Optimal. Leaf size=14 \[ -\frac {2 x}{\sqrt {x \left (x^2+1\right )}} \]
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Rubi [A] time = 0.15, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6719, 449} \[ -\frac {2 x}{\sqrt {x \left (x^2+1\right )}} \]
Antiderivative was successfully verified.
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Rule 449
Rule 6719
Rubi steps
\begin {align*} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x \left (1+x^2\right )}} \, dx &=\frac {\left (\sqrt {x} \sqrt {1+x^2}\right ) \int \frac {-1+x^2}{\sqrt {x} \left (1+x^2\right )^{3/2}} \, dx}{\sqrt {x \left (1+x^2\right )}}\\ &=-\frac {2 x}{\sqrt {x \left (1+x^2\right )}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 12, normalized size = 0.86 \[ -\frac {2 x}{\sqrt {x^3+x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 16, normalized size = 1.14 \[ -\frac {2 \, \sqrt {x^{3} + x}}{x^{2} + 1} \]
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 \[ \int \frac {x^{2} - 1}{\sqrt {{\left (x^{2} + 1\right )} x} {\left (x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 13, normalized size = 0.93 \[ -\frac {2 x}{\sqrt {\left (x^{2}+1\right ) x}} \]
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 \[ \int \frac {x^{2} - 1}{\sqrt {{\left (x^{2} + 1\right )} x} {\left (x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.38, size = 138, normalized size = 9.86 \[ -\frac {2\,x}{\sqrt {x^3+x}}-\frac {\sqrt {1-x\,1{}\mathrm {i}}\,\sqrt {\frac {1}{2}+\frac {x\,1{}\mathrm {i}}{2}}\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {1-x\,1{}\mathrm {i}}\right )\middle |\frac {1}{2}\right )\,\sqrt {x\,1{}\mathrm {i}}\,2{}\mathrm {i}}{\sqrt {x^3+x}}+\frac {\sqrt {1-x\,1{}\mathrm {i}}\,\sqrt {\frac {1}{2}+\frac {x\,1{}\mathrm {i}}{2}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {1-x\,1{}\mathrm {i}}\right )\middle |\frac {1}{2}\right )\,\sqrt {x\,1{}\mathrm {i}}\,2{}\mathrm {i}}{\sqrt {x^3+x}}-\frac {\sqrt {1-x\,1{}\mathrm {i}}\,\sqrt {1+x\,1{}\mathrm {i}}\,\sqrt {-x\,1{}\mathrm {i}}\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {-x\,1{}\mathrm {i}}\right )\middle |-1\right )\,1{}\mathrm {i}}{\sqrt {x^3+x}} \]
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 \[ \int \frac {\left (x - 1\right ) \left (x + 1\right )}{\sqrt {x \left (x^{2} + 1\right )} \left (x^{2} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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