### 3.283 $$\int \log (e^{a+b x^n}) \, dx$$

Optimal. Leaf size=27 $x \log \left (e^{a+b x^n}\right )-\frac{b n x^{n+1}}{n+1}$

[Out]

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

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Rubi [A]  time = 0.0085458, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.3, Rules used = {2548, 12, 30} $x \log \left (e^{a+b x^n}\right )-\frac{b n x^{n+1}}{n+1}$

Antiderivative was successfully veriﬁed.

[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 2548

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

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]

Rubi steps

\begin{align*} \int \log \left (e^{a+b x^n}\right ) \, dx &=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]  time = 0.0237789, size = 25, normalized size = 0.93 $x \left (\log \left (e^{a+b x^n}\right )-\frac{b n x^n}{n+1}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[E^(a + b*x^n)],x]

[Out]

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

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Maple [A]  time = 0.007, size = 27, normalized size = 1. \begin{align*} -{\frac{bn{x}^{1+n}}{1+n}}+x\ln \left ({{\rm e}^{a+b{x}^{n}}} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.08307, size = 45, normalized size = 1.67 \begin{align*} \frac{b x x^{n} +{\left (a n + a\right )} x}{n + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x*x^n + (a*n + a)*x)/(n + 1)

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Sympy [A]  time = 2.81085, size = 34, normalized size = 1.26 \begin{align*} \begin{cases} \frac{a n x}{n + 1} + \frac{a x}{n + 1} + \frac{b x x^{n}}{n + 1} & \text{for}\: n \neq -1 \\a x + b \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.37261, size = 22, normalized size = 0.81 \begin{align*} a x + \frac{b x^{n + 1}}{n + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*x^(n + 1)/(n + 1)