Optimal. Leaf size=29 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {c}} \]
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Rubi [A]
time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {281, 211}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 281
Rubi steps
\begin {align*} \int \frac {x}{a+c x^4} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 29, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 19, normalized size = 0.66
method | result | size |
default | \(\frac {\arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{2 \sqrt {a c}}\) | \(19\) |
risch | \(-\frac {\ln \left (x^{2} \sqrt {-a c}-a \right )}{4 \sqrt {-a c}}+\frac {\ln \left (x^{2} \sqrt {-a c}+a \right )}{4 \sqrt {-a c}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 18, normalized size = 0.62 \begin {gather*} \frac {\arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{2 \, \sqrt {a c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 72, normalized size = 2.48 \begin {gather*} \left [-\frac {\sqrt {-a c} \log \left (\frac {c x^{4} - 2 \, \sqrt {-a c} x^{2} - a}{c x^{4} + a}\right )}{4 \, a c}, -\frac {\sqrt {a c} \arctan \left (\frac {\sqrt {a c}}{c x^{2}}\right )}{2 \, a c}\right ] \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. 56 vs.
\(2 (26) = 52\).
time = 0.07, size = 56, normalized size = 1.93 \begin {gather*} - \frac {\sqrt {- \frac {1}{a c}} \log {\left (- a \sqrt {- \frac {1}{a c}} + x^{2} \right )}}{4} + \frac {\sqrt {- \frac {1}{a c}} \log {\left (a \sqrt {- \frac {1}{a c}} + x^{2} \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.54, size = 18, normalized size = 0.62 \begin {gather*} \frac {\arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{2 \, \sqrt {a c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 19, normalized size = 0.66 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{2\,\sqrt {a}\,\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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