3.282 \(\int \log (e^{a+b x}) \, dx\)

Optimal. Leaf size=17 \[ \frac{\log ^2\left (e^{a+b x}\right )}{2 b} \]

[Out]

Log[E^(a + b*x)]^2/(2*b)

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Rubi [A]  time = 0.003246, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2157, 30} \[ \frac{\log ^2\left (e^{a+b x}\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Log[E^(a + b*x)],x]

[Out]

Log[E^(a + b*x)]^2/(2*b)

Rule 2157

Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Dist[1/c, Subst[Int[x^m, x], x, u], x]] /; FreeQ[m,
x] && PiecewiseLinearQ[u, 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}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int x \, dx,x,\log \left (e^{a+b x}\right )\right )}{b}\\ &=\frac{\log ^2\left (e^{a+b x}\right )}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.0026791, size = 17, normalized size = 1. \[ \frac{\log ^2\left (e^{a+b x}\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[E^(a + b*x)],x]

[Out]

Log[E^(a + b*x)]^2/(2*b)

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Maple [A]  time = 0.003, size = 15, normalized size = 0.9 \begin{align*}{\frac{ \left ( \ln \left ({{\rm e}^{bx+a}} \right ) \right ) ^{2}}{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*ln(exp(b*x+a))^2/b

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Maxima [A]  time = 1.08409, size = 14, normalized size = 0.82 \begin{align*} \frac{1}{2} \, b x^{2} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2 + a*x

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Fricas [A]  time = 1.93074, size = 23, normalized size = 1.35 \begin{align*} \frac{1}{2} \, b x^{2} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2 + a*x

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Sympy [A]  time = 0.088424, size = 8, normalized size = 0.47 \begin{align*} a x + \frac{b x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*x**2/2

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Giac [A]  time = 1.2954, size = 14, normalized size = 0.82 \begin{align*} \frac{1}{2} \, b x^{2} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2 + a*x