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

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

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{3/2}}-\frac {1}{b x} \end {gather*}

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 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]

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 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]

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*}

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Mathematica [A]  time = 0.01, size = 34, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{3/2}}-\frac {1}{b x} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{b x^2+c x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b*x^2 + c*x^4)^(-1),x]

[Out]

IntegrateAlgebraic[(b*x^2 + c*x^4)^(-1), x]

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fricas [A]  time = 0.60, size = 82, normalized size = 2.41 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.16, size = 29, normalized size = 0.85 \begin {gather*} -\frac {c \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} b} - \frac {1}{b x} \end {gather*}

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)

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maple [A]  time = 0.00, size = 30, normalized size = 0.88 \begin {gather*} -\frac {c \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}\, b}-\frac {1}{b x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c/b/(b*c)^(1/2)*arctan(1/(b*c)^(1/2)*c*x)-1/b/x

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maxima [A]  time = 2.88, size = 29, normalized size = 0.85 \begin {gather*} -\frac {c \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} b} - \frac {1}{b x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.27, size = 26, normalized size = 0.76 \begin {gather*} -\frac {1}{b\,x}-\frac {\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{b^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2 + c*x^4),x)

[Out]

- 1/(b*x) - (c^(1/2)*atan((c^(1/2)*x)/b^(1/2)))/b^(3/2)

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sympy [B]  time = 0.19, size = 65, normalized size = 1.91 \begin {gather*} \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 {gather*}

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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