Optimal. Leaf size=20 \[ -\frac {\left (x-\sqrt {a+x^2}\right )^b}{b} \]
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Rubi [A]
time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2147, 30}
\begin {gather*} -\frac {\left (x-\sqrt {a+x^2}\right )^b}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2147
Rubi steps
\begin {align*} \int \frac {\left (x-\sqrt {a+x^2}\right )^b}{\sqrt {a+x^2}} \, dx &=-\text {Subst}\left (\int x^{-1+b} \, dx,x,x-\sqrt {a+x^2}\right )\\ &=-\frac {\left (x-\sqrt {a+x^2}\right )^b}{b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 20, normalized size = 1.00 \begin {gather*} -\frac {\left (x-\sqrt {a+x^2}\right )^b}{b} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.56, size = 52, normalized size = 2.60 \begin {gather*} \text {Piecewise}\left [\left \{\left \{-\frac {{\left (x-\sqrt {a+x^2}\right )}^b}{b},b\text {!=}0\right \}\right \},\text {Piecewise}\left [\left \{\left \{\text {ArcSinh}\left [x \sqrt {\frac {1}{a}}\right ],a>0\right \},\left \{\text {ArcCosh}\left [x \sqrt {-\frac {1}{a}}\right ],a<0\right \}\right \}\right ]\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (x -\sqrt {x^{2}+a}\right )^{b}}{\sqrt {x^{2}+a}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 18, normalized size = 0.90 \begin {gather*} -\frac {{\left (x - \sqrt {x^{2} + a}\right )}^{b}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.76, size = 36, normalized size = 1.80 \begin {gather*} \begin {cases} - \frac {\left (x - \sqrt {a + x^{2}}\right )^{b}}{b} & \text {for}\: b \neq 0 \\\begin {cases} \operatorname {asinh}{\left (x \sqrt {\frac {1}{a}} \right )} & \text {for}\: a > 0 \\\operatorname {acosh}{\left (x \sqrt {- \frac {1}{a}} \right )} & \text {for}\: a < 0 \end {cases} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 18, normalized size = 0.90 \begin {gather*} -\frac {\left (x-\sqrt {a+x^{2}}\right )^{b}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.30, size = 18, normalized size = 0.90 \begin {gather*} -\frac {{\left (x-\sqrt {x^2+a}\right )}^b}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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