Optimal. Leaf size=432 \[ \frac {\left (1+\sqrt [4]{-1}\right ) \tan ^{-1}\left (\frac {(-1)^{7/8} \sqrt {2+\sqrt {2}} \sqrt [8]{2 a-1} x \sqrt [4]{a x^8+1}}{\sqrt {a x^8+1}+(-1)^{3/4} \sqrt [4]{2 a-1} x^2}\right )}{8 \sqrt [8]{2 a-1}}-\frac {i \left (\sqrt {2 \left (3-2 \sqrt {2}\right )}-i \sqrt {2}\right ) \tan ^{-1}\left (\frac {(-1)^{7/8} \left (\sqrt {2}-2\right ) \sqrt [8]{2 a-1} x \sqrt [4]{a x^8+1}}{\sqrt {2-\sqrt {2}} \sqrt {a x^8+1}+(-1)^{3/4} \sqrt {2-\sqrt {2}} \sqrt [4]{2 a-1} x^2}\right )}{16 \sqrt [8]{2 a-1}}+\frac {\left (\sqrt {2}+i \sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {(-1)^{7/8} \sqrt [4]{2 a-1} x^2-\sqrt [8]{-1} \sqrt {a x^8+1}}{\sqrt {2-\sqrt {2}} \sqrt [8]{2 a-1} x \sqrt [4]{a x^8+1}}\right )}{16 \sqrt [8]{2 a-1}}+\frac {\left (1+\sqrt [4]{-1}\right ) \tanh ^{-1}\left (\frac {(-1)^{7/8} \sqrt [4]{2 a-1} x^2-\sqrt [8]{-1} \sqrt {a x^8+1}}{\sqrt {2+\sqrt {2}} \sqrt [8]{2 a-1} x \sqrt [4]{a x^8+1}}\right )}{8 \sqrt [8]{2 a-1}} \]
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Rubi [C] time = 0.38, antiderivative size = 160, normalized size of antiderivative = 0.37, number of steps used = 4, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {6728, 429} \begin {gather*} \frac {a \left (1-\frac {2 a+1}{\sqrt {1-4 a^2}}\right ) x F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};-a x^8,-\frac {2 a^2 x^8}{1-\sqrt {1-4 a^2}}\right )}{1-\sqrt {1-4 a^2}}+\frac {a \left (\frac {2 a+1}{\sqrt {1-4 a^2}}+1\right ) x F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};-a x^8,-\frac {2 a^2 x^8}{\sqrt {1-4 a^2}+1}\right )}{\sqrt {1-4 a^2}+1} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 429
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-1+a x^8\right ) \left (1+a x^8\right )^{3/4}}{1+x^8+a^2 x^{16}} \, dx &=\int \left (\frac {\left (a-\frac {a (1+2 a)}{\sqrt {1-4 a^2}}\right ) \left (1+a x^8\right )^{3/4}}{1-\sqrt {1-4 a^2}+2 a^2 x^8}+\frac {\left (a+\frac {a (1+2 a)}{\sqrt {1-4 a^2}}\right ) \left (1+a x^8\right )^{3/4}}{1+\sqrt {1-4 a^2}+2 a^2 x^8}\right ) \, dx\\ &=\left (a \left (1-\frac {1+2 a}{\sqrt {1-4 a^2}}\right )\right ) \int \frac {\left (1+a x^8\right )^{3/4}}{1-\sqrt {1-4 a^2}+2 a^2 x^8} \, dx+\left (a \left (1+\frac {1+2 a}{\sqrt {1-4 a^2}}\right )\right ) \int \frac {\left (1+a x^8\right )^{3/4}}{1+\sqrt {1-4 a^2}+2 a^2 x^8} \, dx\\ &=\frac {a \left (1-\frac {1+2 a}{\sqrt {1-4 a^2}}\right ) x F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};-a x^8,-\frac {2 a^2 x^8}{1-\sqrt {1-4 a^2}}\right )}{1-\sqrt {1-4 a^2}}+\frac {a \left (1+\frac {1+2 a}{\sqrt {1-4 a^2}}\right ) x F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};-a x^8,-\frac {2 a^2 x^8}{1+\sqrt {1-4 a^2}}\right )}{1+\sqrt {1-4 a^2}}\\ \end {align*}
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Mathematica [F] time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+a x^8\right ) \left (1+a x^8\right )^{3/4}}{1+x^8+a^2 x^{16}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 14.48, size = 486, normalized size = 1.12 \begin {gather*} \frac {i \left (i \sqrt {2}+\sqrt {2 \left (3+2 \sqrt {2}\right )}\right ) \tan ^{-1}\left (\frac {\left ((-1+i)-(1+i) (-1)^{3/4}\right ) \sqrt [4]{-1+2 a} x^2+(1+i) \sqrt {1+a x^8}+(1+i) (-1)^{3/4} \sqrt {1+a x^8}}{2 \sqrt [8]{-1+2 a} x \sqrt [4]{1+a x^8}}\right )}{16 \sqrt [8]{-1+2 a}}+\frac {\left (\sqrt {2}+i \sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \tan ^{-1}\left (\frac {2 \sqrt [8]{-1+2 a} x \sqrt [4]{1+a x^8}}{\left ((1-i)+\sqrt {2}\right ) \sqrt [4]{-1+2 a} x^2-(1+i) \sqrt {1+a x^8}-\sqrt {2} \sqrt {1+a x^8}}\right )}{16 \sqrt [8]{-1+2 a}}+\frac {\left (i+(-1)^{3/4}\right ) \tanh ^{-1}\left (\frac {\left ((-2+2 i)-(2+2 i) (-1)^{3/4}\right ) \sqrt [4]{-1+2 a} x^2-(2+2 i) \sqrt {1+a x^8}-(2+2 i) (-1)^{3/4} \sqrt {1+a x^8}}{4 \sqrt [8]{-1+2 a} x \sqrt [4]{1+a x^8}}\right )}{8 \sqrt [8]{-1+2 a}}-\frac {i \left (-i \sqrt {2}+\sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {\left ((1-i)+\sqrt {2}\right ) \sqrt [4]{-1+2 a} x^2+(1+i) \sqrt {1+a x^8}+\sqrt {2} \sqrt {1+a x^8}}{2 \sqrt [8]{-1+2 a} x \sqrt [4]{1+a x^8}}\right )}{16 \sqrt [8]{-1+2 a}} \end {gather*}
Antiderivative was successfully verified.
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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 {{\left (a x^{8} + 1\right )}^{\frac {3}{4}} {\left (a x^{8} - 1\right )}}{a^{2} x^{16} + x^{8} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{8}-1\right ) \left (a \,x^{8}+1\right )^{\frac {3}{4}}}{a^{2} x^{16}+x^{8}+1}\, dx\]
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*} \int \frac {{\left (a x^{8} + 1\right )}^{\frac {3}{4}} {\left (a x^{8} - 1\right )}}{a^{2} x^{16} + x^{8} + 1}\,{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 = -0.00 \begin {gather*} \int \frac {\left (a\,x^8-1\right )\,{\left (a\,x^8+1\right )}^{3/4}}{a^2\,x^{16}+x^8+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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