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

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

[Out]

1/8/(-x^2+1)^4

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Rubi [B]  time = 0.01, antiderivative size = 81, normalized size of antiderivative = 6.23, number of steps used = 1, number of rules used = 0, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \frac {5}{256 (x+1)}+\frac {5}{256 (x+1)^2}+\frac {1}{64 (x+1)^3}+\frac {1}{128 (x+1)^4}+\frac {5}{256 (1-x)}+\frac {5}{256 (1-x)^2}+\frac {1}{64 (1-x)^3}+\frac {1}{128 (1-x)^4} \]

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/(6
4*(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))

Rubi steps

\begin {align*} \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 &=\frac {1}{128 (1-x)^4}+\frac {1}{64 (1-x)^3}+\frac {5}{256 (1-x)^2}+\frac {5}{256 (1-x)}+\frac {1}{128 (1+x)^4}+\frac {1}{64 (1+x)^3}+\frac {5}{256 (1+x)^2}+\frac {5}{256 (1+x)}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 0.85 \[ \frac {1}{8 \left (x^2-1\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[-1/32*1/(-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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fricas [B]  time = 0.64, size = 24, normalized size = 1.85 \[ \frac {1}{8 \, {\left (x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-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, algorithm="fricas")

[Out]

1/8/(x^8 - 4*x^6 + 6*x^4 - 4*x^2 + 1)

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giac [B]  time = 0.31, size = 57, normalized size = 4.38 \[ \frac {5}{256 \, {\left (x + 1\right )}} - \frac {5}{256 \, {\left (x - 1\right )}} + \frac {5}{256 \, {\left (x + 1\right )}^{2}} + \frac {5}{256 \, {\left (x - 1\right )}^{2}} + \frac {1}{64 \, {\left (x + 1\right )}^{3}} - \frac {1}{64 \, {\left (x - 1\right )}^{3}} + \frac {1}{128 \, {\left (x + 1\right )}^{4}} + \frac {1}{128 \, {\left (x - 1\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-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, 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

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maple [B]  time = 0.00, size = 58, normalized size = 4.46 \[ \frac {1}{128 \left (x -1\right )^{4}}-\frac {1}{64 \left (x -1\right )^{3}}+\frac {5}{256 \left (x -1\right )^{2}}-\frac {5}{256 \left (x -1\right )}+\frac {1}{128 \left (x +1\right )^{4}}+\frac {1}{64 \left (x +1\right )^{3}}+\frac {5}{256 \left (x +1\right )^{2}}+\frac {5}{256 \left (x +1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.94, size = 57, normalized size = 4.38 \[ \frac {5}{256 \, {\left (x + 1\right )}} - \frac {5}{256 \, {\left (x - 1\right )}} + \frac {5}{256 \, {\left (x + 1\right )}^{2}} + \frac {5}{256 \, {\left (x - 1\right )}^{2}} + \frac {1}{64 \, {\left (x + 1\right )}^{3}} - \frac {1}{64 \, {\left (x - 1\right )}^{3}} + \frac {1}{128 \, {\left (x + 1\right )}^{4}} + \frac {1}{128 \, {\left (x - 1\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-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, 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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mupad [B]  time = 0.03, size = 9, normalized size = 0.69 \[ \frac {1}{8\,{\left (x^2-1\right )}^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(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)

[Out]

1/(8*(x^2 - 1)^4)

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sympy [B]  time = 0.31, 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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