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

Optimal. Leaf size=285 \[ \frac{(d+e x)^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{2 e}+\frac{b d \log \left (c^{2/3} x^2+1\right )}{2 \sqrt [3]{c}}-\frac{b d \log \left (c^{4/3} x^4-c^{2/3} x^2+1\right )}{4 \sqrt [3]{c}}+\frac{\sqrt{3} b d \tan ^{-1}\left (\frac{1-2 c^{2/3} x^2}{\sqrt{3}}\right )}{2 \sqrt [3]{c}}-\frac{\sqrt{3} b e \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )}{8 c^{2/3}}+\frac{\sqrt{3} b e \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )}{8 c^{2/3}}-\frac{b e \tan ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac{b e \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac{b e \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right )}{4 c^{2/3}}-\frac{b d^2 \tan ^{-1}\left (c x^3\right )}{2 e} \]

[Out]

-(b*e*ArcTan[c^(1/3)*x])/(2*c^(2/3)) - (b*d^2*ArcTan[c*x^3])/(2*e) + ((d + e*x)^2*(a + b*ArcTan[c*x^3]))/(2*e)
+ (b*e*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(4*c^(2/3)) - (b*e*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(4*c^(2/3)) + (Sqrt[3
]*b*d*ArcTan[(1 - 2*c^(2/3)*x^2)/Sqrt[3]])/(2*c^(1/3)) + (b*d*Log[1 + c^(2/3)*x^2])/(2*c^(1/3)) - (Sqrt[3]*b*e
*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) + (Sqrt[3]*b*e*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2]
)/(8*c^(2/3)) - (b*d*Log[1 - c^(2/3)*x^2 + c^(4/3)*x^4])/(4*c^(1/3))

________________________________________________________________________________________

Rubi [A]  time = 0.60264, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 13, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.812, Rules used = {5205, 12, 1831, 275, 203, 292, 31, 634, 617, 204, 628, 295, 618} \[ \frac{(d+e x)^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{2 e}+\frac{b d \log \left (c^{2/3} x^2+1\right )}{2 \sqrt [3]{c}}-\frac{b d \log \left (c^{4/3} x^4-c^{2/3} x^2+1\right )}{4 \sqrt [3]{c}}+\frac{\sqrt{3} b d \tan ^{-1}\left (\frac{1-2 c^{2/3} x^2}{\sqrt{3}}\right )}{2 \sqrt [3]{c}}-\frac{\sqrt{3} b e \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )}{8 c^{2/3}}+\frac{\sqrt{3} b e \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )}{8 c^{2/3}}-\frac{b e \tan ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac{b e \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac{b e \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right )}{4 c^{2/3}}-\frac{b d^2 \tan ^{-1}\left (c x^3\right )}{2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*e*ArcTan[c^(1/3)*x])/(2*c^(2/3)) - (b*d^2*ArcTan[c*x^3])/(2*e) + ((d + e*x)^2*(a + b*ArcTan[c*x^3]))/(2*e)
+ (b*e*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(4*c^(2/3)) - (b*e*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(4*c^(2/3)) + (Sqrt[3
]*b*d*ArcTan[(1 - 2*c^(2/3)*x^2)/Sqrt[3]])/(2*c^(1/3)) + (b*d*Log[1 + c^(2/3)*x^2])/(2*c^(1/3)) - (Sqrt[3]*b*e
*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) + (Sqrt[3]*b*e*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2]
)/(8*c^(2/3)) - (b*d*Log[1 - c^(2/3)*x^2 + c^(4/3)*x^4])/(4*c^(1/3))

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 1831

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

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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 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 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 295

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[((2*k - 1)*m*Pi)/n] - s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(
(2*k - 1)*Pi)/n]*x + s^2*x^2), x] + Int[(r*Cos[((2*k - 1)*m*Pi)/n] + s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 +
2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*(-1)^(m/2)*r^(m + 2)*Int[1/(r^2 + s^2*x^2), x])/(a*n*s^m) +
Dist[(2*r^(m + 1))/(a*n*s^m), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] &&
IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]

