3.261 \(\int \frac{\log (c (d+e x^3)^p)}{f+g x^2} \, dx\)

Optimal. Leaf size=749 \[ \frac{i p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt [3]{d} \sqrt{g}+i \sqrt [3]{e} \sqrt{f}\right )}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{i p \text{PolyLog}\left (2,1+\frac{2 i \sqrt{f} \sqrt{g} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt [6]{-1} \sqrt [3]{d} \sqrt{g}+\sqrt [3]{e} \sqrt{f}\right )}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{i p \text{PolyLog}\left (2,1-\frac{2 (-1)^{5/6} \sqrt{f} \sqrt{g} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left ((-1)^{5/6} \sqrt [3]{d} \sqrt{g}+\sqrt [3]{e} \sqrt{f}\right )}\right )}{2 \sqrt{f} \sqrt{g}}-\frac{3 i p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt [3]{d} \sqrt{g}+i \sqrt [3]{e} \sqrt{f}\right )}\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 i \sqrt{f} \sqrt{g} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt [6]{-1} \sqrt [3]{d} \sqrt{g}+\sqrt [3]{e} \sqrt{f}\right )}\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 (-1)^{5/6} \sqrt{f} \sqrt{g} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left ((-1)^{5/6} \sqrt [3]{d} \sqrt{g}+\sqrt [3]{e} \sqrt{f}\right )}\right )}{\sqrt{f} \sqrt{g}}+\frac{3 p \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} \sqrt{g}} \]

[Out]

(3*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(Sqrt[f]*Sqrt[g]) - (p*ArcTan[(Sqrt
[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(d^(1/3) + e^(1/3)*x))/((I*e^(1/3)*Sqrt[f] + d^(1/3)*Sqrt[g])*(Sqrt[f]
- I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[((-2*I)*Sqrt[f]*Sqrt[g]*((-1)^(2/3)*d
^(1/3) + e^(1/3)*x))/((e^(1/3)*Sqrt[f] + (-1)^(1/6)*d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[
g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*(-1)^(5/6)*Sqrt[f]*Sqrt[g]*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((e^(
1/3)*Sqrt[f] + (-1)^(5/6)*d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g]) + (ArcTan[(Sqrt[g]*x)/
Sqrt[f]]*Log[c*(d + e*x^3)^p])/(Sqrt[f]*Sqrt[g]) - (((3*I)/2)*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g
]*x)])/(Sqrt[f]*Sqrt[g]) + ((I/2)*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(d^(1/3) + e^(1/3)*x))/((I*e^(1/3)*Sqrt[
f] + d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g]) + ((I/2)*p*PolyLog[2, 1 + ((2*I)*Sqrt[f]*Sq
rt[g]*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((e^(1/3)*Sqrt[f] + (-1)^(1/6)*d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x
))])/(Sqrt[f]*Sqrt[g]) + ((I/2)*p*PolyLog[2, 1 - (2*(-1)^(5/6)*Sqrt[f]*Sqrt[g]*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x
))/((e^(1/3)*Sqrt[f] + (-1)^(5/6)*d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g])

