3.3.26 \(\int \frac {\log (c (b x^n)^p)}{x} \, dx\) [226]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 22, 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 = {2338, 2495} \begin {gather*} \frac {\log ^2\left (c \left (b x^n\right )^p\right )}{2 n p} \end {gather*}

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 2338

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 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} \, dx &=\text {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*}

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Mathematica [A]
time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} \frac {\log ^2\left (c \left (b x^n\right )^p\right )}{2 n p} \end {gather*}

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 = 0.03, size = 21, normalized size = 0.95

method	result	size
derivativedivides	\(\frac {\ln \left (c \left (b \,x^{n}\right )^{p}\right )^{2}}{2 p n}\)	\(21\)
default	\(\frac {\ln \left (c \left (b \,x^{n}\right )^{p}\right )^{2}}{2 p n}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 20, normalized size = 0.91 \begin {gather*} \frac {\log \left (\left (b x^{n}\right )^{p} c\right )^{2}}{2 \, n p} \end {gather*}

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 = 0.35, size = 19, normalized size = 0.86 \begin {gather*} \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 {gather*}

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 = 1.00, size = 37, normalized size = 1.68 \begin {gather*} - \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 {gather*}

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 = 3.76, size = 20, normalized size = 0.91 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 3.80, size = 20, normalized size = 0.91 \begin {gather*} \frac {{\ln \left (c\,{\left (b\,x^n\right )}^p\right )}^2}{2\,n\,p} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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