3.1053 \(\int \frac{1}{(a+b x) (a c-b c x)} \, dx\)

Optimal. Leaf size=17 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{a b c} \]

[Out]

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

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Rubi [A]  time = 0.0085704, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {35, 208} \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{a b c} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)*(a*c - b*c*x)),x]

[Out]

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

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

Mathematica [A]  time = 0.007866, size = 17, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{a b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.004, size = 38, normalized size = 2.2 \begin{align*}{\frac{\ln \left ( bx+a \right ) }{2\,bca}}-{\frac{\ln \left ( bx-a \right ) }{2\,bca}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.03954, size = 50, normalized size = 2.94 \begin{align*} \frac{\log \left (b x + a\right )}{2 \, a b c} - \frac{\log \left (b x - a\right )}{2 \, a b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.42231, size = 58, normalized size = 3.41 \begin{align*} \frac{\log \left (b x + a\right ) - \log \left (b x - a\right )}{2 \, a b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.158794, size = 22, normalized size = 1.29 \begin{align*} - \frac{\frac{\log{\left (- \frac{a}{b} + x \right )}}{2} - \frac{\log{\left (\frac{a}{b} + x \right )}}{2}}{a b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.06458, size = 53, normalized size = 3.12 \begin{align*} \frac{\log \left ({\left | b x + a \right |}\right )}{2 \, a b c} - \frac{\log \left ({\left | b x - a \right |}\right )}{2 \, a b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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