________________________________________________________________________________________

Rubi [A]  time = 0.930986, antiderivative size = 749, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {205, 2470, 12, 260, 6725, 4856, 2402, 2315, 2447} \[ \frac{i p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt [3]{d} \sqrt{g}+i \sqrt [3]{e} \sqrt{f}\right )}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{i p \text{PolyLog}\left (2,1+\frac{2 i \sqrt{f} \sqrt{g} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt [6]{-1} \sqrt [3]{d} \sqrt{g}+\sqrt [3]{e} \sqrt{f}\right )}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{i p \text{PolyLog}\left (2,1-\frac{2 (-1)^{5/6} \sqrt{f} \sqrt{g} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left ((-1)^{5/6} \sqrt [3]{d} \sqrt{g}+\sqrt [3]{e} \sqrt{f}\right )}\right )}{2 \sqrt{f} \sqrt{g}}-\frac{3 i p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt [3]{d} \sqrt{g}+i \sqrt [3]{e} \sqrt{f}\right )}\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 i \sqrt{f} \sqrt{g} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt [6]{-1} \sqrt [3]{d} \sqrt{g}+\sqrt [3]{e} \sqrt{f}\right )}\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 (-1)^{5/6} \sqrt{f} \sqrt{g} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left ((-1)^{5/6} \sqrt [3]{d} \sqrt{g}+\sqrt [3]{e} \sqrt{f}\right )}\right )}{\sqrt{f} \sqrt{g}}+\frac{3 p \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} \sqrt{g}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(Sqrt[f]*Sqrt[g]) - (p*ArcTan[(Sqrt
[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(d^(1/3) + e^(1/3)*x))/((I*e^(1/3)*Sqrt[f] + d^(1/3)*Sqrt[g])*(Sqrt[f]
- I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[((-2*I)*Sqrt[f]*Sqrt[g]*((-1)^(2/3)*d
^(1/3) + e^(1/3)*x))/((e^(1/3)*Sqrt[f] + (-1)^(1/6)*d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[
g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*(-1)^(5/6)*Sqrt[f]*Sqrt[g]*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((e^(
1/3)*Sqrt[f] + (-1)^(5/6)*d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g]) + (ArcTan[(Sqrt[g]*x)/
Sqrt[f]]*Log[c*(d + e*x^3)^p])/(Sqrt[f]*Sqrt[g]) - (((3*I)/2)*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g
]*x)])/(Sqrt[f]*Sqrt[g]) + ((I/2)*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(d^(1/3) + e^(1/3)*x))/((I*e^(1/3)*Sqrt[
f] + d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g]) + ((I/2)*p*PolyLog[2, 1 + ((2*I)*Sqrt[f]*Sq
rt[g]*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((e^(1/3)*Sqrt[f] + (-1)^(1/6)*d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x
))])/(Sqrt[f]*Sqrt[g]) + ((I/2)*p*PolyLog[2, 1 - (2*(-1)^(5/6)*Sqrt[f]*Sqrt[g]*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x
))/((e^(1/3)*Sqrt[f] + (-1)^(5/6)*d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g])

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 2470

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 12

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

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 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 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{\log \left (c \left (d+e x^3\right )^p\right )}{f+g x^2} \, dx &=\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt{f} \sqrt{g}}-(3 e p) \int \frac{x^2 \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} \sqrt{g} \left (d+e x^3\right )} \, dx\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt{f} \sqrt{g}}-\frac{(3 e p) \int \frac{x^2 \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{d+e x^3} \, dx}{\sqrt{f} \sqrt{g}}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt{f} \sqrt{g}}-\frac{(3 e p) \int \left (\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx}{\sqrt{f} \sqrt{g}}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt{f} \sqrt{g}}-\frac{\left (\sqrt [3]{e} p\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt{f} \sqrt{g}}-\frac{\left (\sqrt [3]{e} p\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt{f} \sqrt{g}}-\frac{\left (\sqrt [3]{e} p\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt{f} \sqrt{g}}\\ &=\frac{3 p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (i \sqrt [3]{e} \sqrt{f}+\sqrt [3]{d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 i \sqrt{f} \sqrt{g} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\sqrt [3]{e} \sqrt{f}+\sqrt [6]{-1} \sqrt [3]{d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 (-1)^{5/6} \sqrt{f} \sqrt{g} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (\sqrt [3]{e} \sqrt{f}+(-1)^{5/6} \sqrt [3]{d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{f} \sqrt{g}}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt{f} \sqrt{g}}-3 \frac{p \int \frac{\log \left (\frac{2}{1-\frac{i \sqrt{g} x}{\sqrt{f}}}\right )}{1+\frac{g x^2}{f}} \, dx}{f}+\frac{p \int \frac{\log \left (\frac{2 \sqrt{g} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt{f} \left (i \sqrt [3]{e}+\frac{\sqrt [3]{d} \sqrt{g}}{\sqrt{f}}\right ) \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right )}\right )}{1+\frac{g x^2}{f}} \, dx}{f}+\frac{p \int \frac{\log \left (\frac{2 \sqrt{g} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt{f} \left (i \sqrt [3]{e}-\frac{\sqrt [3]{-1} \sqrt [3]{d} \sqrt{g}}{\sqrt{f}}\right ) \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right )}\right )}{1+\frac{g x^2}{f}} \, dx}{f}+\frac{p \int \frac{\log \left (\frac{2 \sqrt{g} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt{f} \left (i \sqrt [3]{e}+\frac{(-1)^{2/3} \sqrt [3]{d} \sqrt{g}}{\sqrt{f}}\right ) \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right )}\right )}{1+\frac{g x^2}{f}} \, dx}{f}\\ &=\frac{3 p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (i \sqrt [3]{e} \sqrt{f}+\sqrt [3]{d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 i \sqrt{f} \sqrt{g} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\sqrt [3]{e} \sqrt{f}+\sqrt [6]{-1} \sqrt [3]{d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 (-1)^{5/6} \sqrt{f} \sqrt{g} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (\sqrt [3]{e} \sqrt{f}+(-1)^{5/6} \sqrt [3]{d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{f} \sqrt{g}}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt{f} \sqrt{g}}+\frac{i p \text{Li}_2\left (1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (i \sqrt [3]{e} \sqrt{f}+\sqrt [3]{d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{i p \text{Li}_2\left (1+\frac{2 i \sqrt{f} \sqrt{g} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\sqrt [3]{e} \sqrt{f}+\sqrt [6]{-1} \sqrt [3]{d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{i p \text{Li}_2\left (1-\frac{2 (-1)^{5/6} \sqrt{f} \sqrt{g} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (\sqrt [3]{e} \sqrt{f}+(-1)^{5/6} \sqrt [3]{d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{f} \sqrt{g}}-3 \frac{(i p) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{i \sqrt{g} x}{\sqrt{f}}}\right )}{\sqrt{f} \sqrt{g}}\\ &=\frac{3 p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (i \sqrt [3]{e} \sqrt{f}+\sqrt [3]{d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 i \sqrt{f} \sqrt{g} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\sqrt [3]{e} \sqrt{f}+\sqrt [6]{-1} \sqrt [3]{d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 (-1)^{5/6} \sqrt{f} \sqrt{g} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (\sqrt [3]{e} \sqrt{f}+(-1)^{5/6} \sqrt [3]{d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{f} \sqrt{g}}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt{f} \sqrt{g}}-\frac{3 i p \text{Li}_2\left (1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{i p \text{Li}_2\left (1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (i \sqrt [3]{e} \sqrt{f}+\sqrt [3]{d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{i p \text{Li}_2\left (1+\frac{2 i \sqrt{f} \sqrt{g} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\sqrt [3]{e} \sqrt{f}+\sqrt [6]{-1} \sqrt [3]{d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{i p \text{Li}_2\left (1-\frac{2 (-1)^{5/6} \sqrt{f} \sqrt{g} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (\sqrt [3]{e} \sqrt{f}+(-1)^{5/6} \sqrt [3]{d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{f} \sqrt{g}}\\ \end{align*}

