3.4.16 \(\int (-2+(-2+(-2+(-2+x^2)^2)^2)^2) \, dx\) [316]

Optimal. Leaf size=54 \[ 2 x-\frac {64 x^3}{3}+\frac {336 x^5}{5}-96 x^7+\frac {220 x^9}{3}-32 x^{11}+8 x^{13}-\frac {16 x^{15}}{15}+\frac {x^{17}}{17} \]

[Out]

2*x-64/3*x^3+336/5*x^5-96*x^7+220/3*x^9-32*x^11+8*x^13-16/15*x^15+1/17*x^17

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Rubi [A]
time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2086} \begin {gather*} \frac {x^{17}}{17}-\frac {16 x^{15}}{15}+8 x^{13}-32 x^{11}+\frac {220 x^9}{3}-96 x^7+\frac {336 x^5}{5}-\frac {64 x^3}{3}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-2 + (-2 + (-2 + (-2 + x^2)^2)^2)^2,x]

[Out]

2*x - (64*x^3)/3 + (336*x^5)/5 - 96*x^7 + (220*x^9)/3 - 32*x^11 + 8*x^13 - (16*x^15)/15 + x^17/17

Rule 2086

Int[(P_)^(p_), x_Symbol] :> Int[ExpandToSum[P^p, x], x] /; PolyQ[P, x] && IGtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=-2 x+\int \left (-2+\left (-2+\left (-2+x^2\right )^2\right )^2\right )^2 \, dx\\ &=-2 x+\int \left (4-64 x^2+336 x^4-672 x^6+660 x^8-352 x^{10}+104 x^{12}-16 x^{14}+x^{16}\right ) \, dx\\ &=2 x-\frac {64 x^3}{3}+\frac {336 x^5}{5}-96 x^7+\frac {220 x^9}{3}-32 x^{11}+8 x^{13}-\frac {16 x^{15}}{15}+\frac {x^{17}}{17}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 54, normalized size = 1.00 \begin {gather*} 2 x-\frac {64 x^3}{3}+\frac {336 x^5}{5}-96 x^7+\frac {220 x^9}{3}-32 x^{11}+8 x^{13}-\frac {16 x^{15}}{15}+\frac {x^{17}}{17} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-2 + (-2 + (-2 + (-2 + x^2)^2)^2)^2,x]

[Out]

2*x - (64*x^3)/3 + (336*x^5)/5 - 96*x^7 + (220*x^9)/3 - 32*x^11 + 8*x^13 - (16*x^15)/15 + x^17/17

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Maple [A]
time = 0.08, size = 45, normalized size = 0.83

method	result	size
default	\(2 x -\frac {64}{3} x^{3}+\frac {336}{5} x^{5}-96 x^{7}+\frac {220}{3} x^{9}-32 x^{11}+8 x^{13}-\frac {16}{15} x^{15}+\frac {1}{17} x^{17}\)	\(45\)
norman	\(2 x -\frac {64}{3} x^{3}+\frac {336}{5} x^{5}-96 x^{7}+\frac {220}{3} x^{9}-32 x^{11}+8 x^{13}-\frac {16}{15} x^{15}+\frac {1}{17} x^{17}\)	\(45\)
risch	\(2 x -\frac {64}{3} x^{3}+\frac {336}{5} x^{5}-96 x^{7}+\frac {220}{3} x^{9}-32 x^{11}+8 x^{13}-\frac {16}{15} x^{15}+\frac {1}{17} x^{17}\)	\(45\)
parallelrisch	\(2 x -\frac {64}{3} x^{3}+\frac {336}{5} x^{5}-96 x^{7}+\frac {220}{3} x^{9}-32 x^{11}+8 x^{13}-\frac {16}{15} x^{15}+\frac {1}{17} x^{17}\)	\(45\)
parts	\(2 x -\frac {64}{3} x^{3}+\frac {336}{5} x^{5}-96 x^{7}+\frac {220}{3} x^{9}-32 x^{11}+8 x^{13}-\frac {16}{15} x^{15}+\frac {1}{17} x^{17}\)	\(45\)
gosper	\(\frac {x \left (15 x^{16}-272 x^{14}+2040 x^{12}-8160 x^{10}+18700 x^{8}-24480 x^{6}+17136 x^{4}-5440 x^{2}+510\right )}{255}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2-2)^2-2)^2-2)^2-2,x,method=_RETURNVERBOSE)

