3.3.29 \(\int \frac {\log (c (b x^n)^p)}{x^4} \, dx\) [229]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = {2341, 2495} \begin {gather*} -\frac {\log \left (c \left (b x^n\right )^p\right )}{3 x^3}-\frac {n p}{9 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*(n*p)/x^3 - Log[c*(b*x^n)^p]/(3*x^3)

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 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 \left (c \left (b x^n\right )^p\right )}{x^4} \, dx &=\text {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*}

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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} -\frac {n p}{9 x^3}-\frac {\log \left (c \left (b x^n\right )^p\right )}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*(n*p)/x^3 - Log[c*(b*x^n)^p]/(3*x^3)

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (b \,x^{n}\right )^{p}\right )}{x^{4}}\, dx\]

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 = 0.29, size = 23, normalized size = 0.85 \begin {gather*} -\frac {n p}{9 \, x^{3}} - \frac {\log \left (\left (b x^{n}\right )^{p} c\right )}{3 \, x^{3}} \end {gather*}

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 = 0.36, size = 24, normalized size = 0.89 \begin {gather*} -\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 {gather*}

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 = 0.89, size = 24, normalized size = 0.89 \begin {gather*} - \frac {n p}{9 x^{3}} - \frac {\log {\left (c \left (b x^{n}\right )^{p} \right )}}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(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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Giac [A]
time = 3.02, size = 28, normalized size = 1.04 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 3.87, size = 23, normalized size = 0.85 \begin {gather*} -\frac {\ln \left (c\,{\left (b\,x^n\right )}^p\right )}{3\,x^3}-\frac {n\,p}{9\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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