Optimal. Leaf size=309 \[ \frac{x}{2 \sqrt{\sqrt{2}-1}}-\frac{1}{8} \sqrt{\sqrt{2}-1} \log \left (2 \tan ^2(x)-2 \sqrt{\sqrt{2}-1} \tan (x)+\sqrt{2}\right )+\frac{1}{8} \sqrt{\sqrt{2}-1} \log \left (\sqrt{2} \tan ^2(x)+\sqrt{2 \left (\sqrt{2}-1\right )} \tan (x)+1\right )+\frac{\tan ^{-1}\left (\frac{-2 \sqrt{\sqrt{2}-1} \cos ^2(x)-\left (\sqrt{2}-2\right ) \sin (x) \cos (x)+\sqrt{\sqrt{2}-1}}{\left (\sqrt{2}-2\right ) \cos ^2(x)-2 \sqrt{\sqrt{2}-1} \sin (x) \cos (x)+\sqrt{1+\sqrt{2}}+2}\right )}{4 \sqrt{\sqrt{2}-1}}-\frac{\tan ^{-1}\left (\frac{-2 \sqrt{\sqrt{2}-1} \cos ^2(x)+\left (\sqrt{2}-2\right ) \sin (x) \cos (x)+\sqrt{\sqrt{2}-1}}{\left (\sqrt{2}-2\right ) \cos ^2(x)+2 \sqrt{\sqrt{2}-1} \sin (x) \cos (x)+\sqrt{1+\sqrt{2}}+2}\right )}{4 \sqrt{\sqrt{2}-1}} \]
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Rubi [A] time = 0.202052, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {3209, 1169, 634, 618, 204, 628} \[ \frac{x}{2 \sqrt{\sqrt{2}-1}}-\frac{1}{8} \sqrt{\sqrt{2}-1} \log \left (2 \tan ^2(x)-2 \sqrt{\sqrt{2}-1} \tan (x)+\sqrt{2}\right )+\frac{1}{8} \sqrt{\sqrt{2}-1} \log \left (\sqrt{2} \tan ^2(x)+\sqrt{2 \left (\sqrt{2}-1\right )} \tan (x)+1\right )+\frac{\tan ^{-1}\left (\frac{-2 \sqrt{\sqrt{2}-1} \cos ^2(x)-\left (\sqrt{2}-2\right ) \sin (x) \cos (x)+\sqrt{\sqrt{2}-1}}{\left (\sqrt{2}-2\right ) \cos ^2(x)-2 \sqrt{\sqrt{2}-1} \sin (x) \cos (x)+\sqrt{1+\sqrt{2}}+2}\right )}{4 \sqrt{\sqrt{2}-1}}-\frac{\tan ^{-1}\left (\frac{-2 \sqrt{\sqrt{2}-1} \cos ^2(x)+\left (\sqrt{2}-2\right ) \sin (x) \cos (x)+\sqrt{\sqrt{2}-1}}{\left (\sqrt{2}-2\right ) \cos ^2(x)+2 \sqrt{\sqrt{2}-1} \sin (x) \cos (x)+\sqrt{1+\sqrt{2}}+2}\right )}{4 \sqrt{\sqrt{2}-1}} \]
Antiderivative was successfully verified.
