3.1155 \(\int \frac{x^2 (a+b \tan ^{-1}(c x))}{d+e x^2} \, dx\)

Optimal. Leaf size=555 \[ \frac{i b \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} (-c x+i)}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 e^{3/2}}-\frac{i b \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} (1-i c x)}{\sqrt{e}+i c \sqrt{-d}}\right )}{4 e^{3/2}}-\frac{i b \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} (1+i c x)}{\sqrt{e}+i c \sqrt{-d}}\right )}{4 e^{3/2}}+\frac{i b \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} (c x+i)}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 e^{3/2}}-\frac{a \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{3/2}}+\frac{a x}{e}-\frac{b \log \left (c^2 x^2+1\right )}{2 c e}-\frac{i b \sqrt{-d} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 e^{3/2}}+\frac{i b \sqrt{-d} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 e^{3/2}}-\frac{i b \sqrt{-d} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 e^{3/2}}+\frac{i b \sqrt{-d} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 e^{3/2}}+\frac{b x \tan ^{-1}(c x)}{e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.630897, antiderivative size = 555, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {4916, 4846, 260, 4910, 205, 4908, 2409, 2394, 2393, 2391} \[ \frac{i b \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} (-c x+i)}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 e^{3/2}}-\frac{i b \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} (1-i c x)}{\sqrt{e}+i c \sqrt{-d}}\right )}{4 e^{3/2}}-\frac{i b \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} (1+i c x)}{\sqrt{e}+i c \sqrt{-d}}\right )}{4 e^{3/2}}+\frac{i b \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} (c x+i)}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 e^{3/2}}-\frac{a \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{3/2}}+\frac{a x}{e}-\frac{b \log \left (c^2 x^2+1\right )}{2 c e}-\frac{i b \sqrt{-d} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 e^{3/2}}+\frac{i b \sqrt{-d} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 e^{3/2}}-\frac{i b \sqrt{-d} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 e^{3/2}}+\frac{i b \sqrt{-d} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 e^{3/2}}+\frac{b x \tan ^{-1}(c x)}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4916

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(d*f^2)/e, Int[((f*x)^(m - 2)*(a + b*ArcTan[c*x])^p)
/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]

Rule 4846

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x])^p, x] - Dist[b*c*p, Int[
(x*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 4910

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

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

Mathematica [A]  time = 3.38658, size = 776, normalized size = 1.4 \[ \frac{b \left (\frac{c^2 d \left (i \left (\text{PolyLog}\left (2,\frac{\left (2 i \sqrt{-c^2 d e}+c^2 d+e\right ) \left (x \sqrt{-c^2 d e}+c d\right )}{\left (c^2 d-e\right ) \left (c d-x \sqrt{-c^2 d e}\right )}\right )-\text{PolyLog}\left (2,\frac{\left (-2 i \sqrt{-c^2 d e}+c^2 d+e\right ) \left (x \sqrt{-c^2 d e}+c d\right )}{\left (c^2 d-e\right ) \left (c d-x \sqrt{-c^2 d e}\right )}\right )\right )-2 \cos ^{-1}\left (-\frac{c^2 d+e}{c^2 d-e}\right ) \tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )-4 \tan ^{-1}(c x) \tanh ^{-1}\left (\frac{c d}{x \sqrt{-c^2 d e}}\right )-\log \left (\frac{2 c d (c x-i) \left (\sqrt{-c^2 d e}+i e\right )}{\left (c^2 d-e\right ) \left (x \sqrt{-c^2 d e}-c d\right )}\right ) \left (\cos ^{-1}\left (-\frac{c^2 d+e}{c^2 d-e}\right )-2 i \tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )\right )-\log \left (\frac{2 c d (c x+i) \left (\sqrt{-c^2 d e}-i e\right )}{\left (c^2 d-e\right ) \left (x \sqrt{-c^2 d e}-c d\right )}\right ) \left (\cos ^{-1}\left (-\frac{c^2 d+e}{c^2 d-e}\right )+2 i \tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )\right )+\left (2 i \tanh ^{-1}\left (\frac{c d}{x \sqrt{-c^2 d e}}\right )+2 i \tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )+\cos ^{-1}\left (-\frac{c^2 d+e}{c^2 d-e}\right )\right ) \log \left (\frac{\sqrt{2} \sqrt{-c^2 d e} e^{-i \tan ^{-1}(c x)}}{\sqrt{e-c^2 d} \sqrt{\left (e-c^2 d\right ) \cos \left (2 \tan ^{-1}(c x)\right )+c^2 (-d)-e}}\right )+\left (-2 i \tanh ^{-1}\left (\frac{c d}{x \sqrt{-c^2 d e}}\right )-2 i \tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )+\cos ^{-1}\left (-\frac{c^2 d+e}{c^2 d-e}\right )\right ) \log \left (\frac{\sqrt{2} \sqrt{-c^2 d e} e^{i \tan ^{-1}(c x)}}{\sqrt{e-c^2 d} \sqrt{\left (e-c^2 d\right ) \cos \left (2 \tan ^{-1}(c x)\right )+c^2 (-d)-e}}\right )\right )}{\sqrt{-c^2 d e}}-2 \log \left (c^2 x^2+1\right )+4 c x \tan ^{-1}(c x)\right )}{4 c e}-\frac{a \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{3/2}}+\frac{a x}{e} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*x)/e - (a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/e^(3/2) + (b*(4*c*x*ArcTan[c*x] - 2*Log[1 + c^2*x^2] + (c^2*
d*(-4*ArcTan[c*x]*ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] - 2*ArcCos[-((c^2*d + e)/(c^2*d - e))]*ArcTanh[(c*e*x)/S
qrt[-(c^2*d*e)]] - (ArcCos[-((c^2*d + e)/(c^2*d - e))] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c*d*(
I*e + Sqrt[-(c^2*d*e)])*(-I + c*x))/((c^2*d - e)*(-(c*d) + 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*d*((-I)*e + Sqrt[-(c^2*d*e)])*(I + c*x))/((c^2*d
- e)*(-(c*d) + 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)] + (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)])/(Sqrt[-(c^2*d) + e]*E^(I*Ar
cTan[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)] - (2*I)*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*(-PolyL
og[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(c*d + Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c*d - Sqrt[-(c^2*d*e)]*x
))] + PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(c*d + Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c*d - Sqrt[-(
c^2*d*e)]*x))])))/Sqrt[-(c^2*d*e)]))/(4*c*e)

