3.3.36 \(\int \frac {\log ^2(c (b x^n)^p)}{x^4} \, dx\) [236]

Optimal. Leaf size=52 \[ -\frac {2 n^2 p^2}{27 x^3}-\frac {2 n p \log \left (c \left (b x^n\right )^p\right )}{9 x^3}-\frac {\log ^2\left (c \left (b x^n\right )^p\right )}{3 x^3} \]

[Out]

-2/27*n^2*p^2/x^3-2/9*n*p*ln(c*(b*x^n)^p)/x^3-1/3*ln(c*(b*x^n)^p)^2/x^3

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Rubi [A]
time = 0.05, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2342, 2341, 2495} \begin {gather*} -\frac {\log ^2\left (c \left (b x^n\right )^p\right )}{3 x^3}-\frac {2 n p \log \left (c \left (b x^n\right )^p\right )}{9 x^3}-\frac {2 n^2 p^2}{27 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(b*x^n)^p]^2/x^4,x]

[Out]

(-2*n^2*p^2)/(27*x^3) - (2*n*p*Log[c*(b*x^n)^p])/(9*x^3) - Log[c*(b*x^n)^p]^2/(3*x^3)

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

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

Rule 2495

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

\begin {align*} \int \frac {\log ^2\left (c \left (b x^n\right )^p\right )}{x^4} \, dx &=\text {Subst}\left (\int \frac {\log ^2\left (b^p c x^{n p}\right )}{x^4} \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=-\frac {\log ^2\left (c \left (b x^n\right )^p\right )}{3 x^3}+\text {Subst}\left (\frac {1}{3} (2 n p) \int \frac {\log \left (b^p c x^{n p}\right )}{x^4} \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=-\frac {2 n^2 p^2}{27 x^3}-\frac {2 n p \log \left (c \left (b x^n\right )^p\right )}{9 x^3}-\frac {\log ^2\left (c \left (b x^n\right )^p\right )}{3 x^3}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 52, normalized size = 1.00 \begin {gather*} -\frac {2 n^2 p^2}{27 x^3}-\frac {2 n p \log \left (c \left (b x^n\right )^p\right )}{9 x^3}-\frac {\log ^2\left (c \left (b x^n\right )^p\right )}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(b*x^n)^p]^2/x^4,x]

[Out]

(-2*n^2*p^2)/(27*x^3) - (2*n*p*Log[c*(b*x^n)^p])/(9*x^3) - Log[c*(b*x^n)^p]^2/(3*x^3)

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (b \,x^{n}\right )^{p}\right )^{2}}{x^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x^n)^p)^2/x^4,x)

[Out]

int(ln(c*(b*x^n)^p)^2/x^4,x)

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Maxima [A]
time = 0.28, size = 46, normalized size = 0.88 \begin {gather*} -\frac {2 \, n^{2} p^{2}}{27 \, x^{3}} - \frac {2 \, n p \log \left (\left (b x^{n}\right )^{p} c\right )}{9 \, x^{3}} - \frac {\log \left (\left (b x^{n}\right )^{p} c\right )^{2}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^n)^p)^2/x^4,x, algorithm="maxima")

[Out]

-2/27*n^2*p^2/x^3 - 2/9*n*p*log((b*x^n)^p*c)/x^3 - 1/3*log((b*x^n)^p*c)^2/x^3

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Fricas [A]
time = 0.33, size = 88, normalized size = 1.69 \begin {gather*} -\frac {9 \, n^{2} p^{2} \log \left (x\right )^{2} + 2 \, n^{2} p^{2} + 6 \, n p^{2} \log \left (b\right ) + 9 \, p^{2} \log \left (b\right )^{2} + 6 \, {\left (n p + 3 \, p \log \left (b\right )\right )} \log \left (c\right ) + 9 \, \log \left (c\right )^{2} + 6 \, {\left (n^{2} p^{2} + 3 \, n p^{2} \log \left (b\right ) + 3 \, n p \log \left (c\right )\right )} \log \left (x\right )}{27 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^n)^p)^2/x^4,x, algorithm="fricas")

[Out]

-1/27*(9*n^2*p^2*log(x)^2 + 2*n^2*p^2 + 6*n*p^2*log(b) + 9*p^2*log(b)^2 + 6*(n*p + 3*p*log(b))*log(c) + 9*log(
c)^2 + 6*(n^2*p^2 + 3*n*p^2*log(b) + 3*n*p*log(c))*log(x))/x^3

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Sympy [A]
time = 0.89, size = 51, normalized size = 0.98 \begin {gather*} - \frac {2 n^{2} p^{2}}{27 x^{3}} - \frac {2 n p \log {\left (c \left (b x^{n}\right )^{p} \right )}}{9 x^{3}} - \frac {\log {\left (c \left (b x^{n}\right )^{p} \right )}^{2}}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(b*x**n)**p)**2/x**4,x)

[Out]

-2*n**2*p**2/(27*x**3) - 2*n*p*log(c*(b*x**n)**p)/(9*x**3) - log(c*(b*x**n)**p)**2/(3*x**3)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (46) = 92\).
time = 4.67, size = 95, normalized size = 1.83 \begin {gather*} -\frac {n^{2} p^{2} \log \left (x\right )^{2}}{3 \, x^{3}} - \frac {2 \, {\left (n^{2} p^{2} + 3 \, n p^{2} \log \left (b\right ) + 3 \, n p \log \left (c\right )\right )} \log \left (x\right )}{9 \, x^{3}} - \frac {2 \, n^{2} p^{2} + 6 \, n p^{2} \log \left (b\right ) + 9 \, p^{2} \log \left (b\right )^{2} + 6 \, n p \log \left (c\right ) + 18 \, p \log \left (b\right ) \log \left (c\right ) + 9 \, \log \left (c\right )^{2}}{27 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^n)^p)^2/x^4,x, algorithm="giac")

[Out]

-1/3*n^2*p^2*log(x)^2/x^3 - 2/9*(n^2*p^2 + 3*n*p^2*log(b) + 3*n*p*log(c))*log(x)/x^3 - 1/27*(2*n^2*p^2 + 6*n*p
^2*log(b) + 9*p^2*log(b)^2 + 6*n*p*log(c) + 18*p*log(b)*log(c) + 9*log(c)^2)/x^3

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Mupad [B]
time = 3.88, size = 46, normalized size = 0.88 \begin {gather*} -\frac {{\ln \left (c\,{\left (b\,x^n\right )}^p\right )}^2}{3\,x^3}-\frac {2\,n^2\,p^2}{27\,x^3}-\frac {2\,n\,p\,\ln \left (c\,{\left (b\,x^n\right )}^p\right )}{9\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(b*x^n)^p)^2/x^4,x)

[Out]

- log(c*(b*x^n)^p)^2/(3*x^3) - (2*n^2*p^2)/(27*x^3) - (2*n*p*log(c*(b*x^n)^p))/(9*x^3)

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