Rule 618

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

Rubi steps

\begin{align*} \int (d+e x) \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \, dx &=\frac{(d+e x)^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{2 e}-\frac{b \int \frac{3 c x^2 (d+e x)^2}{1+c^2 x^6} \, dx}{2 e}\\ &=\frac{(d+e x)^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{2 e}-\frac{(3 b c) \int \frac{x^2 (d+e x)^2}{1+c^2 x^6} \, dx}{2 e}\\ &=\frac{(d+e x)^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{2 e}-\frac{(3 b c) \int \left (\frac{d^2 x^2}{1+c^2 x^6}+\frac{2 d e x^3}{1+c^2 x^6}+\frac{e^2 x^4}{1+c^2 x^6}\right ) \, dx}{2 e}\\ &=\frac{(d+e x)^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{2 e}-(3 b c d) \int \frac{x^3}{1+c^2 x^6} \, dx-\frac{\left (3 b c d^2\right ) \int \frac{x^2}{1+c^2 x^6} \, dx}{2 e}-\frac{1}{2} (3 b c e) \int \frac{x^4}{1+c^2 x^6} \, dx\\ &=\frac{(d+e x)^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{2 e}-\frac{1}{2} (3 b c d) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x^3} \, dx,x,x^2\right )-\frac{\left (b c d^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x^2} \, dx,x,x^3\right )}{2 e}-\frac{(b e) \int \frac{1}{1+c^{2/3} x^2} \, dx}{2 \sqrt [3]{c}}-\frac{(b e) \int \frac{-\frac{1}{2}+\frac{1}{2} \sqrt{3} \sqrt [3]{c} x}{1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{2 \sqrt [3]{c}}-\frac{(b e) \int \frac{-\frac{1}{2}-\frac{1}{2} \sqrt{3} \sqrt [3]{c} x}{1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{2 \sqrt [3]{c}}\\ &=-\frac{b e \tan ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac{b d^2 \tan ^{-1}\left (c x^3\right )}{2 e}+\frac{(d+e x)^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{2 e}+\frac{1}{2} \left (b \sqrt [3]{c} d\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^{2/3} x} \, dx,x,x^2\right )-\frac{1}{2} \left (b \sqrt [3]{c} d\right ) \operatorname{Subst}\left (\int \frac{1+c^{2/3} x}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )-\frac{\left (\sqrt{3} b e\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}{8 c^{2/3}}+\frac{\left (\sqrt{3} b e\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}{8 c^{2/3}}-\frac{(b e) \int \frac{1}{1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 \sqrt [3]{c}}-\frac{(b e) \int \frac{1}{1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 \sqrt [3]{c}}\\ &=-\frac{b e \tan ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac{b d^2 \tan ^{-1}\left (c x^3\right )}{2 e}+\frac{(d+e x)^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{2 e}+\frac{b d \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac{\sqrt{3} b e \log \left (1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}+\frac{\sqrt{3} b e \log \left (1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac{(b d) \operatorname{Subst}\left (\int \frac{-c^{2/3}+2 c^{4/3} x}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )}{4 \sqrt [3]{c}}-\frac{1}{4} \left (3 b \sqrt [3]{c} d\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )-\frac{(b e) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1-\frac{2 \sqrt [3]{c} x}{\sqrt{3}}\right )}{4 \sqrt{3} c^{2/3}}+\frac{(b e) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1+\frac{2 \sqrt [3]{c} x}{\sqrt{3}}\right )}{4 \sqrt{3} c^{2/3}}\\ &=-\frac{b e \tan ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac{b d^2 \tan ^{-1}\left (c x^3\right )}{2 e}+\frac{(d+e x)^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{2 e}+\frac{b e \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac{b e \tan ^{-1}\left (\sqrt{3}+2 \sqrt [3]{c} x\right )}{4 c^{2/3}}+\frac{b d \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac{\sqrt{3} b e \log \left (1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}+\frac{\sqrt{3} b e \log \left (1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac{b d \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}-\frac{(3 b d) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-2 c^{2/3} x^2\right )}{2 \sqrt [3]{c}}\\ &=-\frac{b e \tan ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac{b d^2 \tan ^{-1}\left (c x^3\right )}{2 e}+\frac{(d+e x)^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{2 e}+\frac{b e \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac{b e \tan ^{-1}\left (\sqrt{3}+2 \sqrt [3]{c} x\right )}{4 c^{2/3}}+\frac{\sqrt{3} b d \tan ^{-1}\left (\frac{1-2 c^{2/3} x^2}{\sqrt{3}}\right )}{2 \sqrt [3]{c}}+\frac{b d \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac{\sqrt{3} b e \log \left (1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}+\frac{\sqrt{3} b e \log \left (1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac{b d \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}\\ \end{align*}

