Integrand size = 33, antiderivative size = 40 \[ \int \frac {1}{\left (a+b x^4\right ) \sqrt {c x^2+d \sqrt {a+b x^4}}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {c x^2+d \sqrt {a+b x^4}}}\right )}{a \sqrt {c}} \]
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Time = 0.10 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2153, 212} \[ \int \frac {1}{\left (a+b x^4\right ) \sqrt {c x^2+d \sqrt {a+b x^4}}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {d \sqrt {a+b x^4}+c x^2}}\right )}{a \sqrt {c}} \]
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Rule 212
Rule 2153
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {c x^2+d \sqrt {a+b x^4}}}\right )}{a} \\ & = \frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {c x^2+d \sqrt {a+b x^4}}}\right )}{a \sqrt {c}} \\ \end{align*}
Time = 0.83 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (a+b x^4\right ) \sqrt {c x^2+d \sqrt {a+b x^4}}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {c x^2+d \sqrt {a+b x^4}}}{\sqrt {c} x}\right )}{a \sqrt {c}} \]
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\[\int \frac {1}{\left (b \,x^{4}+a \right ) \sqrt {c \,x^{2}+d \sqrt {b \,x^{4}+a}}}d x\]
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Timed out. \[ \int \frac {1}{\left (a+b x^4\right ) \sqrt {c x^2+d \sqrt {a+b x^4}}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (a+b x^4\right ) \sqrt {c x^2+d \sqrt {a+b x^4}}} \, dx=\int \frac {1}{\left (a + b x^{4}\right ) \sqrt {c x^{2} + d \sqrt {a + b x^{4}}}}\, dx \]
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\[ \int \frac {1}{\left (a+b x^4\right ) \sqrt {c x^2+d \sqrt {a+b x^4}}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )} \sqrt {c x^{2} + \sqrt {b x^{4} + a} d}} \,d x } \]
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\[ \int \frac {1}{\left (a+b x^4\right ) \sqrt {c x^2+d \sqrt {a+b x^4}}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )} \sqrt {c x^{2} + \sqrt {b x^{4} + a} d}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b x^4\right ) \sqrt {c x^2+d \sqrt {a+b x^4}}} \, dx=\int \frac {1}{\left (b\,x^4+a\right )\,\sqrt {d\,\sqrt {b\,x^4+a}+c\,x^2}} \,d x \]
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