Integrand size = 15, antiderivative size = 43 \[ \int \frac {\sqrt {a+\frac {b}{x^4}}}{x} \, dx=-\frac {1}{2} \sqrt {a+\frac {b}{x^4}}+\frac {1}{2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 52, 65, 214} \[ \int \frac {\sqrt {a+\frac {b}{x^4}}}{x} \, dx=\frac {1}{2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )-\frac {1}{2} \sqrt {a+\frac {b}{x^4}} \]
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Rule 52
Rule 65
Rule 214
Rule 272
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{4} \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\frac {1}{x^4}\right )\right ) \\ & = -\frac {1}{2} \sqrt {a+\frac {b}{x^4}}-\frac {1}{4} a \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^4}\right ) \\ & = -\frac {1}{2} \sqrt {a+\frac {b}{x^4}}-\frac {a \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^4}}\right )}{2 b} \\ & = -\frac {1}{2} \sqrt {a+\frac {b}{x^4}}+\frac {1}{2} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right ) \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.67 \[ \int \frac {\sqrt {a+\frac {b}{x^4}}}{x} \, dx=-\frac {1}{2} \sqrt {a+\frac {b}{x^4}}+\frac {\sqrt {a} \sqrt {a+\frac {b}{x^4}} x^2 \log \left (\sqrt {a} x^2+\sqrt {b+a x^4}\right )}{2 \sqrt {b+a x^4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(64\) vs. \(2(31)=62\).
Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.51
method | result | size |
risch | \(-\frac {\sqrt {\frac {a \,x^{4}+b}{x^{4}}}}{2}+\frac {\sqrt {a}\, \ln \left (x^{2} \sqrt {a}+\sqrt {a \,x^{4}+b}\right ) \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, x^{2}}{2 \sqrt {a \,x^{4}+b}}\) | \(65\) |
default | \(\frac {\sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, \left (a \,x^{4} \sqrt {a \,x^{4}+b}+\sqrt {a}\, \ln \left (x^{2} \sqrt {a}+\sqrt {a \,x^{4}+b}\right ) b \,x^{2}-\left (a \,x^{4}+b \right )^{\frac {3}{2}}\right )}{2 \sqrt {a \,x^{4}+b}\, b}\) | \(80\) |
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Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.53 \[ \int \frac {\sqrt {a+\frac {b}{x^4}}}{x} \, dx=\left [\frac {1}{4} \, \sqrt {a} \log \left (-2 \, a x^{4} - 2 \, \sqrt {a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - b\right ) - \frac {1}{2} \, \sqrt {\frac {a x^{4} + b}{x^{4}}}, -\frac {1}{2} \, \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right ) - \frac {1}{2} \, \sqrt {\frac {a x^{4} + b}{x^{4}}}\right ] \]
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Time = 0.85 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.53 \[ \int \frac {\sqrt {a+\frac {b}{x^4}}}{x} \, dx=\frac {\sqrt {a} \operatorname {asinh}{\left (\frac {\sqrt {a} x^{2}}{\sqrt {b}} \right )}}{2} - \frac {a x^{2}}{2 \sqrt {b} \sqrt {\frac {a x^{4}}{b} + 1}} - \frac {\sqrt {b}}{2 x^{2} \sqrt {\frac {a x^{4}}{b} + 1}} \]
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Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {a+\frac {b}{x^4}}}{x} \, dx=-\frac {1}{4} \, \sqrt {a} \log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{4}}} + \sqrt {a}}\right ) - \frac {1}{2} \, \sqrt {a + \frac {b}{x^{4}}} \]
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Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {a+\frac {b}{x^4}}}{x} \, dx=-\frac {1}{4} \, \sqrt {a} \log \left ({\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2}\right ) + \frac {\sqrt {a} b}{{\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2} - b} \]
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Time = 6.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {a+\frac {b}{x^4}}}{x} \, dx=\frac {\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{2}-\frac {\sqrt {a+\frac {b}{x^4}}}{2} \]
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