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*ln(c*x^n)/d/n

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, 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)

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fricas [A]  time = 0.48, size = 20, normalized size = 0.62 \[ \frac {{\left (n \log \relax (x) + \log \relax (c) - 1\right )} d^{n - 1} x^{n}}{n} \]

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

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giac [A]  time = 0.36, size = 50, normalized size = 1.56 \[ \frac {\frac {1}{d}^{n} x^{n} {\left | d \right |}^{2 \, n} \log \relax (c)}{d n} + \frac {d^{n} x^{n} \log \relax (x)}{d} - \frac {d^{n} x^{n}}{d n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/d)^n*x^n*abs(d)^(2*n)*log(c)/(d*n) + d^n*x^n*log(x)/d - d^n*x^n/(d*n)

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maple [C]  time = 0.14, size = 263, normalized size = 8.22 \[ \frac {x \,{\mathrm e}^{\frac {\left (n -1\right ) \left (-i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )+i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )^{2}-i \pi \mathrm {csgn}\left (i d x \right )^{3}+2 \ln \relax (d )+2 \ln \relax (x )\right )}{2}} \ln \left (x^{n}\right )}{n}+\frac {\left (-i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 \ln \relax (c )-2\right ) x \,{\mathrm e}^{\frac {\left (n -1\right ) \left (-i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )+i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )^{2}-i \pi \mathrm {csgn}\left (i d x \right )^{3}+2 \ln \relax (d )+2 \ln \relax (x )\right )}{2}}}{2 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.67, size = 32, normalized size = 1.00 \[ -\frac {d^{n - 1} x^{n}}{n} + \frac {\left (d x\right )^{n} \log \left (c x^{n}\right )}{d n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \ln \left (c\,x^n\right )\,{\left (d\,x\right )}^{n-1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*x^n)*(d*x)^(n - 1),x)

[Out]

int(log(c*x^n)*(d*x)^(n - 1), x)

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sympy [A]  time = 13.72, size = 68, normalized size = 2.12 \[ \begin {cases} \tilde {\infty } x \log {\relax (c )} & \text {for}\: d = 0 \wedge n = 0 \\\frac {\log {\relax (c )} \log {\relax (x )}}{d} & \text {for}\: n = 0 \\0^{n - 1} \left (n x \log {\relax (x )} - n x + x \log {\relax (c )}\right ) & \text {for}\: d = 0 \\\frac {d^{n} x^{n} \log {\relax (x )}}{d} + \frac {d^{n} x^{n} \log {\relax (c )}}{d n} - \frac {d^{n} x^{n}}{d n} & \text {otherwise} \end {cases} \]

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))

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