Optimal. Leaf size=12 \[ \frac {1}{\sqrt {1+x^2}}+\sinh ^{-1}(x) \]
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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 = {1828, 221}
\begin {gather*} \frac {1}{\sqrt {x^2+1}}+\sinh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 1828
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 = 0.11, size = 22, normalized size = 1.83 \begin {gather*} \frac {1}{\sqrt {1+x^2}}+\tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 11, normalized size = 0.92
method | result | size |
default | \(\arcsinh \left (x \right )+\frac {1}{\sqrt {x^{2}+1}}\) | \(11\) |
risch | \(\arcsinh \left (x \right )+\frac {1}{\sqrt {x^{2}+1}}\) | \(11\) |
trager | \(\frac {1}{\sqrt {x^{2}+1}}+\ln \left (x +\sqrt {x^{2}+1}\right )\) | \(19\) |
meijerg | \(\frac {x}{\sqrt {x^{2}+1}}+\frac {-\frac {\sqrt {\pi }\, x}{\sqrt {x^{2}+1}}+\sqrt {\pi }\, \arcsinh \left (x \right )}{\sqrt {\pi }}-\frac {\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {x^{2}+1}}}{\sqrt {\pi }}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.70, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{\sqrt {x^{2} + 1}} + \operatorname {arsinh}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs.
\(2 (10) = 20\).
time = 0.51, size = 37, normalized size = 3.08 \begin {gather*} -\frac {{\left (x^{2} + 1\right )} \log \left (-x + \sqrt {x^{2} + 1}\right ) - \sqrt {x^{2} + 1}}{x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs.
\(2 (12) = 24\).
time = 5.72, size = 29, normalized size = 2.42 \begin {gather*} \frac {x^{2} \operatorname {asinh}{\left (x \right )}}{x^{2} + 1} + \frac {\operatorname {asinh}{\left (x \right )}}{x^{2} + 1} + \frac {1}{\sqrt {x^{2} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 22 vs.
\(2 (10) = 20\).
time = 0.86, size = 22, normalized size = 1.83 \begin {gather*} \frac {1}{\sqrt {x^{2} + 1}} - \log \left (-x + \sqrt {x^{2} + 1}\right ) \end {gather*}
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 \begin {gather*} \frac {\mathrm {asinh}\left (x\right )+x^2\,\mathrm {asinh}\left (x\right )+\sqrt {x^2+1}}{x^2+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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