3.3.33 \(\int \frac {\log ^2(c (b x^n)^p)}{x} \, dx\) [233]

Optimal. Leaf size=22 \[ \frac {\log ^3\left (c \left (b x^n\right )^p\right )}{3 n p} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2339, 30, 2495} \begin {gather*} \frac {\log ^3\left (c \left (b x^n\right )^p\right )}{3 n p} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, 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 ^2\left (c \left (b x^n\right )^p\right )}{x} \, dx &=\text {Subst}\left (\int \frac {\log ^2\left (b^p c x^{n p}\right )}{x} \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=\text {Subst}\left (\frac {\text {Subst}\left (\int x^2 \, dx,x,\log \left (b^p c x^{n p}\right )\right )}{n p},b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=\frac {\log ^3\left (c \left (b x^n\right )^p\right )}{3 n p}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} \frac {\log ^3\left (c \left (b x^n\right )^p\right )}{3 n p} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.03, size = 21, normalized size = 0.95

method	result	size
derivativedivides	\(\frac {\ln \left (c \left (b \,x^{n}\right )^{p}\right )^{3}}{3 p n}\)	\(21\)
default	\(\frac {\ln \left (c \left (b \,x^{n}\right )^{p}\right )^{3}}{3 p n}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x^n)^p)^2/x,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 20, normalized size = 0.91 \begin {gather*} \frac {\log \left (\left (b x^{n}\right )^{p} c\right )^{3}}{3 \, n p} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^n)^p)^2/x,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (20) = 40\).
time = 0.42, size = 54, normalized size = 2.45 \begin {gather*} \frac {1}{3} \, n^{2} p^{2} \log \left (x\right )^{3} + {\left (n p^{2} \log \left (b\right ) + n p \log \left (c\right )\right )} \log \left (x\right )^{2} + {\left (p^{2} \log \left (b\right )^{2} + 2 \, p \log \left (b\right ) \log \left (c\right ) + \log \left (c\right )^{2}\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)^2/x,x, algorithm="fricas")

[Out]

1/3*n^2*p^2*log(x)^3 + (n*p^2*log(b) + n*p*log(c))*log(x)^2 + (p^2*log(b)^2 + 2*p*log(b)*log(c) + log(c)^2)*lo
g(x)

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Sympy [A]
time = 0.78, size = 41, normalized size = 1.86 \begin {gather*} - \begin {cases} - \log {\left (x \right )} \log {\left (b^{p} c \right )}^{2} & \text {for}\: n = 0 \\- \log {\left (c \right )}^{2} \log {\left (x \right )} & \text {for}\: p = 0 \\- \frac {\log {\left (c \left (b x^{n}\right )^{p} \right )}^{3}}{3 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)**2/x,x)

[Out]

-Piecewise((-log(x)*log(b**p*c)**2, Eq(n, 0)), (-log(c)**2*log(x), Eq(p, 0)), (-log(c*(b*x**n)**p)**3/(3*n*p),
True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (20) = 40\).
time = 5.35, size = 59, normalized size = 2.68 \begin {gather*} \frac {1}{3} \, n^{2} p^{2} \log \left (x\right )^{3} + n p^{2} \log \left (b\right ) \log \left (x\right )^{2} + p^{2} \log \left (b\right )^{2} \log \left (x\right ) + n p \log \left (c\right ) \log \left (x\right )^{2} + 2 \, p \log \left (b\right ) \log \left (c\right ) \log \left (x\right ) + \log \left (c\right )^{2} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^n)^p)^2/x,x, algorithm="giac")

[Out]

1/3*n^2*p^2*log(x)^3 + n*p^2*log(b)*log(x)^2 + p^2*log(b)^2*log(x) + n*p*log(c)*log(x)^2 + 2*p*log(b)*log(c)*l
og(x) + log(c)^2*log(x)

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Mupad [B]
time = 3.75, size = 20, normalized size = 0.91 \begin {gather*} \frac {{\ln \left (c\,{\left (b\,x^n\right )}^p\right )}^3}{3\,n\,p} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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