\(\int (a+b \arctan (c x)) (d+e \log (f+g x^2)) \, dx\) [1298]

Optimal result

Integrand size = 21, antiderivative size = 656 \[ \int (a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=-2 a e x-2 b e x \arctan (c x)+\frac {2 a e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}+\frac {i b e \sqrt {-f} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}}+\frac {b e \log \left (1+c^2 x^2\right )}{c}+x (a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )-\frac {b \log \left (-\frac {g \left (1+c^2 x^2\right )}{c^2 f-g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}-\frac {i b e \sqrt {-f} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (i-c x)}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (1-i c x)}{i c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (1+i c x)}{i c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (i+c x)}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \operatorname {PolyLog}\left (2,\frac {c^2 \left (f+g x^2\right )}{c^2 f-g}\right )}{2 c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5129, 2525, 2441, 2440, 2438, 5036, 4930, 266, 5030, 211, 5028, 2456} \[ \int (a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=x (a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )+\frac {2 a e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-2 a e x-2 b e x \arctan (c x)-\frac {b \log \left (-\frac {g \left (c^2 x^2+1\right )}{c^2 f-g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}-\frac {b e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{c^2 f-g}\right )}{2 c}+\frac {b e \log \left (c^2 x^2+1\right )}{c}-\frac {i b e \sqrt {-f} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (i-c x)}{\sqrt {-f} c+i \sqrt {g}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (1-i c x)}{i \sqrt {-f} c+\sqrt {g}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (i c x+1)}{i \sqrt {-f} c+\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (c x+i)}{\sqrt {-f} c+i \sqrt {g}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}} \]

[In]

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

[Out]

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

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 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 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 2456

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rule 2525

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q,
x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rule 4930

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

Rule 5028

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

Rule 5030

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

Rule 5036

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x], x] - Dist[d*(f^2/e), Int[(f*x)^(m - 2)*((a + b*ArcTan[c*x])^p/
(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]

Rule 5129

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]*(e_.)), x_Symbol] :> Simp[x*(d + e*L
og[f + g*x^2])*(a + b*ArcTan[c*x]), x] + (-Dist[b*c, Int[x*((d + e*Log[f + g*x^2])/(1 + c^2*x^2)), x], x] - Di
st[2*e*g, Int[x^2*((a + b*ArcTan[c*x])/(f + g*x^2)), x], x]) /; FreeQ[{a, b, c, d, e, f, g}, x]

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1352\) vs. \(2(656)=1312\).

