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

Optimal. Leaf size=266 \[ -\frac{p \text{PolyLog}\left (2,-\frac{\sqrt{g} \left (d+e x^n\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f n}-\frac{p \text{PolyLog}\left (2,\frac{\sqrt{g} \left (d+e x^n\right )}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 f n}+\frac{p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{f n}-\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x^n\right )}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 f n}-\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x^n\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f n}+\frac{\log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n} \]

[Out]

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

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Rubi [A]  time = 0.423664, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.407, Rules used = {2475, 266, 36, 29, 31, 2416, 2394, 2315, 260, 2393, 2391} \[ -\frac{p \text{PolyLog}\left (2,-\frac{\sqrt{g} \left (d+e x^n\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f n}-\frac{p \text{PolyLog}\left (2,\frac{\sqrt{g} \left (d+e x^n\right )}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 f n}+\frac{p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{f n}-\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x^n\right )}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 f n}-\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x^n\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f n}+\frac{\log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2475

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 266

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

Rule 36

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2394

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 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 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 2393

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 2391

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

Rubi steps

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

Mathematica [F]  time = 4.93591, size = 0, normalized size = 0. \[ \int \frac{\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{2 n}\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [C]  time = 1.066, size = 695, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/n*ln((d+e*x^n)^p)/f*ln(f+g*(x^n)^2)+1/n*ln((d+e*x^n)^p)/f*ln(x^n)-1/f*p/n*dilog((d+e*x^n)/d)-1/n*p/f*ln(x
^n)*ln((d+e*x^n)/d)+1/2/n*p/f*ln(d+e*x^n)*ln(f+g*(x^n)^2)-1/2/n*p/f*ln(d+e*x^n)*ln((e*(-f*g)^(1/2)-(d+e*x^n)*g
+d*g)/(e*(-f*g)^(1/2)+d*g))-1/2/n*p/f*ln(d+e*x^n)*ln((e*(-f*g)^(1/2)+(d+e*x^n)*g-d*g)/(e*(-f*g)^(1/2)-d*g))-1/
2/n*p/f*dilog((e*(-f*g)^(1/2)-(d+e*x^n)*g+d*g)/(e*(-f*g)^(1/2)+d*g))-1/2/n*p/f*dilog((e*(-f*g)^(1/2)+(d+e*x^n)
*g-d*g)/(e*(-f*g)^(1/2)-d*g))-1/2*I/n*Pi*csgn(I*c*(d+e*x^n)^p)^3/f*ln(x^n)+1/4*I/n*Pi*csgn(I*c*(d+e*x^n)^p)^3/
f*ln(f+g*(x^n)^2)+1/2*I/n*Pi*csgn(I*c*(d+e*x^n)^p)^2*csgn(I*c)/f*ln(x^n)+1/2*I/n*Pi*csgn(I*(d+e*x^n)^p)*csgn(I
*c*(d+e*x^n)^p)^2/f*ln(x^n)-1/2*I/n*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)*csgn(I*c)/f*ln(x^n)-1/4*I/n*P
i*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)^2/f*ln(f+g*(x^n)^2)+1/4*I/n*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x
^n)^p)*csgn(I*c)/f*ln(f+g*(x^n)^2)-1/4*I/n*Pi*csgn(I*c*(d+e*x^n)^p)^2*csgn(I*c)/f*ln(f+g*(x^n)^2)-1/2/n*ln(c)/
f*ln(f+g*(x^n)^2)+1/n*ln(c)/f*ln(x^n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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