3.1301 \(\int \frac{(a+b \tan ^{-1}(c x)) (d+e \log (f+g x^2))}{x^3} \, dx\)

Optimal. Leaf size=528 \[ \frac{i b e \left (c^2 f-g\right ) \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )}{2 f}-\frac{i b e \left (c^2 f-g\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(1-i c x) \left (c \sqrt{-f}-i \sqrt{g}\right )}\right )}{4 f}-\frac{i b e \left (c^2 f-g\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(1-i c x) \left (c \sqrt{-f}+i \sqrt{g}\right )}\right )}{4 f}+\frac{i b e g \text{PolyLog}(2,-i c x)}{2 f}-\frac{i b e g \text{PolyLog}(2,i c x)}{2 f}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac{a e g \log \left (f+g x^2\right )}{2 f}+\frac{a e g \log (x)}{f}-\frac{1}{2} b c^2 \tan ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{b e \left (c^2 f-g\right ) \log \left (\frac{2}{1-i c x}\right ) \tan ^{-1}(c x)}{f}+\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(1-i c x) \left (c \sqrt{-f}-i \sqrt{g}\right )}\right )}{2 f}+\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(1-i c x) \left (c \sqrt{-f}+i \sqrt{g}\right )}\right )}{2 f}-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac{b c e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.769275, antiderivative size = 528, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 18, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {4852, 325, 203, 5021, 801, 635, 205, 260, 446, 72, 6725, 4848, 2391, 4928, 4856, 2402, 2315, 2447} \[ \frac{i b e \left (c^2 f-g\right ) \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )}{2 f}-\frac{i b e \left (c^2 f-g\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(1-i c x) \left (c \sqrt{-f}-i \sqrt{g}\right )}\right )}{4 f}-\frac{i b e \left (c^2 f-g\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(1-i c x) \left (c \sqrt{-f}+i \sqrt{g}\right )}\right )}{4 f}+\frac{i b e g \text{PolyLog}(2,-i c x)}{2 f}-\frac{i b e g \text{PolyLog}(2,i c x)}{2 f}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac{a e g \log \left (f+g x^2\right )}{2 f}+\frac{a e g \log (x)}{f}-\frac{1}{2} b c^2 \tan ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{b e \left (c^2 f-g\right ) \log \left (\frac{2}{1-i c x}\right ) \tan ^{-1}(c x)}{f}+\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(1-i c x) \left (c \sqrt{-f}-i \sqrt{g}\right )}\right )}{2 f}+\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(1-i c x) \left (c \sqrt{-f}+i \sqrt{g}\right )}\right )}{2 f}-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac{b c e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4852

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
n[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 325

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

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 5021

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

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 635

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

Rule 205

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

Rule 260

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

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 4848

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

Rule 2391

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 4928

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 4856

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 2402

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 2315

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

Rule 2447

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]]

