Optimal. Leaf size=41 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {-c x^2+d \sqrt {a+b x^4}}}\right )}{a \sqrt {c}} \]
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Rubi [A]
time = 0.10, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2153, 209}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d \sqrt {a+b x^4}-c x^2}}\right )}{a \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 2153
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 {\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 {\tan ^{-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 = 0.84, size = 44, normalized size = 1.07 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {-c x^2+d \sqrt {a+b x^4}}}{\sqrt {c} x}\right )}{a \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b \,x^{4}+a \right ) \sqrt {-c \,x^{2}+d \sqrt {b \,x^{4}+a}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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}{\left (a + b x^{4}\right ) \sqrt {- c x^{2} + d \sqrt {a + b x^{4}}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.02 \begin {gather*} \int \frac {1}{\left (b\,x^4+a\right )\,\sqrt {d\,\sqrt {b\,x^4+a}-c\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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