Optimal. Leaf size=43 \[ \frac {1}{2} \sqrt {a+c x^4}-\frac {1}{2} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 52, 65,
214} \begin {gather*} \frac {1}{2} \sqrt {a+c x^4}-\frac {1}{2} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 214
Rule 272
Rubi steps
\begin {align*} \int \frac {\sqrt {a+c x^4}}{x} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {\sqrt {a+c x}}{x} \, dx,x,x^4\right )\\ &=\frac {1}{2} \sqrt {a+c x^4}+\frac {1}{4} a \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^4\right )\\ &=\frac {1}{2} \sqrt {a+c x^4}+\frac {a \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^4}\right )}{2 c}\\ &=\frac {1}{2} \sqrt {a+c x^4}-\frac {1}{2} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 43, normalized size = 1.00 \begin {gather*} \frac {1}{2} \sqrt {a+c x^4}-\frac {1}{2} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 41, normalized size = 0.95
method | result | size |
default | \(\frac {\sqrt {x^{4} c +a}}{2}-\frac {\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {x^{4} c +a}}{x^{2}}\right )}{2}\) | \(41\) |
elliptic | \(\frac {\sqrt {x^{4} c +a}}{2}-\frac {\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {x^{4} c +a}}{x^{2}}\right )}{2}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 49, normalized size = 1.14 \begin {gather*} \frac {1}{4} \, \sqrt {a} \log \left (\frac {\sqrt {c x^{4} + a} - \sqrt {a}}{\sqrt {c x^{4} + a} + \sqrt {a}}\right ) + \frac {1}{2} \, \sqrt {c x^{4} + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 84, normalized size = 1.95 \begin {gather*} \left [\frac {1}{4} \, \sqrt {a} \log \left (\frac {c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) + \frac {1}{2} \, \sqrt {c x^{4} + a}, \frac {1}{2} \, \sqrt {-a} \arctan \left (\frac {\sqrt {c x^{4} + a} \sqrt {-a}}{a}\right ) + \frac {1}{2} \, \sqrt {c x^{4} + a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.67, size = 66, normalized size = 1.53 \begin {gather*} - \frac {\sqrt {a} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x^{2}} \right )}}{2} + \frac {a}{2 \sqrt {c} x^{2} \sqrt {\frac {a}{c x^{4}} + 1}} + \frac {\sqrt {c} x^{2}}{2 \sqrt {\frac {a}{c x^{4}} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.59, size = 36, normalized size = 0.84 \begin {gather*} \frac {a \arctan \left (\frac {\sqrt {c x^{4} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a}} + \frac {1}{2} \, \sqrt {c x^{4} + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 31, normalized size = 0.72 \begin {gather*} \frac {\sqrt {c\,x^4+a}}{2}-\frac {\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^4+a}}{\sqrt {a}}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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