Rubi steps

\begin{align*} \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx &=-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{1}{2} b c^2 \tan ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-(2 e g) \int \left (\frac{-a-b c x}{2 x \left (f+g x^2\right )}-\frac{b \left (1+c^2 x^2\right ) \tan ^{-1}(c x)}{2 x \left (f+g x^2\right )}\right ) \, dx\\ &=-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{1}{2} b c^2 \tan ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-(e g) \int \frac{-a-b c x}{x \left (f+g x^2\right )} \, dx+(b e g) \int \frac{\left (1+c^2 x^2\right ) \tan ^{-1}(c x)}{x \left (f+g x^2\right )} \, dx\\ &=-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{1}{2} b c^2 \tan ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-(e g) \int \left (-\frac{a}{f x}+\frac{-b c f+a g x}{f \left (f+g x^2\right )}\right ) \, dx+(b e g) \int \left (\frac{\tan ^{-1}(c x)}{f x}+\frac{\left (c^2 f-g\right ) x \tan ^{-1}(c x)}{f \left (f+g x^2\right )}\right ) \, dx\\ &=\frac{a e g \log (x)}{f}-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{1}{2} b c^2 \tan ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac{(e g) \int \frac{-b c f+a g x}{f+g x^2} \, dx}{f}+\frac{(b e g) \int \frac{\tan ^{-1}(c x)}{x} \, dx}{f}+\frac{\left (b e \left (c^2 f-g\right ) g\right ) \int \frac{x \tan ^{-1}(c x)}{f+g x^2} \, dx}{f}\\ &=\frac{a e g \log (x)}{f}-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{1}{2} b c^2 \tan ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+(b c e g) \int \frac{1}{f+g x^2} \, dx+\frac{(i b e g) \int \frac{\log (1-i c x)}{x} \, dx}{2 f}-\frac{(i b e g) \int \frac{\log (1+i c x)}{x} \, dx}{2 f}+\frac{\left (b e \left (c^2 f-g\right ) g\right ) \int \left (-\frac{\tan ^{-1}(c x)}{2 \sqrt{g} \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{\tan ^{-1}(c x)}{2 \sqrt{g} \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx}{f}-\frac{\left (a e g^2\right ) \int \frac{x}{f+g x^2} \, dx}{f}\\ &=\frac{b c e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}+\frac{a e g \log (x)}{f}-\frac{a e g \log \left (f+g x^2\right )}{2 f}-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{1}{2} b c^2 \tan ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac{i b e g \text{Li}_2(-i c x)}{2 f}-\frac{i b e g \text{Li}_2(i c x)}{2 f}-\frac{\left (b e \left (c^2 f-g\right ) \sqrt{g}\right ) \int \frac{\tan ^{-1}(c x)}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 f}+\frac{\left (b e \left (c^2 f-g\right ) \sqrt{g}\right ) \int \frac{\tan ^{-1}(c x)}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 f}\\ &=\frac{b c e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}+\frac{a e g \log (x)}{f}-\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(c x) \log \left (\frac{2}{1-i c x}\right )}{f}+\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(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 f}+\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(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 f}-\frac{a e g \log \left (f+g x^2\right )}{2 f}-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{1}{2} b c^2 \tan ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac{i b e g \text{Li}_2(-i c x)}{2 f}-\frac{i b e g \text{Li}_2(i c x)}{2 f}+2 \frac{\left (b c e \left (c^2 f-g\right )\right ) \int \frac{\log \left (\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{2 f}-\frac{\left (b c e \left (c^2 f-g\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 f}-\frac{\left (b c e \left (c^2 f-g\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 f}\\ &=\frac{b c e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}+\frac{a e g \log (x)}{f}-\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(c x) \log \left (\frac{2}{1-i c x}\right )}{f}+\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(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 f}+\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(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 f}-\frac{a e g \log \left (f+g x^2\right )}{2 f}-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{1}{2} b c^2 \tan ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac{i b e g \text{Li}_2(-i c x)}{2 f}-\frac{i b e g \text{Li}_2(i c x)}{2 f}-\frac{i b e \left (c^2 f-g\right ) \text{Li}_2\left (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 f}-\frac{i b e \left (c^2 f-g\right ) \text{Li}_2\left (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 f}+2 \frac{\left (i b e \left (c^2 f-g\right )\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i c x}\right )}{2 f}\\ &=\frac{b c e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}+\frac{a e g \log (x)}{f}-\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(c x) \log \left (\frac{2}{1-i c x}\right )}{f}+\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(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 f}+\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(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 f}-\frac{a e g \log \left (f+g x^2\right )}{2 f}-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{1}{2} b c^2 \tan ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac{i b e g \text{Li}_2(-i c x)}{2 f}-\frac{i b e g \text{Li}_2(i c x)}{2 f}+\frac{i b e \left (c^2 f-g\right ) \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{2 f}-\frac{i b e \left (c^2 f-g\right ) \text{Li}_2\left (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 f}-\frac{i b e \left (c^2 f-g\right ) \text{Li}_2\left (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 f}\\ \end{align*}

