Optimal. Leaf size=15 \[ -\frac {2 x}{\sqrt {1+x^2}}+\sinh ^{-1}(x) \]
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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {393, 221}
\begin {gather*} \sinh ^{-1}(x)-\frac {2 x}{\sqrt {x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 393
Rubi steps
\begin {align*} \int \frac {-1+x^2}{\left (1+x^2\right )^{3/2}} \, dx &=-\frac {2 x}{\sqrt {1+x^2}}+\int \frac {1}{\sqrt {1+x^2}} \, dx\\ &=-\frac {2 x}{\sqrt {1+x^2}}+\sinh ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 25, normalized size = 1.67 \begin {gather*} -\frac {2 x}{\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 = 0.10, size = 14, normalized size = 0.93
method | result | size |
default | \(\arcsinh \left (x \right )-\frac {2 x}{\sqrt {x^{2}+1}}\) | \(14\) |
risch | \(\arcsinh \left (x \right )-\frac {2 x}{\sqrt {x^{2}+1}}\) | \(14\) |
trager | \(-\frac {2 x}{\sqrt {x^{2}+1}}+\ln \left (\sqrt {x^{2}+1}+x \right )\) | \(22\) |
meijerg | \(-\frac {x}{\sqrt {x^{2}+1}}+\frac {-\frac {\sqrt {\pi }\, x}{\sqrt {x^{2}+1}}+\sqrt {\pi }\, \arcsinh \left (x \right )}{\sqrt {\pi }}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 13, normalized size = 0.87 \begin {gather*} -\frac {2 \, x}{\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. 44 vs.
\(2 (13) = 26\).
time = 0.87, size = 44, normalized size = 2.93 \begin {gather*} -\frac {2 \, x^{2} + {\left (x^{2} + 1\right )} \log \left (-x + \sqrt {x^{2} + 1}\right ) + 2 \, \sqrt {x^{2} + 1} x + 2}{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. 31 vs.
\(2 (14) = 28\).
time = 2.34, size = 31, normalized size = 2.07 \begin {gather*} \frac {x^{2} \operatorname {asinh}{\left (x \right )}}{x^{2} + 1} - \frac {2 x}{\sqrt {x^{2} + 1}} + \frac {\operatorname {asinh}{\left (x \right )}}{x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.58, size = 25, normalized size = 1.67 \begin {gather*} -\frac {2 \, x}{\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 = 0.04, size = 27, normalized size = 1.80 \begin {gather*} \frac {\mathrm {asinh}\left (x\right )+x^2\,\mathrm {asinh}\left (x\right )-2\,x\,\sqrt {x^2+1}}{x^2+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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