3.167 \(\int (f x)^{-1-n} \log ^2(c (d+e x^n)^p) \, dx\)

Optimal. Leaf size=124 \[ \frac{2 e p^2 x^{n+1} (f x)^{-n-1} \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{d n}-\frac{x (f x)^{-n-1} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{d n}+\frac{2 e p x^{n+1} (f x)^{-n-1} \log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d n} \]

[Out]

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

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Rubi [A]  time = 0.111278, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {2456, 2454, 2397, 2394, 2315} \[ \frac{2 e p^2 x^{n+1} (f x)^{-n-1} \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{d n}-\frac{x (f x)^{-n-1} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{d n}+\frac{2 e p x^{n+1} (f x)^{-n-1} \log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d n} \]

Antiderivative was successfully verified.

[In]

Int[(f*x)^(-1 - n)*Log[c*(d + e*x^n)^p]^2,x]

[Out]

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

Rule 2456

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

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2397

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

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]

Rubi steps

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

Mathematica [A]  time = 0.0891685, size = 148, normalized size = 1.19 \[ -\frac{(f x)^{-n} \left (2 e p^2 x^n \text{PolyLog}\left (2,\frac{d x^{-n}}{e}+1\right )+d \log ^2\left (c \left (d+e x^n\right )^p\right )+2 e p x^n \log \left (-d x^{-n}-e\right ) \log \left (c \left (d+e x^n\right )^p\right )-e p^2 x^n \log ^2\left (-d x^{-n}-e\right )+2 e p^2 x^n \log \left (-\frac{d x^{-n}}{e}\right ) \log \left (-d x^{-n}-e\right )\right )}{d f n} \]

Antiderivative was successfully verified.

[In]

Integrate[(f*x)^(-1 - n)*Log[c*(d + e*x^n)^p]^2,x]

[Out]

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

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Maple [F]  time = 2.043, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{-1-n} \left ( \ln \left ( c \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^(-1-n)*ln(c*(d+e*x^n)^p)^2,x)

[Out]

int((f*x)^(-1-n)*ln(c*(d+e*x^n)^p)^2,x)

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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((f*x)^(-1-n)*log(c*(d+e*x^n)^p)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.44405, size = 446, normalized size = 3.6 \begin{align*} -\frac{2 \, e f^{-n - 1} n p^{2} x^{n} \log \left (x\right ) \log \left (\frac{e x^{n} + d}{d}\right ) - 2 \, e f^{-n - 1} n p x^{n} \log \left (c\right ) \log \left (x\right ) + 2 \, e f^{-n - 1} p^{2} x^{n}{\rm Li}_2\left (-\frac{e x^{n} + d}{d} + 1\right ) + d f^{-n - 1} \log \left (c\right )^{2} +{\left (e f^{-n - 1} p^{2} x^{n} + d f^{-n - 1} p^{2}\right )} \log \left (e x^{n} + d\right )^{2} + 2 \,{\left (d f^{-n - 1} p \log \left (c\right ) -{\left (e n p^{2} \log \left (x\right ) - e p \log \left (c\right )\right )} f^{-n - 1} x^{n}\right )} \log \left (e x^{n} + d\right )}{d n x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^(-1-n)*log(c*(d+e*x^n)^p)^2,x, algorithm="fricas")

[Out]

-(2*e*f^(-n - 1)*n*p^2*x^n*log(x)*log((e*x^n + d)/d) - 2*e*f^(-n - 1)*n*p*x^n*log(c)*log(x) + 2*e*f^(-n - 1)*p
^2*x^n*dilog(-(e*x^n + d)/d + 1) + d*f^(-n - 1)*log(c)^2 + (e*f^(-n - 1)*p^2*x^n + d*f^(-n - 1)*p^2)*log(e*x^n
+ d)^2 + 2*(d*f^(-n - 1)*p*log(c) - (e*n*p^2*log(x) - e*p*log(c))*f^(-n - 1)*x^n)*log(e*x^n + d))/(d*n*x^n)

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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((f*x)**(-1-n)*ln(c*(d+e*x**n)**p)**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (f x\right )^{-n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^(-1-n)*log(c*(d+e*x^n)^p)^2,x, algorithm="giac")

[Out]

integrate((f*x)^(-n - 1)*log((e*x^n + d)^p*c)^2, x)