Optimal. Leaf size=152 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^8+1}}\right )}{4 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^8+1}}\right )}{4 \sqrt [4]{2}}+\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^8+1}}{\sqrt {2} x^2-\sqrt {x^8+1}}\right )}{4\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{x^8+1}}{\sqrt {2} \sqrt {x^8+1}+2 x^2}\right )}{4\ 2^{3/4}} \]
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Rubi [C] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 0.14, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {429} \begin {gather*} -x F_1\left (\frac {1}{8};1,-\frac {3}{4};\frac {9}{8};x^8,-x^8\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 429
Rubi steps
\begin {align*} \int \frac {\left (1+x^8\right )^{3/4}}{-1+x^8} \, dx &=-x F_1\left (\frac {1}{8};1,-\frac {3}{4};\frac {9}{8};x^8,-x^8\right )\\ \end {align*}
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Mathematica [C] time = 0.12, size = 110, normalized size = 0.72 \begin {gather*} \frac {9 x \left (x^8+1\right )^{3/4} F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};-x^8,x^8\right )}{\left (x^8-1\right ) \left (2 x^8 \left (4 F_1\left (\frac {9}{8};-\frac {3}{4},2;\frac {17}{8};-x^8,x^8\right )+3 F_1\left (\frac {9}{8};\frac {1}{4},1;\frac {17}{8};-x^8,x^8\right )\right )+9 F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};-x^8,x^8\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.52, size = 152, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^8+1}}\right )}{4 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^8+1}}\right )}{4 \sqrt [4]{2}}+\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^8+1}}{\sqrt {2} x^2-\sqrt {x^8+1}}\right )}{4\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{x^8+1}}{\sqrt {2} \sqrt {x^8+1}+2 x^2}\right )}{4\ 2^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 16.98, size = 677, normalized size = 4.45 \begin {gather*} -\frac {1}{8} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x^{2}}{x^{9} + x}\right ) - \frac {1}{32} \cdot 2^{\frac {3}{4}} \log \left (-\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x + 4 \, \sqrt {x^{8} + 1} x^{2} + \sqrt {2} {\left (x^{8} + 2 \, x^{4} + 1\right )}}{x^{8} - 2 \, x^{4} + 1}\right ) + \frac {1}{32} \cdot 2^{\frac {3}{4}} \log \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x - 4 \, \sqrt {x^{8} + 1} x^{2} - \sqrt {2} {\left (x^{8} + 2 \, x^{4} + 1\right )}}{x^{8} - 2 \, x^{4} + 1}\right ) - \frac {1}{8} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} - 4 \, \sqrt {x^{8} + 1} x^{2} + 2 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x - \sqrt {2} {\left (x^{8} + 2 \, x^{4} + 1\right )}\right )} \sqrt {\frac {x^{8} + 2 \, x^{4} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {2} \sqrt {x^{8} + 1} x^{2} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x + 1}{x^{8} + 2 \, x^{4} + 1}}}{2 \, {\left (x^{8} - 2 \, x^{4} + 1\right )}}\right ) - \frac {1}{8} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{8} + 1} x^{2} + 2 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (x^{8} + 2 \, x^{4} + 1\right )}\right )} \sqrt {\frac {x^{8} + 2 \, x^{4} - 4 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {2} \sqrt {x^{8} + 1} x^{2} - 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x + 1}{x^{8} + 2 \, x^{4} + 1}}}{2 \, {\left (x^{8} - 2 \, x^{4} + 1\right )}}\right ) - \frac {1}{32} \cdot 2^{\frac {1}{4}} \log \left (\frac {2 \, {\left (x^{8} + 2 \, x^{4} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {2} \sqrt {x^{8} + 1} x^{2} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x + 1\right )}}{x^{8} + 2 \, x^{4} + 1}\right ) + \frac {1}{32} \cdot 2^{\frac {1}{4}} \log \left (\frac {2 \, {\left (x^{8} + 2 \, x^{4} - 4 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {2} \sqrt {x^{8} + 1} x^{2} - 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x + 1\right )}}{x^{8} + 2 \, x^{4} + 1}\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 {{\left (x^{8} + 1\right )}^{\frac {3}{4}}}{x^{8} - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 16.95, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{8}+1\right )^{\frac {3}{4}}}{x^{8}-1}\, 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 {{\left (x^{8} + 1\right )}^{\frac {3}{4}}}{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 = -0.01 \begin {gather*} \int \frac {{\left (x^8+1\right )}^{3/4}}{x^8-1} \,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 {\left (x^{8} + 1\right )^{\frac {3}{4}}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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