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

Optimal. Leaf size=906 \[ \frac{b c^2 \tan ^{-1}\left (c x^3\right ) d^5}{e \left (c^2 d^6+e^6\right )}+\frac{b c^{5/3} \log \left (c^{2/3} x^2+1\right ) d^4}{2 \left (c^2 d^6+e^6\right )}+\frac{3 b c e^2 \log (d+e x) d^2}{c^2 d^6+e^6}-\frac{b c e^2 \log \left (c^2 x^6+1\right ) d^2}{2 \left (c^2 d^6+e^6\right )}-\frac{b c^{2/3} e^3 \tan ^{-1}\left (\sqrt [3]{c} x\right ) d}{c^2 d^6+e^6}+\frac{b c^{2/3} \left (\sqrt{3} c d^3+e^3\right ) \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right ) d}{2 \left (c^2 d^6+e^6\right )}+\frac{b c^{2/3} \left (\sqrt{3} c d^3-e^3\right ) \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right ) d}{2 \left (c^2 d^6+e^6\right )}-\frac{b c^{2/3} \left (c d^3-\sqrt{3} e^3\right ) \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right ) d}{4 \left (c^2 d^6+e^6\right )}-\frac{b c^{2/3} \left (c d^3+\sqrt{3} e^3\right ) \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right ) d}{4 \left (c^2 d^6+e^6\right )}-\frac{a+b \tan ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{\sqrt{3} b c^{5/3} e \left (\sqrt{-c^2} d^3+e^3\right ) \tan ^{-1}\left (\frac{\frac{2 c^{2/3} x}{\sqrt [6]{-c^2}}+1}{\sqrt{3}}\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac{\sqrt{3} b c^{5/3} e \left (\sqrt{-c^2} d^3-e^3\right ) \tan ^{-1}\left (\frac{c^{4/3}+2 \left (-c^2\right )^{5/6} x}{\sqrt{3} c^{4/3}}\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac{b c^{5/3} e \left (\sqrt{-c^2} d^3+e^3\right ) \log \left (\sqrt [6]{-c^2}-c^{2/3} x\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac{b c^{5/3} e \left (\sqrt{-c^2} d^3-e^3\right ) \log \left (c^{2/3} x+\sqrt [6]{-c^2}\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac{b c^{5/3} e \left (\sqrt{-c^2} d^3-e^3\right ) \log \left (c^{4/3} x^2-c^{2/3} \sqrt [6]{-c^2} x+\sqrt [3]{-c^2}\right )}{4 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac{b c^{5/3} e \left (\sqrt{-c^2} d^3+e^3\right ) \log \left (c^{4/3} x^2+c^{2/3} \sqrt [6]{-c^2} x+\sqrt [3]{-c^2}\right )}{4 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )} \]

[Out]

-((b*c^(2/3)*d*e^3*ArcTan[c^(1/3)*x])/(c^2*d^6 + e^6)) + (b*c^2*d^5*ArcTan[c*x^3])/(e*(c^2*d^6 + e^6)) - (a +
b*ArcTan[c*x^3])/(e*(d + e*x)) + (b*c^(2/3)*d*(Sqrt[3]*c*d^3 + e^3)*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(2*(c^2*d^6
+ e^6)) + (b*c^(2/3)*d*(Sqrt[3]*c*d^3 - e^3)*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(2*(c^2*d^6 + e^6)) + (Sqrt[3]*b*
c^(5/3)*e*(Sqrt[-c^2]*d^3 + e^3)*ArcTan[(1 + (2*c^(2/3)*x)/(-c^2)^(1/6))/Sqrt[3]])/(2*(-c^2)^(2/3)*(c^2*d^6 +
e^6)) - (Sqrt[3]*b*c^(5/3)*e*(Sqrt[-c^2]*d^3 - e^3)*ArcTan[(c^(4/3) + 2*(-c^2)^(5/6)*x)/(Sqrt[3]*c^(4/3))])/(2
*(-c^2)^(2/3)*(c^2*d^6 + e^6)) + (b*c^(5/3)*e*(Sqrt[-c^2]*d^3 + e^3)*Log[(-c^2)^(1/6) - c^(2/3)*x])/(2*(-c^2)^
(2/3)*(c^2*d^6 + e^6)) - (b*c^(5/3)*e*(Sqrt[-c^2]*d^3 - e^3)*Log[(-c^2)^(1/6) + c^(2/3)*x])/(2*(-c^2)^(2/3)*(c
^2*d^6 + e^6)) + (3*b*c*d^2*e^2*Log[d + e*x])/(c^2*d^6 + e^6) + (b*c^(5/3)*d^4*Log[1 + c^(2/3)*x^2])/(2*(c^2*d
^6 + e^6)) - (b*c^(2/3)*d*(c*d^3 - Sqrt[3]*e^3)*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(4*(c^2*d^6 + e^6))
- (b*c^(2/3)*d*(c*d^3 + Sqrt[3]*e^3)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(4*(c^2*d^6 + e^6)) + (b*c^(5/3
)*e*(Sqrt[-c^2]*d^3 - e^3)*Log[(-c^2)^(1/3) - c^(2/3)*(-c^2)^(1/6)*x + c^(4/3)*x^2])/(4*(-c^2)^(2/3)*(c^2*d^6
+ e^6)) - (b*c^(5/3)*e*(Sqrt[-c^2]*d^3 + e^3)*Log[(-c^2)^(1/3) + c^(2/3)*(-c^2)^(1/6)*x + c^(4/3)*x^2])/(4*(-c
^2)^(2/3)*(c^2*d^6 + e^6)) - (b*c*d^2*e^2*Log[1 + c^2*x^6])/(2*(c^2*d^6 + e^6))

