Optimal. Leaf size=203 \[ x F_1\left (\frac{1}{2};1,\frac{1}{6};\frac{3}{2};x^2,-x^2\right )+\frac{\log \left (\sqrt [3]{x^2+1}-\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}\right )}{4 \sqrt [6]{2}}-\frac{\log \left (\sqrt [3]{x^2+1}+\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}\right )}{4 \sqrt [6]{2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-2^{5/6} \sqrt [6]{x^2+1}}{\sqrt{3}}\right )}{2 \sqrt [6]{2}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2^{5/6} \sqrt [6]{x^2+1}+1}{\sqrt{3}}\right )}{2 \sqrt [6]{2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{x^2+1}}{\sqrt [6]{2}}\right )}{\sqrt [6]{2}} \]
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Rubi [A] time = 0.367675, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {757, 429, 444, 63, 296, 634, 618, 204, 628, 206} \[ x F_1\left (\frac{1}{2};1,\frac{1}{6};\frac{3}{2};x^2,-x^2\right )+\frac{\log \left (\sqrt [3]{x^2+1}-\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}\right )}{4 \sqrt [6]{2}}-\frac{\log \left (\sqrt [3]{x^2+1}+\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}\right )}{4 \sqrt [6]{2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-2^{5/6} \sqrt [6]{x^2+1}}{\sqrt{3}}\right )}{2 \sqrt [6]{2}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2^{5/6} \sqrt [6]{x^2+1}+1}{\sqrt{3}}\right )}{2 \sqrt [6]{2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{x^2+1}}{\sqrt [6]{2}}\right )}{\sqrt [6]{2}} \]
Antiderivative was successfully verified.
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Rule 757
Rule 429
Rule 444
Rule 63
Rule 296
Rule 634
Rule 618
Rule 204
Rule 628
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(1+x) \sqrt [6]{1+x^2}} \, dx &=\int \left (\frac{1}{\left (1-x^2\right ) \sqrt [6]{1+x^2}}+\frac{x}{\left (-1+x^2\right ) \sqrt [6]{1+x^2}}\right ) \, dx\\ &=\int \frac{1}{\left (1-x^2\right ) \sqrt [6]{1+x^2}} \, dx+\int \frac{x}{\left (-1+x^2\right ) \sqrt [6]{1+x^2}} \, dx\\ &=x F_1\left (\frac{1}{2};1,\frac{1}{6};\frac{3}{2};x^2,-x^2\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt [6]{1+x}} \, dx,x,x^2\right )\\ &=x F_1\left (\frac{1}{2};1,\frac{1}{6};\frac{3}{2};x^2,-x^2\right )+3 \operatorname{Subst}\left (\int \frac{x^4}{-2+x^6} \, dx,x,\sqrt [6]{1+x^2}\right )\\ &=x F_1\left (\frac{1}{2};1,\frac{1}{6};\frac{3}{2};x^2,-x^2\right )-\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{2^{5/6}}-\frac{x}{2}}{\sqrt [3]{2}-\sqrt [6]{2} x+x^2} \, dx,x,\sqrt [6]{1+x^2}\right )}{\sqrt [6]{2}}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{2^{5/6}}+\frac{x}{2}}{\sqrt [3]{2}+\sqrt [6]{2} x+x^2} \, dx,x,\sqrt [6]{1+x^2}\right )}{\sqrt [6]{2}}-\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{2}-x^2} \, dx,x,\sqrt [6]{1+x^2}\right )\\ &=x F_1\left (\frac{1}{2};1,\frac{1}{6};\frac{3}{2};x^2,-x^2\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{1+x^2}}{\sqrt [6]{2}}\right )}{\sqrt [6]{2}}+\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{2}-\sqrt [6]{2} x+x^2} \, dx,x,\sqrt [6]{1+x^2}\right )+\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{2}+\sqrt [6]{2} x+x^2} \, dx,x,\sqrt [6]{1+x^2}\right )+\frac{\operatorname{Subst}\left (\int \frac{-\sqrt [6]{2}+2 x}{\sqrt [3]{2}-\sqrt [6]{2} x+x^2} \, dx,x,\sqrt [6]{1+x^2}\right )}{4 \sqrt [6]{2}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt [6]{2}+2 x}{\sqrt [3]{2}+\sqrt [6]{2} x+x^2} \, dx,x,\sqrt [6]{1+x^2}\right )}{4 \sqrt [6]{2}}\\ &=x F_1\left (\frac{1}{2};1,\frac{1}{6};\frac{3}{2};x^2,-x^2\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{1+x^2}}{\sqrt [6]{2}}\right )}{\sqrt [6]{2}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [6]{2} \sqrt [6]{1+x^2}+\sqrt [3]{1+x^2}\right )}{4 \sqrt [6]{2}}-\frac{\log \left (\sqrt [3]{2}+\sqrt [6]{2} \sqrt [6]{1+x^2}+\sqrt [3]{1+x^2}\right )}{4 \sqrt [6]{2}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-2^{5/6} \sqrt [6]{1+x^2}\right )}{2 \sqrt [6]{2}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2^{5/6} \sqrt [6]{1+x^2}\right )}{2 \sqrt [6]{2}}\\ &=x F_1\left (\frac{1}{2};1,\frac{1}{6};\frac{3}{2};x^2,-x^2\right )-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-2^{5/6} \sqrt [6]{1+x^2}}{\sqrt{3}}\right )}{2 \sqrt [6]{2}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+2^{5/6} \sqrt [6]{1+x^2}}{\sqrt{3}}\right )}{2 \sqrt [6]{2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{1+x^2}}{\sqrt [6]{2}}\right )}{\sqrt [6]{2}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [6]{2} \sqrt [6]{1+x^2}+\sqrt [3]{1+x^2}\right )}{4 \sqrt [6]{2}}-\frac{\log \left (\sqrt [3]{2}+\sqrt [6]{2} \sqrt [6]{1+x^2}+\sqrt [3]{1+x^2}\right )}{4 \sqrt [6]{2}}\\ \end{align*}
Mathematica [C] time = 0.0385877, size = 72, normalized size = 0.35 \[ -\frac{3 \sqrt [6]{\frac{x-i}{x+1}} \sqrt [6]{\frac{x+i}{x+1}} F_1\left (\frac{1}{3};\frac{1}{6},\frac{1}{6};\frac{4}{3};\frac{1-i}{x+1},\frac{1+i}{x+1}\right )}{\sqrt [6]{x^2+1}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.366, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{1+x}{\frac{1}{\sqrt [6]{{x}^{2}+1}}}}\, dx \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}{{\left (x^{2} + 1\right )}^{\frac{1}{6}}{\left (x + 1\right )}}\,{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (x + 1\right ) \sqrt [6]{x^{2} + 1}}\, dx \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}{{\left (x^{2} + 1\right )}^{\frac{1}{6}}{\left (x + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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