\(\int (d+e x) (a+b \arctan (c x^3)) \, dx\) [29]

Optimal result

Integrand size = 16, antiderivative size = 285 \[ \int (d+e x) \left (a+b \arctan \left (c x^3\right )\right ) \, dx=-\frac {b e \arctan \left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d^2 \arctan \left (c x^3\right )}{2 e}+\frac {(d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right )}{2 e}+\frac {b e \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {b e \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d \arctan \left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}+\frac {b d \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {\sqrt {3} b e \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}+\frac {\sqrt {3} b e \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac {b d \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {4980, 1845, 281, 209, 298, 31, 648, 631, 210, 642, 301, 632} \[ \int (d+e x) \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {(d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right )}{2 e}+\frac {\sqrt {3} b d \arctan \left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {b e \arctan \left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {b e \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {b e \arctan \left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{4 c^{2/3}}-\frac {b d^2 \arctan \left (c x^3\right )}{2 e}+\frac {b d \log \left (c^{2/3} x^2+1\right )}{2 \sqrt [3]{c}}-\frac {b d \log \left (c^{4/3} x^4-c^{2/3} x^2+1\right )}{4 \sqrt [3]{c}}-\frac {\sqrt {3} b e \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{8 c^{2/3}}+\frac {\sqrt {3} b e \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{8 c^{2/3}} \]

[In]

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

[Out]

-1/2*(b*e*ArcTan[c^(1/3)*x])/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 31

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

Rule 209

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

Rule 210

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

Rule 281

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 298

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 301

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)/(a*n*s^m))*Int[1/(r^2 + s^2*x^2), x] +
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 631

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 632

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]

Rule 642

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 648

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 1845

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 4980

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

Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^2 \left (a+b \arctan \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 \arctan \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 \arctan \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 \arctan \left (c x^3\right )\right )}{2 e}-\frac {1}{2} (3 b c d) \text {Subst}\left (\int \frac {x}{1+c^2 x^3} \, dx,x,x^2\right )-\frac {\left (b c d^2\right ) \text {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 \arctan \left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d^2 \arctan \left (c x^3\right )}{2 e}+\frac {(d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right )}{2 e}+\frac {1}{2} \left (b \sqrt [3]{c} d\right ) \text {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 ) \text {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 \arctan \left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d^2 \arctan \left (c x^3\right )}{2 e}+\frac {(d+e x)^2 \left (a+b \arctan \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) \text {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 ) \text {Subst}\left (\int \frac {1}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )-\frac {(b e) \text {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) \text {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 \arctan \left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d^2 \arctan \left (c x^3\right )}{2 e}+\frac {(d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right )}{2 e}+\frac {b e \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {b e \arctan \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) \text {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 \arctan \left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d^2 \arctan \left (c x^3\right )}{2 e}+\frac {(d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right )}{2 e}+\frac {b e \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {b e \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d \arctan \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] (verified)

Time = 0.08 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.09 \[ \int (d+e x) \left (a+b \arctan \left (c x^3\right )\right ) \, dx=a d x+\frac {1}{2} a e x^2-\frac {b e \arctan \left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+b d x \arctan \left (c x^3\right )+\frac {1}{2} b e x^2 \arctan \left (c x^3\right )+\frac {b e \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {b e \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {\sqrt {3} b e \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}+\frac {\sqrt {3} b e \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac {b d \left (-2 \sqrt {3} \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )-2 \sqrt {3} \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )-2 \log \left (1+c^{2/3} x^2\right )+\log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )+\log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )\right )}{4 \sqrt [3]{c}} \]

[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))

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.07

method	result	size
default	\(a \left (\frac {1}{2} e \,x^{2}+d x \right )+b \left (\frac {\arctan \left (c \,x^{3}\right ) x^{2} e}{2}+\arctan \left (c \,x^{3}\right ) d x -\frac {3 c \left (-\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} e}{12}+\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d}{6}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) e}{6 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, d}{3}+\frac {c^{2} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {5}{3}} d}{6}+\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} e}{12}+\frac {c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {5}{3}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, d}{3}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) e}{6 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d \ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{3}+\frac {e \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{3 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{2}\right )\)	\(305\)
parts	\(a \left (\frac {1}{2} e \,x^{2}+d x \right )+b \left (\frac {\arctan \left (c \,x^{3}\right ) x^{2} e}{2}+\arctan \left (c \,x^{3}\right ) d x -\frac {3 c \left (-\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} e}{12}+\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d}{6}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) e}{6 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, d}{3}+\frac {c^{2} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {5}{3}} d}{6}+\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} e}{12}+\frac {c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {5}{3}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, d}{3}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) e}{6 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d \ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{3}+\frac {e \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{3 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{2}\right )\)	\(305\)

