\(\int x (a+b \arctan (c x)) (d+e \log (f+g x^2)) \, dx\) [1297]

Optimal result

Integrand size = 22, antiderivative size = 562 \[ \int x (a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=-\frac {b (d-e) x}{2 c}+\frac {b e x}{c}+\frac {b (d-e) \arctan (c x)}{2 c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))-\frac {1}{2} e x^2 (a+b \arctan (c x))-\frac {b e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c \sqrt {g}}-\frac {b e \left (c^2 f-g\right ) \arctan (c x) \log \left (\frac {2}{1-i c x}\right )}{c^2 g}+\frac {b e \left (c^2 f-g\right ) \arctan (c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-i \sqrt {g}\right ) (1-i c x)}\right )}{2 c^2 g}+\frac {b e \left (c^2 f-g\right ) \arctan (c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+i \sqrt {g}\right ) (1-i c x)}\right )}{2 c^2 g}-\frac {b e x \log \left (f+g x^2\right )}{2 c}-\frac {b e \left (c^2 f-g\right ) \arctan (c x) \log \left (f+g x^2\right )}{2 c^2 g}+\frac {e \left (f+g x^2\right ) (a+b \arctan (c x)) \log \left (f+g x^2\right )}{2 g}+\frac {i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 c^2 g}-\frac {i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-i \sqrt {g}\right ) (1-i c x)}\right )}{4 c^2 g}-\frac {i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+i \sqrt {g}\right ) (1-i c x)}\right )}{4 c^2 g} \]

[Out]

-1/2*b*(d-e)*x/c+b*e*x/c+1/2*b*(d-e)*arctan(c*x)/c^2+1/2*d*x^2*(a+b*arctan(c*x))-1/2*e*x^2*(a+b*arctan(c*x))-b
*e*(c^2*f-g)*arctan(c*x)*ln(2/(1-I*c*x))/c^2/g-1/2*b*e*x*ln(g*x^2+f)/c-1/2*b*e*(c^2*f-g)*arctan(c*x)*ln(g*x^2+
f)/c^2/g+1/2*e*(g*x^2+f)*(a+b*arctan(c*x))*ln(g*x^2+f)/g+1/2*b*e*(c^2*f-g)*arctan(c*x)*ln(2*c*((-f)^(1/2)-x*g^
(1/2))/(1-I*c*x)/(c*(-f)^(1/2)-I*g^(1/2)))/c^2/g+1/2*b*e*(c^2*f-g)*arctan(c*x)*ln(2*c*((-f)^(1/2)+x*g^(1/2))/(
1-I*c*x)/(c*(-f)^(1/2)+I*g^(1/2)))/c^2/g+1/2*I*b*e*(c^2*f-g)*polylog(2,1-2/(1-I*c*x))/c^2/g-1/4*I*b*e*(c^2*f-g
)*polylog(2,1-2*c*((-f)^(1/2)-x*g^(1/2))/(1-I*c*x)/(c*(-f)^(1/2)-I*g^(1/2)))/c^2/g-1/4*I*b*e*(c^2*f-g)*polylog
(2,1-2*c*((-f)^(1/2)+x*g^(1/2))/(1-I*c*x)/(c*(-f)^(1/2)+I*g^(1/2)))/c^2/g-b*e*arctan(x*g^(1/2)/f^(1/2))*f^(1/2
)/c/g^(1/2)

