Optimal. Leaf size=101 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a+b x^2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a+b x^2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}} \]
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Rubi [A]
time = 0.01, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {407}
\begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b x^2-a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b x^2-a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 407
Rubi steps
\begin {align*} \int \frac {1}{\left (-2 a+b x^2\right ) \sqrt [4]{-a+b x^2}} \, dx &=-\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a+b x^2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a+b x^2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 88, normalized size = 0.87 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a+b x^2}}{\sqrt {b} x}\right )-\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a+b x^2}}{\sqrt {b} x}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b \,x^{2}-2 a \right ) \left (b \,x^{2}-a \right )^{\frac {1}{4}}}\, dx\]
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 338 vs.
\(2 (71) = 142\).
time = 26.00, size = 338, normalized size = 3.35 \begin {gather*} -\left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, {\left (\sqrt {\frac {1}{2}} {\left (2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{3} b \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} \sqrt {b x^{2} - a} a \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}}\right )} \sqrt {a b \sqrt {\frac {1}{a^{3} b^{2}}}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (b x^{2} - a\right )}^{\frac {1}{4}} a \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}}\right )}}{x}\right ) - \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {b x^{2} - a} a^{2} b^{2} x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} + {\left (b x^{2} - a\right )}^{\frac {1}{4}} a^{2} b \sqrt {\frac {1}{a^{3} b^{2}}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} + {\left (b x^{2} - a\right )}^{\frac {3}{4}}}{b x^{2} - 2 \, a}\right ) + \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (-\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {b x^{2} - a} a^{2} b^{2} x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} - {\left (b x^{2} - a\right )}^{\frac {1}{4}} a^{2} b \sqrt {\frac {1}{a^{3} b^{2}}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} - {\left (b x^{2} - a\right )}^{\frac {3}{4}}}{b x^{2} - 2 \, a}\right ) \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 (- 2 a + b x^{2}\right ) \sqrt [4]{- a + b x^{2}}}\, 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 = -0.01 \begin {gather*} -\int \frac {1}{{\left (b\,x^2-a\right )}^{1/4}\,\left (2\,a-b\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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