________________________________________________________________________________________

Rubi [A]  time = 1.4838, antiderivative size = 906, normalized size of antiderivative = 1., number of steps used = 35, number of rules used = 16, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.889, Rules used = {5205, 12, 6725, 1876, 1416, 635, 203, 260, 634, 617, 204, 628, 1511, 292, 31, 1469} \[ \frac{b c^2 \tan ^{-1}\left (c x^3\right ) d^5}{e \left (c^2 d^6+e^6\right )}+\frac{b c^{5/3} \log \left (c^{2/3} x^2+1\right ) d^4}{2 \left (c^2 d^6+e^6\right )}+\frac{3 b c e^2 \log (d+e x) d^2}{c^2 d^6+e^6}-\frac{b c e^2 \log \left (c^2 x^6+1\right ) d^2}{2 \left (c^2 d^6+e^6\right )}-\frac{b c^{2/3} e^3 \tan ^{-1}\left (\sqrt [3]{c} x\right ) d}{c^2 d^6+e^6}+\frac{b c^{2/3} \left (\sqrt{3} c d^3+e^3\right ) \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right ) d}{2 \left (c^2 d^6+e^6\right )}+\frac{b c^{2/3} \left (\sqrt{3} c d^3-e^3\right ) \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right ) d}{2 \left (c^2 d^6+e^6\right )}-\frac{b c^{2/3} \left (c d^3-\sqrt{3} e^3\right ) \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right ) d}{4 \left (c^2 d^6+e^6\right )}-\frac{b c^{2/3} \left (c d^3+\sqrt{3} e^3\right ) \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right ) d}{4 \left (c^2 d^6+e^6\right )}-\frac{a+b \tan ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{\sqrt{3} b c^{5/3} e \left (\sqrt{-c^2} d^3+e^3\right ) \tan ^{-1}\left (\frac{\frac{2 c^{2/3} x}{\sqrt [6]{-c^2}}+1}{\sqrt{3}}\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac{\sqrt{3} b c^{5/3} e \left (\sqrt{-c^2} d^3-e^3\right ) \tan ^{-1}\left (\frac{c^{4/3}+2 \left (-c^2\right )^{5/6} x}{\sqrt{3} c^{4/3}}\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac{b c^{5/3} e \left (\sqrt{-c^2} d^3+e^3\right ) \log \left (\sqrt [6]{-c^2}-c^{2/3} x\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac{b c^{5/3} e \left (\sqrt{-c^2} d^3-e^3\right ) \log \left (c^{2/3} x+\sqrt [6]{-c^2}\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac{b c^{5/3} e \left (\sqrt{-c^2} d^3-e^3\right ) \log \left (c^{4/3} x^2-c^{2/3} \sqrt [6]{-c^2} x+\sqrt [3]{-c^2}\right )}{4 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac{b c^{5/3} e \left (\sqrt{-c^2} d^3+e^3\right ) \log \left (c^{4/3} x^2+c^{2/3} \sqrt [6]{-c^2} x+\sqrt [3]{-c^2}\right )}{4 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*c^(2/3)*d*e^3*ArcTan[c^(1/3)*x])/(c^2*d^6 + e^6)) + (b*c^2*d^5*ArcTan[c*x^3])/(e*(c^2*d^6 + e^6)) - (a +
b*ArcTan[c*x^3])/(e*(d + e*x)) + (b*c^(2/3)*d*(Sqrt[3]*c*d^3 + e^3)*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(2*(c^2*d^6
+ e^6)) + (b*c^(2/3)*d*(Sqrt[3]*c*d^3 - e^3)*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(2*(c^2*d^6 + e^6)) + (Sqrt[3]*b*
c^(5/3)*e*(Sqrt[-c^2]*d^3 + e^3)*ArcTan[(1 + (2*c^(2/3)*x)/(-c^2)^(1/6))/Sqrt[3]])/(2*(-c^2)^(2/3)*(c^2*d^6 +
e^6)) - (Sqrt[3]*b*c^(5/3)*e*(Sqrt[-c^2]*d^3 - e^3)*ArcTan[(c^(4/3) + 2*(-c^2)^(5/6)*x)/(Sqrt[3]*c^(4/3))])/(2
*(-c^2)^(2/3)*(c^2*d^6 + e^6)) + (b*c^(5/3)*e*(Sqrt[-c^2]*d^3 + e^3)*Log[(-c^2)^(1/6) - c^(2/3)*x])/(2*(-c^2)^
(2/3)*(c^2*d^6 + e^6)) - (b*c^(5/3)*e*(Sqrt[-c^2]*d^3 - e^3)*Log[(-c^2)^(1/6) + c^(2/3)*x])/(2*(-c^2)^(2/3)*(c
^2*d^6 + e^6)) + (3*b*c*d^2*e^2*Log[d + e*x])/(c^2*d^6 + e^6) + (b*c^(5/3)*d^4*Log[1 + c^(2/3)*x^2])/(2*(c^2*d
^6 + e^6)) - (b*c^(2/3)*d*(c*d^3 - Sqrt[3]*e^3)*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(4*(c^2*d^6 + e^6))
- (b*c^(2/3)*d*(c*d^3 + Sqrt[3]*e^3)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(4*(c^2*d^6 + e^6)) + (b*c^(5/3
)*e*(Sqrt[-c^2]*d^3 - e^3)*Log[(-c^2)^(1/3) - c^(2/3)*(-c^2)^(1/6)*x + c^(4/3)*x^2])/(4*(-c^2)^(2/3)*(c^2*d^6
+ e^6)) - (b*c^(5/3)*e*(Sqrt[-c^2]*d^3 + e^3)*Log[(-c^2)^(1/3) + c^(2/3)*(-c^2)^(1/6)*x + c^(4/3)*x^2])/(4*(-c
^2)^(2/3)*(c^2*d^6 + e^6)) - (b*c*d^2*e^2*Log[1 + c^2*x^6])/(2*(c^2*d^6 + e^6))

