Optimal. Leaf size=32 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x^2}+b x^2}}\right )}{2 \sqrt {b}} \]
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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2004, 2033,
212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x^2}+b x^2}}\right )}{2 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2004
Rule 2033
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\frac {a+b x^4}{x^2}}} \, dx &=\int \frac {1}{\sqrt {\frac {a}{x^2}+b x^2}} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {\frac {a}{x^2}+b x^2}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x^2}+b x^2}}\right )}{2 \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 59, normalized size = 1.84 \begin {gather*} \frac {\sqrt {a+b x^4} \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {b} x^2}\right )}{2 \sqrt {b} x \sqrt {\frac {a+b x^4}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 49, normalized size = 1.53
method | result | size |
default | \(\frac {\sqrt {b \,x^{4}+a}\, \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{2 \sqrt {\frac {b \,x^{4}+a}{x^{2}}}\, x \sqrt {b}}\) | \(49\) |
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 = 1.68, size = 80, normalized size = 2.50 \begin {gather*} \left [\frac {\log \left (-2 \, b x^{4} - 2 \, \sqrt {b} x^{3} \sqrt {\frac {b x^{4} + a}{x^{2}}} - a\right )}{4 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {\sqrt {-b} x^{3} \sqrt {\frac {b x^{4} + a}{x^{2}}}}{b x^{4} + a}\right )}{2 \, b}\right ] \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 {\frac {a + b x^{4}}{x^{2}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.54, size = 40, normalized size = 1.25 \begin {gather*} \frac {\log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{4 \, \sqrt {b}} - \frac {\log \left ({\left | -\sqrt {b} x^{2} + \sqrt {b x^{4} + a} \right |}\right )}{2 \, \sqrt {b} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {\frac {b\,x^4+a}{x^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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