3.753 \(\int \frac{1}{(a+b x) (a^2-b^2 x^2)} \, dx\)

Optimal. Leaf size=35 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b}-\frac{1}{2 a b (a+b x)} \]

[Out]

-1/(2*a*b*(a + b*x)) + ArcTanh[(b*x)/a]/(2*a^2*b)

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Rubi [A]  time = 0.0308144, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {627, 44, 208} \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b}-\frac{1}{2 a b (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)*(a^2 - b^2*x^2)),x]

[Out]

-1/(2*a*b*(a + b*x)) + ArcTanh[(b*x)/a]/(2*a^2*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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 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) \left (a^2-b^2 x^2\right )} \, dx &=\int \frac{1}{(a-b x) (a+b x)^2} \, dx\\ &=\int \left (\frac{1}{2 a (a+b x)^2}+\frac{1}{2 a \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=-\frac{1}{2 a b (a+b x)}+\frac{\int \frac{1}{a^2-b^2 x^2} \, dx}{2 a}\\ &=-\frac{1}{2 a b (a+b x)}+\frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b}\\ \end{align*}

Mathematica [A]  time = 0.0114628, size = 47, normalized size = 1.34 \[ \frac{-(a+b x) \log (a-b x)+(a+b x) \log (a+b x)-2 a}{4 a^2 b (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)*(a^2 - b^2*x^2)),x]

[Out]

(-2*a - (a + b*x)*Log[a - b*x] + (a + b*x)*Log[a + b*x])/(4*a^2*b*(a + b*x))

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Maple [A]  time = 0.047, size = 47, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( bx+a \right ) }{4\,b{a}^{2}}}-{\frac{1}{2\,ab \left ( bx+a \right ) }}-{\frac{\ln \left ( bx-a \right ) }{4\,b{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)/(-b^2*x^2+a^2),x)

[Out]

1/4/b/a^2*ln(b*x+a)-1/2/a/b/(b*x+a)-1/4/b/a^2*ln(b*x-a)

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Maxima [A]  time = 1.34774, size = 63, normalized size = 1.8 \begin{align*} -\frac{1}{2 \,{\left (a b^{2} x + a^{2} b\right )}} + \frac{\log \left (b x + a\right )}{4 \, a^{2} b} - \frac{\log \left (b x - a\right )}{4 \, a^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(-b^2*x^2+a^2),x, algorithm="maxima")

[Out]

-1/2/(a*b^2*x + a^2*b) + 1/4*log(b*x + a)/(a^2*b) - 1/4*log(b*x - a)/(a^2*b)

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Fricas [A]  time = 1.73136, size = 109, normalized size = 3.11 \begin{align*} \frac{{\left (b x + a\right )} \log \left (b x + a\right ) -{\left (b x + a\right )} \log \left (b x - a\right ) - 2 \, a}{4 \,{\left (a^{2} b^{2} x + a^{3} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(-b^2*x^2+a^2),x, algorithm="fricas")

[Out]

1/4*((b*x + a)*log(b*x + a) - (b*x + a)*log(b*x - a) - 2*a)/(a^2*b^2*x + a^3*b)

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Sympy [A]  time = 0.395931, size = 39, normalized size = 1.11 \begin{align*} - \frac{1}{2 a^{2} b + 2 a b^{2} x} - \frac{\frac{\log{\left (- \frac{a}{b} + x \right )}}{4} - \frac{\log{\left (\frac{a}{b} + x \right )}}{4}}{a^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(-b**2*x**2+a**2),x)

[Out]

-1/(2*a**2*b + 2*a*b**2*x) - (log(-a/b + x)/4 - log(a/b + x)/4)/(a**2*b)

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Giac [A]  time = 1.20726, size = 65, normalized size = 1.86 \begin{align*} \frac{\log \left ({\left | b x + a \right |}\right )}{4 \, a^{2} b} - \frac{\log \left ({\left | b x - a \right |}\right )}{4 \, a^{2} b} - \frac{1}{2 \,{\left (b x + a\right )} a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(-b^2*x^2+a^2),x, algorithm="giac")

[Out]

1/4*log(abs(b*x + a))/(a^2*b) - 1/4*log(abs(b*x - a))/(a^2*b) - 1/2/((b*x + a)*a*b)