3.3.34 \(\int \frac {\log ^2(c (b x^n)^p)}{x^2} \, dx\) [234]

Optimal. Leaf size=46 \[ -\frac {2 n^2 p^2}{x}-\frac {2 n p \log \left (c \left (b x^n\right )^p\right )}{x}-\frac {\log ^2\left (c \left (b x^n\right )^p\right )}{x} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 46, 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 )}{x}-\frac {2 n p \log \left (c \left (b x^n\right )^p\right )}{x}-\frac {2 n^2 p^2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.01, size = 40, normalized size = 0.87 \begin {gather*} -\frac {2 n^2 p^2+2 n p \log \left (c \left (b x^n\right )^p\right )+\log ^2\left (c \left (b x^n\right )^p\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 46, normalized size = 1.00 \begin {gather*} -\frac {2 \, n^{2} p^{2}}{x} - \frac {2 \, n p \log \left (\left (b x^{n}\right )^{p} c\right )}{x} - \frac {\log \left (\left (b x^{n}\right )^{p} c\right )^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.41, size = 81, normalized size = 1.76 \begin {gather*} -\frac {n^{2} p^{2} \log \left (x\right )^{2} + 2 \, n^{2} p^{2} + 2 \, n p^{2} \log \left (b\right ) + p^{2} \log \left (b\right )^{2} + 2 \, {\left (n p + p \log \left (b\right )\right )} \log \left (c\right ) + \log \left (c\right )^{2} + 2 \, {\left (n^{2} p^{2} + n p^{2} \log \left (b\right ) + n p \log \left (c\right )\right )} \log \left (x\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.22, size = 41, normalized size = 0.89 \begin {gather*} - \frac {2 n^{2} p^{2}}{x} - \frac {2 n p \log {\left (c \left (b x^{n}\right )^{p} \right )}}{x} - \frac {\log {\left (c \left (b x^{n}\right )^{p} \right )}^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.84, size = 90, normalized size = 1.96 \begin {gather*} -\frac {n^{2} p^{2} \log \left (x\right )^{2}}{x} - \frac {2 \, {\left (n^{2} p^{2} + n p^{2} \log \left (b\right ) + n p \log \left (c\right )\right )} \log \left (x\right )}{x} - \frac {2 \, n^{2} p^{2} + 2 \, n p^{2} \log \left (b\right ) + p^{2} \log \left (b\right )^{2} + 2 \, n p \log \left (c\right ) + 2 \, p \log \left (b\right ) \log \left (c\right ) + \log \left (c\right )^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 3.85, size = 40, normalized size = 0.87 \begin {gather*} -\frac {2\,n^2\,p^2+2\,n\,p\,\ln \left (c\,{\left (b\,x^n\right )}^p\right )+{\ln \left (c\,{\left (b\,x^n\right )}^p\right )}^2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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