Integrand size = 22, antiderivative size = 60 \[ \int \frac {1}{\sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {\sqrt {a} \sqrt {1+\frac {b x^2}{a}} \text {arcsinh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} \sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \]
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Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1103, 221} \[ \int \frac {1}{\sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {\sqrt {a} \sqrt {\frac {b x^2}{a}+1} \text {arcsinh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} \sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \]
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Rule 221
Rule 1103
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+\frac {b x^2}{a}} \int \frac {1}{\sqrt {1+\frac {b x^2}{a}}} \, dx}{\sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {\sqrt {a} \sqrt {1+\frac {b x^2}{a}} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} \sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {\sqrt {a+b x^2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} \sqrt [4]{\left (a+b x^2\right )^2}} \]
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\[\int \frac {1}{\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {1}{4}}}d x\]
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none
Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.50 \[ \int \frac {1}{\sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \, dx=\left [\frac {\log \left (-2 \, b x^{2} - 2 \, {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {1}{4}} \sqrt {b} x - a\right )}{2 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {1}{4}} \sqrt {-b} x}{b x^{2} + a}\right )}{b}\right ] \]
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\[ \int \frac {1}{\sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \, dx=\int \frac {1}{\sqrt [4]{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}\, dx \]
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\[ \int \frac {1}{\sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \, dx=\int { \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \, dx=\int { \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \, dx=\int \frac {1}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{1/4}} \,d x \]
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