Optimal. Leaf size=25 \[ \frac {x \left (a^2+x^2\right )}{a^2 \sqrt {\left (a^2+x^2\right )^3}} \]
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Rubi [A]
time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1973, 197}
\begin {gather*} \frac {x \left (a^2+x^2\right )}{a^2 \sqrt {\left (a^2+x^2\right )^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 1973
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\left (a^2+x^2\right )^3}} \, dx &=\frac {\left (a^2+x^2\right )^{3/2} \int \frac {1}{\left (a^2+x^2\right )^{3/2}} \, dx}{\sqrt {\left (a^2+x^2\right )^3}}\\ &=\frac {x \left (a^2+x^2\right )}{a^2 \sqrt {\left (a^2+x^2\right )^3}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 25, normalized size = 1.00 \begin {gather*} \frac {x \left (a^2+x^2\right )}{a^2 \sqrt {\left (a^2+x^2\right )^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 24, normalized size = 0.96
method | result | size |
gosper | \(\frac {x \left (a^{2}+x^{2}\right )}{a^{2} \sqrt {\left (a^{2}+x^{2}\right )^{3}}}\) | \(24\) |
default | \(\frac {x \left (a^{2}+x^{2}\right )}{a^{2} \sqrt {\left (a^{2}+x^{2}\right )^{3}}}\) | \(24\) |
risch | \(\frac {x \left (a^{2}+x^{2}\right )}{a^{2} \sqrt {\left (a^{2}+x^{2}\right )^{3}}}\) | \(24\) |
trager | \(\frac {x \sqrt {a^{6}+3 a^{4} x^{2}+3 a^{2} x^{4}+x^{6}}}{a^{2} \left (a^{2}+x^{2}\right )^{2}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 14, normalized size = 0.56 \begin {gather*} \frac {x}{\sqrt {a^{2} + x^{2}} a^{2}} \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. 64 vs.
\(2 (23) = 46\).
time = 0.42, size = 64, normalized size = 2.56 \begin {gather*} \frac {a^{4} + 2 \, a^{2} x^{2} + x^{4} + \sqrt {a^{6} + 3 \, a^{4} x^{2} + 3 \, a^{2} x^{4} + x^{6}} x}{a^{6} + 2 \, a^{4} x^{2} + a^{2} x^{4}} \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 {1}{\sqrt {\left (a^{2} + x^{2}\right )^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.46, size = 14, normalized size = 0.56 \begin {gather*} \frac {x}{\sqrt {a^{2} + x^{2}} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.51, size = 25, normalized size = 1.00 \begin {gather*} \frac {x\,\sqrt {{\left (a^2+x^2\right )}^3}}{a^2\,{\left (a^2+x^2\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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