3.184 \(\int \frac{1}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=34 \[ -\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]

[Out]

-(1/(b*x)) - (Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/b^(3/2)

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Rubi [A]  time = 0.0142081, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1593, 325, 205} \[ -\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^(-1),x]

[Out]

-(1/(b*x)) - (Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/b^(3/2)

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{b x^2+c x^4} \, dx &=\int \frac{1}{x^2 \left (b+c x^2\right )} \, dx\\ &=-\frac{1}{b x}-\frac{c \int \frac{1}{b+c x^2} \, dx}{b}\\ &=-\frac{1}{b x}-\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0123175, size = 34, normalized size = 1. \[ -\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^(-1),x]

[Out]

-(1/(b*x)) - (Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/b^(3/2)

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Maple [A]  time = 0.046, size = 30, normalized size = 0.9 \begin{align*} -{\frac{1}{bx}}-{\frac{c}{b}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x^4+b*x^2),x)

[Out]

-1/b/x-c/b/(b*c)^(1/2)*arctan(x*c/(b*c)^(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/(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50907, size = 173, normalized size = 5.09 \begin{align*} \left [\frac{x \sqrt{-\frac{c}{b}} \log \left (\frac{c x^{2} - 2 \, b x \sqrt{-\frac{c}{b}} - b}{c x^{2} + b}\right ) - 2}{2 \, b x}, -\frac{x \sqrt{\frac{c}{b}} \arctan \left (x \sqrt{\frac{c}{b}}\right ) + 1}{b x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.376627, size = 65, normalized size = 1.91 \begin{align*} \frac{\sqrt{- \frac{c}{b^{3}}} \log{\left (- \frac{b^{2} \sqrt{- \frac{c}{b^{3}}}}{c} + x \right )}}{2} - \frac{\sqrt{- \frac{c}{b^{3}}} \log{\left (\frac{b^{2} \sqrt{- \frac{c}{b^{3}}}}{c} + x \right )}}{2} - \frac{1}{b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x**4+b*x**2),x)

[Out]

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

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Giac [A]  time = 1.21555, size = 39, normalized size = 1.15 \begin{align*} -\frac{c \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} b} - \frac{1}{b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^4+b*x^2),x, algorithm="giac")

[Out]

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