Rule 5205

Int[((a_.) + ArcTan[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcTan[
u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(1 + u^2), x], x]
, x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(m +
1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 1876

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[(x^ii*(Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii
]*x^(n/2)))/(a + b*x^n), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ
[n/2, 0] && Expon[Pq, x] < n

Rule 1416

Int[((d_) + (e_.)*(x_)^3)/((a_) + (c_.)*(x_)^6), x_Symbol] :> With[{q = Rt[c/a, 6]}, Dist[1/(3*a*q^2), Int[(q^
2*d - e*x)/(1 + q^2*x^2), x], x] + (Dist[1/(6*a*q^2), Int[(2*q^2*d - (Sqrt[3]*q^3*d - e)*x)/(1 - Sqrt[3]*q*x +
q^2*x^2), x], x] + Dist[1/(6*a*q^2), Int[(2*q^2*d + (Sqrt[3]*q^3*d + e)*x)/(1 + Sqrt[3]*q*x + q^2*x^2), x], x
])] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && PosQ[c/a]

Rule 635

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 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 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1511

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[-(a*c),
2]}, -Dist[e/2 + (c*d)/(2*q), Int[(f*x)^m/(q - c*x^n), x], x] + Dist[e/2 - (c*d)/(2*q), Int[(f*x)^m/(q + c*x^
n), x], x]] /; FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 31

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

Rule 1469

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[I
nt[(d + e*x)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[Sim
plify[m - n + 1], 0]

