Optimal. Leaf size=162 \[ \frac {\text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {1}{4} \text {RootSum}\left [3 a^2-b-5 a \text {$\#$1}^4+2 \text {$\#$1}^8\& ,\frac {3 a \log (x)-3 a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-2 \log (x) \text {$\#$1}^4+2 \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{5 a \text {$\#$1}-4 \text {$\#$1}^5}\& \right ] \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(561\) vs. \(2(162)=324\).
time = 1.05, antiderivative size = 561, normalized size of antiderivative = 3.46, number of steps
used = 16, number of rules used = 8, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6860, 246,
218, 212, 209, 385, 214, 211} \begin {gather*} -\frac {\left (a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \text {ArcTan}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4-b}}\right )}{2 \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+8 b}+a^2-2 b}}-\frac {\left (\frac {a^2+4 b}{\sqrt {a^2+8 b}}+a\right ) \text {ArcTan}\left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4-b}}\right )}{2 \left (\sqrt {a^2+8 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+8 b}+a^2-2 b}}-\frac {\left (a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \tanh ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4-b}}\right )}{2 \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+8 b}+a^2-2 b}}-\frac {\left (\frac {a^2+4 b}{\sqrt {a^2+8 b}}+a\right ) \tanh ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4-b}}\right )}{2 \left (\sqrt {a^2+8 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+8 b}+a^2-2 b}}+\frac {\text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 211
Rule 212
Rule 214
Rule 218
Rule 246
Rule 385
Rule 6860
Rubi steps
\begin {align*} \int \frac {-2 b-a x^4+2 x^8}{\sqrt [4]{-b+a x^4} \left (-2 b-a x^4+x^8\right )} \, dx &=\int \left (\frac {2}{\sqrt [4]{-b+a x^4}}+\frac {2 b+a x^4}{\sqrt [4]{-b+a x^4} \left (-2 b-a x^4+x^8\right )}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt [4]{-b+a x^4}} \, dx+\int \frac {2 b+a x^4}{\sqrt [4]{-b+a x^4} \left (-2 b-a x^4+x^8\right )} \, dx\\ &=2 \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\int \left (\frac {a+\frac {a^2+4 b}{\sqrt {a^2+8 b}}}{\left (-a-\sqrt {a^2+8 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}}+\frac {a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}}{\left (-a+\sqrt {a^2+8 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}}\right ) \, dx\\ &=\left (a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \int \frac {1}{\left (-a+\sqrt {a^2+8 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx+\left (a+\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \int \frac {1}{\left (-a-\sqrt {a^2+8 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx+\text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}+\left (a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \text {Subst}\left (\int \frac {1}{-a+\sqrt {a^2+8 b}-\left (2 b+a \left (-a+\sqrt {a^2+8 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\left (a+\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \text {Subst}\left (\int \frac {1}{-a-\sqrt {a^2+8 b}-\left (2 b+a \left (-a-\sqrt {a^2+8 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {\left (a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+8 b}}-\sqrt {a^2-2 b-a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {a-\sqrt {a^2+8 b}}}-\frac {\left (a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+8 b}}+\sqrt {a^2-2 b-a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {a-\sqrt {a^2+8 b}}}-\frac {\left (a+\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+8 b}}-\sqrt {a^2-2 b+a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {a+\sqrt {a^2+8 b}}}-\frac {\left (a+\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+8 b}}+\sqrt {a^2-2 b+a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {a+\sqrt {a^2+8 b}}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {\left (a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+8 b}} x}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{-b+a x^4}}\right )}{2 \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+8 b}}}-\frac {\left (a+\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+8 b}} x}{\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt [4]{-b+a x^4}}\right )}{2 \left (a+\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+8 b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {\left (a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+8 b}} x}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{-b+a x^4}}\right )}{2 \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+8 b}}}-\frac {\left (a+\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+8 b}} x}{\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt [4]{-b+a x^4}}\right )}{2 \left (a+\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+8 b}}}\\ \end {align*}
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Mathematica [A]
time = 0.82, size = 157, normalized size = 0.97 \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {1}{4} \text {RootSum}\left [3 a^2-b-5 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-3 a \log (x)+3 a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-5 a \text {$\#$1}+4 \text {$\#$1}^5}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {2 x^{8}-a \,x^{4}-2 b}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (x^{8}-a \,x^{4}-2 b \right )}\, dx\]
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 [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 1.11, size = 5282, normalized size = 32.60 \begin {gather*} \text {Too large to display} \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 {- a x^{4} - 2 b + 2 x^{8}}{\sqrt [4]{a x^{4} - b} \left (- a x^{4} - 2 b + x^{8}\right )}\, 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 {-2\,x^8+a\,x^4+2\,b}{{\left (a\,x^4-b\right )}^{1/4}\,\left (-x^8+a\,x^4+2\,b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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