3.229 \(\int \frac{\log (c (b x^n)^p)}{x^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0296407, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2304, 2445} \[ -\frac{\log \left (c \left (b x^n\right )^p\right )}{3 x^3}-\frac{n p}{9 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 \left (c \left (b x^n\right )^p\right )}{x^4} \, dx &=\operatorname{Subst}\left (\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{n p}{9 x^3}-\frac{\log \left (c \left (b x^n\right )^p\right )}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0012108, size = 27, normalized size = 1. \[ -\frac{\log \left (c \left (b x^n\right )^p\right )}{3 x^3}-\frac{n p}{9 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( c \left ( b{x}^{n} \right ) ^{p} \right ) }{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.15947, size = 31, normalized size = 1.15 \begin{align*} -\frac{n p}{9 \, x^{3}} - \frac{\log \left (\left (b x^{n}\right )^{p} c\right )}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.879031, size = 74, normalized size = 2.74 \begin{align*} -\frac{3 \, n p \log \left (x\right ) + n p + 3 \, p \log \left (b\right ) + 3 \, \log \left (c\right )}{9 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 7.92444, size = 39, normalized size = 1.44 \begin{align*} - \frac{n p \log{\left (x \right )}}{3 x^{3}} - \frac{n p}{9 x^{3}} - \frac{p \log{\left (b \right )}}{3 x^{3}} - \frac{\log{\left (c \right )}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.31124, size = 38, normalized size = 1.41 \begin{align*} -\frac{n p \log \left (x\right )}{3 \, x^{3}} - \frac{n p + 3 \, p \log \left (b\right ) + 3 \, \log \left (c\right )}{9 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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