Optimal. Leaf size=12 \[ \frac {1}{\sqrt {x^2+1}}+\sinh ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1814, 215} \[ \frac {1}{\sqrt {x^2+1}}+\sinh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 215
Rule 1814
Rubi steps
\begin {align*} \int \frac {1-x+x^2}{\left (1+x^2\right )^{3/2}} \, dx &=\frac {1}{\sqrt {1+x^2}}+\int \frac {1}{\sqrt {1+x^2}} \, dx\\ &=\frac {1}{\sqrt {1+x^2}}+\sinh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ \frac {1}{\sqrt {x^2+1}}+\sinh ^{-1}(x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.17, size = 26, normalized size = 2.17 \[ \frac {1}{\sqrt {x^2+1}}-\log \left (\sqrt {x^2+1}-x\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 37, normalized size = 3.08 \[ -\frac {{\left (x^{2} + 1\right )} \log \left (-x + \sqrt {x^{2} + 1}\right ) - \sqrt {x^{2} + 1}}{x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.67, size = 22, normalized size = 1.83 \[ \frac {1}{\sqrt {x^{2} + 1}} - \log \left (-x + \sqrt {x^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 11, normalized size = 0.92
method | result | size |
default | \(\arcsinh \relax (x )+\frac {1}{\sqrt {x^{2}+1}}\) | \(11\) |
risch | \(\arcsinh \relax (x )+\frac {1}{\sqrt {x^{2}+1}}\) | \(11\) |
trager | \(\frac {1}{\sqrt {x^{2}+1}}-\ln \left (x -\sqrt {x^{2}+1}\right )\) | \(23\) |
meijerg | \(\frac {x}{\sqrt {x^{2}+1}}+\frac {-\frac {\sqrt {\pi }\, x}{\sqrt {x^{2}+1}}+\sqrt {\pi }\, \arcsinh \relax (x )}{\sqrt {\pi }}-\frac {\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {x^{2}+1}}}{\sqrt {\pi }}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 10, normalized size = 0.83 \[ \frac {1}{\sqrt {x^{2} + 1}} + \operatorname {arsinh}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 24, normalized size = 2.00 \[ \frac {\mathrm {asinh}\relax (x)+x^2\,\mathrm {asinh}\relax (x)+\sqrt {x^2+1}}{x^2+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 12.85, size = 29, normalized size = 2.42 \[ \frac {x^{2} \operatorname {asinh}{\relax (x )}}{x^{2} + 1} + \frac {\operatorname {asinh}{\relax (x )}}{x^{2} + 1} + \frac {1}{\sqrt {x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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