\(\int \log (e^{a+b x^n}) \, dx\) [283]

Optimal result

Integrand size = 10, antiderivative size = 27 \[ \int \log \left (e^{a+b x^n}\right ) \, dx=-\frac {b n x^{1+n}}{1+n}+x \log \left (e^{a+b x^n}\right ) \]

[Out]

-b*n*x^(1+n)/(1+n)+x*ln(exp(a+b*x^n))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2628, 12, 30} \[ \int \log \left (e^{a+b x^n}\right ) \, dx=x \log \left (e^{a+b x^n}\right )-\frac {b n x^{n+1}}{n+1} \]

[In]

Int[Log[E^(a + b*x^n)],x]

[Out]

-((b*n*x^(1 + n))/(1 + n)) + x*Log[E^(a + b*x^n)]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

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

Rule 2628

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/u), x], x] /; InverseFunctionFr
eeQ[u, x]

Rubi steps \begin{align*} \text {integral}& = x \log \left (e^{a+b x^n}\right )-\int b n x^n \, dx \\ & = x \log \left (e^{a+b x^n}\right )-(b n) \int x^n \, dx \\ & = -\frac {b n x^{1+n}}{1+n}+x \log \left (e^{a+b x^n}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \log \left (e^{a+b x^n}\right ) \, dx=x \left (-\frac {b n x^n}{1+n}+\log \left (e^{a+b x^n}\right )\right ) \]

[In]

Integrate[Log[E^(a + b*x^n)],x]

[Out]

x*(-((b*n*x^n)/(1 + n)) + Log[E^(a + b*x^n)])

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96

method	result	size
risch	\(x \ln \left ({\mathrm e}^{a +b \,x^{n}}\right )-\frac {b n x \,x^{n}}{1+n}\)	\(26\)
default	\(-\frac {b n \,x^{1+n}}{1+n}+x \ln \left ({\mathrm e}^{a +b \,x^{n}}\right )\)	\(27\)
parts	\(-\frac {b n \,x^{1+n}}{1+n}+x \ln \left ({\mathrm e}^{a +b \,x^{n}}\right )\)	\(27\)
parallelrisch	\(-\frac {b n \,x^{n} x -\ln \left ({\mathrm e}^{a +b \,x^{n}}\right ) x n -x \ln \left ({\mathrm e}^{a +b \,x^{n}}\right )}{1+n}\)	\(41\)

[In]

int(ln(exp(a+b*x^n)),x,method=_RETURNVERBOSE)

[Out]

x*ln(exp(a+b*x^n))-b*n/(1+n)*x*x^n

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \log \left (e^{a+b x^n}\right ) \, dx=\frac {b x x^{n} + {\left (a n + a\right )} x}{n + 1} \]

[In]

integrate(log(exp(a+b*x^n)),x, algorithm="fricas")

[Out]

(b*x*x^n + (a*n + a)*x)/(n + 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (22) = 44\).

Time = 0.42 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.41 \[ \int \log \left (e^{a+b x^n}\right ) \, dx=\begin {cases} - \frac {b n x x^{n}}{n + 1} + \frac {n x \log {\left (e^{a} e^{b x^{n}} \right )}}{n + 1} + \frac {x \log {\left (e^{a} e^{b x^{n}} \right )}}{n + 1} & \text {for}\: n \neq -1 \\b \log {\left (x \right )} + x \log {\left (e^{a} e^{\frac {b}{x}} \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(ln(exp(a+b*x**n)),x)

[Out]

Piecewise((-b*n*x*x**n/(n + 1) + n*x*log(exp(a)*exp(b*x**n))/(n + 1) + x*log(exp(a)*exp(b*x**n))/(n + 1), Ne(n
, -1)), (b*log(x) + x*log(exp(a)*exp(b/x)), True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.59 \[ \int \log \left (e^{a+b x^n}\right ) \, dx=a x + \frac {b x^{n + 1}}{n + 1} \]

[In]

integrate(log(exp(a+b*x^n)),x, algorithm="maxima")

[Out]

a*x + b*x^(n + 1)/(n + 1)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.59 \[ \int \log \left (e^{a+b x^n}\right ) \, dx=a x + \frac {b x^{n + 1}}{n + 1} \]

[In]

integrate(log(exp(a+b*x^n)),x, algorithm="giac")

[Out]

a*x + b*x^(n + 1)/(n + 1)

Mupad [B] (verification not implemented)

Time = 1.73 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \log \left (e^{a+b x^n}\right ) \, dx=\left \{\begin {array}{cl} x\,\ln \left ({\mathrm {e}}^{a+\frac {b}{x}}\right )+b\,\ln \left (x\right ) & \text {\ if\ \ }n=-1\\ x\,\ln \left ({\mathrm {e}}^{a+b\,x^n}\right )-\frac {b\,n\,x^{n+1}}{n+1} & \text {\ if\ \ }n\neq -1 \end {array}\right . \]

[In]

int(log(exp(a + b*x^n)),x)

[Out]

piecewise(n == -1, x*log(exp(a + b/x)) + b*log(x), n ~= -1, x*log(exp(a + b*x^n)) - (b*n*x^(n + 1))/(n + 1))