Mathematica [A]  time = 0.0893988, size = 310, normalized size = 1.09 \[ a d x+\frac{1}{2} a e x^2-\frac{b d \left (-2 \log \left (c^{2/3} x^2+1\right )+\log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )+\log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )-2 \sqrt{3} \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )-2 \sqrt{3} \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right )\right )}{4 \sqrt [3]{c}}-\frac{\sqrt{3} b e \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )}{8 c^{2/3}}+\frac{\sqrt{3} b e \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )}{8 c^{2/3}}-\frac{b e \tan ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac{b e \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac{b e \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right )}{4 c^{2/3}}+b d x \tan ^{-1}\left (c x^3\right )+\frac{1}{2} b e x^2 \tan ^{-1}\left (c x^3\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*d*x + (a*e*x^2)/2 - (b*e*ArcTan[c^(1/3)*x])/(2*c^(2/3)) + b*d*x*ArcTan[c*x^3] + (b*e*x^2*ArcTan[c*x^3])/2 +
(b*e*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(4*c^(2/3)) - (b*e*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(4*c^(2/3)) - (Sqrt[3]*b
*e*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) + (Sqrt[3]*b*e*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^
2])/(8*c^(2/3)) - (b*d*(-2*Sqrt[3]*ArcTan[Sqrt[3] - 2*c^(1/3)*x] - 2*Sqrt[3]*ArcTan[Sqrt[3] + 2*c^(1/3)*x] - 2
*Log[1 + c^(2/3)*x^2] + Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2] + Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2]))/
(4*c^(1/3))

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Maple [A]  time = 0.069, size = 314, normalized size = 1.1 \begin{align*}{\frac{a{x}^{2}e}{2}}+adx+{\frac{b\arctan \left ( c{x}^{3} \right ){x}^{2}e}{2}}+b\arctan \left ( c{x}^{3} \right ) dx+{\frac{bc\sqrt{3}e}{8}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{c}^{-2}}x+\sqrt [3]{{c}^{-2}} \right ) \left ({c}^{-2} \right ) ^{{\frac{5}{6}}}}-{\frac{bcd}{4}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{c}^{-2}}x+\sqrt [3]{{c}^{-2}} \right ) \left ({c}^{-2} \right ) ^{{\frac{2}{3}}}}-{\frac{be}{4\,c}\arctan \left ( 2\,{\frac{x}{\sqrt [6]{{c}^{-2}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{c}^{-2}}}}}+{\frac{bc\sqrt{3}d}{2} \left ({c}^{-2} \right ) ^{{\frac{2}{3}}}\arctan \left ( 2\,{\frac{x}{\sqrt [6]{{c}^{-2}}}}+\sqrt{3} \right ) }-{\frac{b{c}^{3}d}{4}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{c}^{-2}}x+\sqrt [3]{{c}^{-2}} \right ) \left ({c}^{-2} \right ) ^{{\frac{5}{3}}}}-{\frac{bc\sqrt{3}e}{8}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{c}^{-2}}x+\sqrt [3]{{c}^{-2}} \right ) \left ({c}^{-2} \right ) ^{{\frac{5}{6}}}}-{\frac{b{c}^{3}\sqrt{3}d}{2} \left ({c}^{-2} \right ) ^{{\frac{5}{3}}}\arctan \left ( 2\,{\frac{x}{\sqrt [6]{{c}^{-2}}}}-\sqrt{3} \right ) }-{\frac{be}{4\,c}\arctan \left ( 2\,{\frac{x}{\sqrt [6]{{c}^{-2}}}}-\sqrt{3} \right ){\frac{1}{\sqrt [6]{{c}^{-2}}}}}+{\frac{bcd}{2} \left ({c}^{-2} \right ) ^{{\frac{2}{3}}}\ln \left ({x}^{2}+\sqrt [3]{{c}^{-2}} \right ) }-{\frac{be}{2\,c}\arctan \left ({x{\frac{1}{\sqrt [6]{{c}^{-2}}}}} \right ){\frac{1}{\sqrt [6]{{c}^{-2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2*e+a*d*x+1/2*b*arctan(c*x^3)*x^2*e+b*arctan(c*x^3)*d*x+1/8*b*c*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)*e-1/4*b*c*ln(x^2+3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*(1/c^2)^(2/3)*d-1/4*b/c/
(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2))*e+1/2*b*c*(1/c^2)^(2/3)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2))*3^(1
/2)*d-1/4*b*c^3*ln(x^2-3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*(1/c^2)^(5/3)*d-1/8*b*c*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)*e-1/2*b*c^3*(1/c^2)^(5/3)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))*3^(1/2
)*d-1/4*b/c/(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))*e+1/2*b*c*(1/c^2)^(2/3)*d*ln(x^2+(1/c^2)^(1/3))-1/
2*b/c*e/(1/c^2)^(1/6)*arctan(x/(1/c^2)^(1/6))

