Optimal. Leaf size=89 \[ \frac{x}{4 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{1-i} \tan (x)\right )}{4 \sqrt{1-i}}+\frac{\tan ^{-1}\left (\sqrt{1+i} \tan (x)\right )}{4 \sqrt{1+i}}+\frac{\tan (x)}{4}+\frac{\tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right )}{4 \sqrt{2}} \]
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Rubi [A] time = 0.0770086, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {3211, 3181, 203, 3175, 3767, 8} \[ \frac{x}{4 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{1-i} \tan (x)\right )}{4 \sqrt{1-i}}+\frac{\tan ^{-1}\left (\sqrt{1+i} \tan (x)\right )}{4 \sqrt{1+i}}+\frac{\tan (x)}{4}+\frac{\tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 3211
Rule 3181
Rule 203
Rule 3175
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{1-\sin ^8(x)} \, dx &=\frac{1}{4} \int \frac{1}{1-\sin ^2(x)} \, dx+\frac{1}{4} \int \frac{1}{1-i \sin ^2(x)} \, dx+\frac{1}{4} \int \frac{1}{1+i \sin ^2(x)} \, dx+\frac{1}{4} \int \frac{1}{1+\sin ^2(x)} \, dx\\ &=\frac{1}{4} \int \sec ^2(x) \, dx+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1+(1-i) x^2} \, dx,x,\tan (x)\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1+(1+i) x^2} \, dx,x,\tan (x)\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\tan (x)\right )\\ &=\frac{x}{4 \sqrt{2}}+\frac{\tan ^{-1}\left (\frac{\cos (x) \sin (x)}{1+\sqrt{2}+\sin ^2(x)}\right )}{4 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{1-i} \tan (x)\right )}{4 \sqrt{1-i}}+\frac{\tan ^{-1}\left (\sqrt{1+i} \tan (x)\right )}{4 \sqrt{1+i}}-\frac{1}{4} \operatorname{Subst}(\int 1 \, dx,x,-\tan (x))\\ &=\frac{x}{4 \sqrt{2}}+\frac{\tan ^{-1}\left (\frac{\cos (x) \sin (x)}{1+\sqrt{2}+\sin ^2(x)}\right )}{4 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{1-i} \tan (x)\right )}{4 \sqrt{1-i}}+\frac{\tan ^{-1}\left (\sqrt{1+i} \tan (x)\right )}{4 \sqrt{1+i}}+\frac{\tan (x)}{4}\\ \end{align*}
Mathematica [A] time = 0.166119, size = 64, normalized size = 0.72 \[ \frac{1}{8} \left (\frac{2 \tan ^{-1}\left (\sqrt{1-i} \tan (x)\right )}{\sqrt{1-i}}+\frac{2 \tan ^{-1}\left (\sqrt{1+i} \tan (x)\right )}{\sqrt{1+i}}+\sqrt{2} \tan ^{-1}\left (\sqrt{2} \tan (x)\right )+2 \tan (x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.078, size = 255, normalized size = 2.9 \begin{align*}{\frac{\tan \left ( x \right ) }{4}}+{\frac{\arctan \left ( \sqrt{2}\tan \left ( x \right ) \right ) \sqrt{2}}{8}}+{\frac{\sqrt{2}\sqrt{-2+2\,\sqrt{2}}\ln \left ( \sqrt{2}+\sqrt{-2+2\,\sqrt{2}}\sqrt{2}\tan \left ( x \right ) +2\, \left ( \tan \left ( x \right ) \right ) ^{2} \right ) }{32}}+{\frac{\sqrt{2}}{8\,\sqrt{1+\sqrt{2}}}\arctan \left ({\frac{\sqrt{2}\sqrt{-2+2\,\sqrt{2}}+4\,\tan \left ( x \right ) }{2\,\sqrt{1+\sqrt{2}}}} \right ) }+{\frac{1}{8\,\sqrt{1+\sqrt{2}}}\arctan \left ({\frac{\sqrt{2}\sqrt{-2+2\,\sqrt{2}}+4\,\tan \left ( x \right ) }{2\,\sqrt{1+\sqrt{2}}}} \right ) }-{\frac{\sqrt{2}\sqrt{-2+2\,\sqrt{2}}\ln \left ( -\sqrt{-2+2\,\sqrt{2}}\sqrt{2}\tan \left ( x \right ) +2\, \left ( \tan \left ( x \right ) \right ) ^{2}+\sqrt{2} \right ) }{32}}+{\frac{\sqrt{2}}{8\,\sqrt{1+\sqrt{2}}}\arctan \left ({\frac{-\sqrt{2}\sqrt{-2+2\,\sqrt{2}}+4\,\tan \left ( x \right ) }{2\,\sqrt{1+\sqrt{2}}}} \right ) }+{\frac{1}{8\,\sqrt{1+\sqrt{2}}}\arctan \left ({\frac{-\sqrt{2}\sqrt{-2+2\,\sqrt{2}}+4\,\tan \left ( x \right ) }{2\,\sqrt{1+\sqrt{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{1}{\sin \left (x\right )^{8} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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