3.3.35 \(\int \frac {\log ^2(c (b x^n)^p)}{x^3} \, dx\) [235]

Optimal. Leaf size=52 \[ -\frac {n^2 p^2}{4 x^2}-\frac {n p \log \left (c \left (b x^n\right )^p\right )}{2 x^2}-\frac {\log ^2\left (c \left (b x^n\right )^p\right )}{2 x^2} \]

[Out]

-1/4*n^2*p^2/x^2-1/2*n*p*ln(c*(b*x^n)^p)/x^2-1/2*ln(c*(b*x^n)^p)^2/x^2

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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 )}{2 x^2}-\frac {n p \log \left (c \left (b x^n\right )^p\right )}{2 x^2}-\frac {n^2 p^2}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(b*x^n)^p]^2/x^3,x]

[Out]

-1/4*(n^2*p^2)/x^2 - (n*p*Log[c*(b*x^n)^p])/(2*x^2) - Log[c*(b*x^n)^p]^2/(2*x^2)

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

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Mathematica [A]
time = 0.00, size = 43, normalized size = 0.83 \begin {gather*} -\frac {n^2 p^2+2 n p \log \left (c \left (b x^n\right )^p\right )+2 \log ^2\left (c \left (b x^n\right )^p\right )}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(b*x^n)^p]^2/x^3,x]

[Out]

-1/4*(n^2*p^2 + 2*n*p*Log[c*(b*x^n)^p] + 2*Log[c*(b*x^n)^p]^2)/x^2

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (b \,x^{n}\right )^{p}\right )^{2}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x^n)^p)^2/x^3,x)

[Out]

int(ln(c*(b*x^n)^p)^2/x^3,x)

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Maxima [A]
time = 0.28, size = 46, normalized size = 0.88 \begin {gather*} -\frac {n^{2} p^{2}}{4 \, x^{2}} - \frac {n p \log \left (\left (b x^{n}\right )^{p} c\right )}{2 \, x^{2}} - \frac {\log \left (\left (b x^{n}\right )^{p} c\right )^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^n)^p)^2/x^3,x, algorithm="maxima")

[Out]

-1/4*n^2*p^2/x^2 - 1/2*n*p*log((b*x^n)^p*c)/x^2 - 1/2*log((b*x^n)^p*c)^2/x^2

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Fricas [A]
time = 0.34, size = 87, normalized size = 1.67 \begin {gather*} -\frac {2 \, n^{2} p^{2} \log \left (x\right )^{2} + n^{2} p^{2} + 2 \, n p^{2} \log \left (b\right ) + 2 \, p^{2} \log \left (b\right )^{2} + 2 \, {\left (n p + 2 \, p \log \left (b\right )\right )} \log \left (c\right ) + 2 \, \log \left (c\right )^{2} + 2 \, {\left (n^{2} p^{2} + 2 \, n p^{2} \log \left (b\right ) + 2 \, n p \log \left (c\right )\right )} \log \left (x\right )}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^n)^p)^2/x^3,x, algorithm="fricas")

[Out]

-1/4*(2*n^2*p^2*log(x)^2 + n^2*p^2 + 2*n*p^2*log(b) + 2*p^2*log(b)^2 + 2*(n*p + 2*p*log(b))*log(c) + 2*log(c)^
2 + 2*(n^2*p^2 + 2*n*p^2*log(b) + 2*n*p*log(c))*log(x))/x^2

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Sympy [A]
time = 0.49, size = 48, normalized size = 0.92 \begin {gather*} - \frac {n^{2} p^{2}}{4 x^{2}} - \frac {n p \log {\left (c \left (b x^{n}\right )^{p} \right )}}{2 x^{2}} - \frac {\log {\left (c \left (b x^{n}\right )^{p} \right )}^{2}}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(b*x**n)**p)**2/x**3,x)

[Out]

-n**2*p**2/(4*x**2) - n*p*log(c*(b*x**n)**p)/(2*x**2) - log(c*(b*x**n)**p)**2/(2*x**2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (46) = 92\).
time = 5.07, size = 94, normalized size = 1.81 \begin {gather*} -\frac {n^{2} p^{2} \log \left (x\right )^{2}}{2 \, x^{2}} - \frac {{\left (n^{2} p^{2} + 2 \, n p^{2} \log \left (b\right ) + 2 \, n p \log \left (c\right )\right )} \log \left (x\right )}{2 \, x^{2}} - \frac {n^{2} p^{2} + 2 \, n p^{2} \log \left (b\right ) + 2 \, p^{2} \log \left (b\right )^{2} + 2 \, n p \log \left (c\right ) + 4 \, p \log \left (b\right ) \log \left (c\right ) + 2 \, \log \left (c\right )^{2}}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^n)^p)^2/x^3,x, algorithm="giac")

[Out]

-1/2*n^2*p^2*log(x)^2/x^2 - 1/2*(n^2*p^2 + 2*n*p^2*log(b) + 2*n*p*log(c))*log(x)/x^2 - 1/4*(n^2*p^2 + 2*n*p^2*
log(b) + 2*p^2*log(b)^2 + 2*n*p*log(c) + 4*p*log(b)*log(c) + 2*log(c)^2)/x^2

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Mupad [B]
time = 3.79, size = 46, normalized size = 0.88 \begin {gather*} -\frac {{\ln \left (c\,{\left (b\,x^n\right )}^p\right )}^2}{2\,x^2}-\frac {n^2\,p^2}{4\,x^2}-\frac {n\,p\,\ln \left (c\,{\left (b\,x^n\right )}^p\right )}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(b*x^n)^p)^2/x^3,x)

[Out]

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

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