### 3.752 $$\int \frac{a+b x}{a^2-b^2 x^2} \, dx$$

Optimal. Leaf size=12 $-\frac{\log (a-b x)}{b}$

[Out]

-(Log[a - b*x]/b)

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Rubi [A]  time = 0.0044518, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.1, Rules used = {627, 31} $-\frac{\log (a-b x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x)/(a^2 - b^2*x^2),x]

[Out]

-(Log[a - b*x]/b)

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{a+b x}{a^2-b^2 x^2} \, dx &=\int \frac{1}{a-b x} \, dx\\ &=-\frac{\log (a-b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0011062, size = 12, normalized size = 1. $-\frac{\log (a-b x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x)/(a^2 - b^2*x^2),x]

[Out]

-(Log[a - b*x]/b)

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Maple [A]  time = 0.038, size = 14, normalized size = 1.2 \begin{align*} -{\frac{\ln \left ( bx-a \right ) }{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/(-b^2*x^2+a^2),x)

[Out]

-1/b*ln(b*x-a)

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Maxima [A]  time = 1.40966, size = 18, normalized size = 1.5 \begin{align*} -\frac{\log \left (b x - a\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b^2*x^2+a^2),x, algorithm="maxima")

[Out]

-log(b*x - a)/b

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Fricas [A]  time = 1.73668, size = 23, normalized size = 1.92 \begin{align*} -\frac{\log \left (b x - a\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b^2*x^2+a^2),x, algorithm="fricas")

[Out]

-log(b*x - a)/b

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Sympy [A]  time = 0.067708, size = 8, normalized size = 0.67 \begin{align*} - \frac{\log{\left (- a + b x \right )}}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b**2*x**2+a**2),x)

[Out]

-log(-a + b*x)/b

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Giac [A]  time = 1.20268, size = 19, normalized size = 1.58 \begin{align*} -\frac{\log \left ({\left | b x - a \right |}\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b^2*x^2+a^2),x, algorithm="giac")

[Out]

-log(abs(b*x - a))/b