Time = 3.26 (sec) , antiderivative size = 1352, normalized size of antiderivative = 2.06 \[ \int (a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=a d x-2 a e x+b d x \arctan (c x)+\frac {2 a e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-\frac {b d \log \left (1+c^2 x^2\right )}{2 c}+a e x \log \left (f+g x^2\right )+b e \left (x \arctan (c x)-\frac {\log \left (1+c^2 x^2\right )}{2 c}\right ) \log \left (f+g x^2\right )+\frac {b e g \left (\frac {\left (-\log \left (-\frac {i}{c}+x\right )-\log \left (\frac {i}{c}+x\right )+\log \left (1+c^2 x^2\right )\right ) \log \left (f+g x^2\right )}{2 g}+\frac {\log \left (-\frac {i}{c}+x\right ) \log \left (1-\frac {\sqrt {g} \left (-\frac {i}{c}+x\right )}{-i \sqrt {f}-\frac {i \sqrt {g}}{c}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {g} \left (-\frac {i}{c}+x\right )}{-i \sqrt {f}-\frac {i \sqrt {g}}{c}}\right )}{2 g}+\frac {\log \left (-\frac {i}{c}+x\right ) \log \left (1-\frac {\sqrt {g} \left (-\frac {i}{c}+x\right )}{i \sqrt {f}-\frac {i \sqrt {g}}{c}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {g} \left (-\frac {i}{c}+x\right )}{i \sqrt {f}-\frac {i \sqrt {g}}{c}}\right )}{2 g}+\frac {\log \left (\frac {i}{c}+x\right ) \log \left (1-\frac {\sqrt {g} \left (\frac {i}{c}+x\right )}{-i \sqrt {f}+\frac {i \sqrt {g}}{c}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {g} \left (\frac {i}{c}+x\right )}{-i \sqrt {f}+\frac {i \sqrt {g}}{c}}\right )}{2 g}+\frac {\log \left (\frac {i}{c}+x\right ) \log \left (1-\frac {\sqrt {g} \left (\frac {i}{c}+x\right )}{i \sqrt {f}+\frac {i \sqrt {g}}{c}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {g} \left (\frac {i}{c}+x\right )}{i \sqrt {f}+\frac {i \sqrt {g}}{c}}\right )}{2 g}\right )}{c}-\frac {b e \left (4 c x \arctan (c x)+4 \log \left (\frac {1}{\sqrt {1+c^2 x^2}}\right )+\frac {c^2 f \left (4 \arctan (c x) \text {arctanh}\left (\frac {\sqrt {-c^2 f g}}{c g x}\right )-2 \arccos \left (\frac {c^2 f+g}{-c^2 f+g}\right ) \text {arctanh}\left (\frac {c g x}{\sqrt {-c^2 f g}}\right )-\left (\arccos \left (\frac {c^2 f+g}{-c^2 f+g}\right )-2 i \text {arctanh}\left (\frac {c g x}{\sqrt {-c^2 f g}}\right )\right ) \log \left (-\frac {2 c^2 f \left (i g+\sqrt {-c^2 f g}\right ) (-i+c x)}{\left (c^2 f-g\right ) \left (c^2 f-c \sqrt {-c^2 f g} x\right )}\right )-\left (\arccos \left (\frac {c^2 f+g}{-c^2 f+g}\right )+2 i \text {arctanh}\left (\frac {c g x}{\sqrt {-c^2 f g}}\right )\right ) \log \left (\frac {2 i c^2 f \left (g+i \sqrt {-c^2 f g}\right ) (i+c x)}{\left (c^2 f-g\right ) \left (c^2 f-c \sqrt {-c^2 f g} x\right )}\right )+\left (\arccos \left (\frac {c^2 f+g}{-c^2 f+g}\right )-2 i \text {arctanh}\left (\frac {\sqrt {-c^2 f g}}{c g x}\right )+2 i \text {arctanh}\left (\frac {c g x}{\sqrt {-c^2 f g}}\right )\right ) \log \left (\frac {\sqrt {2} e^{-i \arctan (c x)} \sqrt {-c^2 f g}}{\sqrt {-c^2 f+g} \sqrt {-c^2 f-g+\left (-c^2 f+g\right ) \cos (2 \arctan (c x))}}\right )+\left (\arccos \left (\frac {c^2 f+g}{-c^2 f+g}\right )+2 i \text {arctanh}\left (\frac {\sqrt {-c^2 f g}}{c g x}\right )-2 i \text {arctanh}\left (\frac {c g x}{\sqrt {-c^2 f g}}\right )\right ) \log \left (\frac {\sqrt {2} e^{i \arctan (c x)} \sqrt {-c^2 f g}}{\sqrt {-c^2 f+g} \sqrt {-c^2 f-g+\left (-c^2 f+g\right ) \cos (2 \arctan (c x))}}\right )+i \left (-\operatorname {PolyLog}\left (2,\frac {\left (c^2 f+g-2 i \sqrt {-c^2 f g}\right ) \left (c^2 f+c \sqrt {-c^2 f g} x\right )}{\left (c^2 f-g\right ) \left (c^2 f-c \sqrt {-c^2 f g} x\right )}\right )+\operatorname {PolyLog}\left (2,\frac {\left (c^2 f+g+2 i \sqrt {-c^2 f g}\right ) \left (c^2 f+c \sqrt {-c^2 f g} x\right )}{\left (c^2 f-g\right ) \left (c^2 f-c \sqrt {-c^2 f g} x\right )}\right )\right )\right )}{\sqrt {-c^2 f g}}\right )}{2 c} \]

[In]

