3.158 \(\int (d x)^{-1+n} \log (c x^n) \, dx\)

Optimal. Leaf size=32 \[ \frac{(d x)^n \log \left (c x^n\right )}{d n}-\frac{(d x)^n}{d n} \]

[Out]

-((d*x)^n/(d*n)) + ((d*x)^n*Log[c*x^n])/(d*n)

________________________________________________________________________________________

Rubi [A]  time = 0.0110705, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2304} \[ \frac{(d x)^n \log \left (c x^n\right )}{d n}-\frac{(d x)^n}{d n} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(-1 + n)*Log[c*x^n],x]

[Out]

-((d*x)^n/(d*n)) + ((d*x)^n*Log[c*x^n])/(d*n)

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin{align*} \int (d x)^{-1+n} \log \left (c x^n\right ) \, dx &=-\frac{(d x)^n}{d n}+\frac{(d x)^n \log \left (c x^n\right )}{d n}\\ \end{align*}

Mathematica [A]  time = 0.0041659, size = 20, normalized size = 0.62 \[ \frac{(d x)^n \left (\log \left (c x^n\right )-1\right )}{d n} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(-1 + n)*Log[c*x^n],x]

[Out]

((d*x)^n*(-1 + Log[c*x^n]))/(d*n)

________________________________________________________________________________________

Maple [C]  time = 0.088, size = 263, normalized size = 8.2 \begin{align*}{\frac{x\ln \left ({x}^{n} \right ) }{n}{{\rm e}^{{\frac{ \left ( -1+n \right ) \left ( -i \left ({\it csgn} \left ( idx \right ) \right ) ^{3}\pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( id \right ) \pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) \pi -i\pi \,{\it csgn} \left ( idx \right ){\it csgn} \left ( id \right ){\it csgn} \left ( ix \right ) +2\,\ln \left ( x \right ) +2\,\ln \left ( d \right ) \right ) }{2}}}}}+{\frac{ \left ( i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +2\,\ln \left ( c \right ) -2 \right ) x}{2\,n}{{\rm e}^{{\frac{ \left ( -1+n \right ) \left ( -i \left ({\it csgn} \left ( idx \right ) \right ) ^{3}\pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( id \right ) \pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) \pi -i\pi \,{\it csgn} \left ( idx \right ){\it csgn} \left ( id \right ){\it csgn} \left ( ix \right ) +2\,\ln \left ( x \right ) +2\,\ln \left ( d \right ) \right ) }{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(-1+n)*ln(c*x^n),x)

[Out]

1/n*x*exp(1/2*(-1+n)*(-I*csgn(I*d*x)^3*Pi+I*csgn(I*d*x)^2*csgn(I*d)*Pi+I*csgn(I*d*x)^2*csgn(I*x)*Pi-I*csgn(I*d
*x)*csgn(I*d)*csgn(I*x)*Pi+2*ln(x)+2*ln(d)))*ln(x^n)+1/2*(I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*csgn(I*x^n)*cs
gn(I*c*x^n)*csgn(I*c)-I*Pi*csgn(I*c*x^n)^3+I*Pi*csgn(I*c*x^n)^2*csgn(I*c)+2*ln(c)-2)/n*x*exp(1/2*(-1+n)*(-I*cs
gn(I*d*x)^3*Pi+I*csgn(I*d*x)^2*csgn(I*d)*Pi+I*csgn(I*d*x)^2*csgn(I*x)*Pi-I*csgn(I*d*x)*csgn(I*d)*csgn(I*x)*Pi+
2*ln(x)+2*ln(d)))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(-1+n)*log(c*x^n),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 0.993306, size = 55, normalized size = 1.72 \begin{align*} \frac{{\left (n \log \left (x\right ) + \log \left (c\right ) - 1\right )} d^{n - 1} x^{n}}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(-1+n)*log(c*x^n),x, algorithm="fricas")

[Out]

(n*log(x) + log(c) - 1)*d^(n - 1)*x^n/n

________________________________________________________________________________________

Sympy [A]  time = 39.6049, size = 68, normalized size = 2.12 \begin{align*} \begin{cases} \tilde{\infty } x \log{\left (c \right )} & \text{for}\: d = 0 \wedge n = 0 \\\frac{\log{\left (c \right )} \log{\left (x \right )}}{d} & \text{for}\: n = 0 \\0^{n - 1} \left (n x \log{\left (x \right )} - n x + x \log{\left (c \right )}\right ) & \text{for}\: d = 0 \\\frac{d^{n} x^{n} \log{\left (x \right )}}{d} + \frac{d^{n} x^{n} \log{\left (c \right )}}{d n} - \frac{d^{n} x^{n}}{d n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(-1+n)*ln(c*x**n),x)

[Out]

Piecewise((zoo*x*log(c), Eq(d, 0) & Eq(n, 0)), (log(c)*log(x)/d, Eq(n, 0)), (0**(n - 1)*(n*x*log(x) - n*x + x*
log(c)), Eq(d, 0)), (d**n*x**n*log(x)/d + d**n*x**n*log(c)/(d*n) - d**n*x**n/(d*n), True))

________________________________________________________________________________________

Giac [A]  time = 1.38578, size = 57, normalized size = 1.78 \begin{align*} \frac{d^{n} x^{n} \log \left (x\right )}{d} + \frac{d^{n} x^{n} \log \left (c\right )}{d n} - \frac{d^{n} x^{n}}{d n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(-1+n)*log(c*x^n),x, algorithm="giac")

[Out]

d^n*x^n*log(x)/d + d^n*x^n*log(c)/(d*n) - d^n*x^n/(d*n)