3.234 \(\int \frac{\log ^2(c (b x^n)^p)}{x^2} \, dx\)

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

[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

________________________________________________________________________________________

Rubi [A]  time = 0.0701019, antiderivative size = 46, normalized size of antiderivative = 1., 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 = {2305, 2304, 2445} \[ -\frac{\log ^2\left (c \left (b x^n\right )^p\right )}{x}-\frac{2 n p \log \left (c \left (b x^n\right )^p\right )}{x}-\frac{2 n^2 p^2}{x} \]

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 2305

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 2304

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 2445

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 &=\operatorname{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}+\operatorname{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*}

Mathematica [A]  time = 0.0043859, size = 40, normalized size = 0.87 \[ -\frac{\log ^2\left (c \left (b x^n\right )^p\right )+2 n p \log \left (c \left (b x^n\right )^p\right )+2 n^2 p^2}{x} \]

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)

________________________________________________________________________________________

Maple [F]  time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( c \left ( b{x}^{n} \right ) ^{p} \right ) \right ) ^{2}}{{x}^{2}}}\, dx \end{align*}

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)

________________________________________________________________________________________

Maxima [A]  time = 1.13829, size = 62, normalized size = 1.35 \begin{align*} -\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{align*}

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

________________________________________________________________________________________

Fricas [A]  time = 0.766316, size = 209, normalized size = 4.54 \begin{align*} -\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{align*}

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

________________________________________________________________________________________

Sympy [B]  time = 1.2047, size = 117, normalized size = 2.54 \begin{align*} - \frac{n^{2} p^{2} \log{\left (x \right )}^{2}}{x} - \frac{2 n^{2} p^{2} \log{\left (x \right )}}{x} - \frac{2 n^{2} p^{2}}{x} - \frac{2 n p^{2} \log{\left (b \right )} \log{\left (x \right )}}{x} - \frac{2 n p^{2} \log{\left (b \right )}}{x} - \frac{2 n p \log{\left (c \right )} \log{\left (x \right )}}{x} - \frac{2 n p \log{\left (c \right )}}{x} - \frac{p^{2} \log{\left (b \right )}^{2}}{x} - \frac{2 p \log{\left (b \right )} \log{\left (c \right )}}{x} - \frac{\log{\left (c \right )}^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.27003, size = 122, normalized size = 2.65 \begin{align*} -\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{align*}

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