3.711 \(\int \frac{x}{2 a+2 b+x^4} \, dx\)

Optimal. Leaf size=33 \[ \frac{\tan ^{-1}\left (\frac{x^2}{\sqrt{2} \sqrt{a+b}}\right )}{2 \sqrt{2} \sqrt{a+b}} \]

[Out]

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

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Rubi [A]  time = 0.0223402, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {275, 203} \[ \frac{\tan ^{-1}\left (\frac{x^2}{\sqrt{2} \sqrt{a+b}}\right )}{2 \sqrt{2} \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

Int[x/(2*a + 2*b + x^4),x]

[Out]

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

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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0082419, size = 33, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{x^2}{\sqrt{2} \sqrt{a+b}}\right )}{2 \sqrt{2} \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(2*a + 2*b + x^4),x]

[Out]

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

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Maple [A]  time = 0.003, size = 26, normalized size = 0.8 \begin{align*}{\frac{1}{2}\arctan \left ({{x}^{2}{\frac{1}{\sqrt{2\,a+2\,b}}}} \right ){\frac{1}{\sqrt{2\,a+2\,b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.49796, size = 230, normalized size = 6.97 \begin{align*} \left [-\frac{\sqrt{-2 \, a - 2 \, b} \log \left (\frac{x^{4} - 2 \, \sqrt{-2 \, a - 2 \, b} x^{2} - 2 \, a - 2 \, b}{x^{4} + 2 \, a + 2 \, b}\right )}{8 \,{\left (a + b\right )}}, \frac{\sqrt{2 \, a + 2 \, b} \arctan \left (\frac{\sqrt{2 \, a + 2 \, b} x^{2}}{2 \,{\left (a + b\right )}}\right )}{4 \,{\left (a + b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.205193, size = 110, normalized size = 3.33 \begin{align*} - \frac{\sqrt{2} \sqrt{- \frac{1}{a + b}} \log{\left (- \sqrt{2} a \sqrt{- \frac{1}{a + b}} - \sqrt{2} b \sqrt{- \frac{1}{a + b}} + x^{2} \right )}}{8} + \frac{\sqrt{2} \sqrt{- \frac{1}{a + b}} \log{\left (\sqrt{2} a \sqrt{- \frac{1}{a + b}} + \sqrt{2} b \sqrt{- \frac{1}{a + b}} + x^{2} \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*sqrt(-1/(a + b))*log(-sqrt(2)*a*sqrt(-1/(a + b)) - sqrt(2)*b*sqrt(-1/(a + b)) + x**2)/8 + sqrt(2)*sqr
t(-1/(a + b))*log(sqrt(2)*a*sqrt(-1/(a + b)) + sqrt(2)*b*sqrt(-1/(a + b)) + x**2)/8

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Giac [A]  time = 1.11739, size = 32, normalized size = 0.97 \begin{align*} \frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} x^{2}}{2 \, \sqrt{a + b}}\right )}{4 \, \sqrt{a + b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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