3.11.53 \(\int \frac {1}{(a+b x) (a c-b c x)} \, dx\) [1053]

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 = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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, 214} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{a b c} \end {gather*}

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 214

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

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{a b c} \end {gather*}

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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Mathics [A]
time = 1.80, size = 31, normalized size = 1.82 \begin {gather*} \frac {\text {Log}\left [\frac {a}{b}+x\right ]-\text {Log}\left [-\frac {a}{b}+x\right ]}{2 a b c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[a / b + x] - Log[-a / b + x]) / (2 a b c)

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Maple [A]
time = 0.16, size = 35, normalized size = 2.06

method	result	size
default	\(\frac {-\frac {\ln \left (-b x +a \right )}{2 a b}+\frac {\ln \left (b x +a \right )}{2 a b}}{c}\)	\(35\)
norman	\(-\frac {\ln \left (-b x +a \right )}{2 a b c}+\frac {\ln \left (b x +a \right )}{2 a b c}\)	\(37\)
risch	\(-\frac {\ln \left (-b x +a \right )}{2 a b c}+\frac {\ln \left (b x +a \right )}{2 a b c}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (17) = 34\).
time = 0.30, size = 37, normalized size = 2.18 \begin {gather*} \frac {\log \left (b x + a\right )}{2 \, a b c} - \frac {\log \left (b x - a\right )}{2 \, a b c} \end {gather*}

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 = 0.30, size = 28, normalized size = 1.65 \begin {gather*} \frac {\log \left (b x + a\right ) - \log \left (b x - a\right )}{2 \, a b c} \end {gather*}

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 [A]
time = 0.09, size = 22, normalized size = 1.29 \begin {gather*} - \frac {\frac {\log {\left (- \frac {a}{b} + x \right )}}{2} - \frac {\log {\left (\frac {a}{b} + x \right )}}{2}}{a b c} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (17) = 34\).
time = 0.00, size = 37, normalized size = 2.18 \begin {gather*} -\frac {b \ln \left |x b-a\right |}{2 b^{2} a c}+\frac {b \ln \left |x b+a\right |}{2 b^{2} a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.17, size = 17, normalized size = 1.00 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{a\,b\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*c - b*c*x)*(a + b*x)),x)

[Out]

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

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