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

Optimal. Leaf size=33 \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a x} \]

[Out]

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

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Rubi [A]  time = 0.0121497, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {325, 208} \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a - b*x^2)),x]

[Out]

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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

Mathematica [A]  time = 0.0109404, size = 33, normalized size = 1. \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a - b*x^2)),x]

[Out]

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

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Maple [A]  time = 0.004, size = 29, normalized size = 0.9 \begin{align*}{\frac{b}{a}{\it Artanh} \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{ax}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.2156, size = 173, normalized size = 5.24 \begin{align*} \left [\frac{x \sqrt{\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{\frac{b}{a}} + a}{b x^{2} - a}\right ) - 2}{2 \, a x}, -\frac{x \sqrt{-\frac{b}{a}} \arctan \left (x \sqrt{-\frac{b}{a}}\right ) + 1}{a x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.329602, size = 58, normalized size = 1.76 \begin{align*} - \frac{\sqrt{\frac{b}{a^{3}}} \log{\left (- \frac{a^{2} \sqrt{\frac{b}{a^{3}}}}{b} + x \right )}}{2} + \frac{\sqrt{\frac{b}{a^{3}}} \log{\left (\frac{a^{2} \sqrt{\frac{b}{a^{3}}}}{b} + x \right )}}{2} - \frac{1}{a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.93039, size = 42, normalized size = 1.27 \begin{align*} -\frac{b \arctan \left (\frac{b x}{\sqrt{-a b}}\right )}{\sqrt{-a b} a} - \frac{1}{a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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