3.2.68 \(\int (f x)^{-1-2 n} \log ^2(c (d+e x^n)^p) \, dx\) [168]

Optimal. Leaf size=200 \[ \frac {e^2 p^2 x^{1+2 n} (f x)^{-1-2 n} \log (x)}{d^2}-\frac {e p x^{1+n} (f x)^{-1-2 n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d^2 n}-\frac {x (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{2 n}-\frac {e^2 p x^{1+2 n} (f x)^{-1-2 n} \log \left (c \left (d+e x^n\right )^p\right ) \log \left (1-\frac {d}{d+e x^n}\right )}{d^2 n}+\frac {e^2 p^2 x^{1+2 n} (f x)^{-1-2 n} \text {Li}_2\left (\frac {d}{d+e x^n}\right )}{d^2 n} \]

[Out]

e^2*p^2*x^(1+2*n)*(f*x)^(-1-2*n)*ln(x)/d^2-e*p*x^(1+n)*(f*x)^(-1-2*n)*(d+e*x^n)*ln(c*(d+e*x^n)^p)/d^2/n-1/2*x*
(f*x)^(-1-2*n)*ln(c*(d+e*x^n)^p)^2/n-e^2*p*x^(1+2*n)*(f*x)^(-1-2*n)*ln(c*(d+e*x^n)^p)*ln(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 = 0.18, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2506, 2504, 2445, 2458, 2389, 2379, 2438, 2351, 31} \begin {gather*} \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} \end {gather*}

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) - (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)

Rule 31

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

Rule 2351

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 2379

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

Rule 2389

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 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 2445

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 2458

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 2504

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 2506

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])

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.20, size = 288, normalized size = 1.44 \begin {gather*} \frac {(f x)^{-2 n} \left (e^2 n^2 p^2 x^{2 n} \log ^2(x)+e^2 p^2 x^{2 n} \log ^2\left (e+d x^{-n}\right )-2 e^2 p^2 x^{2 n} \log \left (e-e x^{-n}\right )-2 e^2 p^2 x^{2 n} \log \left (e+d x^{-n}\right ) \log \left (e-e x^{-n}\right )-2 d e p x^n \log \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 )-d^2 \log ^2\left (c \left (d+e x^n\right )^p\right )+2 e^2 n p x^{2 n} \log (x) \left (p+p \log \left (e+d x^{-n}\right )-p \log \left (e-e x^{-n}\right )-\log \left (c \left (d+e x^n\right )^p\right )+p \log \left (1+\frac {e x^n}{d}\right )\right )+2 e^2 p^2 x^{2 n} \text {Li}_2\left (-\frac {e x^n}{d}\right )\right )}{2 d^2 f n} \end {gather*}

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 = 0.07, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{-1-2 n} \ln \left (c \left (d +e \,x^{n}\right )^{p}\right )^{2}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume((-2*n)-1>0)', see `assume?` fo
r more detai

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Fricas [A]
time = 0.38, size = 278, normalized size = 1.39 \begin {gather*} \frac {2 \, f^{-2 \, n - 1} n p^{2} x^{2 \, n} e^{2} \log \left (x\right ) \log \left (\frac {x^{n} e + d}{d}\right ) + 2 \, f^{-2 \, n - 1} p^{2} x^{2 \, n} {\rm Li}_2\left (-\frac {x^{n} e + d}{d} + 1\right ) e^{2} - 2 \, d f^{-2 \, n - 1} p x^{n} e \log \left (c\right ) - d^{2} f^{-2 \, n - 1} \log \left (c\right )^{2} + 2 \, {\left (n p^{2} e^{2} - n p e^{2} \log \left (c\right )\right )} f^{-2 \, n - 1} x^{2 \, n} \log \left (x\right ) - {\left (d^{2} f^{-2 \, n - 1} p^{2} - f^{-2 \, n - 1} p^{2} x^{2 \, n} e^{2}\right )} \log \left (x^{n} e + d\right )^{2} - 2 \, {\left (d f^{-2 \, n - 1} p^{2} x^{n} e + d^{2} f^{-2 \, n - 1} p \log \left (c\right ) + {\left (n p^{2} e^{2} \log \left (x\right ) + p^{2} e^{2} - p e^{2} \log \left (c\right )\right )} f^{-2 \, n - 1} x^{2 \, n}\right )} \log \left (x^{n} e + d\right )}{2 \, d^{2} n x^{2 \, n}} \end {gather*}

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

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

[Out]

Timed out

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Giac [F]
time = 0.00, 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((f*x)^(-1-2*n)*log(c*(d+e*x^n)^p)^2,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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