Optimal. Leaf size=17 \[ \frac {\sqrt {1+x^2}}{1-x} \]
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Rubi [A]
time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {817}
\begin {gather*} \frac {\sqrt {x^2+1}}{1-x} \end {gather*}
Antiderivative was successfully verified.
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Rule 817
Rubi steps
\begin {align*} \int \frac {1+x}{(1-x)^2 \sqrt {1+x^2}} \, dx &=\frac {\sqrt {1+x^2}}{1-x}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 17, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^2}}{1-x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 19, normalized size = 1.12
method | result | size |
gosper | \(-\frac {\sqrt {x^{2}+1}}{-1+x}\) | \(15\) |
trager | \(-\frac {\sqrt {x^{2}+1}}{-1+x}\) | \(15\) |
risch | \(-\frac {\sqrt {x^{2}+1}}{-1+x}\) | \(15\) |
default | \(-\frac {\sqrt {\left (-1+x \right )^{2}+2 x}}{-1+x}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.68, size = 14, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {x^{2} + 1}}{x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.71, size = 17, normalized size = 1.00 \begin {gather*} -\frac {x + \sqrt {x^{2} + 1} - 1}{x - 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 {x + 1}{\left (x - 1\right )^{2} \sqrt {x^{2} + 1}}\, dx \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. 35 vs.
\(2 (14) = 28\).
time = 0.73, size = 35, normalized size = 2.06 \begin {gather*} -\frac {\sqrt {\frac {2}{x - 1} + \frac {2}{{\left (x - 1\right )}^{2}} + 1}}{\mathrm {sgn}\left (\frac {1}{x - 1}\right )} + \mathrm {sgn}\left (\frac {1}{x - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 14, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {x^2+1}}{x-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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