Rubi steps

\begin{align*} \int \frac{a+b \tan ^{-1}\left (c x^3\right )}{(d+e x)^2} \, dx &=-\frac{a+b \tan ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{b \int \frac{3 c x^2}{(d+e x) \left (1+c^2 x^6\right )} \, dx}{e}\\ &=-\frac{a+b \tan ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{(3 b c) \int \frac{x^2}{(d+e x) \left (1+c^2 x^6\right )} \, dx}{e}\\ &=-\frac{a+b \tan ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{(3 b c) \int \left (\frac{d^2 e^4}{\left (c^2 d^6+e^6\right ) (d+e x)}+\frac{(d-e x) \left (-e^4+c^2 d^4 x^2+c^2 d^2 e^2 x^4\right )}{\left (c^2 d^6+e^6\right ) \left (1+c^2 x^6\right )}\right ) \, dx}{e}\\ &=-\frac{a+b \tan ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{3 b c d^2 e^2 \log (d+e x)}{c^2 d^6+e^6}+\frac{(3 b c) \int \frac{(d-e x) \left (-e^4+c^2 d^4 x^2+c^2 d^2 e^2 x^4\right )}{1+c^2 x^6} \, dx}{e \left (c^2 d^6+e^6\right )}\\ &=-\frac{a+b \tan ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{3 b c d^2 e^2 \log (d+e x)}{c^2 d^6+e^6}+\frac{(3 b c) \int \left (\frac{-d e^4-c^2 d^4 e x^3}{1+c^2 x^6}+\frac{x \left (e^5+c^2 d^3 e^2 x^3\right )}{1+c^2 x^6}+\frac{x^2 \left (c^2 d^5-c^2 d^2 e^3 x^3\right )}{1+c^2 x^6}\right ) \, dx}{e \left (c^2 d^6+e^6\right )}\\ &=-\frac{a+b \tan ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{3 b c d^2 e^2 \log (d+e x)}{c^2 d^6+e^6}+\frac{(3 b c) \int \frac{-d e^4-c^2 d^4 e x^3}{1+c^2 x^6} \, dx}{e \left (c^2 d^6+e^6\right )}+\frac{(3 b c) \int \frac{x \left (e^5+c^2 d^3 e^2 x^3\right )}{1+c^2 x^6} \, dx}{e \left (c^2 d^6+e^6\right )}+\frac{(3 b c) \int \frac{x^2 \left (c^2 d^5-c^2 d^2 e^3 x^3\right )}{1+c^2 x^6} \, dx}{e \left (c^2 d^6+e^6\right )}\\ &=-\frac{a+b \tan ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{3 b c d^2 e^2 \log (d+e x)}{c^2 d^6+e^6}+\frac{\left (b \sqrt [3]{c}\right ) \int \frac{-2 c^{2/3} d e^4-\left (c^2 d^4 e-\sqrt{3} c d e^4\right ) x}{1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{2 e \left (c^2 d^6+e^6\right )}+\frac{\left (b \sqrt [3]{c}\right ) \int \frac{-2 c^{2/3} d e^4+\left (-c^2 d^4 e-\sqrt{3} c d e^4\right ) x}{1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{2 e \left (c^2 d^6+e^6\right )}+\frac{\left (b \sqrt [3]{c}\right ) \int \frac{-c^{2/3} d e^4+c^2 d^4 e x}{1+c^{2/3} x^2} \, dx}{e \left (c^2 d^6+e^6\right )}+\frac{(b c) \operatorname{Subst}\left (\int \frac{c^2 d^5-c^2 d^2 e^3 x}{1+c^2 x^2} \, dx,x,x^3\right )}{e \left (c^2 d^6+e^6\right )}-\frac{\left (3 b c^3 e \left (d^3+\frac{e^3}{\sqrt{-c^2}}\right )\right ) \int \frac{x}{\sqrt{-c^2}-c^2 x^3} \, dx}{2 \left (c^2 d^6+e^6\right )}+\frac{\left (3 b c e \left (c^2 d^3+\sqrt{-c^2} e^3\right )\right ) \int \frac{x}{\sqrt{-c^2}+c^2 x^3} \, dx}{2 \left (c^2 d^6+e^6\right )}\\ &=-\frac{a+b \tan ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{3 b c d^2 e^2 \log (d+e x)}{c^2 d^6+e^6}+\frac{\left (b c^{7/3} d^4\right ) \int \frac{x}{1+c^{2/3} x^2} \, dx}{c^2 d^6+e^6}+\frac{\left (b c^3 d^5\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x^2} \, dx,x,x^3\right )}{e \left (c^2 d^6+e^6\right )}-\frac{\left (b c^3 d^2 e^2\right ) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x^2} \, dx,x,x^3\right )}{c^2 d^6+e^6}-\frac{\left (b c d e^3\right ) \int \frac{1}{1+c^{2/3} x^2} \, dx}{c^2 d^6+e^6}+\frac{\left (b c d \left (\sqrt{3} c d^3-e^3\right )\right ) \int \frac{1}{1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \left (c^2 d^6+e^6\right )}+\frac{\left (b \sqrt [3]{c} \sqrt [3]{-c^2} e \left (\sqrt{-c^2} d^3-e^3\right )\right ) \int \frac{1}{\sqrt [6]{-c^2}+c^{2/3} x} \, dx}{2 \left (c^2 d^6+e^6\right )}-\frac{\left (b \sqrt [3]{c} \sqrt [3]{-c^2} e \left (\sqrt{-c^2} d^3-e^3\right )\right ) \int \frac{\sqrt [6]{-c^2}+c^{2/3} x}{\sqrt [3]{-c^2}-c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2} \, dx}{2 \left (c^2 d^6+e^6\right )}-\frac{\left (b c d \left (\sqrt{3} c d^3+e^3\right )\right ) \int \frac{1}{1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \left (c^2 d^6+e^6\right )}-\frac{\left (b c^{7/3} e \left (\sqrt{-c^2} d^3+e^3\right )\right ) \int \frac{1}{\sqrt [6]{-c^2}-c^{2/3} x} \, dx}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac{\left (b c^{7/3} e \left (\sqrt{-c^2} d^3+e^3\right )\right ) \int \frac{\sqrt [6]{-c^2}-c^{2/3} x}{\sqrt [3]{-c^2}+c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2} \, dx}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac{\left (b c^{2/3} d \left (c d^3-\sqrt{3} e^3\right )\right ) \int \frac{-\sqrt{3} \sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \left (c^2 d^6+e^6\right )}-\frac{\left (b c^{2/3} d \left (c d^3+\sqrt{3} e^3\right )\right ) \int \frac{\sqrt{3} \sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \left (c^2 d^6+e^6\right )}\\ &=-\frac{b c^{2/3} d e^3 \tan ^{-1}\left (\sqrt [3]{c} x\right )}{c^2 d^6+e^6}+\frac{b c^2 d^5 \tan ^{-1}\left (c x^3\right )}{e \left (c^2 d^6+e^6\right )}-\frac{a+b \tan ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{b c^{5/3} e \left (\sqrt{-c^2} d^3+e^3\right ) \log \left (\sqrt [6]{-c^2}-c^{2/3} x\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac{b \sqrt [3]{-c^2} e \left (\sqrt{-c^2} d^3-e^3\right ) \log \left (\sqrt [6]{-c^2}+c^{2/3} x\right )}{2 \sqrt [3]{c} \left (c^2 d^6+e^6\right )}+\frac{3 b c d^2 e^2 \log (d+e x)}{c^2 d^6+e^6}+\frac{b c^{5/3} d^4 \log \left (1+c^{2/3} x^2\right )}{2 \left (c^2 d^6+e^6\right )}-\frac{b c^{2/3} d \left (c d^3-\sqrt{3} e^3\right ) \log \left (1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c^2 d^6+e^6\right )}-\frac{b c^{2/3} d \left (c d^3+\sqrt{3} e^3\right ) \log \left (1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c^2 d^6+e^6\right )}-\frac{b c d^2 e^2 \log \left (1+c^2 x^6\right )}{2 \left (c^2 d^6+e^6\right )}-\frac{\left (b \sqrt [3]{-c^2} e \left (\sqrt{-c^2} d^3-e^3\right )\right ) \int \frac{-c^{2/3} \sqrt [6]{-c^2}+2 c^{4/3} x}{\sqrt [3]{-c^2}-c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2} \, dx}{4 \sqrt [3]{c} \left (c^2 d^6+e^6\right )}-\frac{\left (b c^{5/3} e \left (\sqrt{-c^2} d^3+e^3\right )\right ) \int \frac{c^{2/3} \sqrt [6]{-c^2}+2 c^{4/3} x}{\sqrt [3]{-c^2}+c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2} \, dx}{4 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac{\left (3 b c^{7/3} e \left (\sqrt{-c^2} d^3+e^3\right )\right ) \int \frac{1}{\sqrt [3]{-c^2}+c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2} \, dx}{4 \sqrt{-c^2} \left (c^2 d^6+e^6\right )}-\frac{\left (b c^{2/3} d \left (3 c d^3-\sqrt{3} e^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1+\frac{2 \sqrt [3]{c} x}{\sqrt{3}}\right )}{6 \left (c^2 d^6+e^6\right )}-\frac{\left (b c^{2/3} d \left (3 c d^3+\sqrt{3} e^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1-\frac{2 \sqrt [3]{c} x}{\sqrt{3}}\right )}{6 \left (c^2 d^6+e^6\right )}+\frac{\left (3 b \sqrt [3]{c} e \left (c^2 d^3+\sqrt{-c^2} e^3\right )\right ) \int \frac{1}{\sqrt [3]{-c^2}-c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2} \, dx}{4 \left (c^2 d^6+e^6\right )}\\ &=-\frac{b c^{2/3} d e^3 \tan ^{-1}\left (\sqrt [3]{c} x\right )}{c^2 d^6+e^6}+\frac{b c^2 d^5 \tan ^{-1}\left (c x^3\right )}{e \left (c^2 d^6+e^6\right )}-\frac{a+b \tan ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{b c^{2/3} d \left (\sqrt{3} c d^3+e^3\right ) \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )}{2 \left (c^2 d^6+e^6\right )}+\frac{b c^{2/3} d \left (\sqrt{3} c d^3-e^3\right ) \tan ^{-1}\left (\sqrt{3}+2 \sqrt [3]{c} x\right )}{2 \left (c^2 d^6+e^6\right )}+\frac{b c^{5/3} e \left (\sqrt{-c^2} d^3+e^3\right ) \log \left (\sqrt [6]{-c^2}-c^{2/3} x\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac{b \sqrt [3]{-c^2} e \left (\sqrt{-c^2} d^3-e^3\right ) \log \left (\sqrt [6]{-c^2}+c^{2/3} x\right )}{2 \sqrt [3]{c} \left (c^2 d^6+e^6\right )}+\frac{3 b c d^2 e^2 \log (d+e x)}{c^2 d^6+e^6}+\frac{b c^{5/3} d^4 \log \left (1+c^{2/3} x^2\right )}{2 \left (c^2 d^6+e^6\right )}-\frac{b c^{2/3} d \left (c d^3-\sqrt{3} e^3\right ) \log \left (1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c^2 d^6+e^6\right )}-\frac{b c^{2/3} d \left (c d^3+\sqrt{3} e^3\right ) \log \left (1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c^2 d^6+e^6\right )}-\frac{b \sqrt [3]{-c^2} e \left (\sqrt{-c^2} d^3-e^3\right ) \log \left (\sqrt [3]{-c^2}-c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2\right )}{4 \sqrt [3]{c} \left (c^2 d^6+e^6\right )}-\frac{b c^{5/3} e \left (\sqrt{-c^2} d^3+e^3\right ) \log \left (\sqrt [3]{-c^2}+c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2\right )}{4 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac{b c d^2 e^2 \log \left (1+c^2 x^6\right )}{2 \left (c^2 d^6+e^6\right )}-\frac{\left (3 b \sqrt [3]{-c^2} e \left (\sqrt{-c^2} d^3-e^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 c^{2/3} x}{\sqrt [6]{-c^2}}\right )}{2 \sqrt [3]{c} \left (c^2 d^6+e^6\right )}-\frac{\left (3 b c^{5/3} e \left (\sqrt{-c^2} d^3+e^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 c^{2/3} x}{\sqrt [6]{-c^2}}\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}\\ &=-\frac{b c^{2/3} d e^3 \tan ^{-1}\left (\sqrt [3]{c} x\right )}{c^2 d^6+e^6}+\frac{b c^2 d^5 \tan ^{-1}\left (c x^3\right )}{e \left (c^2 d^6+e^6\right )}-\frac{a+b \tan ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac{b c^{2/3} d \left (\sqrt{3} c d^3+e^3\right ) \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )}{2 \left (c^2 d^6+e^6\right )}+\frac{b c^{2/3} d \left (\sqrt{3} c d^3-e^3\right ) \tan ^{-1}\left (\sqrt{3}+2 \sqrt [3]{c} x\right )}{2 \left (c^2 d^6+e^6\right )}+\frac{\sqrt{3} b c^{5/3} e \left (\sqrt{-c^2} d^3+e^3\right ) \tan ^{-1}\left (\frac{1+\frac{2 c^{2/3} x}{\sqrt [6]{-c^2}}}{\sqrt{3}}\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac{\sqrt{3} b \sqrt [3]{-c^2} e \left (\sqrt{-c^2} d^3-e^3\right ) \tan ^{-1}\left (\frac{c^{4/3}+2 \left (-c^2\right )^{5/6} x}{\sqrt{3} c^{4/3}}\right )}{2 \sqrt [3]{c} \left (c^2 d^6+e^6\right )}+\frac{b c^{5/3} e \left (\sqrt{-c^2} d^3+e^3\right ) \log \left (\sqrt [6]{-c^2}-c^{2/3} x\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac{b \sqrt [3]{-c^2} e \left (\sqrt{-c^2} d^3-e^3\right ) \log \left (\sqrt [6]{-c^2}+c^{2/3} x\right )}{2 \sqrt [3]{c} \left (c^2 d^6+e^6\right )}+\frac{3 b c d^2 e^2 \log (d+e x)}{c^2 d^6+e^6}+\frac{b c^{5/3} d^4 \log \left (1+c^{2/3} x^2\right )}{2 \left (c^2 d^6+e^6\right )}-\frac{b c^{2/3} d \left (c d^3-\sqrt{3} e^3\right ) \log \left (1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c^2 d^6+e^6\right )}-\frac{b c^{2/3} d \left (c d^3+\sqrt{3} e^3\right ) \log \left (1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c^2 d^6+e^6\right )}-\frac{b \sqrt [3]{-c^2} e \left (\sqrt{-c^2} d^3-e^3\right ) \log \left (\sqrt [3]{-c^2}-c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2\right )}{4 \sqrt [3]{c} \left (c^2 d^6+e^6\right )}-\frac{b c^{5/3} e \left (\sqrt{-c^2} d^3+e^3\right ) \log \left (\sqrt [3]{-c^2}+c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2\right )}{4 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac{b c d^2 e^2 \log \left (1+c^2 x^6\right )}{2 \left (c^2 d^6+e^6\right )}\\ \end{align*}

