^3*b*(c^2*e*d)^(
1/2)/d/e/(c^4*d^2-2*c^2*d*e+e^2)*arctan(c*x)^2+2*c^3*b/(c^4*d^2-2*c^2*d*e+e^2)*e/d/(c^2*d-e)*ln((1+I*c*x)/(c^2
*x^2+1)^(1/2))+3/8*c^8*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*e*arctan(c*x)*x^3+5/8*c^8*b/(c^4*d^2-2*c^
2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*d*arctan(c*x)*x

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x))/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \arctan \left (c x\right ) + a}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x))/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

integral((b*arctan(c*x) + a)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atan(c*x))/(e*x**2+d)**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x))/(e*x^2+d)^3,x, algorithm="giac")

[Out]

integrate((b*arctan(c*x) + a)/(e*x^2 + d)^3, x)