[Out]

2*x-64/3*x^3+336/5*x^5-96*x^7+220/3*x^9-32*x^11+8*x^13-16/15*x^15+1/17*x^17

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Maxima [A]
time = 0.34, size = 44, normalized size = 0.81 \begin {gather*} \frac {1}{17} \, x^{17} - \frac {16}{15} \, x^{15} + 8 \, x^{13} - 32 \, x^{11} + \frac {220}{3} \, x^{9} - 96 \, x^{7} + \frac {336}{5} \, x^{5} - \frac {64}{3} \, x^{3} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-2)^2-2)^2-2)^2-2,x, algorithm="maxima")

[Out]

1/17*x^17 - 16/15*x^15 + 8*x^13 - 32*x^11 + 220/3*x^9 - 96*x^7 + 336/5*x^5 - 64/3*x^3 + 2*x

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Fricas [A]
time = 0.56, size = 44, normalized size = 0.81 \begin {gather*} \frac {1}{17} \, x^{17} - \frac {16}{15} \, x^{15} + 8 \, x^{13} - 32 \, x^{11} + \frac {220}{3} \, x^{9} - 96 \, x^{7} + \frac {336}{5} \, x^{5} - \frac {64}{3} \, x^{3} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-2)^2-2)^2-2)^2-2,x, algorithm="fricas")

[Out]

1/17*x^17 - 16/15*x^15 + 8*x^13 - 32*x^11 + 220/3*x^9 - 96*x^7 + 336/5*x^5 - 64/3*x^3 + 2*x

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Sympy [A]
time = 0.01, size = 49, normalized size = 0.91 \begin {gather*} \frac {x^{17}}{17} - \frac {16 x^{15}}{15} + 8 x^{13} - 32 x^{11} + \frac {220 x^{9}}{3} - 96 x^{7} + \frac {336 x^{5}}{5} - \frac {64 x^{3}}{3} + 2 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**2-2)**2-2)**2-2)**2-2,x)

[Out]

x**17/17 - 16*x**15/15 + 8*x**13 - 32*x**11 + 220*x**9/3 - 96*x**7 + 336*x**5/5 - 64*x**3/3 + 2*x

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Giac [A]
time = 0.45, size = 44, normalized size = 0.81 \begin {gather*} \frac {1}{17} \, x^{17} - \frac {16}{15} \, x^{15} + 8 \, x^{13} - 32 \, x^{11} + \frac {220}{3} \, x^{9} - 96 \, x^{7} + \frac {336}{5} \, x^{5} - \frac {64}{3} \, x^{3} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-2)^2-2)^2-2)^2-2,x, algorithm="giac")

[Out]

1/17*x^17 - 16/15*x^15 + 8*x^13 - 32*x^11 + 220/3*x^9 - 96*x^7 + 336/5*x^5 - 64/3*x^3 + 2*x

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Mupad [B]
time = 0.10, size = 44, normalized size = 0.81 \begin {gather*} \frac {x^{17}}{17}-\frac {16\,x^{15}}{15}+8\,x^{13}-32\,x^{11}+\frac {220\,x^9}{3}-96\,x^7+\frac {336\,x^5}{5}-\frac {64\,x^3}{3}+2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2 - 2)^2 - 2)^2 - 2)^2 - 2,x)

[Out]

2*x - (64*x^3)/3 + (336*x^5)/5 - 96*x^7 + (220*x^9)/3 - 32*x^11 + 8*x^13 - (16*x^15)/15 + x^17/17

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Chatgpt [F] Failed to verify
time = 1.00, size = 25, normalized size = 0.46 \begin {gather*} \frac {\left (x^{2}-2\right )^{5}}{10}-\frac {4 \left (x^{2}-2\right )^{3}}{3}+2 x^{2}-4 \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int((((x^2-2)^2-2)^2-2)^2-2,x)

[Out]

1/10*(x^2-2)^5-4/3*(x^2-2)^3+2*x^2-4

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