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Maxima [A]  time = 1.47298, size = 383, normalized size = 1.34 \begin{align*} \frac{1}{2} \, a e x^{2} - \frac{1}{4} \,{\left (c{\left (\frac{2 \, \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{3}}{\left (2 \, x^{2} - \frac{1}{c^{2}}^{\frac{1}{3}}\right )}\right )}{c^{2}} + \frac{{\left (c^{2}\right )}^{\frac{1}{3}} \log \left (x^{4} - \frac{1}{c^{2}}^{\frac{1}{3}} x^{2} + \frac{1}{c^{2}}^{\frac{2}{3}}\right )}{c^{2}} - \frac{2 \,{\left (c^{2}\right )}^{\frac{1}{3}} \log \left (x^{2} + \frac{1}{c^{2}}^{\frac{1}{3}}\right )}{c^{2}}\right )} - 4 \, x \arctan \left (c x^{3}\right )\right )} b d + \frac{1}{8} \,{\left (4 \, x^{2} \arctan \left (c x^{3}\right ) + c{\left (\frac{\sqrt{3} \log \left ({\left (c^{2}\right )}^{\frac{1}{3}} x^{2} + \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{6}} x + 1\right )}{{\left (c^{2}\right )}^{\frac{5}{6}}} - \frac{\sqrt{3} \log \left ({\left (c^{2}\right )}^{\frac{1}{3}} x^{2} - \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{6}} x + 1\right )}{{\left (c^{2}\right )}^{\frac{5}{6}}} - \frac{4 \, \arctan \left ({\left (c^{2}\right )}^{\frac{1}{6}} x\right )}{{\left (c^{2}\right )}^{\frac{5}{6}}} - \frac{2 \,{\left (c^{2}\right )}^{\frac{1}{6}} \arctan \left (\frac{2 \,{\left (c^{2}\right )}^{\frac{1}{3}} x + \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{6}}}{{\left (c^{2}\right )}^{\frac{1}{6}}}\right )}{c^{2}} - \frac{2 \,{\left (c^{2}\right )}^{\frac{1}{6}} \arctan \left (\frac{2 \,{\left (c^{2}\right )}^{\frac{1}{3}} x - \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{6}}}{{\left (c^{2}\right )}^{\frac{1}{6}}}\right )}{c^{2}}\right )}\right )} b e + a d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*e*x^2 - 1/4*(c*(2*sqrt(3)*(c^2)^(1/3)*arctan(1/3*sqrt(3)*(c^2)^(1/3)*(2*x^2 - (c^(-2))^(1/3)))/c^2 + (c^
2)^(1/3)*log(x^4 - (c^(-2))^(1/3)*x^2 + (c^(-2))^(2/3))/c^2 - 2*(c^2)^(1/3)*log(x^2 + (c^(-2))^(1/3))/c^2) - 4
*x*arctan(c*x^3))*b*d + 1/8*(4*x^2*arctan(c*x^3) + c*(sqrt(3)*log((c^2)^(1/3)*x^2 + sqrt(3)*(c^2)^(1/6)*x + 1)
/(c^2)^(5/6) - sqrt(3)*log((c^2)^(1/3)*x^2 - sqrt(3)*(c^2)^(1/6)*x + 1)/(c^2)^(5/6) - 4*arctan((c^2)^(1/6)*x)/
(c^2)^(5/6) - 2*(c^2)^(1/6)*arctan((2*(c^2)^(1/3)*x + sqrt(3)*(c^2)^(1/6))/(c^2)^(1/6))/c^2 - 2*(c^2)^(1/6)*ar
ctan((2*(c^2)^(1/3)*x - sqrt(3)*(c^2)^(1/6))/(c^2)^(1/6))/c^2))*b*e + a*d*x

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

[Out]