[In]

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

[Out]

a*(1/2*e*x^2+d*x)+b*(1/2*arctan(c*x^3)*x^2*e+arctan(c*x^3)*d*x-3/2*c*(-1/12*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+1/6*ln(x^2+3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*(1/c^2)^(2/3)*d+1/6/c^2/
(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2))*e-1/3*(1/c^2)^(2/3)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2))*3^(1/2)*
d+1/6*c^2*ln(x^2-3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*(1/c^2)^(5/3)*d+1/12*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+1/3*c^2*(1/c^2)^(5/3)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))*3^(1/2)*d+1/6/c^2
/(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))*e-1/3*(1/c^2)^(2/3)*d*ln(x^2+(1/c^2)^(1/3))+1/3/c^2*e/(1/c^2)
^(1/6)*arctan(x/(1/c^2)^(1/6))))

Fricas [F(-2)]

Exception generated. \[ \int (d+e x) \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> no explicit roots found

Sympy [A] (verification not implemented)

Time = 10.83 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.36 \[ \int (d+e x) \left (a+b \arctan \left (c x^3\right )\right ) \, dx=a d x + \frac {a e x^{2}}{2} - 3 b c d \operatorname {RootSum} {\left (216 t^{3} c^{4} + 1, \left ( t \mapsto t \log {\left (36 t^{2} c^{2} + x^{2} \right )} \right )\right )} - \frac {3 b c e \operatorname {RootSum} {\left (46656 t^{6} c^{10} + 1, \left ( t \mapsto t \log {\left (7776 t^{5} c^{8} + x \right )} \right )\right )}}{2} + b d x \operatorname {atan}{\left (c x^{3} \right )} + \frac {b e x^{2} \operatorname {atan}{\left (c x^{3} \right )}}{2} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.81 \[ \int (d+e x) \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {1}{2} \, a e x^{2} - \frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {4}{3}} x^{2} - c^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}} + \frac {\log \left (c^{\frac {4}{3}} x^{4} - c^{\frac {2}{3}} x^{2} + 1\right )}{c^{\frac {4}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {2}{3}} x^{2} + 1}{c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}}\right )} - 4 \, x \arctan \left (c x^{3}\right )\right )} b d + \frac {1}{8} \, {\left (4 \, x^{2} \arctan \left (c x^{3}\right ) + c {\left (\frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} + \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} - \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {4 \, \arctan \left (c^{\frac {1}{3}} x\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x + \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x - \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}}\right )}\right )} b e + a d x \]

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

Giac [A] (verification not implemented)

