\(\int \frac {a+b \arctan (c x^3)}{d+e x} \, dx\) [30]

Optimal result

Integrand size = 18, antiderivative size = 739 \[ \int \frac {a+b \arctan \left (c x^3\right )}{d+e x} \, dx=\frac {\left (a+b \arctan \left (c x^3\right )\right ) \log (d+e x)}{e}+\frac {b c \log \left (\frac {e \left (1-\sqrt [6]{-c^2} x\right )}{\sqrt [6]{-c^2} d+e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {b c \log \left (-\frac {e \left (1+\sqrt [6]{-c^2} x\right )}{\sqrt [6]{-c^2} d-e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}+\frac {b c \log \left (-\frac {e \left (\sqrt [3]{-1}+\sqrt [6]{-c^2} x\right )}{\sqrt [6]{-c^2} d-\sqrt [3]{-1} e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {b c \log \left (-\frac {e \left ((-1)^{2/3}+\sqrt [6]{-c^2} x\right )}{\sqrt [6]{-c^2} d-(-1)^{2/3} e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}+\frac {b c \log \left (\frac {(-1)^{2/3} e \left (1+\sqrt [3]{-1} \sqrt [6]{-c^2} x\right )}{\sqrt [6]{-c^2} d+(-1)^{2/3} e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {b c \log \left (\frac {\sqrt [3]{-1} e \left (1+(-1)^{2/3} \sqrt [6]{-c^2} x\right )}{\sqrt [6]{-c^2} d+\sqrt [3]{-1} e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d-e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d+e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d-\sqrt [3]{-1} e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d+\sqrt [3]{-1} e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d-(-1)^{2/3} e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d+(-1)^{2/3} e}\right )}{2 \sqrt {-c^2} e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 739, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4976, 281, 209, 2463, 266, 2441, 2440, 2438} \[ \int \frac {a+b \arctan \left (c x^3\right )}{d+e x} \, dx=\frac {\log (d+e x) \left (a+b \arctan \left (c x^3\right )\right )}{e}-\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d-e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d+e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d-\sqrt [3]{-1} e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d+\sqrt [3]{-1} e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d-(-1)^{2/3} e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d+(-1)^{2/3} e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \log (d+e x) \log \left (\frac {e \left (1-\sqrt [6]{-c^2} x\right )}{\sqrt [6]{-c^2} d+e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \log (d+e x) \log \left (-\frac {e \left (\sqrt [6]{-c^2} x+1\right )}{\sqrt [6]{-c^2} d-e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \log (d+e x) \log \left (-\frac {e \left (\sqrt [6]{-c^2} x+\sqrt [3]{-1}\right )}{\sqrt [6]{-c^2} d-\sqrt [3]{-1} e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \log (d+e x) \log \left (-\frac {e \left (\sqrt [6]{-c^2} x+(-1)^{2/3}\right )}{\sqrt [6]{-c^2} d-(-1)^{2/3} e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \log (d+e x) \log \left (\frac {(-1)^{2/3} e \left (\sqrt [3]{-1} \sqrt [6]{-c^2} x+1\right )}{\sqrt [6]{-c^2} d+(-1)^{2/3} e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [6]{-c^2} x+1\right )}{\sqrt [6]{-c^2} d+\sqrt [3]{-1} e}\right )}{2 \sqrt {-c^2} e} \]

[In]

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

[Out]

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

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 266

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 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 2438

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]

Rule 2440

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 2441

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 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 4976

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.98 (sec) , antiderivative size = 522, normalized size of antiderivative = 0.71 \[ \int \frac {a+b \arctan \left (c x^3\right )}{d+e x} \, dx=\frac {a \log (d+e x)}{e}+\frac {b \left (2 \arctan \left (c x^3\right ) \log (d+e x)-i \left (\log \left (\frac {e \left (-i+\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 \sqrt [3]{c} d+\left (-i+\sqrt {3}\right ) e}\right ) \log (d+e x)-\log \left (\frac {e \left (i+\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 \sqrt [3]{c} d+\left (i+\sqrt {3}\right ) e}\right ) \log (d+e x)+\log \left (\frac {e \left (i-\sqrt [3]{c} x\right )}{\sqrt [3]{c} d+i e}\right ) \log (d+e x)-\log \left (-\frac {e \left (i+\sqrt [3]{c} x\right )}{\sqrt [3]{c} d-i e}\right ) \log (d+e x)-\log \left (\frac {e \left (-i+\sqrt {3}+2 \sqrt [3]{c} x\right )}{-2 \sqrt [3]{c} d+\left (-i+\sqrt {3}\right ) e}\right ) \log (d+e x)+\log \left (\frac {e \left (i+\sqrt {3}+2 \sqrt [3]{c} x\right )}{-2 \sqrt [3]{c} d+\left (i+\sqrt {3}\right ) e}\right ) \log (d+e x)-\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-i e}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+i e}\right )-\operatorname {PolyLog}\left (2,\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d+i e-\sqrt {3} e}\right )+\operatorname {PolyLog}\left (2,\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d+\left (-i+\sqrt {3}\right ) e}\right )+\operatorname {PolyLog}\left (2,\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d-\left (i+\sqrt {3}\right ) e}\right )-\operatorname {PolyLog}\left (2,\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d+\left (i+\sqrt {3}\right ) e}\right )\right )\right )}{2 e} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.24 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.23

