3.54 \(\int \frac{1}{(a+c x^2)^2} \, dx\)

Optimal. Leaf size=45 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}+\frac{x}{2 a \left (a+c x^2\right )} \]

[Out]

x/(2*a*(a + c*x^2)) + ArcTan[(Sqrt[c]*x)/Sqrt[a]]/(2*a^(3/2)*Sqrt[c])

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Rubi [A]  time = 0.009972, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {199, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}+\frac{x}{2 a \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(-2),x]

[Out]

x/(2*a*(a + c*x^2)) + ArcTan[(Sqrt[c]*x)/Sqrt[a]]/(2*a^(3/2)*Sqrt[c])

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

Mathematica [A]  time = 0.026729, size = 45, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}+\frac{x}{2 a \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^(-2),x]

[Out]

x/(2*a*(a + c*x^2)) + ArcTan[(Sqrt[c]*x)/Sqrt[a]]/(2*a^(3/2)*Sqrt[c])

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Maple [A]  time = 0.054, size = 36, normalized size = 0.8 \begin{align*}{\frac{x}{2\,a \left ( c{x}^{2}+a \right ) }}+{\frac{1}{2\,a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.16386, size = 261, normalized size = 5.8 \begin{align*} \left [\frac{2 \, a c x -{\left (c x^{2} + a\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right )}{4 \,{\left (a^{2} c^{2} x^{2} + a^{3} c\right )}}, \frac{a c x +{\left (c x^{2} + a\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right )}{2 \,{\left (a^{2} c^{2} x^{2} + a^{3} c\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.392589, size = 78, normalized size = 1.73 \begin{align*} \frac{x}{2 a^{2} + 2 a c x^{2}} - \frac{\sqrt{- \frac{1}{a^{3} c}} \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} c}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} c}} \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} c}} + x \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x**2+a)**2,x)

[Out]

x/(2*a**2 + 2*a*c*x**2) - sqrt(-1/(a**3*c))*log(-a**2*sqrt(-1/(a**3*c)) + x)/4 + sqrt(-1/(a**3*c))*log(a**2*sq
rt(-1/(a**3*c)) + x)/4

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Giac [A]  time = 1.29076, size = 47, normalized size = 1.04 \begin{align*} \frac{\arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a} + \frac{x}{2 \,{\left (c x^{2} + a\right )} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a) + 1/2*x/((c*x^2 + a)*a)