3.1533 \(\int \frac {c+d x}{(a-b x) (a+b x)} \, dx\)

Optimal. Leaf size=49 \[ \frac {(b c-a d) \log (a+b x)}{2 a b^2}-\frac {(a d+b c) \log (a-b x)}{2 a b^2} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {72} \[ \frac {(b c-a d) \log (a+b x)}{2 a b^2}-\frac {(a d+b c) \log (a-b x)}{2 a b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/((a - b*x)*(a + b*x)),x]

[Out]

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

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 37, normalized size = 0.76 \[ \frac {c \tanh ^{-1}\left (\frac {b x}{a}\right )}{a b}-\frac {d \log \left (a^2-b^2 x^2\right )}{2 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*ArcTanh[(b*x)/a])/(a*b) - (d*Log[a^2 - b^2*x^2])/(2*b^2)

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fricas [A]  time = 0.81, size = 41, normalized size = 0.84 \[ \frac {{\left (b c - a d\right )} \log \left (b x + a\right ) - {\left (b c + a d\right )} \log \left (b x - a\right )}{2 \, a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.93, size = 48, normalized size = 0.98 \[ \frac {{\left (b c - a d\right )} \log \left ({\left | b x + a \right |}\right )}{2 \, a b^{2}} - \frac {{\left (b c + a d\right )} \log \left ({\left | b x - a \right |}\right )}{2 \, a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 60, normalized size = 1.22 \[ -\frac {c \ln \left (b x -a \right )}{2 a b}+\frac {c \ln \left (b x +a \right )}{2 a b}-\frac {d \ln \left (b x -a \right )}{2 b^{2}}-\frac {d \ln \left (b x +a \right )}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.48, size = 46, normalized size = 0.94 \[ \frac {{\left (b c - a d\right )} \log \left (b x + a\right )}{2 \, a b^{2}} - \frac {{\left (b c + a d\right )} \log \left (b x - a\right )}{2 \, a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.11, size = 45, normalized size = 0.92 \[ -\frac {\ln \left (a+b\,x\right )\,\left (a\,d-b\,c\right )}{2\,a\,b^2}-\frac {\ln \left (a-b\,x\right )\,\left (a\,d+b\,c\right )}{2\,a\,b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.32, size = 71, normalized size = 1.45 \[ - \frac {\left (a d - b c\right ) \log {\left (x + \frac {a^{2} d - a \left (a d - b c\right )}{b^{2} c} \right )}}{2 a b^{2}} - \frac {\left (a d + b c\right ) \log {\left (x + \frac {a^{2} d - a \left (a d + b c\right )}{b^{2} c} \right )}}{2 a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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