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]

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

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 30

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

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]

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*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ \frac {\log ^2\left (e^{a+b x}\right )}{2 b} \]

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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fricas [A]  time = 0.39, size = 10, normalized size = 0.59 \[ \frac {1}{2} \, b x^{2} + a x \]

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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giac [A]  time = 0.17, size = 10, normalized size = 0.59 \[ \frac {1}{2} \, b x^{2} + a x \]

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

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maple [A]  time = 0.06, size = 15, normalized size = 0.88 \[ \frac {\ln \left ({\mathrm e}^{b x +a}\right )^{2}}{2 b} \]

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 = 0.62, size = 10, normalized size = 0.59 \[ \frac {1}{2} \, b x^{2} + a x \]

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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mupad [B]  time = 0.07, size = 17, normalized size = 1.00 \[ x\,\ln \left ({\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\right )-\frac {b\,x^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(exp(a + b*x)),x)

[Out]

x*log(exp(b*x)*exp(a)) - (b*x^2)/2

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sympy [A]  time = 0.08, size = 8, normalized size = 0.47 \[ a x + \frac {b x^{2}}{2} \]

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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