3.3.65 \(\int \frac {\log (c (d+\frac {e}{x^2})^p)}{f+g x^2} \, dx\) [265]

Optimal. Leaf size=597 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {2 p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}-\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}+\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}+\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}-\frac {i p \text {Li}_2\left (\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}-\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}-\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}+\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}+\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.57, antiderivative size = 597, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {211, 2520, 12, 266, 6820, 5048, 4940, 2438, 4966, 2449, 2352, 2497} \begin {gather*} \frac {i p \text {PolyLog}\left (2,1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {e} \sqrt {g}+i \sqrt {-d} \sqrt {f}\right )}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d} x+\sqrt {e}\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {e} \sqrt {g}+i \sqrt {-d} \sqrt {f}\right )}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \text {PolyLog}\left (2,-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}-\frac {i p \text {PolyLog}\left (2,\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}-\frac {i p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}+\frac {\text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}-\frac {p \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {e} \sqrt {g}+i \sqrt {-d} \sqrt {f}\right )}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d} x+\sqrt {e}\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {e} \sqrt {g}+i \sqrt {-d} \sqrt {f}\right )}\right )}{\sqrt {f} \sqrt {g}}+\frac {2 p \text {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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e/x^2)^p])/(Sqrt[f]*Sqrt[g]) + (2*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]
*(Sqrt[e] - Sqrt[-d]*x))/((I*Sqrt[-d]*Sqrt[f] - Sqrt[e]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g])
- (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(Sqrt[e] + Sqrt[-d]*x))/((I*Sqrt[-d]*Sqrt[f] + Sqrt[e]
*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g]) + (I*p*PolyLog[2, ((-I)*Sqrt[g]*x)/Sqrt[f]])/(Sqrt[f]*S
qrt[g]) - (I*p*PolyLog[2, (I*Sqrt[g]*x)/Sqrt[f]])/(Sqrt[f]*Sqrt[g]) - (I*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]*(Sqrt[e] - Sqrt[-d]*x))/((I*S
qrt[-d]*Sqrt[f] - Sqrt[e]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g]) + ((I/2)*p*PolyLog[2, 1 - (2*S
qrt[f]*Sqrt[g]*(Sqrt[e] + Sqrt[-d]*x))/((I*Sqrt[-d]*Sqrt[f] + Sqrt[e]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqr
t[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 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 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 4940

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 4966

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

Rule 5048

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

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.27, size = 706, normalized size = 1.18 \begin {gather*} \frac {\log \left (c \left (d+\frac {e}{x^2}\right )^p\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )+2 p \log \left (\frac {\sqrt {g} x}{\sqrt {-f}}\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )-p \log \left (\frac {\sqrt {g} \left (-\sqrt {e}+\sqrt {-d} x\right )}{\sqrt {-d} \sqrt {-f}-\sqrt {e} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )-p \log \left (\frac {\sqrt {g} \left (\sqrt {e}+\sqrt {-d} x\right )}{\sqrt {-d} \sqrt {-f}+\sqrt {e} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )-\log \left (c \left (d+\frac {e}{x^2}\right )^p\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )-2 p \log \left (\frac {f \sqrt {g} x}{(-f)^{3/2}}\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )+p \log \left (\frac {\sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\sqrt {-d} \sqrt {-f}+\sqrt {e} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )+p \log \left (-\frac {\sqrt {g} \left (\sqrt {e}+\sqrt {-d} x\right )}{\sqrt {-d} \sqrt {-f}-\sqrt {e} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )-p \text {Li}_2\left (\frac {\sqrt {-d} \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {-d} \sqrt {-f}-\sqrt {e} \sqrt {g}}\right )-p \text {Li}_2\left (\frac {\sqrt {-d} \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {-d} \sqrt {-f}+\sqrt {e} \sqrt {g}}\right )+p \text {Li}_2\left (\frac {\sqrt {-d} \left (\sqrt {-f}+\sqrt {g} x\right )}{\sqrt {-d} \sqrt {-f}-\sqrt {e} \sqrt {g}}\right )+p \text {Li}_2\left (\frac {\sqrt {-d} \left (\sqrt {-f}+\sqrt {g} x\right )}{\sqrt {-d} \sqrt {-f}+\sqrt {e} \sqrt {g}}\right )-2 p \text {Li}_2\left (1+\frac {\sqrt {g} x}{\sqrt {-f}}\right )+2 p \text {Li}_2\left (1+\frac {f \sqrt {g} x}{(-f)^{3/2}}\right )}{2 \sqrt {-f} \sqrt {g}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (d +\frac {e}{x^{2}}\right )^{p}\right )}{g \,x^{2}+f}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (c\,{\left (d+\frac {e}{x^2}\right )}^p\right )}{g\,x^2+f} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________