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

Optimal. Leaf size=29 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.0104272, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {275, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + c*x^4),x]

[Out]

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

Mathematica [A]  time = 0.0050101, size = 29, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + c*x^4),x]

[Out]

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

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Maple [A]  time = 0.001, size = 19, normalized size = 0.7 \begin{align*}{\frac{1}{2}\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),x)

[Out]

1/2/(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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.74437, size = 166, normalized size = 5.72 \begin{align*} \left [-\frac{\sqrt{-a c} \log \left (\frac{c x^{4} - 2 \, \sqrt{-a c} x^{2} - a}{c x^{4} + a}\right )}{4 \, a c}, -\frac{\sqrt{a c} \arctan \left (\frac{\sqrt{a c}}{c x^{2}}\right )}{2 \, a c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.216372, size = 56, normalized size = 1.93 \begin{align*} - \frac{\sqrt{- \frac{1}{a c}} \log{\left (- a \sqrt{- \frac{1}{a c}} + x^{2} \right )}}{4} + \frac{\sqrt{- \frac{1}{a c}} \log{\left (a \sqrt{- \frac{1}{a c}} + x^{2} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12171, size = 24, normalized size = 0.83 \begin{align*} \frac{\arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{2 \, \sqrt{a c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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