3.14.95 \(\int \frac {1}{(a-b x) (a+b x)} \, dx\)

Optimal. Leaf size=14 \[ \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{a b} \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {35, 208} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{a b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a - b*x)*(a + b*x)),x]

[Out]

ArcTanh[(b*x)/a]/(a*b)

Rule 35

Int[1/(((a_) + (b_.)*(x_))*((c_) + (d_.)*(x_))), x_Symbol] :> Int[1/(a*c + b*d*x^2), x] /; FreeQ[{a, b, c, d},
x] && EqQ[b*c + a*d, 0]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{a b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a - b*x)*(a + b*x)),x]

[Out]

ArcTanh[(b*x)/a]/(a*b)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(a-b x) (a+b x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/((a - b*x)*(a + b*x)),x]

[Out]

IntegrateAlgebraic[1/((a - b*x)*(a + b*x)), x]

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fricas [A]  time = 1.14, size = 25, normalized size = 1.79 \begin {gather*} \frac {\log \left (b x + a\right ) - \log \left (b x - a\right )}{2 \, a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x+a)/(b*x+a),x, algorithm="fricas")

[Out]

1/2*(log(b*x + a) - log(b*x - a))/(a*b)

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giac [B]  time = 0.96, size = 33, normalized size = 2.36 \begin {gather*} \frac {\log \left ({\left | b x + a \right |}\right )}{2 \, a b} - \frac {\log \left ({\left | b x - a \right |}\right )}{2 \, a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x+a)/(b*x+a),x, algorithm="giac")

[Out]

1/2*log(abs(b*x + a))/(a*b) - 1/2*log(abs(b*x - a))/(a*b)

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maple [B]  time = 0.01, size = 32, normalized size = 2.29 \begin {gather*} -\frac {\ln \left (b x -a \right )}{2 a b}+\frac {\ln \left (b x +a \right )}{2 a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-b*x+a)/(b*x+a),x)

[Out]

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

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maxima [B]  time = 0.53, size = 31, normalized size = 2.21 \begin {gather*} \frac {\log \left (b x + a\right )}{2 \, a b} - \frac {\log \left (b x - a\right )}{2 \, a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x+a)/(b*x+a),x, algorithm="maxima")

[Out]

1/2*log(b*x + a)/(a*b) - 1/2*log(b*x - a)/(a*b)

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mupad [B]  time = 0.04, size = 14, normalized size = 1.00 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{a\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x)*(a - b*x)),x)

[Out]

atanh((b*x)/a)/(a*b)

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sympy [B]  time = 0.15, size = 20, normalized size = 1.43 \begin {gather*} - \frac {\frac {\log {\left (- \frac {a}{b} + x \right )}}{2} - \frac {\log {\left (\frac {a}{b} + x \right )}}{2}}{a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x+a)/(b*x+a),x)

[Out]

-(log(-a/b + x)/2 - log(a/b + x)/2)/(a*b)

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