Optimal. Leaf size=96 \[ -\frac {\text {RootSum}\left [b+2 a \text {$\#$1}^4-2 \text {$\#$1}^8\& ,\frac {a \log (x)-a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}-2 \text {$\#$1}^5}\& \right ]}{8 b} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(441\) vs. \(2(96)=192\).
time = 0.47, antiderivative size = 441, normalized size of antiderivative = 4.59, number of steps
used = 9, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1442, 385, 218,
214, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+2 b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+2 b}} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt {a^2+2 b} \left (a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+2 b}+a^2+b}}-\frac {\text {ArcTan}\left (\frac {x \sqrt [4]{a \sqrt {a^2+2 b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+2 b}+a} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt {a^2+2 b} \left (\sqrt {a^2+2 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+2 b}+a^2+b}}+\frac {\tanh ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+2 b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+2 b}} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt {a^2+2 b} \left (a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+2 b}+a^2+b}}-\frac {\tanh ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+2 b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+2 b}+a} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt {a^2+2 b} \left (\sqrt {a^2+2 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+2 b}+a^2+b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 218
Rule 385
Rule 1442
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b-2 a x^4+x^8\right )} \, dx &=\frac {\int \frac {1}{\left (-2 a-2 \sqrt {a^2+2 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx}{\sqrt {a^2+2 b}}-\frac {\int \frac {1}{\left (-2 a+2 \sqrt {a^2+2 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx}{\sqrt {a^2+2 b}}\\ &=\frac {\text {Subst}\left (\int \frac {1}{-2 a-2 \sqrt {a^2+2 b}-\left (-2 b+a \left (-2 a-2 \sqrt {a^2+2 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+2 b}}-\frac {\text {Subst}\left (\int \frac {1}{-2 a+2 \sqrt {a^2+2 b}-\left (-2 b+a \left (-2 a+2 \sqrt {a^2+2 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+2 b}}\\ &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+2 b}}-\sqrt {a^2+b-a \sqrt {a^2+2 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+2 b} \sqrt {a-\sqrt {a^2+2 b}}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+2 b}}+\sqrt {a^2+b-a \sqrt {a^2+2 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+2 b} \sqrt {a-\sqrt {a^2+2 b}}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+2 b}}-\sqrt {a^2+b+a \sqrt {a^2+2 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+2 b} \sqrt {a+\sqrt {a^2+2 b}}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+2 b}}+\sqrt {a^2+b+a \sqrt {a^2+2 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+2 b} \sqrt {a+\sqrt {a^2+2 b}}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a^2+b-a \sqrt {a^2+2 b}} x}{\sqrt [4]{a-\sqrt {a^2+2 b}} \sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+2 b} \left (a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{a^2+b-a \sqrt {a^2+2 b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a^2+b+a \sqrt {a^2+2 b}} x}{\sqrt [4]{a+\sqrt {a^2+2 b}} \sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+2 b} \left (a+\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{a^2+b+a \sqrt {a^2+2 b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a^2+b-a \sqrt {a^2+2 b}} x}{\sqrt [4]{a-\sqrt {a^2+2 b}} \sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+2 b} \left (a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{a^2+b-a \sqrt {a^2+2 b}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a^2+b+a \sqrt {a^2+2 b}} x}{\sqrt [4]{a+\sqrt {a^2+2 b}} \sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+2 b} \left (a+\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{a^2+b+a \sqrt {a^2+2 b}}}\\ \end {align*}
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Mathematica [A]
time = 0.42, size = 97, normalized size = 1.01 \begin {gather*} -\frac {\text {RootSum}\left [b+2 a \text {$\#$1}^4-2 \text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]}{8 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{4}+b \right )^{\frac {1}{4}} \left (x^{8}-2 a \,x^{4}-2 b \right )}\, dx\]
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 [F(-1)] Timed out
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{a x^{4} + b} \left (- 2 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 {1}{{\left (a\,x^4+b\right )}^{1/4}\,\left (-x^8+2\,a\,x^4+2\,b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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