Optimal. Leaf size=129 \[ -\frac {\tan ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt {b x^2+2}+2\ 2^{3/4}}{2 \sqrt {b} x \sqrt [4]{b x^2+2}}\right )}{2\ 2^{3/4} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {b x^2+2}}{2 \sqrt {b} x \sqrt [4]{b x^2+2}}\right )}{2\ 2^{3/4} \sqrt {b}} \]
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Rubi [A] time = 0.02, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {397} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt {b x^2+2}+2\ 2^{3/4}}{2 \sqrt {b} x \sqrt [4]{b x^2+2}}\right )}{2\ 2^{3/4} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {b x^2+2}}{2 \sqrt {b} x \sqrt [4]{b x^2+2}}\right )}{2\ 2^{3/4} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 397
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{2+b x^2} \left (4+b x^2\right )} \, dx &=-\frac {\tan ^{-1}\left (\frac {2\ 2^{3/4}+2 \sqrt [4]{2} \sqrt {2+b x^2}}{2 \sqrt {b} x \sqrt [4]{2+b x^2}}\right )}{2\ 2^{3/4} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {2+b x^2}}{2 \sqrt {b} x \sqrt [4]{2+b x^2}}\right )}{2\ 2^{3/4} \sqrt {b}}\\ \end {align*}
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Mathematica [C] time = 0.14, size = 144, normalized size = 1.12 \begin {gather*} -\frac {12 x F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};-\frac {b x^2}{2},-\frac {b x^2}{4}\right )}{\sqrt [4]{b x^2+2} \left (b x^2+4\right ) \left (b x^2 \left (2 F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};-\frac {b x^2}{2},-\frac {b x^2}{4}\right )+F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};-\frac {b x^2}{2},-\frac {b x^2}{4}\right )\right )-12 F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};-\frac {b x^2}{2},-\frac {b x^2}{4}\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.27, size = 137, normalized size = 1.06 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\frac {\sqrt {b} x^2}{2 \sqrt [4]{2}}-\frac {\sqrt [4]{2} \sqrt {b x^2+2}}{\sqrt {b}}}{x \sqrt [4]{b x^2+2}}\right )}{4\ 2^{3/4} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4} \sqrt {b} x \sqrt [4]{b x^2+2}}{\sqrt {2} b x^2+4 \sqrt {b x^2+2}}\right )}{4\ 2^{3/4} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 28.02, size = 755, normalized size = 5.85
result too large to display
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}{{\left (b x^{2} + 4\right )} {\left (b x^{2} + 2\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b \,x^{2}+2\right )^{\frac {1}{4}} \left (b \,x^{2}+4\right )}\, dx \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}{{\left (b x^{2} + 4\right )} {\left (b x^{2} + 2\right )}^{\frac {1}{4}}}\,{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}{{\left (b\,x^2+2\right )}^{1/4}\,\left (b\,x^2+4\right )} \,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 [4]{b x^{2} + 2} \left (b x^{2} + 4\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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