Optimal. Leaf size=103 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{2}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{1+x^4}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{1+x^4}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1913, 385,
218, 212, 209, 455, 65, 304} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}-\frac {\text {ArcTan}\left (\frac {\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 212
Rule 218
Rule 304
Rule 385
Rule 455
Rule 1913
Rubi steps
\begin {align*} \int \frac {1+x^3}{\left (1-x^4\right ) \sqrt [4]{1+x^4}} \, dx &=\int \frac {1}{\left (1-x^4\right ) \sqrt [4]{1+x^4}} \, dx+\int \frac {x^3}{\left (1-x^4\right ) \sqrt [4]{1+x^4}} \, dx\\ &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{(1-x) \sqrt [4]{1+x}} \, dx,x,x^4\right )+\text {Subst}\left (\int \frac {1}{1-2 x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\text {Subst}\left (\int \frac {x^2}{2-x^4} \, dx,x,\sqrt [4]{1+x^4}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{2}}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {2}-x^2} \, dx,x,\sqrt [4]{1+x^4}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {2}+x^2} \, dx,x,\sqrt [4]{1+x^4}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{2}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{1+x^4}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{1+x^4}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 10.13, size = 93, normalized size = 0.90 \begin {gather*} \frac {1}{4} x^4 F_1\left (1;\frac {1}{4},1;2;-x^4,x^4\right )+\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )-\log \left (1-\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )+\log \left (1+\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{4 \sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{3}+1}{\left (-x^{4}+1\right ) \left (x^{4}+1\right )^{\frac {1}{4}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 \left (- \frac {x}{x^{3} \sqrt [4]{x^{4} + 1} - x^{2} \sqrt [4]{x^{4} + 1} + x \sqrt [4]{x^{4} + 1} - \sqrt [4]{x^{4} + 1}}\right )\, dx - \int \frac {x^{2}}{x^{3} \sqrt [4]{x^{4} + 1} - x^{2} \sqrt [4]{x^{4} + 1} + x \sqrt [4]{x^{4} + 1} - \sqrt [4]{x^{4} + 1}}\, dx - \int \frac {1}{x^{3} \sqrt [4]{x^{4} + 1} - x^{2} \sqrt [4]{x^{4} + 1} + x \sqrt [4]{x^{4} + 1} - \sqrt [4]{x^{4} + 1}}\, dx \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*} \text {could not integrate} \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 {x^3+1}{\left (x^4-1\right )\,{\left (x^4+1\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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