3.3.28 \(\int \frac {\log (c (b x^n)^p)}{x^3} \, dx\) [228]

Optimal. Leaf size=27 \[ -\frac {n p}{4 x^2}-\frac {\log \left (c \left (b x^n\right )^p\right )}{2 x^2} \]

[Out]

-1/4*n*p/x^2-1/2*ln(c*(b*x^n)^p)/x^2

________________________________________________________________________________________

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 )}{2 x^2}-\frac {n p}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(n*p)/x^2 - Log[c*(b*x^n)^p]/(2*x^2)

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^3} \, dx &=\text {Subst}\left (\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 p}{4 x^2}-\frac {\log \left (c \left (b x^n\right )^p\right )}{2 x^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} -\frac {n p}{4 x^2}-\frac {\log \left (c \left (b x^n\right )^p\right )}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(n*p)/x^2 - Log[c*(b*x^n)^p]/(2*x^2)

________________________________________________________________________________________

Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (b \,x^{n}\right )^{p}\right )}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x^n)^p)/x^3,x)

[Out]

int(ln(c*(b*x^n)^p)/x^3,x)

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 23, normalized size = 0.85 \begin {gather*} -\frac {n p}{4 \, x^{2}} - \frac {\log \left (\left (b x^{n}\right )^{p} c\right )}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^n)^p)/x^3,x, algorithm="maxima")

[Out]

-1/4*n*p/x^2 - 1/2*log((b*x^n)^p*c)/x^2

________________________________________________________________________________________

Fricas [A]
time = 0.33, size = 24, normalized size = 0.89 \begin {gather*} -\frac {2 \, n p \log \left (x\right ) + n p + 2 \, p \log \left (b\right ) + 2 \, \log \left (c\right )}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^n)^p)/x^3,x, algorithm="fricas")

[Out]

-1/4*(2*n*p*log(x) + n*p + 2*p*log(b) + 2*log(c))/x^2

________________________________________________________________________________________

Sympy [A]
time = 0.49, size = 24, normalized size = 0.89 \begin {gather*} - \frac {n p}{4 x^{2}} - \frac {\log {\left (c \left (b x^{n}\right )^{p} \right )}}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(b*x**n)**p)/x**3,x)

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 5.65, size = 28, normalized size = 1.04 \begin {gather*} -\frac {n p \log \left (x\right )}{2 \, x^{2}} - \frac {n p + 2 \, p \log \left (b\right ) + 2 \, \log \left (c\right )}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^n)^p)/x^3,x, algorithm="giac")

[Out]

-1/2*n*p*log(x)/x^2 - 1/4*(n*p + 2*p*log(b) + 2*log(c))/x^2

________________________________________________________________________________________

Mupad [B]
time = 3.84, size = 23, normalized size = 0.85 \begin {gather*} -\frac {\ln \left (c\,{\left (b\,x^n\right )}^p\right )}{2\,x^2}-\frac {n\,p}{4\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(b*x^n)^p)/x^3,x)

[Out]

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

________________________________________________________________________________________