Optimal. Leaf size=203 \[ x F_1\left (\frac {1}{2};1,\frac {1}{6};\frac {3}{2};x^2,-x^2\right )-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-2^{5/6} \sqrt [6]{1+x^2}}{\sqrt {3}}\right )}{2 \sqrt [6]{2}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+2^{5/6} \sqrt [6]{1+x^2}}{\sqrt {3}}\right )}{2 \sqrt [6]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{1+x^2}}{\sqrt [6]{2}}\right )}{\sqrt [6]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [6]{2} \sqrt [6]{1+x^2}+\sqrt [3]{1+x^2}\right )}{4 \sqrt [6]{2}}-\frac {\log \left (\sqrt [3]{2}+\sqrt [6]{2} \sqrt [6]{1+x^2}+\sqrt [3]{1+x^2}\right )}{4 \sqrt [6]{2}} \]
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Rubi [A]
time = 0.25, antiderivative size = 203, normalized size of antiderivative = 1.00, 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 = {771, 440,
455, 65, 302, 648, 632, 210, 642, 212} \begin {gather*} x F_1\left (\frac {1}{2};1,\frac {1}{6};\frac {3}{2};x^2,-x^2\right )-\frac {\sqrt {3} \text {ArcTan}\left (\frac {1-2^{5/6} \sqrt [6]{x^2+1}}{\sqrt {3}}\right )}{2 \sqrt [6]{2}}+\frac {\sqrt {3} \text {ArcTan}\left (\frac {2^{5/6} \sqrt [6]{x^2+1}+1}{\sqrt {3}}\right )}{2 \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 {\log \left (\sqrt [3]{x^2+1}+\sqrt [6]{2} \sqrt [6]{x^2+1}+\sqrt [3]{2}\right )}{4 \sqrt [6]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{x^2+1}}{\sqrt [6]{2}}\right )}{\sqrt [6]{2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 210
Rule 212
Rule 302
Rule 440
Rule 455
Rule 632
Rule 642
Rule 648
Rule 771
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} \text {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 \text {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 {\text {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 {\text {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}}-\text {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} \text {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} \text {Subst}\left (\int \frac {1}{\sqrt [3]{2}+\sqrt [6]{2} x+x^2} \, dx,x,\sqrt [6]{1+x^2}\right )+\frac {\text {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 {\text {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 \text {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 \text {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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 15.29, size = 72, normalized size = 0.35 \begin {gather*} -\frac {3 \sqrt [6]{\frac {-i+x}{1+x}} \sqrt [6]{\frac {i+x}{1+x}} F_1\left (\frac {1}{3};\frac {1}{6},\frac {1}{6};\frac {4}{3};\frac {1-i}{1+x},\frac {1+i}{1+x}\right )}{\sqrt [6]{1+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (x +1\right ) \left (x^{2}+1\right )^{\frac {1}{6}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (x + 1\right ) \sqrt [6]{x^{2} + 1}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (x^2+1\right )}^{1/6}\,\left (x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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