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {2504, 2436, 2332, 5139, 327, 209, 2608, 2498, 211, 2520, 12, 5048, 4966, 2449, 2352, 2497} \[ \int x (a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\frac {1}{2} d x^2 (a+b \arctan (c x))+\frac {e \left (f+g x^2\right ) \log \left (f+g x^2\right ) (a+b \arctan (c x))}{2 g}-\frac {1}{2} e x^2 (a+b \arctan (c x))+\frac {b (d-e) \arctan (c x)}{2 c^2}-\frac {b e \arctan (c x) \left (c^2 f-g\right ) \log \left (f+g x^2\right )}{2 c^2 g}-\frac {b e \arctan (c x) \left (c^2 f-g\right ) \log \left (\frac {2}{1-i c x}\right )}{c^2 g}+\frac {b e \arctan (c x) \left (c^2 f-g\right ) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(1-i c x) \left (c \sqrt {-f}-i \sqrt {g}\right )}\right )}{2 c^2 g}+\frac {b e \arctan (c x) \left (c^2 f-g\right ) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(1-i c x) \left (c \sqrt {-f}+i \sqrt {g}\right )}\right )}{2 c^2 g}-\frac {b e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c \sqrt {g}}+\frac {i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 c^2 g}-\frac {i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-i \sqrt {g}\right ) (1-i c x)}\right )}{4 c^2 g}-\frac {i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {g} x+\sqrt {-f}\right )}{\left (\sqrt {-f} c+i \sqrt {g}\right ) (1-i c x)}\right )}{4 c^2 g}-\frac {b x (d-e)}{2 c}-\frac {b e x \log \left (f+g x^2\right )}{2 c}+\frac {b e x}{c} \]

[In]

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

[Out]

-1/2*(b*(d - e)*x)/c + (b*e*x)/c + (b*(d - e)*ArcTan[c*x])/(2*c^2) + (d*x^2*(a + b*ArcTan[c*x]))/2 - (e*x^2*(a
+ b*ArcTan[c*x]))/2 - (b*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/(c*Sqrt[g]) - (b*e*(c^2*f - g)*ArcTan[c*x]*Lo
g[2/(1 - I*c*x)])/(c^2*g) + (b*e*(c^2*f - g)*ArcTan[c*x]*Log[(2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - I*Sqr
t[g])*(1 - I*c*x))])/(2*c^2*g) + (b*e*(c^2*f - g)*ArcTan[c*x]*Log[(2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] +
I*Sqrt[g])*(1 - I*c*x))])/(2*c^2*g) - (b*e*x*Log[f + g*x^2])/(2*c) - (b*e*(c^2*f - g)*ArcTan[c*x]*Log[f + g*x^
2])/(2*c^2*g) + (e*(f + g*x^2)*(a + b*ArcTan[c*x])*Log[f + g*x^2])/(2*g) + ((I/2)*b*e*(c^2*f - g)*PolyLog[2, 1
- 2/(1 - I*c*x)])/(c^2*g) - ((I/4)*b*e*(c^2*f - g)*PolyLog[2, 1 - (2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] -
I*Sqrt[g])*(1 - I*c*x))])/(c^2*g) - ((I/4)*b*e*(c^2*f - g)*PolyLog[2, 1 - (2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sq
rt[-f] + I*Sqrt[g])*(1 - I*c*x))])/(c^2*g)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[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 211

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

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2436

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

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 2498

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2520

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[u*(x^(n - 1)/(d + e*x^n)
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 2608

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 4966

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

Rule 5048

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTan[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[a,
0])