Mathematica [A]  time = 0.596954, size = 867, normalized size = 1.16 \[ \frac{-p \log \left (\frac{\sqrt{g} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\sqrt [3]{e} \sqrt{-f}+\sqrt [3]{d} \sqrt{g}}\right ) \log \left (\sqrt{-f}-\sqrt{g} x\right )-p \log \left (\frac{\sqrt{g} \left (\sqrt [3]{e} x-\sqrt [3]{-1} \sqrt [3]{d}\right )}{\sqrt [3]{e} \sqrt{-f}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt{g}}\right ) \log \left (\sqrt{-f}-\sqrt{g} x\right )-p \log \left (\frac{\sqrt{g} \left (\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}\right )}{\sqrt [3]{e} \sqrt{-f}+(-1)^{2/3} \sqrt [3]{d} \sqrt{g}}\right ) \log \left (\sqrt{-f}-\sqrt{g} x\right )+\log \left (c \left (e x^3+d\right )^p\right ) \log \left (\sqrt{-f}-\sqrt{g} x\right )+p \log \left (-\frac{\sqrt{g} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\sqrt [3]{e} \sqrt{-f}-\sqrt [3]{d} \sqrt{g}}\right ) \log \left (\sqrt{g} x+\sqrt{-f}\right )+p \log \left (\frac{\sqrt{g} \left (\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}\right )}{(-1)^{2/3} \sqrt [3]{d} \sqrt{g}-\sqrt [3]{e} \sqrt{-f}}\right ) \log \left (\sqrt{g} x+\sqrt{-f}\right )+p \log \left (\frac{\sqrt [3]{-1} \sqrt{g} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}{\sqrt [3]{e} \sqrt{-f}+\sqrt [3]{-1} \sqrt [3]{d} \sqrt{g}}\right ) \log \left (\sqrt{g} x+\sqrt{-f}\right )-\log \left (\sqrt{g} x+\sqrt{-f}\right ) \log \left (c \left (e x^3+d\right )^p\right )-p \text{PolyLog}\left (2,\frac{\sqrt [3]{e} \left (\sqrt{-f}-\sqrt{g} x\right )}{\sqrt [3]{e} \sqrt{-f}+\sqrt [3]{d} \sqrt{g}}\right )-p \text{PolyLog}\left (2,\frac{\sqrt [3]{e} \left (\sqrt{-f}-\sqrt{g} x\right )}{\sqrt [3]{e} \sqrt{-f}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt{g}}\right )-p \text{PolyLog}\left (2,\frac{\sqrt [3]{e} \left (\sqrt{-f}-\sqrt{g} x\right )}{\sqrt [3]{e} \sqrt{-f}+(-1)^{2/3} \sqrt [3]{d} \sqrt{g}}\right )+p \text{PolyLog}\left (2,\frac{\sqrt [3]{e} \left (\sqrt{g} x+\sqrt{-f}\right )}{\sqrt [3]{e} \sqrt{-f}-\sqrt [3]{d} \sqrt{g}}\right )+p \text{PolyLog}\left (2,\frac{\sqrt [3]{e} \left (\sqrt{g} x+\sqrt{-f}\right )}{\sqrt [3]{e} \sqrt{-f}+\sqrt [3]{-1} \sqrt [3]{d} \sqrt{g}}\right )+p \text{PolyLog}\left (2,\frac{\sqrt [3]{e} \left (\sqrt{g} x+\sqrt{-f}\right )}{\sqrt [3]{e} \sqrt{-f}-(-1)^{2/3} \sqrt [3]{d} \sqrt{g}}\right )}{2 \sqrt{-f} \sqrt{g}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(p*Log[(Sqrt[g]*(d^(1/3) + e^(1/3)*x))/(e^(1/3)*Sqrt[-f] + d^(1/3)*Sqrt[g])]*Log[Sqrt[-f] - Sqrt[g]*x]) - p*
Log[(Sqrt[g]*(-((-1)^(1/3)*d^(1/3)) + e^(1/3)*x))/(e^(1/3)*Sqrt[-f] - (-1)^(1/3)*d^(1/3)*Sqrt[g])]*Log[Sqrt[-f
] - Sqrt[g]*x] - p*Log[(Sqrt[g]*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/(e^(1/3)*Sqrt[-f] + (-1)^(2/3)*d^(1/3)*Sqrt[
g])]*Log[Sqrt[-f] - Sqrt[g]*x] + p*Log[-((Sqrt[g]*(d^(1/3) + e^(1/3)*x))/(e^(1/3)*Sqrt[-f] - d^(1/3)*Sqrt[g]))
]*Log[Sqrt[-f] + Sqrt[g]*x] + p*Log[(Sqrt[g]*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/(-(e^(1/3)*Sqrt[-f]) + (-1)^(2/
3)*d^(1/3)*Sqrt[g])]*Log[Sqrt[-f] + Sqrt[g]*x] + p*Log[((-1)^(1/3)*Sqrt[g]*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/(
e^(1/3)*Sqrt[-f] + (-1)^(1/3)*d^(1/3)*Sqrt[g])]*Log[Sqrt[-f] + Sqrt[g]*x] + Log[Sqrt[-f] - Sqrt[g]*x]*Log[c*(d
+ e*x^3)^p] - Log[Sqrt[-f] + Sqrt[g]*x]*Log[c*(d + e*x^3)^p] - p*PolyLog[2, (e^(1/3)*(Sqrt[-f] - Sqrt[g]*x))/
(e^(1/3)*Sqrt[-f] + d^(1/3)*Sqrt[g])] - p*PolyLog[2, (e^(1/3)*(Sqrt[-f] - Sqrt[g]*x))/(e^(1/3)*Sqrt[-f] - (-1)
^(1/3)*d^(1/3)*Sqrt[g])] - p*PolyLog[2, (e^(1/3)*(Sqrt[-f] - Sqrt[g]*x))/(e^(1/3)*Sqrt[-f] + (-1)^(2/3)*d^(1/3
)*Sqrt[g])] + p*PolyLog[2, (e^(1/3)*(Sqrt[-f] + Sqrt[g]*x))/(e^(1/3)*Sqrt[-f] - d^(1/3)*Sqrt[g])] + p*PolyLog[
2, (e^(1/3)*(Sqrt[-f] + Sqrt[g]*x))/(e^(1/3)*Sqrt[-f] + (-1)^(1/3)*d^(1/3)*Sqrt[g])] + p*PolyLog[2, (e^(1/3)*(
Sqrt[-f] + Sqrt[g]*x))/(e^(1/3)*Sqrt[-f] - (-1)^(2/3)*d^(1/3)*Sqrt[g])])/(2*Sqrt[-f]*Sqrt[g])