Mathematica [A]  time = 14.4625, size = 536, normalized size = 0.59 \[ \frac{-4 a \sqrt [3]{c} \left (c^2 d^6+e^6\right )-2 b c^{4/3} d^2 e^3 \log \left (c^2 x^6+1\right ) (d+e x)+2 b e \left (c^2 d^4+c^{2/3} e^4\right ) \log \left (c^{2/3} x^2+1\right ) (d+e x)-b c^{2/3} e \left (c^{4/3} d^4-\sqrt{3} c d^3 e-\sqrt{3} \sqrt [3]{c} d e^3+e^4\right ) \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right ) (d+e x)-b c^{2/3} e \left (c^{4/3} d^4+\sqrt{3} c d^3 e+\sqrt{3} \sqrt [3]{c} d e^3+e^4\right ) \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right ) (d+e x)-4 b \sqrt [3]{c} \left (c^2 d^6+e^6\right ) \tan ^{-1}\left (c x^3\right )+12 b c^{4/3} d^2 e^3 (d+e x) \log (d+e x)-4 b c d \left (-c^{2/3} d^2 e^2+c^{4/3} d^4+e^4\right ) \tan ^{-1}\left (\sqrt [3]{c} x\right ) (d+e x)-2 b c^{2/3} \left (-\sqrt{3} c^{4/3} d^4 e+2 c^{5/3} d^5+c d^3 e^2-\sqrt [3]{c} d e^4+\sqrt{3} e^5\right ) \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right ) (d+e x)+2 b c^{2/3} \left (\sqrt{3} c^{4/3} d^4 e+2 c^{5/3} d^5+c d^3 e^2-\sqrt [3]{c} d e^4-\sqrt{3} e^5\right ) \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right ) (d+e x)}{4 \sqrt [3]{c} e \left (c^2 d^6+e^6\right ) (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a*c^(1/3)*(c^2*d^6 + e^6) - 4*b*c*d*(c^(4/3)*d^4 - c^(2/3)*d^2*e^2 + e^4)*(d + e*x)*ArcTan[c^(1/3)*x] - 4*
b*c^(1/3)*(c^2*d^6 + e^6)*ArcTan[c*x^3] - 2*b*c^(2/3)*(2*c^(5/3)*d^5 - Sqrt[3]*c^(4/3)*d^4*e + c*d^3*e^2 - c^(
1/3)*d*e^4 + Sqrt[3]*e^5)*(d + e*x)*ArcTan[Sqrt[3] - 2*c^(1/3)*x] + 2*b*c^(2/3)*(2*c^(5/3)*d^5 + Sqrt[3]*c^(4/
3)*d^4*e + c*d^3*e^2 - c^(1/3)*d*e^4 - Sqrt[3]*e^5)*(d + e*x)*ArcTan[Sqrt[3] + 2*c^(1/3)*x] + 12*b*c^(4/3)*d^2
*e^3*(d + e*x)*Log[d + e*x] + 2*b*e*(c^2*d^4 + c^(2/3)*e^4)*(d + e*x)*Log[1 + c^(2/3)*x^2] - b*c^(2/3)*e*(c^(4
/3)*d^4 - Sqrt[3]*c*d^3*e - Sqrt[3]*c^(1/3)*d*e^3 + e^4)*(d + e*x)*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2] -
b*c^(2/3)*e*(c^(4/3)*d^4 + Sqrt[3]*c*d^3*e + Sqrt[3]*c^(1/3)*d*e^3 + e^4)*(d + e*x)*Log[1 + Sqrt[3]*c^(1/3)*x
+ c^(2/3)*x^2] - 2*b*c^(4/3)*d^2*e^3*(d + e*x)*Log[1 + c^2*x^6])/(4*c^(1/3)*e*(c^2*d^6 + e^6)*(d + e*x))