Rule 5139

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]*(e_.))*(x_)^(m_.), x_Symbol] :> With
[{u = IntHide[x^m*(d + e*Log[f + g*x^2]), x]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[b*c, Int[ExpandIntegrand[u
/(1 + c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} d x^2 (a+b \arctan (c x))-\frac {1}{2} e x^2 (a+b \arctan (c x))+\frac {e \left (f+g x^2\right ) (a+b \arctan (c x)) \log \left (f+g x^2\right )}{2 g}-(b c) \int \left (\frac {(d-e) x^2}{2 \left (1+c^2 x^2\right )}+\frac {e \left (f+g x^2\right ) \log \left (f+g x^2\right )}{2 g \left (1+c^2 x^2\right )}\right ) \, dx \\ & = \frac {1}{2} d x^2 (a+b \arctan (c x))-\frac {1}{2} e x^2 (a+b \arctan (c x))+\frac {e \left (f+g x^2\right ) (a+b \arctan (c x)) \log \left (f+g x^2\right )}{2 g}-\frac {1}{2} (b c (d-e)) \int \frac {x^2}{1+c^2 x^2} \, dx-\frac {(b c e) \int \frac {\left (f+g x^2\right ) \log \left (f+g x^2\right )}{1+c^2 x^2} \, dx}{2 g} \\ & = -\frac {b (d-e) x}{2 c}+\frac {1}{2} d x^2 (a+b \arctan (c x))-\frac {1}{2} e x^2 (a+b \arctan (c x))+\frac {e \left (f+g x^2\right ) (a+b \arctan (c x)) \log \left (f+g x^2\right )}{2 g}+\frac {(b (d-e)) \int \frac {1}{1+c^2 x^2} \, dx}{2 c}-\frac {(b c e) \int \left (\frac {g \log \left (f+g x^2\right )}{c^2}+\frac {\left (c^2 f-g\right ) \log \left (f+g x^2\right )}{c^2 \left (1+c^2 x^2\right )}\right ) \, dx}{2 g} \\ & = -\frac {b (d-e) x}{2 c}+\frac {b (d-e) \arctan (c x)}{2 c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))-\frac {1}{2} e x^2 (a+b \arctan (c x))+\frac {e \left (f+g x^2\right ) (a+b \arctan (c x)) \log \left (f+g x^2\right )}{2 g}-\frac {(b e) \int \log \left (f+g x^2\right ) \, dx}{2 c}-\frac {\left (b c e \left (f-\frac {g}{c^2}\right )\right ) \int \frac {\log \left (f+g x^2\right )}{1+c^2 x^2} \, dx}{2 g} \\ & = -\frac {b (d-e) x}{2 c}+\frac {b (d-e) \arctan (c x)}{2 c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))-\frac {1}{2} e x^2 (a+b \arctan (c x))-\frac {b e x \log \left (f+g x^2\right )}{2 c}-\frac {b e \left (f-\frac {g}{c^2}\right ) \arctan (c x) \log \left (f+g x^2\right )}{2 g}+\frac {e \left (f+g x^2\right ) (a+b \arctan (c x)) \log \left (f+g x^2\right )}{2 g}+\frac {(b e g) \int \frac {x^2}{f+g x^2} \, dx}{c}+\left (b c e \left (f-\frac {g}{c^2}\right )\right ) \int \frac {x \arctan (c x)}{c \left (f+g x^2\right )} \, dx \\ & = -\frac {b (d-e) x}{2 c}+\frac {b e x}{c}+\frac {b (d-e) \arctan (c x)}{2 c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))-\frac {1}{2} e x^2 (a+b \arctan (c x))-\frac {b e x \log \left (f+g x^2\right )}{2 c}-\frac {b e \left (f-\frac {g}{c^2}\right ) \arctan (c x) \log \left (f+g x^2\right )}{2 g}+\frac {e \left (f+g x^2\right ) (a+b \arctan (c x)) \log \left (f+g x^2\right )}{2 g}-\frac {(b e f) \int \frac {1}{f+g x^2} \, dx}{c}+\left (b e \left (f-\frac {g}{c^2}\right )\right ) \int \frac {x \arctan (c x)}{f+g x^2} \, dx \\ & = -\frac {b (d-e) x}{2 c}+\frac {b e x}{c}+\frac {b (d-e) \arctan (c x)}{2 c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))-\frac {1}{2} e x^2 (a+b \arctan (c x))-\frac {b e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c \sqrt {g}}-\frac {b e x \log \left (f+g x^2\right )}{2 c}-\frac {b e \left (f-\frac {g}{c^2}\right ) \arctan (c x) \log \left (f+g x^2\right )}{2 g}+\frac {e \left (f+g x^2\right ) (a+b \arctan (c x)) \log \left (f+g x^2\right )}{2 g}+\left (b e \left (f-\frac {g}{c^2}\right )\right ) \int \left (-\frac {\arctan (c x)}{2 \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\arctan (c x)}{2 \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx \\ & = -\frac {b (d-e) x}{2 c}+\frac {b e x}{c}+\frac {b (d-e) \arctan (c x)}{2 c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))-\frac {1}{2} e x^2 (a+b \arctan (c x))-\frac {b e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c \sqrt {g}}-\frac {b e x \log \left (f+g x^2\right )}{2 c}-\frac {b e \left (f-\frac {g}{c^2}\right ) \arctan (c x) \log \left (f+g x^2\right )}{2 g}+\frac {e \left (f+g x^2\right ) (a+b \arctan (c x)) \log \left (f+g x^2\right )}{2 g}-\frac {\left (b e \left (f-\frac {g}{c^2}\right )\right ) \int \frac {\arctan (c x)}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 \sqrt {g}}+\frac {\left (b e \left (f-\frac {g}{c^2}\right )\right ) \int \frac {\arctan (c x)}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 \sqrt {g}} \\ & = -\frac {b (d-e) x}{2 c}+\frac {b e x}{c}+\frac {b (d-e) \arctan (c x)}{2 c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))-\frac {1}{2} e x^2 (a+b \arctan (c x))-\frac {b e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c \sqrt {g}}-\frac {b e \left (f-\frac {g}{c^2}\right ) \arctan (c x) \log \left (\frac {2}{1-i c x}\right )}{g}+\frac {b e \left (f-\frac {g}{c^2}\right ) \arctan (c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-i \sqrt {g}\right ) (1-i c x)}\right )}{2 g}+\frac {b e \left (f-\frac {g}{c^2}\right ) \arctan (c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+i \sqrt {g}\right ) (1-i c x)}\right )}{2 g}-\frac {b e x \log \left (f+g x^2\right )}{2 c}-\frac {b e \left (f-\frac {g}{c^2}\right ) \arctan (c x) \log \left (f+g x^2\right )}{2 g}+\frac {e \left (f+g x^2\right ) (a+b \arctan (c x)) \log \left (f+g x^2\right )}{2 g}+2 \frac {\left (b c e \left (f-\frac {g}{c^2}\right )\right ) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{2 g}-\frac {\left (b c e \left (f-\frac {g}{c^2}\right )\right ) \int \frac {\log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-i \sqrt {g}\right ) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{2 g}-\frac {\left (b c e \left (f-\frac {g}{c^2}\right )\right ) \int \frac {\log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+i \sqrt {g}\right ) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{2 g} \\ & = -\frac {b (d-e) x}{2 c}+\frac {b e x}{c}+\frac {b (d-e) \arctan (c x)}{2 c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))-\frac {1}{2} e x^2 (a+b \arctan (c x))-\frac {b e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c \sqrt {g}}-\frac {b e \left (f-\frac {g}{c^2}\right ) \arctan (c x) \log \left (\frac {2}{1-i c x}\right )}{g}+\frac {b e \left (f-\frac {g}{c^2}\right ) \arctan (c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-i \sqrt {g}\right ) (1-i c x)}\right )}{2 g}+\frac {b e \left (f-\frac {g}{c^2}\right ) \arctan (c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+i \sqrt {g}\right ) (1-i c x)}\right )}{2 g}-\frac {b e x \log \left (f+g x^2\right )}{2 c}-\frac {b e \left (f-\frac {g}{c^2}\right ) \arctan (c x) \log \left (f+g x^2\right )}{2 g}+\frac {e \left (f+g x^2\right ) (a+b \arctan (c x)) \log \left (f+g x^2\right )}{2 g}-\frac {i b e \left (f-\frac {g}{c^2}\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-i \sqrt {g}\right ) (1-i c x)}\right )}{4 g}-\frac {i b e \left (f-\frac {g}{c^2}\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+i \sqrt {g}\right ) (1-i c x)}\right )}{4 g}+2 \frac {\left (i b e \left (f-\frac {g}{c^2}\right )\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{2 g} \\ & = -\frac {b (d-e) x}{2 c}+\frac {b e x}{c}+\frac {b (d-e) \arctan (c x)}{2 c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))-\frac {1}{2} e x^2 (a+b \arctan (c x))-\frac {b e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c \sqrt {g}}-\frac {b e \left (f-\frac {g}{c^2}\right ) \arctan (c x) \log \left (\frac {2}{1-i c x}\right )}{g}+\frac {b e \left (f-\frac {g}{c^2}\right ) \arctan (c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-i \sqrt {g}\right ) (1-i c x)}\right )}{2 g}+\frac {b e \left (f-\frac {g}{c^2}\right ) \arctan (c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+i \sqrt {g}\right ) (1-i c x)}\right )}{2 g}-\frac {b e x \log \left (f+g x^2\right )}{2 c}-\frac {b e \left (f-\frac {g}{c^2}\right ) \arctan (c x) \log \left (f+g x^2\right )}{2 g}+\frac {e \left (f+g x^2\right ) (a+b \arctan (c x)) \log \left (f+g x^2\right )}{2 g}+\frac {i b e \left (f-\frac {g}{c^2}\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 g}-\frac {i b e \left (f-\frac {g}{c^2}\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-i \sqrt {g}\right ) (1-i c x)}\right )}{4 g}-\frac {i b e \left (f-\frac {g}{c^2}\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+i \sqrt {g}\right ) (1-i c x)}\right )}{4 g} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1140\) vs. \(2(562)=1124\).

