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

Optimal. Leaf size=22 \[ \frac{\log ^2\left (c \left (b x^n\right )^p\right )}{2 n p} \]

[Out]

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

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Rubi [A]  time = 0.0299711, antiderivative size = 22, 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 = {2301, 2445} \[ \frac{\log ^2\left (c \left (b x^n\right )^p\right )}{2 n p} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

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

Mathematica [A]  time = 0.0011076, size = 22, normalized size = 1. \[ \frac{\log ^2\left (c \left (b x^n\right )^p\right )}{2 n p} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 21, normalized size = 1. \begin{align*}{\frac{ \left ( \ln \left ( c \left ( b{x}^{n} \right ) ^{p} \right ) \right ) ^{2}}{2\,pn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.14399, size = 27, normalized size = 1.23 \begin{align*} \frac{\log \left (\left (b x^{n}\right )^{p} c\right )^{2}}{2 \, n p} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.968036, size = 63, normalized size = 2.86 \begin{align*} \frac{1}{2} \, n p \log \left (x\right )^{2} +{\left (p \log \left (b\right ) + \log \left (c\right )\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.9949, size = 37, normalized size = 1.68 \begin{align*} - \begin{cases} - \log{\left (x \right )} \log{\left (b^{p} c \right )} & \text{for}\: n = 0 \\- \log{\left (c \right )} \log{\left (x \right )} & \text{for}\: p = 0 \\- \frac{\log{\left (c \left (b x^{n}\right )^{p} \right )}^{2}}{2 n p} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Piecewise((-log(x)*log(b**p*c), Eq(n, 0)), (-log(c)*log(x), Eq(p, 0)), (-log(c*(b*x**n)**p)**2/(2*n*p), True)
)

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Giac [A]  time = 1.29278, size = 27, normalized size = 1.23 \begin{align*} \frac{1}{2} \, n p \log \left (x\right )^{2} + p \log \left (b\right ) \log \left (x\right ) + \log \left (c\right ) \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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