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Maple [A]  time = 0.139, size = 1220, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*b*c*d^2*e^2*ln(e*x+d)/(c^2*d^6+e^6)-1/4*b*e^3*c^3/(c^2*d^6+e^6)*ln(x^2+3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3)
)*3^(1/2)*(1/c^2)^(7/6)*d-1/4*b*e*c^3/(c^2*d^6+e^6)*ln(x^2+3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*3^(1/2)*(1/c
^2)^(5/6)*d^3+1/4*b*e^3*c^3/(c^2*d^6+e^6)*ln(x^2-3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*3^(1/2)*(1/c^2)^(7/6)*
d+1/4*b*e*c^3/(c^2*d^6+e^6)*ln(x^2-3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*3^(1/2)*(1/c^2)^(5/6)*d^3-1/4*b*c^3/
(c^2*d^6+e^6)*ln(x^2+3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*(1/c^2)^(2/3)*d^4-1/4*b*c^3/(c^2*d^6+e^6)*ln(x^2-3
^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*(1/c^2)^(2/3)*d^4+1/2*b*c^3/(c^2*d^6+e^6)*ln(x^2+(1/c^2)^(1/3))*(1/c^2)^
(2/3)*d^4-1/2*b*e^2*c/(c^2*d^6+e^6)*ln(x^2+(1/c^2)^(1/3))*d^2-1/4*b*e^4*c/(c^2*d^6+e^6)*ln(x^2+3^(1/2)*(1/c^2)
^(1/6)*x+(1/c^2)^(1/3))*(1/c^2)^(1/3)-1/2*b*e^2*c/(c^2*d^6+e^6)*ln(x^2+3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*
d^2-1/4*b*e^4*c/(c^2*d^6+e^6)*ln(x^2-3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*(1/c^2)^(1/3)-1/2*b*e^2*c/(c^2*d^6
+e^6)*ln(x^2-3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*d^2+1/2*b*e^4*c/(c^2*d^6+e^6)*ln(x^2+(1/c^2)^(1/3))*(1/c^2
)^(1/3)-a/(e*x+d)/e-1/2*b*c^3/(c^2*d^6+e^6)*(1/c^2)^(2/3)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))*3^(1/2)*d^4+1/2*b*
c^3/(c^2*d^6+e^6)*(1/c^2)^(2/3)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2))*3^(1/2)*d^4-1/2*b*e^3*c^3/(c^2*d^6+e^6)*(1/c
^2)^(7/6)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2))*d-1/2*b*e^3*c^3/(c^2*d^6+e^6)*(1/c^2)^(7/6)*arctan(2*x/(1/c^2)^(1/
6)-3^(1/2))*d+b*e^4*c^3/(c^2*d^6+e^6)*(1/c^2)^(4/3)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))*3^(1/2)-b*e^3*c^3/(c^2*d
^6+e^6)*(1/c^2)^(7/6)*arctan(x/(1/c^2)^(1/6))*d+1/2*b*e*c/(c^2*d^6+e^6)/(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1/6)
-3^(1/2))*d^3-b/(e*x+d)/e*arctan(c*x^3)+1/2*b*e*c/(c^2*d^6+e^6)/(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2)
)*d^3-1/2*b*e^4*c/(c^2*d^6+e^6)*(1/c^2)^(1/3)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2))*3^(1/2)+b/e*c^3/(c^2*d^6+e^6)*
(1/c^2)^(1/2)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2))*d^5-1/2*b*e^4*c/(c^2*d^6+e^6)*(1/c^2)^(1/3)*arctan(2*x/(1/c^2)
^(1/6)-3^(1/2))*3^(1/2)+b/e*c^3/(c^2*d^6+e^6)*(1/c^2)^(1/2)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))*d^5-b/e*c^3/(c^2
*d^6+e^6)*(1/c^2)^(1/2)*arctan(x/(1/c^2)^(1/6))*d^5+b*e*c/(c^2*d^6+e^6)/(1/c^2)^(1/6)*arctan(x/(1/c^2)^(1/6))*
d^3