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Maple [C]  time = 0.716, size = 327, normalized size = 0.4 \begin{align*}{(\ln \left ( \left ( e{x}^{3}+d \right ) ^{p} \right ) -p\ln \left ( e{x}^{3}+d \right ) )\arctan \left ({gx{\frac{1}{\sqrt{fg}}}} \right ){\frac{1}{\sqrt{fg}}}}+{\frac{p}{2\,g}\sum _{{\it \_alpha}={\it RootOf} \left ( g{{\it \_Z}}^{2}+f \right ) } \left ({\frac{1}{{\it \_alpha}} \left ( \ln \left ( x-{\it \_alpha} \right ) \ln \left ( e{x}^{3}+d \right ) -\sum _{{\it \_R1}={\it RootOf} \left ({{\it \_Z}}^{3}eg+3\,{\it \_alpha}\,{{\it \_Z}}^{2}eg-3\,{\it \_Z}\,ef-ef{\it \_alpha}+dg \right ) }\ln \left ( x-{\it \_alpha} \right ) \ln \left ({\frac{{\it \_R1}-x+{\it \_alpha}}{{\it \_R1}}} \right ) +{\it dilog} \left ({\frac{{\it \_R1}-x+{\it \_alpha}}{{\it \_R1}}} \right ) \right ) } \right ) }+{{\frac{i}{2}}\pi \,{\it csgn} \left ( i \left ( e{x}^{3}+d \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( e{x}^{3}+d \right ) ^{p} \right ) \right ) ^{2}\arctan \left ({gx{\frac{1}{\sqrt{fg}}}} \right ){\frac{1}{\sqrt{fg}}}}-{{\frac{i}{2}}\pi \,{\it csgn} \left ( i \left ( e{x}^{3}+d \right ) ^{p} \right ){\it csgn} \left ( ic \left ( e{x}^{3}+d \right ) ^{p} \right ){\it csgn} \left ( ic \right ) \arctan \left ({gx{\frac{1}{\sqrt{fg}}}} \right ){\frac{1}{\sqrt{fg}}}}-{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ic \left ( e{x}^{3}+d \right ) ^{p} \right ) \right ) ^{3}\arctan \left ({gx{\frac{1}{\sqrt{fg}}}} \right ){\frac{1}{\sqrt{fg}}}}+{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ic \left ( e{x}^{3}+d \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) \arctan \left ({gx{\frac{1}{\sqrt{fg}}}} \right ){\frac{1}{\sqrt{fg}}}}+{\ln \left ( c \right ) \arctan \left ({gx{\frac{1}{\sqrt{fg}}}} \right ){\frac{1}{\sqrt{fg}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(ln((e*x^3+d)^p)-p*ln(e*x^3+d))/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))+1/2*p/g*sum(1/_alpha*(ln(x-_alpha)*ln(e*x^
3+d)-sum(ln(x-_alpha)*ln((_R1-x+_alpha)/_R1)+dilog((_R1-x+_alpha)/_R1),_R1=RootOf(_Z^3*e*g+3*_Z^2*_alpha*e*g-3
*_Z*e*f-_alpha*e*f+d*g))),_alpha=RootOf(_Z^2*g+f))+1/2*I/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*Pi*csgn(I*(e*x^3+
d)^p)*csgn(I*c*(e*x^3+d)^p)^2-1/2*I/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*Pi*csgn(I*(e*x^3+d)^p)*csgn(I*c*(e*x^3
+d)^p)*csgn(I*c)-1/2*I/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*Pi*csgn(I*c*(e*x^3+d)^p)^3+1/2*I/(f*g)^(1/2)*arctan
(x*g/(f*g)^(1/2))*Pi*csgn(I*c*(e*x^3+d)^p)^2*csgn(I*c)+1/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*ln(c)

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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(log(c*(e*x^3+d)^p)/(g*x^2+f),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{\log \left ({\left (e x^{3} + d\right )}^{p} c\right )}{g x^{2} + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log((e*x^3 + d)^p*c)/(g*x^2 + f), 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(ln(c*(e*x**3+d)**p)/(g*x**2+f),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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