Optimal. Leaf size=78 \[ \frac {\sqrt {a-b x^2} \sqrt {a+b x^2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a-b x^2}}\right )}{\sqrt {2} a \sqrt {b} \sqrt {a^2-b^2 x^4}} \]
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Rubi [A] time = 0.04, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1152, 377, 205} \begin {gather*} \frac {\sqrt {a-b x^2} \sqrt {a+b x^2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a-b x^2}}\right )}{\sqrt {2} a \sqrt {b} \sqrt {a^2-b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 377
Rule 1152
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b x^2} \sqrt {a^2-b^2 x^4}} \, dx &=\frac {\left (\sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {a-b x^2} \left (a+b x^2\right )} \, dx}{\sqrt {a^2-b^2 x^4}}\\ &=\frac {\left (\sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 a b x^2} \, dx,x,\frac {x}{\sqrt {a-b x^2}}\right )}{\sqrt {a^2-b^2 x^4}}\\ &=\frac {\sqrt {a-b x^2} \sqrt {a+b x^2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a-b x^2}}\right )}{\sqrt {2} a \sqrt {b} \sqrt {a^2-b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 78, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a^2-b^2 x^4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a-b x^2}}\right )}{\sqrt {2} a \sqrt {b} \sqrt {a-b x^2} \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 1.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a+b x^2} \sqrt {a^2-b^2 x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.85, size = 152, normalized size = 1.95 \begin {gather*} \left [-\frac {\sqrt {2} \sqrt {-b} \log \left (-\frac {3 \, b^{2} x^{4} + 2 \, a b x^{2} - 2 \, \sqrt {2} \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a} \sqrt {-b} x - a^{2}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \, a b}, -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a} \sqrt {b}}{2 \, {\left (b^{2} x^{3} + a b x\right )}}\right )}{2 \, a \sqrt {b}}\right ] \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*} \int \frac {1}{\sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 249, normalized size = 3.19 \begin {gather*} \frac {\sqrt {-b^{2} x^{4}+a^{2}}\, \left (\sqrt {2}\, \sqrt {a}\, \sqrt {b}\, \ln \left (\frac {2 \left (a -\sqrt {-a b}\, x +\sqrt {2}\, \sqrt {-b \,x^{2}+a}\, \sqrt {a}\right ) b}{b x -\sqrt {-a b}}\right )-\sqrt {2}\, \sqrt {a}\, \sqrt {b}\, \ln \left (\frac {2 \left (a +\sqrt {-a b}\, x +\sqrt {2}\, \sqrt {-b \,x^{2}+a}\, \sqrt {a}\right ) b}{b x +\sqrt {-a b}}\right )-2 \sqrt {-a b}\, \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {\frac {\left (-b x +\sqrt {a b}\right ) \left (b x +\sqrt {a b}\right )}{b}}}\right )+2 \sqrt {-a b}\, \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {-b \,x^{2}+a}}\right )\right ) \sqrt {b}}{2 \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, \left (\sqrt {-a b}+\sqrt {a b}\right ) \left (\sqrt {-a b}-\sqrt {a b}\right ) \sqrt {-a b}} \end {gather*}
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*} \int \frac {1}{\sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a}}\,{d x} \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 = -0.01 \begin {gather*} \int \frac {1}{\sqrt {a^2-b^2\,x^4}\,\sqrt {b\,x^2+a}} \,d x \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}{\sqrt {- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )} \sqrt {a + b x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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