3.233 \(\int \frac{\log ^2(c (b x^n)^p)}{x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0496139, antiderivative size = 22, normalized size of antiderivative = 1., 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 = {2302, 30, 2445} \[ \frac{\log ^3\left (c \left (b x^n\right )^p\right )}{3 n p} \]

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 2302

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 30

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

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 ^2\left (c \left (b x^n\right )^p\right )}{x} \, dx &=\operatorname{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 )\\ &=\operatorname{Subst}\left (\frac{\operatorname{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*}

Mathematica [A]  time = 0.001235, size = 22, normalized size = 1. \[ \frac{\log ^3\left (c \left (b x^n\right )^p\right )}{3 n p} \]

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 = 0.005, size = 21, normalized size = 1. \begin{align*}{\frac{ \left ( \ln \left ( c \left ( b{x}^{n} \right ) ^{p} \right ) \right ) ^{3}}{3\,pn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.11672, size = 27, normalized size = 1.23 \begin{align*} \frac{\log \left (\left (b x^{n}\right )^{p} c\right )^{3}}{3 \, n p} \end{align*}

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]  time = 0.831912, size = 157, normalized size = 7.14 \begin{align*} \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{align*}

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 = 1.85619, size = 41, normalized size = 1.86 \begin{align*} - \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{align*}

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]  time = 1.32034, size = 80, normalized size = 3.64 \begin{align*} \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{align*}

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)