none

Time = 0.83 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.83 \[ \int (d+e x) \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {1}{2} \, b e x^{2} \arctan \left (c x^{3}\right ) + \frac {1}{2} \, a e x^{2} + b d x \arctan \left (c x^{3}\right ) + a d x + \frac {b c d \log \left (x^{2} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{2 \, {\left | c \right |}^{\frac {4}{3}}} - \frac {b c e \arctan \left (x {\left | c \right |}^{\frac {1}{3}}\right )}{2 \, {\left | c \right |}^{\frac {5}{3}}} + \frac {{\left (2 \, \sqrt {3} b c d {\left | c \right |}^{\frac {1}{3}} - b c e\right )} \arctan \left ({\left (2 \, x + \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{4 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {{\left (2 \, \sqrt {3} b c d {\left | c \right |}^{\frac {1}{3}} + b c e\right )} \arctan \left ({\left (2 \, x - \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{4 \, {\left | c \right |}^{\frac {5}{3}}} + \frac {{\left (\sqrt {3} b c e - 2 \, b c d {\left | c \right |}^{\frac {1}{3}}\right )} \log \left (x^{2} + \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{8 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {{\left (\sqrt {3} b c e + 2 \, b c d {\left | c \right |}^{\frac {1}{3}}\right )} \log \left (x^{2} - \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{8 \, {\left | c \right |}^{\frac {5}{3}}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.70 \[ \int (d+e x) \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\mathrm {atan}\left (c\,x^3\right )\,\left (\frac {b\,e\,x^2}{2}+b\,d\,x\right )+\left (\sum _{k=1}^6\ln \left (-\mathrm {root}\left (4096\,a^6\,c^4-1024\,a^3\,b^3\,c^3\,d^3+576\,a^2\,b^4\,c^2\,d^2\,e^2-48\,a\,b^5\,c\,d\,e^4+64\,b^6\,c^2\,d^6+b^6\,e^6,a,k\right )\,\left (\mathrm {root}\left (4096\,a^6\,c^4-1024\,a^3\,b^3\,c^3\,d^3+576\,a^2\,b^4\,c^2\,d^2\,e^2-48\,a\,b^5\,c\,d\,e^4+64\,b^6\,c^2\,d^6+b^6\,e^6,a,k\right )\,\left (\mathrm {root}\left (4096\,a^6\,c^4-1024\,a^3\,b^3\,c^3\,d^3+576\,a^2\,b^4\,c^2\,d^2\,e^2-48\,a\,b^5\,c\,d\,e^4+64\,b^6\,c^2\,d^6+b^6\,e^6,a,k\right )\,\left (-486\,b^2\,c^{10}\,e^2\,x+1944\,b^2\,c^{10}\,d\,e+\mathrm {root}\left (4096\,a^6\,c^4-1024\,a^3\,b^3\,c^3\,d^3+576\,a^2\,b^4\,c^2\,d^2\,e^2-48\,a\,b^5\,c\,d\,e^4+64\,b^6\,c^2\,d^6+b^6\,e^6,a,k\right )\,b\,c^{11}\,d\,x\,3888\right )-\frac {243\,b^3\,c^9\,e^3}{2}\right )-486\,b^4\,c^{10}\,d^4\,x\right )-\frac {243\,b^5\,c^9\,d^4\,e}{2}-\frac {243\,b^5\,c^9\,d^3\,e^2\,x}{4}\right )\,\mathrm {root}\left (4096\,a^6\,c^4-1024\,a^3\,b^3\,c^3\,d^3+576\,a^2\,b^4\,c^2\,d^2\,e^2-48\,a\,b^5\,c\,d\,e^4+64\,b^6\,c^2\,d^6+b^6\,e^6,a,k\right )\right )+a\,d\,x+\frac {a\,e\,x^2}{2} \]

[In]

int((a + b*atan(c*x^3))*(d + e*x),x)

[Out]

atan(c*x^3)*(b*d*x + (b*e*x^2)/2) + symsum(log(- root(4096*a^6*c^4 - 1024*a^3*b^3*c^3*d^3 + 576*a^2*b^4*c^2*d^
2*e^2 - 48*a*b^5*c*d*e^4 + 64*b^6*c^2*d^6 + b^6*e^6, a, k)*(root(4096*a^6*c^4 - 1024*a^3*b^3*c^3*d^3 + 576*a^2
*b^4*c^2*d^2*e^2 - 48*a*b^5*c*d*e^4 + 64*b^6*c^2*d^6 + b^6*e^6, a, k)*(root(4096*a^6*c^4 - 1024*a^3*b^3*c^3*d^
3 + 576*a^2*b^4*c^2*d^2*e^2 - 48*a*b^5*c*d*e^4 + 64*b^6*c^2*d^6 + b^6*e^6, a, k)*(1944*b^2*c^10*d*e - 486*b^2*
c^10*e^2*x + 3888*root(4096*a^6*c^4 - 1024*a^3*b^3*c^3*d^3 + 576*a^2*b^4*c^2*d^2*e^2 - 48*a*b^5*c*d*e^4 + 64*b
^6*c^2*d^6 + b^6*e^6, a, k)*b*c^11*d*x) - (243*b^3*c^9*e^3)/2) - 486*b^4*c^10*d^4*x) - (243*b^5*c^9*d^4*e)/2 -
(243*b^5*c^9*d^3*e^2*x)/4)*root(4096*a^6*c^4 - 1024*a^3*b^3*c^3*d^3 + 576*a^2*b^4*c^2*d^2*e^2 - 48*a*b^5*c*d*
e^4 + 64*b^6*c^2*d^6 + b^6*e^6, a, k), k, 1, 6) + a*d*x + (a*e*x^2)/2