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*(1 - x^2)^4)

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Rubi [B]  time = 0.0118387, 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., 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*}

Mathematica [A]  time = 0.0015173, 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 = 0.003, size = 58, normalized size = 4.5 \begin{align*}{\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\,x-256}}+{\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}} \end{align*}

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

[Out]

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

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Maxima [B]  time = 0.965265, size = 77, normalized size = 5.92 \begin{align*} \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}} \end{align*}

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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Fricas [B]  time = 1.01074, size = 53, normalized size = 4.08 \begin{align*} \frac{1}{8 \,{\left (x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1\right )}} \end{align*}

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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Sympy [B]  time = 0.276525, size = 22, normalized size = 1.69 \begin{align*} \frac{1}{8 x^{8} - 32 x^{6} + 48 x^{4} - 32 x^{2} + 8} \end{align*}

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 [B]  time = 1.10273, size = 77, normalized size = 5.92 \begin{align*} \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}} \end{align*}

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