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

Optimal. Leaf size=46 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{x}{2 a \left (a-b x^2\right )} \]

[Out]

x/(2*a*(a - b*x^2)) + ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(2*a^(3/2)*Sqrt[b])

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Rubi [A]  time = 0.0122472, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {199, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{x}{2 a \left (a-b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^2)^(-2),x]

[Out]

x/(2*a*(a - b*x^2)) + ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(2*a^(3/2)*Sqrt[b])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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}{\left (a-b x^2\right )^2} \, dx &=\frac{x}{2 a \left (a-b x^2\right )}+\frac{\int \frac{1}{a-b x^2} \, dx}{2 a}\\ &=\frac{x}{2 a \left (a-b x^2\right )}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0163193, size = 47, normalized size = 1.02 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}-\frac{x}{2 a \left (b x^2-a\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^2)^(-2),x]

[Out]

-x/(2*a*(-a + b*x^2)) + ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(2*a^(3/2)*Sqrt[b])

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Maple [A]  time = 0.003, size = 38, normalized size = 0.8 \begin{align*} -{\frac{x}{2\,a \left ( b{x}^{2}-a \right ) }}+{\frac{1}{2\,a}{\it Artanh} \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.25269, size = 263, normalized size = 5.72 \begin{align*} \left [-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{a b} \log \left (\frac{b x^{2} + 2 \, \sqrt{a b} x + a}{b x^{2} - a}\right )}{4 \,{\left (a^{2} b^{2} x^{2} - a^{3} b\right )}}, -\frac{a b x +{\left (b x^{2} - a\right )} \sqrt{-a b} \arctan \left (\frac{\sqrt{-a b} x}{a}\right )}{2 \,{\left (a^{2} b^{2} x^{2} - a^{3} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(2*a*b*x - (b*x^2 - a)*sqrt(a*b)*log((b*x^2 + 2*sqrt(a*b)*x + a)/(b*x^2 - a)))/(a^2*b^2*x^2 - a^3*b), -1
/2*(a*b*x + (b*x^2 - a)*sqrt(-a*b)*arctan(sqrt(-a*b)*x/a))/(a^2*b^2*x^2 - a^3*b)]

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Sympy [A]  time = 0.375821, size = 71, normalized size = 1.54 \begin{align*} - \frac{x}{- 2 a^{2} + 2 a b x^{2}} - \frac{\sqrt{\frac{1}{a^{3} b}} \log{\left (- a^{2} \sqrt{\frac{1}{a^{3} b}} + x \right )}}{4} + \frac{\sqrt{\frac{1}{a^{3} b}} \log{\left (a^{2} \sqrt{\frac{1}{a^{3} b}} + x \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(-2*a**2 + 2*a*b*x**2) - sqrt(1/(a**3*b))*log(-a**2*sqrt(1/(a**3*b)) + x)/4 + sqrt(1/(a**3*b))*log(a**2*sqr
t(1/(a**3*b)) + x)/4

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Giac [A]  time = 2.05377, size = 53, normalized size = 1.15 \begin{align*} -\frac{\arctan \left (\frac{b x}{\sqrt{-a b}}\right )}{2 \, \sqrt{-a b} a} - \frac{x}{2 \,{\left (b x^{2} - a\right )} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arctan(b*x/sqrt(-a*b))/(sqrt(-a*b)*a) - 1/2*x/((b*x^2 - a)*a)