Optimal. Leaf size=20 \[ -\frac {\left (x-\sqrt {a+x^2}\right )^b}{b} \]
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Rubi [A] time = 0.06, 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 = {2122, 30} \[ -\frac {\left (x-\sqrt {a+x^2}\right )^b}{b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2122
Rubi steps
\begin {align*} \int \frac {\left (x-\sqrt {a+x^2}\right )^b}{\sqrt {a+x^2}} \, dx &=-\operatorname {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 = 0.01, size = 20, normalized size = 1.00 \[ -\frac {\left (x-\sqrt {a+x^2}\right )^b}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 18, normalized size = 0.90 \[ -\frac {{\left (x - \sqrt {x^{2} + a}\right )}^{b}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x - \sqrt {x^{2} + a}\right )}^{b}}{\sqrt {x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, 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 \[ \int \frac {{\left (x - \sqrt {x^{2} + a}\right )}^{b}}{\sqrt {x^{2} + a}}\,{d x} \]
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 \[ -\frac {{\left (x-\sqrt {x^2+a}\right )}^b}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.60, size = 36, normalized size = 1.80 \[ \begin {cases} - \frac {\left (x - \sqrt {a + x^{2}}\right )^{b}}{b} & \text {for}\: b \neq 0 \\\begin {cases} \operatorname {asinh}{\left (x \sqrt {\frac {1}{a}} \right )} & \text {for}\: a > 0 \\\operatorname {acosh}{\left (x \sqrt {- \frac {1}{a}} \right )} & \text {for}\: a < 0 \end {cases} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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