Optimal. Leaf size=81 \[ \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )}{\sqrt [4]{2}}+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )}{\sqrt [4]{2}} \]
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Rubi [A] time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {408, 240, 212, 206, 203, 377} \begin {gather*} \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )}{\sqrt [4]{2}}+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )}{\sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 240
Rule 377
Rule 408
Rubi steps
\begin {align*} \int \frac {\left (-1+x^4\right )^{3/4}}{1+x^4} \, dx &=-\left (2 \int \frac {1}{\sqrt [4]{-1+x^4} \left (1+x^4\right )} \, dx\right )+\int \frac {1}{\sqrt [4]{-1+x^4}} \, dx\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{1-2 x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\right )+\operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt [4]{2}}+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt [4]{2}}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 109, normalized size = 1.35 \begin {gather*} -\frac {5 x \left (x^4-1\right )^{3/4} F_1\left (\frac {1}{4};-\frac {3}{4},1;\frac {5}{4};x^4,-x^4\right )}{\left (x^4+1\right ) \left (x^4 \left (4 F_1\left (\frac {5}{4};-\frac {3}{4},2;\frac {9}{4};x^4,-x^4\right )+3 F_1\left (\frac {5}{4};\frac {1}{4},1;\frac {9}{4};x^4,-x^4\right )\right )-5 F_1\left (\frac {1}{4};-\frac {3}{4},1;\frac {5}{4};x^4,-x^4\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.30, size = 81, normalized size = 1.00 \begin {gather*} \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt [4]{2}}+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 149, normalized size = 1.84 \begin {gather*} -2^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {3}{4}} x \sqrt {\frac {\sqrt {2} x^{2} + \sqrt {x^{4} - 1}}{x^{2}}} - 2^{\frac {3}{4}} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {1}{4} \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \cdot 2^{\frac {3}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \arctan \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \log \left (\frac {x + {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \log \left (-\frac {x - {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\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^{4} - 1\right )}^{\frac {3}{4}}}{x^{4} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}-1\right )^{\frac {3}{4}}}{x^{4}+1}\, dx\]
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 (x^{4} - 1\right )}^{\frac {3}{4}}}{x^{4} + 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^4-1\right )}^{3/4}}{x^4+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 (\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )\right )^{\frac {3}{4}}}{x^{4} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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