Optimal. Leaf size=33 \[ \frac {2 x \sqrt {\frac {\left (1-x^2\right )^2}{x+x^3}}}{1-x^2} \]
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Rubi [A]
time = 0.12, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6851, 2081,
460} \begin {gather*} \frac {2 x \sqrt {\frac {\left (1-x^2\right )^2}{x^3+x}}}{1-x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 460
Rule 2081
Rule 6851
Rubi steps
\begin {align*} \int \frac {\sqrt {\frac {\left (-1+x^2\right )^2}{x+x^3}}}{1+x^2} \, dx &=\frac {\left (\sqrt {\frac {\left (-1+x^2\right )^2}{x+x^3}} \sqrt {x+x^3}\right ) \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x+x^3}} \, dx}{-1+x^2}\\ &=\frac {\left (\sqrt {x} \sqrt {1+x^2} \sqrt {\frac {\left (-1+x^2\right )^2}{x+x^3}}\right ) \int \frac {-1+x^2}{\sqrt {x} \left (1+x^2\right )^{3/2}} \, dx}{-1+x^2}\\ &=\frac {2 x \sqrt {\frac {\left (1-x^2\right )^2}{x+x^3}}}{1-x^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 29, normalized size = 0.88 \begin {gather*} -\frac {2 x \sqrt {\frac {\left (-1+x^2\right )^2}{x+x^3}}}{-1+x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 31, normalized size = 0.94
method | result | size |
default | \(-\frac {2 \sqrt {\frac {\left (x^{2}-1\right )^{2}}{x \left (x^{2}+1\right )}}\, x}{x^{2}-1}\) | \(31\) |
risch | \(-\frac {2 \sqrt {\frac {\left (x^{2}-1\right )^{2}}{x \left (x^{2}+1\right )}}\, x}{x^{2}-1}\) | \(31\) |
gosper | \(-\frac {2 x \sqrt {\frac {\left (x^{2}-1\right )^{2}}{x \left (x^{2}+1\right )}}}{\left (1+x \right ) \left (-1+x \right )}\) | \(34\) |
trager | \(-\frac {2 x \sqrt {-\frac {-x^{4}+2 x^{2}-1}{x^{3}+x}}}{x^{2}-1}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 30, normalized size = 0.91 \begin {gather*} -\frac {2 \, x \sqrt {\frac {x^{4} - 2 \, x^{2} + 1}{x^{3} + x}}}{x^{2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\frac {\left (x - 1\right )^{2} \left (x + 1\right )^{2}}{x^{3} + x}}}{x^{2} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.49, size = 43, normalized size = 1.30 \begin {gather*} -\frac {\sqrt {\frac {1}{x^3+x}}\,\left (2\,x^3+2\,x\right )\,\sqrt {{\left (x^2-1\right )}^2}}{\left (x^2-1\right )\,\left (x^2+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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