Optimal. Leaf size=104 \[ \frac {\tan ^{-1}\left (\frac {\left (1-\sqrt [3]{b x^2+1}\right )^2}{3 \sqrt {b} x}\right )}{12 \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{b x^2+1}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{3}\right )}{12 \sqrt {b}} \]
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Rubi [A] time = 0.02, antiderivative size = 104, 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 = {394} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\left (1-\sqrt [3]{b x^2+1}\right )^2}{3 \sqrt {b} x}\right )}{12 \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{b x^2+1}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{3}\right )}{12 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 394
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{1+b x^2} \left (9+b x^2\right )} \, dx &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{3}\right )}{12 \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\left (1-\sqrt [3]{1+b x^2}\right )^2}{3 \sqrt {b} x}\right )}{12 \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{1+b x^2}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} \sqrt {b}}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 137, normalized size = 1.32 \begin {gather*} -\frac {27 x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-b x^2,-\frac {b x^2}{9}\right )}{\sqrt [3]{b x^2+1} \left (b x^2+9\right ) \left (2 b x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};-b x^2,-\frac {b x^2}{9}\right )+3 F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};-b x^2,-\frac {b x^2}{9}\right )\right )-27 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-b x^2,-\frac {b x^2}{9}\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 4.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{1+b x^2} \left (9+b x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [F(-1)] 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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x^{2} + 9\right )} {\left (b x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b \,x^{2}+1\right )^{\frac {1}{3}} \left (b \,x^{2}+9\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} + 9\right )} {\left (b x^{2} + 1\right )}^{\frac {1}{3}}}\,{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+1\right )}^{1/3}\,\left (b\,x^2+9\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 [3]{b x^{2} + 1} \left (b x^{2} + 9\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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