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

Optimal. Leaf size=200 \[ \frac{e^2 p^2 x^{2 n+1} (f x)^{-2 n-1} \text{PolyLog}\left (2,\frac{d}{d+e x^n}\right )}{d^2 n}-\frac{e^2 p x^{2 n+1} (f x)^{-2 n-1} \log \left (1-\frac{d}{d+e x^n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d^2 n}-\frac{e p x^{n+1} (f x)^{-2 n-1} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d^2 n}-\frac{x (f x)^{-2 n-1} \log ^2\left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac{e^2 p^2 x^{2 n+1} \log (x) (f x)^{-2 n-1}}{d^2} \]

[Out]

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

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Rubi [A]  time = 0.316349, antiderivative size = 238, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {2456, 2454, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31} \[ -\frac{e^2 p^2 x^{2 n+1} (f x)^{-2 n-1} \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{d^2 n}+\frac{e^2 x^{2 n+1} (f x)^{-2 n-1} \log ^2\left (c \left (d+e x^n\right )^p\right )}{2 d^2 n}-\frac{e^2 p x^{2 n+1} (f x)^{-2 n-1} \log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d^2 n}-\frac{e p x^{n+1} (f x)^{-2 n-1} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d^2 n}-\frac{x (f x)^{-2 n-1} \log ^2\left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac{e^2 p^2 x^{2 n+1} \log (x) (f x)^{-2 n-1}}{d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^2*p^2*x^(1 + 2*n)*(f*x)^(-1 - 2*n)*Log[x])/d^2 - (e*p*x^(1 + n)*(f*x)^(-1 - 2*n)*(d + e*x^n)*Log[c*(d + e*x
^n)^p])/(d^2*n) - (e^2*p*x^(1 + 2*n)*(f*x)^(-1 - 2*n)*Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/(d^2*n) - (x*(f*
x)^(-1 - 2*n)*Log[c*(d + e*x^n)^p]^2)/(2*n) + (e^2*x^(1 + 2*n)*(f*x)^(-1 - 2*n)*Log[c*(d + e*x^n)^p]^2)/(2*d^2
*n) - (e^2*p^2*x^(1 + 2*n)*(f*x)^(-1 - 2*n)*PolyLog[2, 1 + (e*x^n)/d])/(d^2*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 2398

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

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2344

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2317

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

Rule 2314

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[(x*(d + e*x^r)^(q
+ 1)*(a + b*Log[c*x^n]))/d, x] - Dist[(b*n)/d, Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 31

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

Rubi steps

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

Mathematica [A]  time = 0.262026, size = 288, normalized size = 1.44 \[ \frac{(f x)^{-2 n} \left (2 e^2 p^2 x^{2 n} \text{PolyLog}\left (2,-\frac{e x^n}{d}\right )-d^2 \log ^2\left (c \left (d+e x^n\right )^p\right )+2 e^2 p x^{2 n} \log \left (e-e x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )+2 e^2 n p x^{2 n} \log (x) \left (-\log \left (c \left (d+e x^n\right )^p\right )+p \log \left (d x^{-n}+e\right )+p \log \left (\frac{e x^n}{d}+1\right )-p \log \left (e-e x^{-n}\right )+p\right )-2 d e p x^n \log \left (c \left (d+e x^n\right )^p\right )+e^2 p^2 x^{2 n} \log ^2\left (d x^{-n}+e\right )-2 e^2 p^2 x^{2 n} \log \left (e-e x^{-n}\right ) \log \left (d x^{-n}+e\right )+e^2 n^2 p^2 x^{2 n} \log ^2(x)-2 e^2 p^2 x^{2 n} \log \left (e-e x^{-n}\right )\right )}{2 d^2 f n} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 1.978, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{-1-2\,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)^(-1-2*n)*ln(c*(d+e*x^n)^p)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.23388, size = 645, normalized size = 3.22 \begin{align*} \frac{2 \, e^{2} f^{-2 \, n - 1} n p^{2} x^{2 \, n} \log \left (x\right ) \log \left (\frac{e x^{n} + d}{d}\right ) + 2 \, e^{2} f^{-2 \, n - 1} p^{2} x^{2 \, n}{\rm Li}_2\left (-\frac{e x^{n} + d}{d} + 1\right ) - 2 \, d e f^{-2 \, n - 1} p x^{n} \log \left (c\right ) - d^{2} f^{-2 \, n - 1} \log \left (c\right )^{2} + 2 \,{\left (e^{2} n p^{2} - e^{2} n p \log \left (c\right )\right )} f^{-2 \, n - 1} x^{2 \, n} \log \left (x\right ) +{\left (e^{2} f^{-2 \, n - 1} p^{2} x^{2 \, n} - d^{2} f^{-2 \, n - 1} p^{2}\right )} \log \left (e x^{n} + d\right )^{2} - 2 \,{\left (d e f^{-2 \, n - 1} p^{2} x^{n} + d^{2} f^{-2 \, n - 1} p \log \left (c\right ) +{\left (e^{2} n p^{2} \log \left (x\right ) + e^{2} p^{2} - e^{2} p \log \left (c\right )\right )} f^{-2 \, n - 1} x^{2 \, n}\right )} \log \left (e x^{n} + d\right )}{2 \, d^{2} n x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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