3.112 \(\int \frac{-x^2+2 x^4}{1+2 x^2} \, dx\)

Optimal. Leaf size=25 \[ \frac{x^3}{3}-x+\frac{\tan ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

[Out]

-x + x^3/3 + ArcTan[Sqrt[2]*x]/Sqrt[2]

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Rubi [A]  time = 0.0241169, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {1593, 459, 321, 203} \[ \frac{x^3}{3}-x+\frac{\tan ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(-x^2 + 2*x^4)/(1 + 2*x^2),x]

[Out]

-x + x^3/3 + ArcTan[Sqrt[2]*x]/Sqrt[2]

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]

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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

Mathematica [A]  time = 0.0101867, size = 25, normalized size = 1. \[ \frac{x^3}{3}-x+\frac{\tan ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-x^2 + 2*x^4)/(1 + 2*x^2),x]

[Out]

-x + x^3/3 + ArcTan[Sqrt[2]*x]/Sqrt[2]

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Maple [A]  time = 0.003, size = 21, normalized size = 0.8 \begin{align*} -x+{\frac{{x}^{3}}{3}}+{\frac{\arctan \left ( x\sqrt{2} \right ) \sqrt{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x+1/3*x^3+1/2*arctan(x*2^(1/2))*2^(1/2)

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Maxima [A]  time = 1.52928, size = 27, normalized size = 1.08 \begin{align*} \frac{1}{3} \, x^{3} + \frac{1}{2} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) - x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 + 1/2*sqrt(2)*arctan(sqrt(2)*x) - x

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Fricas [A]  time = 1.40986, size = 61, normalized size = 2.44 \begin{align*} \frac{1}{3} \, x^{3} + \frac{1}{2} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) - x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4-x^2)/(2*x^2+1),x, algorithm="fricas")

[Out]

1/3*x^3 + 1/2*sqrt(2)*arctan(sqrt(2)*x) - x

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Sympy [A]  time = 0.087976, size = 20, normalized size = 0.8 \begin{align*} \frac{x^{3}}{3} - x + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**4-x**2)/(2*x**2+1),x)

[Out]

x**3/3 - x + sqrt(2)*atan(sqrt(2)*x)/2

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Giac [A]  time = 1.17975, size = 27, normalized size = 1.08 \begin{align*} \frac{1}{3} \, x^{3} + \frac{1}{2} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) - x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4-x^2)/(2*x^2+1),x, algorithm="giac")

[Out]

1/3*x^3 + 1/2*sqrt(2)*arctan(sqrt(2)*x) - x