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+arctanh(x*b^(1/2)/a^(1/2))*b^(1/2)/a^(3/2)

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 1.00 \[ \frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x} \]

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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fricas [A]  time = 0.84, size = 82, normalized size = 2.48 \[ \left [\frac {x \sqrt {\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {\frac {b}{a}} + a}{b x^{2} - a}\right ) - 2}{2 \, a x}, -\frac {x \sqrt {-\frac {b}{a}} \arctan \left (x \sqrt {-\frac {b}{a}}\right ) + 1}{a x}\right ] \]

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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giac [A]  time = 0.62, size = 31, normalized size = 0.94 \[ -\frac {b \arctan \left (\frac {b x}{\sqrt {-a b}}\right )}{\sqrt {-a b} a} - \frac {1}{a x} \]

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)

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maple [A]  time = 0.00, size = 29, normalized size = 0.88 \[ \frac {b \arctanh \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}-\frac {1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.89, size = 44, normalized size = 1.33 \[ -\frac {b \log \left (\frac {b x - \sqrt {a b}}{b x + \sqrt {a b}}\right )}{2 \, \sqrt {a b} a} - \frac {1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.61, size = 25, normalized size = 0.76 \[ \frac {\sqrt {b}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a - b*x^2)),x)

[Out]

(b^(1/2)*atanh((b^(1/2)*x)/a^(1/2)))/a^(3/2) - 1/(a*x)

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sympy [B]  time = 0.19, size = 58, normalized size = 1.76 \[ - \frac {\sqrt {\frac {b}{a^{3}}} \log {\left (- \frac {a^{2} \sqrt {\frac {b}{a^{3}}}}{b} + x \right )}}{2} + \frac {\sqrt {\frac {b}{a^{3}}} \log {\left (\frac {a^{2} \sqrt {\frac {b}{a^{3}}}}{b} + x \right )}}{2} - \frac {1}{a x} \]

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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