3.3.70 \(\int (1-3 x^2+x^4)^n \, dx\) [270]

Optimal. Leaf size=99 \[ x \left (1-\frac {2 x^2}{3-\sqrt {5}}\right )^{-n} \left (1-\frac {2 x^2}{3+\sqrt {5}}\right )^{-n} \left (1-3 x^2+x^4\right )^n \operatorname {AppellF1}\left (\frac {1}{2},-n,-n,\frac {3}{2},\frac {2 x^2}{3+\sqrt {5}},\frac {2 x^2}{3-\sqrt {5}}\right ) \]

[Out]

x*(x^4-3*x^2+1)^n*AppellF1(1/2,-n,-n,3/2,2*x^2/(3-5^(1/2)),2*x^2/(3+5^(1/2)))/((1-2*x^2/(3-5^(1/2)))^n)/((1-2*
x^2/(3+5^(1/2)))^n)

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Rubi [A]
time = 0.03, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1119, 440} \begin {gather*} x \left (1-\frac {2 x^2}{3-\sqrt {5}}\right )^{-n} \left (1-\frac {2 x^2}{3+\sqrt {5}}\right )^{-n} \left (x^4-3 x^2+1\right )^n \operatorname {AppellF1}\left (\frac {1}{2},-n,-n,\frac {3}{2},\frac {2 x^2}{3+\sqrt {5}},\frac {2 x^2}{3-\sqrt {5}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 3*x^2 + x^4)^n,x]

[Out]

(x*(1 - 3*x^2 + x^4)^n*AppellF1[1/2, -n, -n, 3/2, (2*x^2)/(3 + Sqrt[5]), (2*x^2)/(3 - Sqrt[5])])/((1 - (2*x^2)
/(3 - Sqrt[5]))^n*(1 - (2*x^2)/(3 + Sqrt[5]))^n)

Rule 440

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 1119

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[a^IntPart[p]*(
(a + b*x^2 + c*x^4)^FracPart[p]/((1 + 2*c*(x^2/(b + q)))^FracPart[p]*(1 + 2*c*(x^2/(b - q)))^FracPart[p])), In
t[(1 + 2*c*(x^2/(b + q)))^p*(1 + 2*c*(x^2/(b - q)))^p, x], x]] /; FreeQ[{a, b, c, p}, x] && NeQ[b^2 - 4*a*c, 0
]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\left (\left (1+\frac {2 x^2}{-3-\sqrt {5}}\right )^{-n} \left (1+\frac {2 x^2}{-3+\sqrt {5}}\right )^{-n} \left (1-3 x^2+x^4\right )^n\right ) \int \left (1+\frac {2 x^2}{-3-\sqrt {5}}\right )^n \left (1+\frac {2 x^2}{-3+\sqrt {5}}\right )^n \, dx\\ &=x \left (1-\frac {2 x^2}{3-\sqrt {5}}\right )^{-n} \left (1-\frac {2 x^2}{3+\sqrt {5}}\right )^{-n} \left (1-3 x^2+x^4\right )^n \operatorname {AppellF1}\left (\frac {1}{2},-n,-n,\frac {3}{2},\frac {2 x^2}{3+\sqrt {5}},\frac {2 x^2}{3-\sqrt {5}}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.09, size = 142, normalized size = 1.43 \begin {gather*} \left (3+\sqrt {5}\right )^n x \left (-\left (3+\sqrt {5}-2 x^2\right )^2\right )^{-n} \left (-3-\sqrt {5}+2 x^2\right )^n \left (-3+\sqrt {5}+2 x^2\right )^n \left (\frac {\left (-3+\sqrt {5}+2 x^2\right )^2}{-3+\sqrt {5}}\right )^{-n} \left (1-3 x^2+x^4\right )^n \operatorname {AppellF1}\left (\frac {1}{2},-n,-n,\frac {3}{2},-\frac {2 x^2}{-3+\sqrt {5}},\frac {2 x^2}{3+\sqrt {5}}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(1 - 3*x^2 + x^4)^n,x]

[Out]

((3 + Sqrt[5])^n*x*(-3 - Sqrt[5] + 2*x^2)^n*(-3 + Sqrt[5] + 2*x^2)^n*(1 - 3*x^2 + x^4)^n*AppellF1[1/2, -n, -n,
3/2, (-2*x^2)/(-3 + Sqrt[5]), (2*x^2)/(3 + Sqrt[5])])/((-(3 + Sqrt[5] - 2*x^2)^2)^n*((-3 + Sqrt[5] + 2*x^2)^2
/(-3 + Sqrt[5]))^n)

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \left (x^{4}-3 x^{2}+1\right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-3*x^2+1)^n,x)

[Out]

int((x^4-3*x^2+1)^n,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3*x^2+1)^n,x, algorithm="maxima")

[Out]

integrate((x^4 - 3*x^2 + 1)^n, x)

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Fricas [F]
time = 0.58, size = 14, normalized size = 0.14 \begin {gather*} {\rm integral}\left ({\left (x^{4} - 3 \, x^{2} + 1\right )}^{n}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3*x^2+1)^n,x, algorithm="fricas")

[Out]

integral((x^4 - 3*x^2 + 1)^n, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (x^{4} - 3 x^{2} + 1\right )^{n}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-3*x**2+1)**n,x)

[Out]

Integral((x**4 - 3*x**2 + 1)**n, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3*x^2+1)^n,x, algorithm="giac")

[Out]

integrate((x^4 - 3*x^2 + 1)^n, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (x^4-3\,x^2+1\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4 - 3*x^2 + 1)^n,x)

[Out]

int((x^4 - 3*x^2 + 1)^n, x)

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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int((x^4-3*x^2+1)^n,x)

[Out]

not solved

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