3.663 \(\int \frac{x}{(a+c x^4)^2} \, dx\)

Optimal. Leaf size=49 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}+\frac{x^2}{4 a \left (a+c x^4\right )} \]

[Out]

x^2/(4*a*(a + c*x^4)) + ArcTan[(Sqrt[c]*x^2)/Sqrt[a]]/(4*a^(3/2)*Sqrt[c])

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Rubi [A]  time = 0.0210504, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {275, 199, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}+\frac{x^2}{4 a \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + c*x^4)^2,x]

[Out]

x^2/(4*a*(a + c*x^4)) + ArcTan[(Sqrt[c]*x^2)/Sqrt[a]]/(4*a^(3/2)*Sqrt[c])

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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

Mathematica [A]  time = 0.0255249, size = 49, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}+\frac{x^2}{4 a \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + c*x^4)^2,x]

[Out]

x^2/(4*a*(a + c*x^4)) + ArcTan[(Sqrt[c]*x^2)/Sqrt[a]]/(4*a^(3/2)*Sqrt[c])

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Maple [A]  time = 0.006, size = 40, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{4\,a \left ( c{x}^{4}+a \right ) }}+{\frac{1}{4\,a}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.80212, size = 274, normalized size = 5.59 \begin{align*} \left [\frac{2 \, a c x^{2} -{\left (c x^{4} + a\right )} \sqrt{-a c} \log \left (\frac{c x^{4} - 2 \, \sqrt{-a c} x^{2} - a}{c x^{4} + a}\right )}{8 \,{\left (a^{2} c^{2} x^{4} + a^{3} c\right )}}, \frac{a c x^{2} -{\left (c x^{4} + a\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c}}{c x^{2}}\right )}{4 \,{\left (a^{2} c^{2} x^{4} + a^{3} c\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(2*a*c*x^2 - (c*x^4 + a)*sqrt(-a*c)*log((c*x^4 - 2*sqrt(-a*c)*x^2 - a)/(c*x^4 + a)))/(a^2*c^2*x^4 + a^3*c
), 1/4*(a*c*x^2 - (c*x^4 + a)*sqrt(a*c)*arctan(sqrt(a*c)/(c*x^2)))/(a^2*c^2*x^4 + a^3*c)]

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Sympy [B]  time = 0.613966, size = 83, normalized size = 1.69 \begin{align*} \frac{x^{2}}{4 a^{2} + 4 a c x^{4}} - \frac{\sqrt{- \frac{1}{a^{3} c}} \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} c}} + x^{2} \right )}}{8} + \frac{\sqrt{- \frac{1}{a^{3} c}} \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} c}} + x^{2} \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(c*x**4+a)**2,x)

[Out]

x**2/(4*a**2 + 4*a*c*x**4) - sqrt(-1/(a**3*c))*log(-a**2*sqrt(-1/(a**3*c)) + x**2)/8 + sqrt(-1/(a**3*c))*log(a
**2*sqrt(-1/(a**3*c)) + x**2)/8

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Giac [A]  time = 1.16712, size = 53, normalized size = 1.08 \begin{align*} \frac{x^{2}}{4 \,{\left (c x^{4} + a\right )} a} + \frac{\arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \, \sqrt{a c} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^2/((c*x^4 + a)*a) + 1/4*arctan(c*x^2/sqrt(a*c))/(sqrt(a*c)*a)