3.424 \(\int \left (-\frac{1}{32 (-1+x)^5}+\frac{3}{64 (-1+x)^4}-\frac{5}{128 (-1+x)^3}+\frac{5}{256 (-1+x)^2}-\frac{1}{32 (1+x)^5}-\frac{3}{64 (1+x)^4}-\frac{5}{128 (1+x)^3}-\frac{5}{256 (1+x)^2}\right ) \, dx\)

Optimal. Leaf size=13 \[ \frac{1}{8 \left (1-x^2\right )^4} \]

[Out]

1/(8*(1 - x^2)^4)

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Rubi [B]  time = 0.031487, antiderivative size = 81, normalized size of antiderivative = 6.23, number of steps used = 1, number of rules used = 0, integrand size = 73, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \frac{5}{256 (x+1)}+\frac{5}{256 (x+1)^2}+\frac{1}{64 (x+1)^3}+\frac{1}{128 (x+1)^4}+\frac{5}{256 (1-x)}+\frac{5}{256 (1-x)^2}+\frac{1}{64 (1-x)^3}+\frac{1}{128 (1-x)^4} \]

Antiderivative was successfully verified.

[In]  Int[-1/(32*(-1 + x)^5) + 3/(64*(-1 + x)^4) - 5/(128*(-1 + x)^3) + 5/(256*(-1 + x)^2) - 1/(32*(1 + x)^5) - 3/(64*(1 + x)^4) - 5/(128*(1 + x)^3) - 5/(256*(1 + x)^2),x]

[Out]

1/(128*(1 - x)^4) + 1/(64*(1 - x)^3) + 5/(256*(1 - x)^2) + 5/(256*(1 - x)) + 1/(
128*(1 + x)^4) + 1/(64*(1 + x)^3) + 5/(256*(1 + x)^2) + 5/(256*(1 + x))

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Rubi in Sympy [A]  time = 5.52815, size = 63, normalized size = 4.85 \[ \frac{5}{256 \left (x + 1\right )} + \frac{5}{256 \left (x + 1\right )^{2}} + \frac{1}{64 \left (x + 1\right )^{3}} + \frac{1}{128 \left (x + 1\right )^{4}} + \frac{5}{256 \left (- x + 1\right )} + \frac{5}{256 \left (- x + 1\right )^{2}} + \frac{1}{64 \left (- x + 1\right )^{3}} + \frac{1}{128 \left (- x + 1\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(-1/32/(-1+x)**5+3/64/(-1+x)**4-5/128/(-1+x)**3+5/256/(-1+x)**2-1/32/(1+x)**5-3/64/(1+x)**4-5/128/(1+x)**3-5/256/(1+x)**2,x)

[Out]

5/(256*(x + 1)) + 5/(256*(x + 1)**2) + 1/(64*(x + 1)**3) + 1/(128*(x + 1)**4) +
5/(256*(-x + 1)) + 5/(256*(-x + 1)**2) + 1/(64*(-x + 1)**3) + 1/(128*(-x + 1)**4
)

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Mathematica [A]  time = 0.00230004, size = 11, normalized size = 0.85 \[ \frac{1}{8 \left (x^2-1\right )^4} \]

Antiderivative was successfully verified.

[In]  Integrate[-1/(32*(-1 + x)^5) + 3/(64*(-1 + x)^4) - 5/(128*(-1 + x)^3) + 5/(256*(-1 + x)^2) - 1/(32*(1 + x)^5) - 3/(64*(1 + x)^4) - 5/(128*(1 + x)^3) - 5/(256*(1 + x)^2),x]

[Out]

1/(8*(-1 + x^2)^4)

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Maple [B]  time = 0.004, size = 58, normalized size = 4.5 \[{\frac{1}{128\, \left ( 1+x \right ) ^{4}}}+{\frac{1}{64\, \left ( 1+x \right ) ^{3}}}+{\frac{5}{256\, \left ( 1+x \right ) ^{2}}}+{\frac{5}{256+256\,x}}+{\frac{1}{128\, \left ( -1+x \right ) ^{4}}}-{\frac{1}{64\, \left ( -1+x \right ) ^{3}}}+{\frac{5}{256\, \left ( -1+x \right ) ^{2}}}-{\frac{5}{-256+256\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(-1/32/(-1+x)^5+3/64/(-1+x)^4-5/128/(-1+x)^3+5/256/(-1+x)^2-1/32/(1+x)^5-3/64/(1+x)^4-5/128/(1+x)^3-5/256/(1+x)^2,x)