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Maple [C]  time = 0.501, size = 2409, normalized size = 4.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*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^2*d-e)+1/8
/c*b/(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)+1/c*b/e*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+2/c*b/(c^2*d-e)*ln((1+I*c*x)/(c^2*x
^2+1)^(1/2))-1/4/c*b/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*b*(d*e)^(1/2)/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))-1/4*b*(d*e)^(1/2)*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^2*d-e)^2-a*d/e/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+a*x/e+b*x*arctan(c*x)/e-2*c*b/e*d/(c^
2*d-e)*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))-1/8*c^2*b*(d*e)^(1/2)/e^3*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))+1/8*c^5*b/e^2*d^3/(c^2*d-e)^2*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)+3/8/c^2*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))-1/4*c*b/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)+1/8*c^3*b/e^2*d^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)+1/8*c^3*b/e*d^2/(c^2*d-e)^2*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^2*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^2*d
-e)+3/8/c*b*e/(c^2*d-e)^2*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)+I/c*b*arctan(c*x)/e+1/4*c*b/e^2*d*sum((_R1^2*c^2*d-_R1^2*e+c^2
*d+3*e)/(_R1^2*c^2*d-_R1^2*e+c^2*d+e)*(I*arctan(c*x)*ln((_R1-(1+I*c*x)/(c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-(1+I
*c*x)/(c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf((c^2*d-e)*_Z^4+(2*c^2*d+2*e)*_Z^2+c^2*d-e))-5/8*c*b*d/(c^2*d-e)^2*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)-1/4*c*b/e^2*d*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)-1/4*c*b/e^2*d*sum((_R1^2*c^2*d-_R1^2*e+c^2*d-e)/(_R1^2
*c^2*d-_R1^2*e+c^2*d+e)*(I*arctan(c*x)*ln((_R1-(1+I*c*x)/(c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-(1+I*c*x)/(c^2*x^2
+1)^(1/2))/_R1)),_R1=RootOf((c^2*d-e)*_Z^4+(2*c^2*d+2*e)*_Z^2+c^2*d-e))+1/8*c^6*b*(d*e)^(1/2)/e^3*d^3*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^2*d-e)^2+3/8/c^2*b*(d*e)^(1/2)/d*e*arc
tanh(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^2*d-e)^2-5/4*c^2*b*(d*e)^(1/2)/e^
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^2*d-e)+1/4*c^4*b*(d*e)^(1/
2)/e^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^2*d-e)^2-1/2*c^2*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^2*d-e)^2

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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(x^2*(a+b*arctan(c*x))/(e*x^2+d),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 x^{2} \arctan \left (c x\right ) + a x^{2}}{e x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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