Optimal. Leaf size=40 \[ -\frac {3}{4 \left (1-x^2\right )^2}+\frac {1}{3 \left (1-x^2\right )^3}+\frac {1}{8 \left (1-x^2\right )^4} \]
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Rubi [A] time = 0.04, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1593, 446, 77} \[ -\frac {3}{4 \left (1-x^2\right )^2}+\frac {1}{3 \left (1-x^2\right )^3}+\frac {1}{8 \left (1-x^2\right )^4} \]
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rule 1593
Rubi steps
\begin {align*} \int \frac {-4 x^3+3 x^5}{\left (-1+x^2\right )^5} \, dx &=\int \frac {x^3 \left (-4+3 x^2\right )}{\left (-1+x^2\right )^5} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x (-4+3 x)}{(-1+x)^5} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {1}{(-1+x)^5}+\frac {2}{(-1+x)^4}+\frac {3}{(-1+x)^3}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{8 \left (1-x^2\right )^4}+\frac {1}{3 \left (1-x^2\right )^3}-\frac {3}{4 \left (1-x^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.58 \[ \frac {-18 x^4+28 x^2-7}{24 \left (x^2-1\right )^4} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.01, size = 23, normalized size = 0.58 \[ \frac {-18 x^4+28 x^2-7}{24 \left (x^2-1\right )^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 36, normalized size = 0.90 \[ -\frac {18 \, x^{4} - 28 \, x^{2} + 7}{24 \, {\left (x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 21, normalized size = 0.52 \[ -\frac {18 \, x^{4} - 28 \, x^{2} + 7}{24 \, {\left (x^{2} - 1\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 21, normalized size = 0.52
method | result | size |
norman | \(\frac {-\frac {3}{4} x^{4}+\frac {7}{6} x^{2}-\frac {7}{24}}{\left (x^{2}-1\right )^{4}}\) | \(21\) |
risch | \(\frac {-\frac {3}{4} x^{4}+\frac {7}{6} x^{2}-\frac {7}{24}}{\left (x^{2}-1\right )^{4}}\) | \(21\) |
gosper | \(-\frac {18 x^{4}-28 x^{2}+7}{24 \left (x^{2}-1\right )^{4}}\) | \(22\) |
meijerg | \(-\frac {x^{6} \left (-x^{2}+4\right )}{8 \left (-x^{2}+1\right )^{4}}+\frac {x^{4} \left (x^{4}-4 x^{2}+6\right )}{6 \left (-x^{2}+1\right )^{4}}\) | \(47\) |
default | \(\frac {1}{128 \left (-1+x \right )^{4}}-\frac {11}{192 \left (-1+x \right )^{3}}-\frac {27}{256 \left (-1+x \right )^{2}}+\frac {27}{256 \left (-1+x \right )}+\frac {1}{128 \left (1+x \right )^{4}}+\frac {11}{192 \left (1+x \right )^{3}}-\frac {27}{256 \left (1+x \right )^{2}}-\frac {27}{256 \left (1+x \right )}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 36, normalized size = 0.90 \[ -\frac {18 \, x^{4} - 28 \, x^{2} + 7}{24 \, {\left (x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 36, normalized size = 0.90 \[ -\frac {\frac {3\,x^4}{4}-\frac {7\,x^2}{6}+\frac {7}{24}}{x^8-4\,x^6+6\,x^4-4\,x^2+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 32, normalized size = 0.80 \[ \frac {- 18 x^{4} + 28 x^{2} - 7}{24 x^{8} - 96 x^{6} + 144 x^{4} - 96 x^{2} + 24} \]
Verification of antiderivative is not currently implemented for this CAS.
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