[Out]

1/128/(1+x)^4+1/64/(1+x)^3+5/256/(1+x)^2+5/256/(1+x)+1/128/(-1+x)^4-1/64/(-1+x)^
3+5/256/(-1+x)^2-5/256/(-1+x)

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Maxima [A]  time = 0.796665, size = 77, normalized size = 5.92 \[ \frac{5}{256 \,{\left (x + 1\right )}} - \frac{5}{256 \,{\left (x - 1\right )}} + \frac{5}{256 \,{\left (x + 1\right )}^{2}} + \frac{5}{256 \,{\left (x - 1\right )}^{2}} + \frac{1}{64 \,{\left (x + 1\right )}^{3}} - \frac{1}{64 \,{\left (x - 1\right )}^{3}} + \frac{1}{128 \,{\left (x + 1\right )}^{4}} + \frac{1}{128 \,{\left (x - 1\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-5/256/(x + 1)^2 + 5/256/(x - 1)^2 - 5/128/(x + 1)^3 - 5/128/(x - 1)^3 - 3/64/(x + 1)^4 + 3/64/(x - 1)^4 - 1/32/(x + 1)^5 - 1/32/(x - 1)^5,x, algorithm="maxima")

[Out]

5/256/(x + 1) - 5/256/(x - 1) + 5/256/(x + 1)^2 + 5/256/(x - 1)^2 + 1/64/(x + 1)
^3 - 1/64/(x - 1)^3 + 1/128/(x + 1)^4 + 1/128/(x - 1)^4

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Fricas [A]  time = 0.265836, size = 32, normalized size = 2.46 \[ \frac{1}{8 \,{\left (x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-5/256/(x + 1)^2 + 5/256/(x - 1)^2 - 5/128/(x + 1)^3 - 5/128/(x - 1)^3 - 3/64/(x + 1)^4 + 3/64/(x - 1)^4 - 1/32/(x + 1)^5 - 1/32/(x - 1)^5,x, algorithm="fricas")

[Out]

1/8/(x^8 - 4*x^6 + 6*x^4 - 4*x^2 + 1)

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Sympy [A]  time = 0.833841, size = 22, normalized size = 1.69 \[ \frac{1}{8 x^{8} - 32 x^{6} + 48 x^{4} - 32 x^{2} + 8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/32/(-1+x)**5+3/64/(-1+x)**4-5/128/(-1+x)**3+5/256/(-1+x)**2-1/32/(1+x)**5-3/64/(1+x)**4-5/128/(1+x)**3-5/256/(1+x)**2,x)

[Out]

1/(8*x**8 - 32*x**6 + 48*x**4 - 32*x**2 + 8)

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GIAC/XCAS [A]  time = 0.259185, size = 77, normalized size = 5.92 \[ \frac{5}{256 \,{\left (x + 1\right )}} - \frac{5}{256 \,{\left (x - 1\right )}} + \frac{5}{256 \,{\left (x + 1\right )}^{2}} + \frac{5}{256 \,{\left (x - 1\right )}^{2}} + \frac{1}{64 \,{\left (x + 1\right )}^{3}} - \frac{1}{64 \,{\left (x - 1\right )}^{3}} + \frac{1}{128 \,{\left (x + 1\right )}^{4}} + \frac{1}{128 \,{\left (x - 1\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-5/256/(x + 1)^2 + 5/256/(x - 1)^2 - 5/128/(x + 1)^3 - 5/128/(x - 1)^3 - 3/64/(x + 1)^4 + 3/64/(x - 1)^4 - 1/32/(x + 1)^5 - 1/32/(x - 1)^5,x, algorithm="giac")

[Out]

5/256/(x + 1) - 5/256/(x - 1) + 5/256/(x + 1)^2 + 5/256/(x - 1)^2 + 1/64/(x + 1)
^3 - 1/64/(x - 1)^3 + 1/128/(x + 1)^4 + 1/128/(x - 1)^4