method	result	size
default	\(\frac {a \ln \left (e x +d \right )}{e}+\frac {b \ln \left (e x +d \right ) \arctan \left (c \,x^{3}\right )}{e}-\frac {b \,e^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c^{2}-6 c^{2} d \,\textit {\_Z}^{5}+15 c^{2} d^{2} \textit {\_Z}^{4}-20 c^{2} d^{3} \textit {\_Z}^{3}+15 c^{2} d^{4} \textit {\_Z}^{2}-6 c^{2} d^{5} \textit {\_Z} +c^{2} d^{6}+e^{6}\right )}{\sum }\frac {\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )}{\textit {\_R1}^{3}-3 \textit {\_R1}^{2} d +3 \textit {\_R1} \,d^{2}-d^{3}}\right )}{2 c}\)	\(172\)
parts	\(\frac {a \ln \left (e x +d \right )}{e}+\frac {b \ln \left (e x +d \right ) \arctan \left (c \,x^{3}\right )}{e}-\frac {b \,e^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c^{2}-6 c^{2} d \,\textit {\_Z}^{5}+15 c^{2} d^{2} \textit {\_Z}^{4}-20 c^{2} d^{3} \textit {\_Z}^{3}+15 c^{2} d^{4} \textit {\_Z}^{2}-6 c^{2} d^{5} \textit {\_Z} +c^{2} d^{6}+e^{6}\right )}{\sum }\frac {\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )}{\textit {\_R1}^{3}-3 \textit {\_R1}^{2} d +3 \textit {\_R1} \,d^{2}-d^{3}}\right )}{2 c}\)	\(172\)
risch	\(\frac {i b \ln \left (e x +d \right ) \ln \left (-i c \,x^{3}+1\right )}{2 e}-\frac {i b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2} c d +3 \textit {\_Z} c \,d^{2}-c \,d^{3}+e^{3} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right )\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{2 e}+\frac {a \ln \left (e x +d \right )}{e}-\frac {i b \ln \left (e x +d \right ) \ln \left (i c \,x^{3}+1\right )}{2 e}+\frac {i b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2} c d +3 \textit {\_Z} c \,d^{2}-c \,d^{3}-e^{3} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right )\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{2 e}\)	\(232\)

[In]

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

[Out]

a*ln(e*x+d)/e+b*ln(e*x+d)/e*arctan(c*x^3)-1/2*b/c*e^2*sum(1/(_R1^3-3*_R1^2*d+3*_R1*d^2-d^3)*(ln(e*x+d)*ln((-e*
x+_R1-d)/_R1)+dilog((-e*x+_R1-d)/_R1)),_R1=RootOf(_Z^6*c^2-6*_Z^5*c^2*d+15*_Z^4*c^2*d^2-20*_Z^3*c^2*d^3+15*_Z^
2*c^2*d^4-6*_Z*c^2*d^5+c^2*d^6+e^6))

Fricas [F]

\[ \int \frac {a+b \arctan \left (c x^3\right )}{d+e x} \, dx=\int { \frac {b \arctan \left (c x^{3}\right ) + a}{e x + d} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \arctan \left (c x^3\right )}{d+e x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {a+b \arctan \left (c x^3\right )}{d+e x} \, dx=\int { \frac {b \arctan \left (c x^{3}\right ) + a}{e x + d} \,d x } \]

[In]

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

[Out]

2*b*integrate(1/2*arctan(c*x^3)/(e*x + d), x) + a*log(e*x + d)/e

Giac [F]

\[ \int \frac {a+b \arctan \left (c x^3\right )}{d+e x} \, dx=\int { \frac {b \arctan \left (c x^{3}\right ) + a}{e x + d} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arctan \left (c x^3\right )}{d+e x} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x^3\right )}{d+e\,x} \,d x \]

[In]

int((a + b*atan(c*x^3))/(d + e*x),x)

[Out]

int((a + b*atan(c*x^3))/(d + e*x), x)