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Maxima [A]  time = 1.56955, size = 1018, normalized size = 1.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((12*d^2*e^2*log(e*x + d)/(c^2*d^6 + e^6) - (2*((c^2)^(1/3)*c^2*d^5 - c^2*d^3*e^2 + (c^2)^(2/3)*d*e^4)*log
(((c^2)^(1/3)*x - sqrt(-(c^2)^(1/3)))/((c^2)^(1/3)*x + sqrt(-(c^2)^(1/3))))/((c^2)^(2/3)*sqrt(-(c^2)^(1/3))) -
2*((c^2)^(1/3)*c^2*d^4*e - c^2*d^2*e^3 + (c^2)^(2/3)*e^5)*log((c^2)^(1/3)*x^2 + 1)/c^2 + (sqrt(3)*sqrt(c^2)*c
^2*d^3*e^2 + (c^2)^(2/3)*c^2*d^4*e + sqrt(3)*(c^2)^(1/6)*c^2*d*e^4 + 2*(c^2)^(1/3)*c^2*d^2*e^3 + c^2*e^5)*log(
(c^2)^(1/3)*x^2 + sqrt(3)*(c^2)^(1/6)*x + 1)/((c^2)^(1/3)*c^2) - (sqrt(3)*sqrt(c^2)*c^2*d^3*e^2 - (c^2)^(2/3)*
c^2*d^4*e + sqrt(3)*(c^2)^(1/6)*c^2*d*e^4 - 2*(c^2)^(1/3)*c^2*d^2*e^3 - c^2*e^5)*log((c^2)^(1/3)*x^2 - sqrt(3)
*(c^2)^(1/6)*x + 1)/((c^2)^(1/3)*c^2) - (2*c^4*d^5 + sqrt(3)*(c^2)^(5/6)*c^2*d^4*e + (c^2)^(2/3)*c^2*d^3*e^2 -
sqrt(3)*(c^2)^(1/6)*c^2*e^5 - (3*(c^2)^(1/3)*c^2 - 2*(c^2)^(4/3))*d*e^4)*log((2*(c^2)^(1/3)*x + sqrt(3)*(c^2)
^(1/6) - sqrt(-(c^2)^(1/3)))/(2*(c^2)^(1/3)*x + sqrt(3)*(c^2)^(1/6) + sqrt(-(c^2)^(1/3))))/((c^2)^(1/3)*c^2*sq
rt(-(c^2)^(1/3))) - (2*c^4*d^5 - sqrt(3)*(c^2)^(5/6)*c^2*d^4*e + (c^2)^(2/3)*c^2*d^3*e^2 + sqrt(3)*(c^2)^(1/6)
*c^2*e^5 - (3*(c^2)^(1/3)*c^2 - 2*(c^2)^(4/3))*d*e^4)*log((2*(c^2)^(1/3)*x - sqrt(3)*(c^2)^(1/6) - sqrt(-(c^2)
^(1/3)))/(2*(c^2)^(1/3)*x - sqrt(3)*(c^2)^(1/6) + sqrt(-(c^2)^(1/3))))/((c^2)^(1/3)*c^2*sqrt(-(c^2)^(1/3))))/(
c^2*d^6*e + e^7))*c - 4*arctan(c*x^3)/(e^2*x + d*e))*b - a/(e^2*x + d*e)

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Fricas [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*arctan(c*x^3))/(e*x+d)^2,x, algorithm="fricas")

[Out]

Timed out

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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**3))/(e*x+d)**2,x)

[Out]

Timed out

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Giac [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*arctan(c*x^3))/(e*x+d)^2,x, algorithm="giac")

[Out]

Timed out