3.235 \(\int \frac{\log ^2(c (b x^n)^p)}{x^3} \, dx\)

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

[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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Rubi [A]  time = 0.0696167, antiderivative size = 52, 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} \[ -\frac{\log ^2\left (c \left (b x^n\right )^p\right )}{2 x^2}-\frac{n p \log \left (c \left (b x^n\right )^p\right )}{2 x^2}-\frac{n^2 p^2}{4 x^2} \]

Antiderivative was successfully verified.

[In]

Int[Log[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)

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^3} \, dx &=\operatorname{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}+\operatorname{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*}

Mathematica [A]  time = 0.0043444, size = 43, normalized size = 0.83 \[ -\frac{2 \log ^2\left (c \left (b x^n\right )^p\right )+2 n p \log \left (c \left (b x^n\right )^p\right )+n^2 p^2}{4 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( c \left ( b{x}^{n} \right ) ^{p} \right ) \right ) ^{2}}{{x}^{3}}}\, dx \end{align*}

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 = 1.15784, size = 62, normalized size = 1.19 \begin{align*} -\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{align*}

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 = 0.836053, size = 231, normalized size = 4.44 \begin{align*} -\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{align*}

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 [B]  time = 3.1379, size = 134, normalized size = 2.58 \begin{align*} - \frac{n^{2} p^{2} \log{\left (x \right )}^{2}}{2 x^{2}} - \frac{n^{2} p^{2} \log{\left (x \right )}}{2 x^{2}} - \frac{n^{2} p^{2}}{4 x^{2}} - \frac{n p^{2} \log{\left (b \right )} \log{\left (x \right )}}{x^{2}} - \frac{n p^{2} \log{\left (b \right )}}{2 x^{2}} - \frac{n p \log{\left (c \right )} \log{\left (x \right )}}{x^{2}} - \frac{n p \log{\left (c \right )}}{2 x^{2}} - \frac{p^{2} \log{\left (b \right )}^{2}}{2 x^{2}} - \frac{p \log{\left (b \right )} \log{\left (c \right )}}{x^{2}} - \frac{\log{\left (c \right )}^{2}}{2 x^{2}} \end{align*}

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

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Giac [B]  time = 1.32222, size = 127, normalized size = 2.44 \begin{align*} -\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{align*}

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