3.3.25 \(\int \log (c (b x^n)^p) \, dx\) [225]

Optimal. Leaf size=18 \[ -n p x+x \log \left (c \left (b x^n\right )^p\right ) \]

[Out]

-n*p*x+x*ln(c*(b*x^n)^p)

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2332, 2495} \begin {gather*} x \log \left (c \left (b x^n\right )^p\right )-n p x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(n*p*x) + x*Log[c*(b*x^n)^p]

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{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 \log \left (c \left (b x^n\right )^p\right ) \, dx &=\text {Subst}\left (\int \log \left (b^p c x^{n p}\right ) \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=-n p x+x \log \left (c \left (b x^n\right )^p\right )\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} -n p x+x \log \left (c \left (b x^n\right )^p\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(n*p*x) + x*Log[c*(b*x^n)^p]

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Maple [A]
time = 0.02, size = 19, normalized size = 1.06

method	result	size
default	\(-n p x +x \ln \left (c \left (b \,x^{n}\right )^{p}\right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-n*p*x+x*ln(c*(b*x^n)^p)

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Maxima [A]
time = 0.29, size = 18, normalized size = 1.00 \begin {gather*} -n p x + x \log \left (\left (b x^{n}\right )^{p} c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-n*p*x + x*log((b*x^n)^p*c)

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Fricas [A]
time = 0.36, size = 21, normalized size = 1.17 \begin {gather*} n p x \log \left (x\right ) - n p x + p x \log \left (b\right ) + x \log \left (c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

n*p*x*log(x) - n*p*x + p*x*log(b) + x*log(c)

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Sympy [A]
time = 0.12, size = 15, normalized size = 0.83 \begin {gather*} - n p x + x \log {\left (c \left (b x^{n}\right )^{p} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-n*p*x + x*log(c*(b*x**n)**p)

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Giac [A]
time = 5.82, size = 21, normalized size = 1.17 \begin {gather*} n p x \log \left (x\right ) - n p x + p x \log \left (b\right ) + x \log \left (c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

n*p*x*log(x) - n*p*x + p*x*log(b) + x*log(c)

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Mupad [B]
time = 0.03, size = 17, normalized size = 0.94 \begin {gather*} x\,\left (\ln \left (c\,{\left (b\,x^n\right )}^p\right )-n\,p\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(log(c*(b*x^n)^p) - n*p)

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