Time = 6.75 (sec) , antiderivative size = 1140, normalized size of antiderivative = 2.03 \[ \int x (a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\frac {-2 b c d g x+6 b c e g x+2 a c^2 d g x^2-2 a c^2 e g x^2+2 b d g \arctan (c x)-2 b e g \arctan (c x)+2 b c^2 d g x^2 \arctan (c x)-2 b c^2 e g x^2 \arctan (c x)-4 b c e \sqrt {f} \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )+4 i b c^2 e f \arcsin \left (\sqrt {\frac {c^2 f}{c^2 f-g}}\right ) \arctan \left (\frac {c g x}{\sqrt {c^2 f g}}\right )-4 i b e g \arcsin \left (\sqrt {\frac {c^2 f}{c^2 f-g}}\right ) \arctan \left (\frac {c g x}{\sqrt {c^2 f g}}\right )-4 b c^2 e f \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+4 b e g \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+2 b c^2 e f \arcsin \left (\sqrt {\frac {c^2 f}{c^2 f-g}}\right ) \log \left (\frac {c^2 \left (1+e^{2 i \arctan (c x)}\right ) f+\left (-1+e^{2 i \arctan (c x)}\right ) g-2 e^{2 i \arctan (c x)} \sqrt {c^2 f g}}{c^2 f-g}\right )-2 b e g \arcsin \left (\sqrt {\frac {c^2 f}{c^2 f-g}}\right ) \log \left (\frac {c^2 \left (1+e^{2 i \arctan (c x)}\right ) f+\left (-1+e^{2 i \arctan (c x)}\right ) g-2 e^{2 i \arctan (c x)} \sqrt {c^2 f g}}{c^2 f-g}\right )+2 b c^2 e f \arctan (c x) \log \left (\frac {c^2 \left (1+e^{2 i \arctan (c x)}\right ) f+\left (-1+e^{2 i \arctan (c x)}\right ) g-2 e^{2 i \arctan (c x)} \sqrt {c^2 f g}}{c^2 f-g}\right )-2 b e g \arctan (c x) \log \left (\frac {c^2 \left (1+e^{2 i \arctan (c x)}\right ) f+\left (-1+e^{2 i \arctan (c x)}\right ) g-2 e^{2 i \arctan (c x)} \sqrt {c^2 f g}}{c^2 f-g}\right )-2 b c^2 e f \arcsin \left (\sqrt {\frac {c^2 f}{c^2 f-g}}\right ) \log \left (1+\frac {e^{2 i \arctan (c x)} \left (c^2 f+g+2 \sqrt {c^2 f g}\right )}{c^2 f-g}\right )+2 b e g \arcsin \left (\sqrt {\frac {c^2 f}{c^2 f-g}}\right ) \log \left (1+\frac {e^{2 i \arctan (c x)} \left (c^2 f+g+2 \sqrt {c^2 f g}\right )}{c^2 f-g}\right )+2 b c^2 e f \arctan (c x) \log \left (1+\frac {e^{2 i \arctan (c x)} \left (c^2 f+g+2 \sqrt {c^2 f g}\right )}{c^2 f-g}\right )-2 b e g \arctan (c x) \log \left (1+\frac {e^{2 i \arctan (c x)} \left (c^2 f+g+2 \sqrt {c^2 f g}\right )}{c^2 f-g}\right )+2 a c^2 e f \log \left (f+g x^2\right )-2 b c e g x \log \left (f+g x^2\right )+2 a c^2 e g x^2 \log \left (f+g x^2\right )+2 b e g \arctan (c x) \log \left (f+g x^2\right )+2 b c^2 e g x^2 \arctan (c x) \log \left (f+g x^2\right )+2 i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,\frac {e^{2 i \arctan (c x)} \left (-c^2 f-g+2 \sqrt {c^2 f g}\right )}{c^2 f-g}\right )-i b c^2 e f \operatorname {PolyLog}\left (2,-\frac {e^{2 i \arctan (c x)} \left (c^2 f+g+2 \sqrt {c^2 f g}\right )}{c^2 f-g}\right )+i b e g \operatorname {PolyLog}\left (2,-\frac {e^{2 i \arctan (c x)} \left (c^2 f+g+2 \sqrt {c^2 f g}\right )}{c^2 f-g}\right )}{4 c^2 g} \]

