\(\int \frac {\log (c (d+e x^3)^p)}{f+g x^2} \, dx\) [261]

Optimal result

Integrand size = 22, antiderivative size = 749 \[ \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{f+g x^2} \, dx=\frac {3 p \arctan \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 \arctan \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 \arctan \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 \arctan \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 {\arctan \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 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1-\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 \operatorname {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 [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 \operatorname {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 [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}} \]

[Out]

arctan(x*g^(1/2)/f^(1/2))*ln(c*(e*x^3+d)^p)/f^(1/2)/g^(1/2)+3*p*arctan(x*g^(1/2)/f^(1/2))*ln(2*f^(1/2)/(f^(1/2
)-I*x*g^(1/2)))/f^(1/2)/g^(1/2)-p*arctan(x*g^(1/2)/f^(1/2))*ln(2*(d^(1/3)+e^(1/3)*x)*f^(1/2)*g^(1/2)/(I*e^(1/3
)*f^(1/2)+d^(1/3)*g^(1/2))/(f^(1/2)-I*x*g^(1/2)))/f^(1/2)/g^(1/2)-p*arctan(x*g^(1/2)/f^(1/2))*ln(-2*I*((-1)^(2
/3)*d^(1/3)+e^(1/3)*x)*f^(1/2)*g^(1/2)/(e^(1/3)*f^(1/2)+(-1)^(1/6)*d^(1/3)*g^(1/2))/(f^(1/2)-I*x*g^(1/2)))/f^(
1/2)/g^(1/2)-p*arctan(x*g^(1/2)/f^(1/2))*ln(2*(-1)^(5/6)*(d^(1/3)+(-1)^(2/3)*e^(1/3)*x)*f^(1/2)*g^(1/2)/(e^(1/
3)*f^(1/2)+(-1)^(5/6)*d^(1/3)*g^(1/2))/(f^(1/2)-I*x*g^(1/2)))/f^(1/2)/g^(1/2)-3/2*I*p*polylog(2,1-2*f^(1/2)/(f
^(1/2)-I*x*g^(1/2)))/f^(1/2)/g^(1/2)+1/2*I*p*polylog(2,1-2*(d^(1/3)+e^(1/3)*x)*f^(1/2)*g^(1/2)/(I*e^(1/3)*f^(1
/2)+d^(1/3)*g^(1/2))/(f^(1/2)-I*x*g^(1/2)))/f^(1/2)/g^(1/2)+1/2*I*p*polylog(2,1+2*I*((-1)^(2/3)*d^(1/3)+e^(1/3
)*x)*f^(1/2)*g^(1/2)/(e^(1/3)*f^(1/2)+(-1)^(1/6)*d^(1/3)*g^(1/2))/(f^(1/2)-I*x*g^(1/2)))/f^(1/2)/g^(1/2)+1/2*I
*p*polylog(2,1-2*(-1)^(5/6)*(d^(1/3)+(-1)^(2/3)*e^(1/3)*x)*f^(1/2)*g^(1/2)/(e^(1/3)*f^(1/2)+(-1)^(5/6)*d^(1/3)
*g^(1/2))/(f^(1/2)-I*x*g^(1/2)))/f^(1/2)/g^(1/2)

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 749, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {211, 2520, 12, 266, 6857, 4966, 2449, 2352, 2497} \[ \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{f+g x^2} \, dx=\frac {\arctan \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 \arctan \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 \arctan \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 \arctan \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 \arctan \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 {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [3]{e} x+\sqrt [3]{d}\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 \operatorname {PolyLog}\left (2,\frac {2 i \sqrt {f} \sqrt {g} \left (\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}\right )}{\left (\sqrt [3]{e} \sqrt {f}+\sqrt [6]{-1} \sqrt [3]{d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+1\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 (-1)^{5/6} \sqrt {f} \sqrt {g} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\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}}-\frac {3 i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 \sqrt {f} \sqrt {g}} \]

[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 12

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

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 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 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 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 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 6857

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]

Rubi steps \begin{align*} \text {integral}& = \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) \text {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] (verified)

Time = 0.50 (sec) , antiderivative size = 867, normalized size of antiderivative = 1.16 \[ \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{f+g x^2} \, dx=\frac {-p \log \left (\frac {\sqrt {g} \left (\sqrt [3]{d}+\sqrt [3]{e} x\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]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\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 ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e} \sqrt {-f}+(-1)^{2/3} \sqrt [3]{d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )+p \log \left (-\frac {\sqrt {g} \left (\sqrt [3]{d}+\sqrt [3]{e} x\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 ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{e} \sqrt {-f}+(-1)^{2/3} \sqrt [3]{d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )+p \log \left (\frac {\sqrt [3]{-1} \sqrt {g} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{e} \sqrt {-f}+\sqrt [3]{-1} \sqrt [3]{d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )+\log \left (\sqrt {-f}-\sqrt {g} x\right ) \log \left (c \left (d+e x^3\right )^p\right )-\log \left (\sqrt {-f}+\sqrt {g} x\right ) \log \left (c \left (d+e x^3\right )^p\right )-p \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 )}{2 \sqrt {-f} \sqrt {g}} \]

[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])

Maple [C] (warning: unable to verify)

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

Time = 1.45 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.36

method	result	size
risch	\(\frac {\left (\ln \left (\left (e \,x^{3}+d \right )^{p}\right )-p \ln \left (e \,x^{3}+d \right )\right ) \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{\sqrt {f g}}+\frac {p \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (g \,\textit {\_Z}^{2}+f \right )}{\sum }\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (e \,x^{3}+d \right )-\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} e g +3 \underline {\hspace {1.25 ex}}\alpha \,\textit {\_Z}^{2} e g -3 \textit {\_Z} e f -e f \underline {\hspace {1.25 ex}}\alpha +d g \right )}{\sum }\left (\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\textit {\_R1} -x +\underline {\hspace {1.25 ex}}\alpha }{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x +\underline {\hspace {1.25 ex}}\alpha }{\textit {\_R1}}\right )\right )\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{2 g}+\frac {\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{3}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{3}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{\sqrt {f g}}\)	\(272\)

[In]

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

[Out]

(ln((e*x^3+d)^p)-p*ln(e*x^3+d))/(f*g)^(1/2)*arctan(g*x/(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*Pi*csgn(I*(e*x^3+d)^p)*csgn(I*c*(e*x^3+d)^p)^2-1/2*I
*Pi*csgn(I*(e*x^3+d)^p)*csgn(I*c*(e*x^3+d)^p)*csgn(I*c)-1/2*I*Pi*csgn(I*c*(e*x^3+d)^p)^3+1/2*I*Pi*csgn(I*c*(e*
x^3+d)^p)^2*csgn(I*c)+ln(c))/(f*g)^(1/2)*arctan(g*x/(f*g)^(1/2))

Fricas [F]

\[ \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{f+g x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{f+g x^2} \, dx=\int \frac {\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}{g\,x^2+f} \,d x \]

[In]

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

[Out]

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