### 3.247 $$\int \frac{\log (a+b x)}{a+b x} \, dx$$

Optimal. Leaf size=15 $\frac{\log ^2(a+b x)}{2 b}$

[Out]

Log[a + b*x]^2/(2*b)

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Rubi [A]  time = 0.0183944, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {2390, 2301} $\frac{\log ^2(a+b x)}{2 b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[a + b*x]/(a + b*x),x]

[Out]

Log[a + b*x]^2/(2*b)

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin{align*} \int \frac{\log (a+b x)}{a+b x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b}\\ &=\frac{\log ^2(a+b x)}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.0016411, size = 15, normalized size = 1. $\frac{\log ^2(a+b x)}{2 b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[a + b*x]/(a + b*x),x]

[Out]

Log[a + b*x]^2/(2*b)

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Maple [A]  time = 0.002, size = 14, normalized size = 0.9 \begin{align*}{\frac{ \left ( \ln \left ( bx+a \right ) \right ) ^{2}}{2\,b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.09039, size = 18, normalized size = 1.2 \begin{align*} \frac{\log \left (b x + a\right )^{2}}{2 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(b*x + a)^2/b

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Fricas [A]  time = 1.78235, size = 30, normalized size = 2. \begin{align*} \frac{\log \left (b x + a\right )^{2}}{2 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(b*x + a)^2/b

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Sympy [A]  time = 0.292422, size = 10, normalized size = 0.67 \begin{align*} \frac{\log{\left (a + b x \right )}^{2}}{2 b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a + b*x)**2/(2*b)

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Giac [A]  time = 1.18051, size = 18, normalized size = 1.2 \begin{align*} \frac{\log \left (b x + a\right )^{2}}{2 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(b*x + a)^2/b