Optimal. Leaf size=40 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d \sqrt {a+b x^4}+c x^2}}\right )}{a \sqrt {c}} \]
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Rubi [A] time = 0.14, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2128, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d \sqrt {a+b x^4}+c x^2}}\right )}{a \sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2128
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^4\right ) \sqrt {c x^2+d \sqrt {a+b x^4}}} \, dx &=\frac {\operatorname {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*}
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Mathematica [A] time = 0.49, size = 50, normalized size = 1.25 \[ \frac {\sqrt {-\frac {1}{c}} \cot ^{-1}\left (\frac {\sqrt {-\frac {1}{c}} \sqrt {d \sqrt {a+b x^4}+c x^2}}{x}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{4} + a\right )} \sqrt {c x^{2} + \sqrt {b x^{4} + a} d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{4}+a \right ) \sqrt {c \,x^{2}+\sqrt {b \,x^{4}+a}\, d}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{4} + a\right )} \sqrt {c x^{2} + \sqrt {b x^{4} + a} d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\left (b\,x^4+a\right )\,\sqrt {d\,\sqrt {b\,x^4+a}+c\,x^2}} \,d x \]
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 \[ \int \frac {1}{\left (a + b x^{4}\right ) \sqrt {c x^{2} + d \sqrt {a + b x^{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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