Optimal. Leaf size=218 \[ \frac {1}{8} \left (\sqrt {1+\sqrt {4-2 \sqrt {2}}}+\sqrt {2+2 \sqrt [4]{2}+2 \sqrt {1+\sqrt {2}}+2 \sqrt {2+\sqrt {2}}}+\sqrt {1+\sqrt {4+2 \sqrt {2}}}\right ) \left (x-\tan ^{-1}(\tan (x))\right )+\frac {\tan ^{-1}\left (\sqrt {1-\sqrt [4]{-1}} \tan (x)\right )}{4 \sqrt {1-\sqrt [4]{-1}}}+\frac {\tan ^{-1}\left (\sqrt {1+\sqrt [4]{-1}} \tan (x)\right )}{4 \sqrt {1+\sqrt [4]{-1}}}+\frac {\tan ^{-1}\left (\sqrt {1-(-1)^{3/4}} \tan (x)\right )}{4 \sqrt {1-(-1)^{3/4}}}+\frac {\tan ^{-1}\left (\sqrt {1+(-1)^{3/4}} \tan (x)\right )}{4 \sqrt {1+(-1)^{3/4}}} \]
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Rubi [A] time = 0.20, antiderivative size = 129, normalized size of antiderivative = 0.59, number of steps used = 9, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3211, 3181, 203} \[ \frac {\tan ^{-1}\left (\sqrt {1-\sqrt [4]{-1}} \tan (x)\right )}{4 \sqrt {1-\sqrt [4]{-1}}}+\frac {\tan ^{-1}\left (\sqrt {1+\sqrt [4]{-1}} \tan (x)\right )}{4 \sqrt {1+\sqrt [4]{-1}}}+\frac {\tan ^{-1}\left (\sqrt {1-(-1)^{3/4}} \tan (x)\right )}{4 \sqrt {1-(-1)^{3/4}}}+\frac {\tan ^{-1}\left (\sqrt {1+(-1)^{3/4}} \tan (x)\right )}{4 \sqrt {1+(-1)^{3/4}}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 3181
Rule 3211
Rubi steps
\begin {align*} \int \frac {1}{1+\sin ^8(x)} \, dx &=\frac {1}{4} \int \frac {1}{1-\sqrt [4]{-1} \sin ^2(x)} \, dx+\frac {1}{4} \int \frac {1}{1+\sqrt [4]{-1} \sin ^2(x)} \, dx+\frac {1}{4} \int \frac {1}{1-(-1)^{3/4} \sin ^2(x)} \, dx+\frac {1}{4} \int \frac {1}{1+(-1)^{3/4} \sin ^2(x)} \, dx\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+\left (1-\sqrt [4]{-1}\right ) x^2} \, dx,x,\tan (x)\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+\left (1+\sqrt [4]{-1}\right ) x^2} \, dx,x,\tan (x)\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+\left (1-(-1)^{3/4}\right ) x^2} \, dx,x,\tan (x)\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+\left (1+(-1)^{3/4}\right ) x^2} \, dx,x,\tan (x)\right )\\ &=\frac {\tan ^{-1}\left (\sqrt {1-\sqrt [4]{-1}} \tan (x)\right )}{4 \sqrt {1-\sqrt [4]{-1}}}+\frac {\tan ^{-1}\left (\sqrt {1+\sqrt [4]{-1}} \tan (x)\right )}{4 \sqrt {1+\sqrt [4]{-1}}}+\frac {\tan ^{-1}\left (\sqrt {1-(-1)^{3/4}} \tan (x)\right )}{4 \sqrt {1-(-1)^{3/4}}}+\frac {\tan ^{-1}\left (\sqrt {1+(-1)^{3/4}} \tan (x)\right )}{4 \sqrt {1+(-1)^{3/4}}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 141, normalized size = 0.65 \[ 8 \text {RootSum}\left [\text {$\#$1}^8-8 \text {$\#$1}^7+28 \text {$\#$1}^6-56 \text {$\#$1}^5+326 \text {$\#$1}^4-56 \text {$\#$1}^3+28 \text {$\#$1}^2-8 \text {$\#$1}+1\& ,\frac {2 \text {$\#$1}^3 \tan ^{-1}\left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right )-i \text {$\#$1}^3 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (2 x)+1\right )}{\text {$\#$1}^7-7 \text {$\#$1}^6+21 \text {$\#$1}^5-35 \text {$\#$1}^4+163 \text {$\#$1}^3-21 \text {$\#$1}^2+7 \text {$\#$1}-1}\& \right ] \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.20, size = 71, normalized size = 0.33 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (2 \textit {\_Z}^{8}+4 \textit {\_Z}^{6}+6 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )}{\sum }\frac {\left (\textit {\_R}^{6}+3 \textit {\_R}^{4}+3 \textit {\_R}^{2}+1\right ) \ln \left (\tan \relax (x )-\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{5}+3 \textit {\_R}^{3}+\textit {\_R}}\right )}{8} \]
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 \[ \int \frac {1}{\sin \relax (x)^{8} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.92, size = 945, normalized size = 4.33 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{\sin ^{8}{\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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