Optimal. Leaf size=511 \[ \frac{\sqrt{2-\sqrt{2-b}} \log \left (-\sqrt{2-\sqrt{2-b}} x+x^2+1\right )}{8 \sqrt{2-b}}-\frac{\sqrt{2-\sqrt{2-b}} \log \left (\sqrt{2-\sqrt{2-b}} x+x^2+1\right )}{8 \sqrt{2-b}}-\frac{\sqrt{\sqrt{2-b}+2} \log \left (-\sqrt{\sqrt{2-b}+2} x+x^2+1\right )}{8 \sqrt{2-b}}+\frac{\sqrt{\sqrt{2-b}+2} \log \left (\sqrt{\sqrt{2-b}+2} x+x^2+1\right )}{8 \sqrt{2-b}}-\frac{\sqrt{b+2} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2-b}}-2 x}{\sqrt{\sqrt{2-b}+2}}\right )}{4 \sqrt{2-\sqrt{2-b}} \sqrt{2-b}}+\frac{\sqrt{b+2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{2-b}+2}-2 x}{\sqrt{2-\sqrt{2-b}}}\right )}{4 \sqrt{\sqrt{2-b}+2} \sqrt{2-b}}+\frac{\sqrt{b+2} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2-b}}+2 x}{\sqrt{\sqrt{2-b}+2}}\right )}{4 \sqrt{2-\sqrt{2-b}} \sqrt{2-b}}-\frac{\sqrt{b+2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{2-b}+2}+2 x}{\sqrt{2-\sqrt{2-b}}}\right )}{4 \sqrt{\sqrt{2-b}+2} \sqrt{2-b}} \]
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Rubi [A] time = 0.358509, antiderivative size = 511, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1421, 1169, 634, 618, 204, 628} \[ \frac{\sqrt{2-\sqrt{2-b}} \log \left (-\sqrt{2-\sqrt{2-b}} x+x^2+1\right )}{8 \sqrt{2-b}}-\frac{\sqrt{2-\sqrt{2-b}} \log \left (\sqrt{2-\sqrt{2-b}} x+x^2+1\right )}{8 \sqrt{2-b}}-\frac{\sqrt{\sqrt{2-b}+2} \log \left (-\sqrt{\sqrt{2-b}+2} x+x^2+1\right )}{8 \sqrt{2-b}}+\frac{\sqrt{\sqrt{2-b}+2} \log \left (\sqrt{\sqrt{2-b}+2} x+x^2+1\right )}{8 \sqrt{2-b}}-\frac{\sqrt{b+2} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2-b}}-2 x}{\sqrt{\sqrt{2-b}+2}}\right )}{4 \sqrt{2-\sqrt{2-b}} \sqrt{2-b}}+\frac{\sqrt{b+2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{2-b}+2}-2 x}{\sqrt{2-\sqrt{2-b}}}\right )}{4 \sqrt{\sqrt{2-b}+2} \sqrt{2-b}}+\frac{\sqrt{b+2} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2-b}}+2 x}{\sqrt{\sqrt{2-b}+2}}\right )}{4 \sqrt{2-\sqrt{2-b}} \sqrt{2-b}}-\frac{\sqrt{b+2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{2-b}+2}+2 x}{\sqrt{2-\sqrt{2-b}}}\right )}{4 \sqrt{\sqrt{2-b}+2} \sqrt{2-b}} \]
Antiderivative was successfully verified.
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Rule 1421
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1-x^4}{1+b x^4+x^8} \, dx &=-\frac{\int \frac{\sqrt{2-b}+2 x^2}{-1-\sqrt{2-b} x^2-x^4} \, dx}{2 \sqrt{2-b}}-\frac{\int \frac{\sqrt{2-b}-2 x^2}{-1+\sqrt{2-b} x^2-x^4} \, dx}{2 \sqrt{2-b}}\\ &=\frac{\int \frac{\sqrt{2-\sqrt{2-b}} \sqrt{2-b}-\left (-2+\sqrt{2-b}\right ) x}{1-\sqrt{2-\sqrt{2-b}} x+x^2} \, dx}{4 \sqrt{2-\sqrt{2-b}} \sqrt{2-b}}+\frac{\int \frac{\sqrt{2-\sqrt{2-b}} \sqrt{2-b}+\left (-2+\sqrt{2-b}\right ) x}{1+\sqrt{2-\sqrt{2-b}} x+x^2} \, dx}{4 \sqrt{2-\sqrt{2-b}} \sqrt{2-b}}+\frac{\int \frac{\sqrt{2+\sqrt{2-b}} \sqrt{2-b}-\left (2+\sqrt{2-b}\right ) x}{1-\sqrt{2+\sqrt{2-b}} x+x^2} \, dx}{4 \sqrt{2+\sqrt{2-b}} \sqrt{2-b}}+\frac{\int \frac{\sqrt{2+\sqrt{2-b}} \sqrt{2-b}+\left (2+\sqrt{2-b}\right ) x}{1+\sqrt{2+\sqrt{2-b}} x+x^2} \, dx}{4 \sqrt{2+\sqrt{2-b}} \sqrt{2-b}}\\ &=-\left (\frac{1}{8} \left (-1+\frac{2}{\sqrt{2-b}}\right ) \int \frac{1}{1-\sqrt{2+\sqrt{2-b}} x+x^2} \, dx\right )-\frac{1}{8} \left (-1+\frac{2}{\sqrt{2-b}}\right ) \int \frac{1}{1+\sqrt{2+\sqrt{2-b}} x+x^2} \, dx+\frac{1}{8} \left (1+\frac{2}{\sqrt{2-b}}\right ) \int \frac{1}{1-\sqrt{2-\sqrt{2-b}} x+x^2} \, dx+\frac{1}{8} \left (1+\frac{2}{\sqrt{2-b}}\right ) \int \frac{1}{1+\sqrt{2-\sqrt{2-b}} x+x^2} \, dx+\frac{\sqrt{2-\sqrt{2-b}} \int \frac{-\sqrt{2-\sqrt{2-b}}+2 x}{1-\sqrt{2-\sqrt{2-b}} x+x^2} \, dx}{8 \sqrt{2-b}}-\frac{\sqrt{2-\sqrt{2-b}} \int \frac{\sqrt{2-\sqrt{2-b}}+2 x}{1+\sqrt{2-\sqrt{2-b}} x+x^2} \, dx}{8 \sqrt{2-b}}-\frac{\sqrt{2+\sqrt{2-b}} \int \frac{-\sqrt{2+\sqrt{2-b}}+2 x}{1-\sqrt{2+\sqrt{2-b}} x+x^2} \, dx}{8 \sqrt{2-b}}+\frac{\sqrt{2+\sqrt{2-b}} \int \frac{\sqrt{2+\sqrt{2-b}}+2 x}{1+\sqrt{2+\sqrt{2-b}} x+x^2} \, dx}{8 \sqrt{2-b}}\\ &=\frac{\sqrt{2-\sqrt{2-b}} \log \left (1-\sqrt{2-\sqrt{2-b}} x+x^2\right )}{8 \sqrt{2-b}}-\frac{\sqrt{2-\sqrt{2-b}} \log \left (1+\sqrt{2-\sqrt{2-b}} x+x^2\right )}{8 \sqrt{2-b}}-\frac{\sqrt{2+\sqrt{2-b}} \log \left (1-\sqrt{2+\sqrt{2-b}} x+x^2\right )}{8 \sqrt{2-b}}+\frac{\sqrt{2+\sqrt{2-b}} \log \left (1+\sqrt{2+\sqrt{2-b}} x+x^2\right )}{8 \sqrt{2-b}}+\frac{1}{4} \left (-1-\frac{2}{\sqrt{2-b}}\right ) \operatorname{Subst}\left (\int \frac{1}{-2-\sqrt{2-b}-x^2} \, dx,x,-\sqrt{2-\sqrt{2-b}}+2 x\right )+\frac{1}{4} \left (-1-\frac{2}{\sqrt{2-b}}\right ) \operatorname{Subst}\left (\int \frac{1}{-2-\sqrt{2-b}-x^2} \, dx,x,\sqrt{2-\sqrt{2-b}}+2 x\right )-\frac{1}{4} \left (1-\frac{2}{\sqrt{2-b}}\right ) \operatorname{Subst}\left (\int \frac{1}{-2+\sqrt{2-b}-x^2} \, dx,x,-\sqrt{2+\sqrt{2-b}}+2 x\right )-\frac{1}{4} \left (1-\frac{2}{\sqrt{2-b}}\right ) \operatorname{Subst}\left (\int \frac{1}{-2+\sqrt{2-b}-x^2} \, dx,x,\sqrt{2+\sqrt{2-b}}+2 x\right )\\ &=-\frac{\sqrt{2+\sqrt{2-b}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2-b}}-2 x}{\sqrt{2+\sqrt{2-b}}}\right )}{4 \sqrt{2-b}}+\frac{\sqrt{2-\sqrt{2-b}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2-b}}-2 x}{\sqrt{2-\sqrt{2-b}}}\right )}{4 \sqrt{2-b}}+\frac{\sqrt{2+\sqrt{2-b}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2-b}}+2 x}{\sqrt{2+\sqrt{2-b}}}\right )}{4 \sqrt{2-b}}-\frac{\sqrt{2-\sqrt{2-b}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2-b}}+2 x}{\sqrt{2-\sqrt{2-b}}}\right )}{4 \sqrt{2-b}}+\frac{\sqrt{2-\sqrt{2-b}} \log \left (1-\sqrt{2-\sqrt{2-b}} x+x^2\right )}{8 \sqrt{2-b}}-\frac{\sqrt{2-\sqrt{2-b}} \log \left (1+\sqrt{2-\sqrt{2-b}} x+x^2\right )}{8 \sqrt{2-b}}-\frac{\sqrt{2+\sqrt{2-b}} \log \left (1-\sqrt{2+\sqrt{2-b}} x+x^2\right )}{8 \sqrt{2-b}}+\frac{\sqrt{2+\sqrt{2-b}} \log \left (1+\sqrt{2+\sqrt{2-b}} x+x^2\right )}{8 \sqrt{2-b}}\\ \end{align*}
Mathematica [C] time = 0.0246006, size = 57, normalized size = 0.11 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^4 b+\text{$\#$1}^8+1\& ,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{\text{$\#$1}^3 b+2 \text{$\#$1}^7}\& \right ] \]
Antiderivative was successfully verified.
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Maple [C] time = 0.003, size = 44, normalized size = 0.1 \begin{align*}{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+{{\it \_Z}}^{4}b+1 \right ) }{\frac{ \left ( -{{\it \_R}}^{4}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}+{{\it \_R}}^{3}b}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x^{4} - 1}{x^{8} + b x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.51297, size = 3725, normalized size = 7.29 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.26308, size = 76, normalized size = 0.15 \begin{align*} - \operatorname{RootSum}{\left (t^{8} \left (65536 b^{4} - 524288 b^{3} + 1572864 b^{2} - 2097152 b + 1048576\right ) + t^{4} \left (256 b^{3} - 1024 b^{2} + 1024 b\right ) + 1, \left ( t \mapsto t \log{\left (1024 t^{5} b^{2} - 4096 t^{5} b + 4096 t^{5} + 4 t b - 4 t + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x^{4} - 1}{x^{8} + b x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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