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

[Out]

a*d*x - 2*a*e*x + b*d*x*ArcTan[c*x] + (2*a*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[g] - (b*d*Log[1 + c^2*x
^2])/(2*c) + a*e*x*Log[f + g*x^2] + b*e*(x*ArcTan[c*x] - Log[1 + c^2*x^2]/(2*c))*Log[f + g*x^2] + (b*e*g*(((-L
og[(-I)/c + x] - Log[I/c + x] + Log[1 + c^2*x^2])*Log[f + g*x^2])/(2*g) + (Log[(-I)/c + x]*Log[1 - (Sqrt[g]*((
-I)/c + x))/((-I)*Sqrt[f] - (I*Sqrt[g])/c)] + PolyLog[2, (Sqrt[g]*((-I)/c + x))/((-I)*Sqrt[f] - (I*Sqrt[g])/c)
])/(2*g) + (Log[(-I)/c + x]*Log[1 - (Sqrt[g]*((-I)/c + x))/(I*Sqrt[f] - (I*Sqrt[g])/c)] + PolyLog[2, (Sqrt[g]*
((-I)/c + x))/(I*Sqrt[f] - (I*Sqrt[g])/c)])/(2*g) + (Log[I/c + x]*Log[1 - (Sqrt[g]*(I/c + x))/((-I)*Sqrt[f] +
(I*Sqrt[g])/c)] + PolyLog[2, (Sqrt[g]*(I/c + x))/((-I)*Sqrt[f] + (I*Sqrt[g])/c)])/(2*g) + (Log[I/c + x]*Log[1
- (Sqrt[g]*(I/c + x))/(I*Sqrt[f] + (I*Sqrt[g])/c)] + PolyLog[2, (Sqrt[g]*(I/c + x))/(I*Sqrt[f] + (I*Sqrt[g])/c
)])/(2*g)))/c - (b*e*(4*c*x*ArcTan[c*x] + 4*Log[1/Sqrt[1 + c^2*x^2]] + (c^2*f*(4*ArcTan[c*x]*ArcTanh[Sqrt[-(c^
2*f*g)]/(c*g*x)] - 2*ArcCos[(c^2*f + g)/(-(c^2*f) + g)]*ArcTanh[(c*g*x)/Sqrt[-(c^2*f*g)]] - (ArcCos[(c^2*f + g
)/(-(c^2*f) + g)] - (2*I)*ArcTanh[(c*g*x)/Sqrt[-(c^2*f*g)]])*Log[(-2*c^2*f*(I*g + Sqrt[-(c^2*f*g)])*(-I + c*x)
)/((c^2*f - g)*(c^2*f - c*Sqrt[-(c^2*f*g)]*x))] - (ArcCos[(c^2*f + g)/(-(c^2*f) + g)] + (2*I)*ArcTanh[(c*g*x)/
Sqrt[-(c^2*f*g)]])*Log[((2*I)*c^2*f*(g + I*Sqrt[-(c^2*f*g)])*(I + c*x))/((c^2*f - g)*(c^2*f - c*Sqrt[-(c^2*f*g
)]*x))] + (ArcCos[(c^2*f + g)/(-(c^2*f) + g)] - (2*I)*ArcTanh[Sqrt[-(c^2*f*g)]/(c*g*x)] + (2*I)*ArcTanh[(c*g*x
)/Sqrt[-(c^2*f*g)]])*Log[(Sqrt[2]*Sqrt[-(c^2*f*g)])/(E^(I*ArcTan[c*x])*Sqrt[-(c^2*f) + g]*Sqrt[-(c^2*f) - g +
(-(c^2*f) + g)*Cos[2*ArcTan[c*x]]])] + (ArcCos[(c^2*f + g)/(-(c^2*f) + g)] + (2*I)*ArcTanh[Sqrt[-(c^2*f*g)]/(c
*g*x)] - (2*I)*ArcTanh[(c*g*x)/Sqrt[-(c^2*f*g)]])*Log[(Sqrt[2]*E^(I*ArcTan[c*x])*Sqrt[-(c^2*f*g)])/(Sqrt[-(c^2
*f) + g]*Sqrt[-(c^2*f) - g + (-(c^2*f) + g)*Cos[2*ArcTan[c*x]]])] + I*(-PolyLog[2, ((c^2*f + g - (2*I)*Sqrt[-(
c^2*f*g)])*(c^2*f + c*Sqrt[-(c^2*f*g)]*x))/((c^2*f - g)*(c^2*f - c*Sqrt[-(c^2*f*g)]*x))] + PolyLog[2, ((c^2*f
+ g + (2*I)*Sqrt[-(c^2*f*g)])*(c^2*f + c*Sqrt[-(c^2*f*g)]*x))/((c^2*f - g)*(c^2*f - c*Sqrt[-(c^2*f*g)]*x))])))
/Sqrt[-(c^2*f*g)]))/(2*c)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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