[In]

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

[Out]

(-2*b*c*d*g*x + 6*b*c*e*g*x + 2*a*c^2*d*g*x^2 - 2*a*c^2*e*g*x^2 + 2*b*d*g*ArcTan[c*x] - 2*b*e*g*ArcTan[c*x] +
2*b*c^2*d*g*x^2*ArcTan[c*x] - 2*b*c^2*e*g*x^2*ArcTan[c*x] - 4*b*c*e*Sqrt[f]*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]
] + (4*I)*b*c^2*e*f*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*ArcTan[(c*g*x)/Sqrt[c^2*f*g]] - (4*I)*b*e*g*ArcSin[Sqrt[
(c^2*f)/(c^2*f - g)]]*ArcTan[(c*g*x)/Sqrt[c^2*f*g]] - 4*b*c^2*e*f*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] +
4*b*e*g*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 2*b*c^2*e*f*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*Log[(c^2*(
1 + E^((2*I)*ArcTan[c*x]))*f + (-1 + E^((2*I)*ArcTan[c*x]))*g - 2*E^((2*I)*ArcTan[c*x])*Sqrt[c^2*f*g])/(c^2*f
- g)] - 2*b*e*g*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*Log[(c^2*(1 + E^((2*I)*ArcTan[c*x]))*f + (-1 + E^((2*I)*ArcT
an[c*x]))*g - 2*E^((2*I)*ArcTan[c*x])*Sqrt[c^2*f*g])/(c^2*f - g)] + 2*b*c^2*e*f*ArcTan[c*x]*Log[(c^2*(1 + E^((
2*I)*ArcTan[c*x]))*f + (-1 + E^((2*I)*ArcTan[c*x]))*g - 2*E^((2*I)*ArcTan[c*x])*Sqrt[c^2*f*g])/(c^2*f - g)] -
2*b*e*g*ArcTan[c*x]*Log[(c^2*(1 + E^((2*I)*ArcTan[c*x]))*f + (-1 + E^((2*I)*ArcTan[c*x]))*g - 2*E^((2*I)*ArcTa
n[c*x])*Sqrt[c^2*f*g])/(c^2*f - g)] - 2*b*c^2*e*f*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*Log[1 + (E^((2*I)*ArcTan[c
*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] + 2*b*e*g*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*Log[1 + (E^((2*I)
*ArcTan[c*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] + 2*b*c^2*e*f*ArcTan[c*x]*Log[1 + (E^((2*I)*ArcTan[c
*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] - 2*b*e*g*ArcTan[c*x]*Log[1 + (E^((2*I)*ArcTan[c*x])*(c^2*f +
g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] + 2*a*c^2*e*f*Log[f + g*x^2] - 2*b*c*e*g*x*Log[f + g*x^2] + 2*a*c^2*e*g*x^
2*Log[f + g*x^2] + 2*b*e*g*ArcTan[c*x]*Log[f + g*x^2] + 2*b*c^2*e*g*x^2*ArcTan[c*x]*Log[f + g*x^2] + (2*I)*b*e
*(c^2*f - g)*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - I*b*e*(c^2*f - g)*PolyLog[2, (E^((2*I)*ArcTan[c*x])*(-(c^2*f
) - g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] - I*b*c^2*e*f*PolyLog[2, -((E^((2*I)*ArcTan[c*x])*(c^2*f + g + 2*Sqrt[c
^2*f*g]))/(c^2*f - g))] + I*b*e*g*PolyLog[2, -((E^((2*I)*ArcTan[c*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f -
g))])/(4*c^2*g)