Mathematica [B]  time = 5.67443, size = 1213, normalized size = 2.3 \[ -\frac{2 b c^2 d f \tan ^{-1}(c x) x^2-4 b c e \sqrt{f} \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) x^2-4 i b c^2 e f \sin ^{-1}\left (\sqrt{\frac{c^2 f}{c^2 f-g}}\right ) \tan ^{-1}\left (\frac{c g x}{\sqrt{c^2 f g}}\right ) x^2+4 i b e g \sin ^{-1}\left (\sqrt{\frac{c^2 f}{c^2 f-g}}\right ) \tan ^{-1}\left (\frac{c g x}{\sqrt{c^2 f g}}\right ) x^2-4 b e g \tan ^{-1}(c x) \log \left (1-e^{2 i \tan ^{-1}(c x)}\right ) x^2+4 b c^2 e f \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right ) x^2-2 b c^2 e f \sin ^{-1}\left (\sqrt{\frac{c^2 f}{c^2 f-g}}\right ) \log \left (\frac{\left (1+e^{2 i \tan ^{-1}(c x)}\right ) f c^2+\left (-1+e^{2 i \tan ^{-1}(c x)}\right ) g-2 e^{2 i \tan ^{-1}(c x)} \sqrt{c^2 f g}}{c^2 f-g}\right ) x^2+2 b e g \sin ^{-1}\left (\sqrt{\frac{c^2 f}{c^2 f-g}}\right ) \log \left (\frac{\left (1+e^{2 i \tan ^{-1}(c x)}\right ) f c^2+\left (-1+e^{2 i \tan ^{-1}(c x)}\right ) g-2 e^{2 i \tan ^{-1}(c x)} \sqrt{c^2 f g}}{c^2 f-g}\right ) x^2-2 b c^2 e f \tan ^{-1}(c x) \log \left (\frac{\left (1+e^{2 i \tan ^{-1}(c x)}\right ) f c^2+\left (-1+e^{2 i \tan ^{-1}(c x)}\right ) g-2 e^{2 i \tan ^{-1}(c x)} \sqrt{c^2 f g}}{c^2 f-g}\right ) x^2+2 b e g \tan ^{-1}(c x) \log \left (\frac{\left (1+e^{2 i \tan ^{-1}(c x)}\right ) f c^2+\left (-1+e^{2 i \tan ^{-1}(c x)}\right ) g-2 e^{2 i \tan ^{-1}(c x)} \sqrt{c^2 f g}}{c^2 f-g}\right ) x^2+2 b c^2 e f \sin ^{-1}\left (\sqrt{\frac{c^2 f}{c^2 f-g}}\right ) \log \left (\frac{e^{2 i \tan ^{-1}(c x)} \left (f c^2+g+2 \sqrt{c^2 f g}\right )}{c^2 f-g}+1\right ) x^2-2 b e g \sin ^{-1}\left (\sqrt{\frac{c^2 f}{c^2 f-g}}\right ) \log \left (\frac{e^{2 i \tan ^{-1}(c x)} \left (f c^2+g+2 \sqrt{c^2 f g}\right )}{c^2 f-g}+1\right ) x^2-2 b c^2 e f \tan ^{-1}(c x) \log \left (\frac{e^{2 i \tan ^{-1}(c x)} \left (f c^2+g+2 \sqrt{c^2 f g}\right )}{c^2 f-g}+1\right ) x^2+2 b e g \tan ^{-1}(c x) \log \left (\frac{e^{2 i \tan ^{-1}(c x)} \left (f c^2+g+2 \sqrt{c^2 f g}\right )}{c^2 f-g}+1\right ) x^2-4 a e g \log (x) x^2+2 a e g \log \left (g x^2+f\right ) x^2+2 b c^2 e f \tan ^{-1}(c x) \log \left (g x^2+f\right ) x^2-2 i b c^2 e f \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right ) x^2+2 i b e g \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(c x)}\right ) x^2+i b c^2 e f \text{PolyLog}\left (2,-\frac{e^{2 i \tan ^{-1}(c x)} \left (f c^2+g-2 \sqrt{c^2 f g}\right )}{c^2 f-g}\right ) x^2-i b e g \text{PolyLog}\left (2,-\frac{e^{2 i \tan ^{-1}(c x)} \left (f c^2+g-2 \sqrt{c^2 f g}\right )}{c^2 f-g}\right ) x^2+i b c^2 e f \text{PolyLog}\left (2,-\frac{e^{2 i \tan ^{-1}(c x)} \left (f c^2+g+2 \sqrt{c^2 f g}\right )}{c^2 f-g}\right ) x^2-i b e g \text{PolyLog}\left (2,-\frac{e^{2 i \tan ^{-1}(c x)} \left (f c^2+g+2 \sqrt{c^2 f g}\right )}{c^2 f-g}\right ) x^2+2 b c d f x+2 b c e f \log \left (g x^2+f\right ) x+2 a d f+2 b d f \tan ^{-1}(c x)+2 a e f \log \left (g x^2+f\right )+2 b e f \tan ^{-1}(c x) \log \left (g x^2+f\right )}{4 f x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 4.83, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\arctan \left ( cx \right ) \right ) \left ( d+e\ln \left ( g{x}^{2}+f \right ) \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b d \arctan \left (c x\right ) + a d +{\left (b e \arctan \left (c x\right ) + a e\right )} \log \left (g x^{2} + f\right )}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*d*arctan(c*x) + a*d + (b*e*arctan(c*x) + a*e)*log(g*x^2 + f))/x^3, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atan(c*x))*(d+e*ln(g*x**2+f))/x**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (c x\right ) + a\right )}{\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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