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Rule 3209
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{1+\sin ^4(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1+x^2}{1+2 x^2+2 x^4} \, dx,x,\tan (x)\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{-1+\sqrt{2}}-\left (1-\frac{1}{\sqrt{2}}\right ) x}{\frac{1}{\sqrt{2}}-\sqrt{-1+\sqrt{2}} x+x^2} \, dx,x,\tan (x)\right )}{2 \sqrt{2 \left (-1+\sqrt{2}\right )}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{-1+\sqrt{2}}+\left (1-\frac{1}{\sqrt{2}}\right ) x}{\frac{1}{\sqrt{2}}+\sqrt{-1+\sqrt{2}} x+x^2} \, dx,x,\tan (x)\right )}{2 \sqrt{2 \left (-1+\sqrt{2}\right )}}\\ &=-\left (\frac{1}{8} \sqrt{-1+\sqrt{2}} \operatorname{Subst}\left (\int \frac{-\sqrt{-1+\sqrt{2}}+2 x}{\frac{1}{\sqrt{2}}-\sqrt{-1+\sqrt{2}} x+x^2} \, dx,x,\tan (x)\right )\right )+\frac{1}{8} \sqrt{-1+\sqrt{2}} \operatorname{Subst}\left (\int \frac{\sqrt{-1+\sqrt{2}}+2 x}{\frac{1}{\sqrt{2}}+\sqrt{-1+\sqrt{2}} x+x^2} \, dx,x,\tan (x)\right )+\frac{1}{8} \sqrt{3+2 \sqrt{2}} \operatorname{Subst}\left (\int \frac{1}{\frac{1}{\sqrt{2}}-\sqrt{-1+\sqrt{2}} x+x^2} \, dx,x,\tan (x)\right )+\frac{1}{8} \sqrt{3+2 \sqrt{2}} \operatorname{Subst}\left (\int \frac{1}{\frac{1}{\sqrt{2}}+\sqrt{-1+\sqrt{2}} x+x^2} \, dx,x,\tan (x)\right )\\ &=-\frac{1}{8} \sqrt{-1+\sqrt{2}} \log \left (\sqrt{2}-2 \sqrt{-1+\sqrt{2}} \tan (x)+2 \tan ^2(x)\right )+\frac{1}{8} \sqrt{-1+\sqrt{2}} \log \left (1+\sqrt{2 \left (-1+\sqrt{2}\right )} \tan (x)+\sqrt{2} \tan ^2(x)\right )-\frac{1}{4} \sqrt{3+2 \sqrt{2}} \operatorname{Subst}\left (\int \frac{1}{-1-\sqrt{2}-x^2} \, dx,x,-\sqrt{-1+\sqrt{2}}+2 \tan (x)\right )-\frac{1}{4} \sqrt{3+2 \sqrt{2}} \operatorname{Subst}\left (\int \frac{1}{-1-\sqrt{2}-x^2} \, dx,x,\sqrt{-1+\sqrt{2}}+2 \tan (x)\right )\\ &=\frac{1}{2} \sqrt{1+\sqrt{2}} x+\frac{1}{4} \sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{-1+\sqrt{2}}-2 \sqrt{-1+\sqrt{2}} \cos ^2(x)+\left (2-\sqrt{2}\right ) \cos (x) \sin (x)}{2+\sqrt{1+\sqrt{2}}-\left (2-\sqrt{2}\right ) \cos ^2(x)-2 \sqrt{-1+\sqrt{2}} \cos (x) \sin (x)}\right )-\frac{1}{4} \sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{-1+\sqrt{2}}-2 \sqrt{-1+\sqrt{2}} \cos ^2(x)-\left (2-\sqrt{2}\right ) \cos (x) \sin (x)}{2+\sqrt{1+\sqrt{2}}-\left (2-\sqrt{2}\right ) \cos ^2(x)+2 \sqrt{-1+\sqrt{2}} \cos (x) \sin (x)}\right )-\frac{1}{8} \sqrt{-1+\sqrt{2}} \log \left (\sqrt{2}-2 \sqrt{-1+\sqrt{2}} \tan (x)+2 \tan ^2(x)\right )+\frac{1}{8} \sqrt{-1+\sqrt{2}} \log \left (1+\sqrt{2 \left (-1+\sqrt{2}\right )} \tan (x)+\sqrt{2} \tan ^2(x)\right )\\ \end{align*}
Mathematica [C] time = 0.0734058, size = 45, normalized size = 0.15 \[ \frac{\tan ^{-1}\left (\sqrt{1-i} \tan (x)\right )}{2 \sqrt{1-i}}+\frac{\tan ^{-1}\left (\sqrt{1+i} \tan (x)\right )}{2 \sqrt{1+i}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.151, size = 239, normalized size = 0.8 \begin{align*}{\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 ) }{16}}+{\frac{\sqrt{2}}{4\,\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}{4\,\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 ) }{16}}+{\frac{\sqrt{2}}{4\,\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}{4\,\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sin \left (x\right )^{4} + 1}\,{d x} \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 )^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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