Timed out

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Sympy [A]  time = 61.3176, size = 104, normalized size = 0.36 \begin{align*} a d x + \frac{a e x^{2}}{2} - 3 b c d \operatorname{RootSum}{\left (216 t^{3} c^{4} + 1, \left ( t \mapsto t \log{\left (36 t^{2} c^{2} + x^{2} \right )} \right )\right )} - \frac{3 b c e \operatorname{RootSum}{\left (46656 t^{6} c^{10} + 1, \left ( t \mapsto t \log{\left (7776 t^{5} c^{8} + x \right )} \right )\right )}}{2} + b d x \operatorname{atan}{\left (c x^{3} \right )} + \frac{b e x^{2} \operatorname{atan}{\left (c x^{3} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d*x + a*e*x**2/2 - 3*b*c*d*RootSum(216*_t**3*c**4 + 1, Lambda(_t, _t*log(36*_t**2*c**2 + x**2))) - 3*b*c*e*R
ootSum(46656*_t**6*c**10 + 1, Lambda(_t, _t*log(7776*_t**5*c**8 + x)))/2 + b*d*x*atan(c*x**3) + b*e*x**2*atan(
c*x**3)/2

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Giac [A]  time = 2.48028, size = 355, normalized size = 1.25 \begin{align*} \frac{1}{8} \, b c^{5}{\left (\frac{\sqrt{3}{\left | c \right |}^{\frac{1}{3}} \log \left (x^{2} + \frac{\sqrt{3} x}{{\left | c \right |}^{\frac{1}{3}}} + \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{c^{6}} - \frac{\sqrt{3}{\left | c \right |}^{\frac{1}{3}} \log \left (x^{2} - \frac{\sqrt{3} x}{{\left | c \right |}^{\frac{1}{3}}} + \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{c^{6}} - \frac{2 \,{\left | c \right |}^{\frac{1}{3}} \arctan \left ({\left (2 \, x + \frac{\sqrt{3}}{{\left | c \right |}^{\frac{1}{3}}}\right )}{\left | c \right |}^{\frac{1}{3}}\right )}{c^{6}} - \frac{2 \,{\left | c \right |}^{\frac{1}{3}} \arctan \left ({\left (2 \, x - \frac{\sqrt{3}}{{\left | c \right |}^{\frac{1}{3}}}\right )}{\left | c \right |}^{\frac{1}{3}}\right )}{c^{6}} - \frac{4 \,{\left | c \right |}^{\frac{1}{3}} \arctan \left (x{\left | c \right |}^{\frac{1}{3}}\right )}{c^{6}}\right )} e - \frac{1}{4} \, b c^{3} d{\left (\frac{2 \, \sqrt{3}{\left | c \right |}^{\frac{2}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{\left | c \right |}^{\frac{2}{3}}\right )}{c^{4}} + \frac{{\left | c \right |}^{\frac{2}{3}} \log \left (x^{4} - \frac{x^{2}}{{\left | c \right |}^{\frac{2}{3}}} + \frac{1}{{\left | c \right |}^{\frac{4}{3}}}\right )}{c^{4}} - \frac{2 \, \log \left (x^{2} + \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{c^{2}{\left | c \right |}^{\frac{4}{3}}}\right )} + \frac{1}{2} \, b x^{2} \arctan \left (c x^{3}\right ) e + b d x \arctan \left (c x^{3}\right ) + \frac{1}{2} \, a x^{2} e + a d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b*c^5*(sqrt(3)*abs(c)^(1/3)*log(x^2 + sqrt(3)*x/abs(c)^(1/3) + 1/abs(c)^(2/3))/c^6 - sqrt(3)*abs(c)^(1/3)*
log(x^2 - sqrt(3)*x/abs(c)^(1/3) + 1/abs(c)^(2/3))/c^6 - 2*abs(c)^(1/3)*arctan((2*x + sqrt(3)/abs(c)^(1/3))*ab
s(c)^(1/3))/c^6 - 2*abs(c)^(1/3)*arctan((2*x - sqrt(3)/abs(c)^(1/3))*abs(c)^(1/3))/c^6 - 4*abs(c)^(1/3)*arctan
(x*abs(c)^(1/3))/c^6)*e - 1/4*b*c^3*d*(2*sqrt(3)*abs(c)^(2/3)*arctan(1/3*sqrt(3)*(2*x^2 - 1/abs(c)^(2/3))*abs(
c)^(2/3))/c^4 + abs(c)^(2/3)*log(x^4 - x^2/abs(c)^(2/3) + 1/abs(c)^(4/3))/c^4 - 2*log(x^2 + 1/abs(c)^(2/3))/(c
^2*abs(c)^(4/3))) + 1/2*b*x^2*arctan(c*x^3)*e + b*d*x*arctan(c*x^3) + 1/2*a*x^2*e + a*d*x