Maple [C] (warning: unable to verify)

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

Time = 13.54 (sec) , antiderivative size = 10102, normalized size of antiderivative = 17.98

method	result	size
default	\(\text {Expression too large to display}\)	\(10102\)
parts	\(\text {Expression too large to display}\)	\(10102\)

[In]

int(x*(a+b*arctan(c*x))*(d+e*ln(g*x^2+f)),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F]

\[ \int x (a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )} x \,d x } \]

[In]

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

[Out]

integral(b*d*x*arctan(c*x) + a*d*x + (b*e*x*arctan(c*x) + a*e*x)*log(g*x^2 + f), x)

Sympy [F(-1)]

Timed out. \[ \int x (a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\text {Timed out} \]

[In]

integrate(x*(a+b*atan(c*x))*(d+e*ln(g*x**2+f)),x)

[Out]

Timed out

Maxima [F]

\[ \int x (a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )} x \,d x } \]

[In]

integrate(x*(a+b*arctan(c*x))*(d+e*log(g*x^2+f)),x, algorithm="maxima")

[Out]

1/2*a*d*x^2 + 1/2*(x^2*arctan(c*x) - c*(x/c^2 - arctan(c*x)/c^3))*b*d - 1/2*(g*x^2 - (g*x^2 + f)*log(g*x^2 + f
) + f)*a*e/g - 1/2*(2*c*f*arctan(g*x/sqrt(f*g)) + (4*c^4*g*integrate(1/2*x^3*arctan(c*x)/(c^2*g*x^2 + c^2*f),
x) + 4*c^2*g*integrate(1/2*x*arctan(c*x)/(c^2*g*x^2 + c^2*f), x) - 2*c*x + (c*x - (c^2*x^2 + 1)*arctan(c*x))*l
og(g*x^2 + f))*sqrt(f*g))*b*e/(sqrt(f*g)*c^2)

Giac [F]

\[ \int x (a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )} x \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int x (a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int x\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right ) \,d x \]

[In]

int(x*(a + b*atan(c*x))*(d + e*log(f + g*x^2)),x)

[Out]

int(x*(a + b*atan(c*x))*(d + e*log(f + g*x^2)), x)