Optimal. Leaf size=56 \[ \frac{\sqrt{x^2+2}}{4}-\frac{1}{16} \sqrt{33} \tanh ^{-1}\left (\frac{8-x}{\sqrt{33} \sqrt{x^2+2}}\right )-\frac{1}{16} \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]
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Rubi [A] time = 0.0327453, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {735, 844, 215, 725, 206} \[ \frac{\sqrt{x^2+2}}{4}-\frac{1}{16} \sqrt{33} \tanh ^{-1}\left (\frac{8-x}{\sqrt{33} \sqrt{x^2+2}}\right )-\frac{1}{16} \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Rule 735
Rule 844
Rule 215
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{2+x^2}}{1+4 x} \, dx &=\frac{\sqrt{2+x^2}}{4}+\frac{1}{4} \int \frac{8-x}{(1+4 x) \sqrt{2+x^2}} \, dx\\ &=\frac{\sqrt{2+x^2}}{4}-\frac{1}{16} \int \frac{1}{\sqrt{2+x^2}} \, dx+\frac{33}{16} \int \frac{1}{(1+4 x) \sqrt{2+x^2}} \, dx\\ &=\frac{\sqrt{2+x^2}}{4}-\frac{1}{16} \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )-\frac{33}{16} \operatorname{Subst}\left (\int \frac{1}{33-x^2} \, dx,x,\frac{8-x}{\sqrt{2+x^2}}\right )\\ &=\frac{\sqrt{2+x^2}}{4}-\frac{1}{16} \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )-\frac{1}{16} \sqrt{33} \tanh ^{-1}\left (\frac{8-x}{\sqrt{33} \sqrt{2+x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0283808, size = 56, normalized size = 1. \[ \frac{\sqrt{x^2+2}}{4}-\frac{1}{16} \sqrt{33} \tanh ^{-1}\left (\frac{8-x}{\sqrt{33} \sqrt{x^2+2}}\right )-\frac{1}{16} \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 57, normalized size = 1. \begin{align*}{\frac{1}{16}\sqrt{16\, \left ( x+1/4 \right ) ^{2}-8\,x+31}}-{\frac{1}{16}{\it Arcsinh} \left ({\frac{x\sqrt{2}}{2}} \right ) }-{\frac{\sqrt{33}}{16}{\it Artanh} \left ({\frac{8\,\sqrt{33}}{33} \left ( 4-{\frac{x}{2}} \right ){\frac{1}{\sqrt{16\, \left ( x+1/4 \right ) ^{2}-8\,x+31}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53476, size = 72, normalized size = 1.29 \begin{align*} \frac{1}{16} \, \sqrt{33} \operatorname{arsinh}\left (\frac{\sqrt{2} x}{2 \,{\left | 4 \, x + 1 \right |}} - \frac{4 \, \sqrt{2}}{{\left | 4 \, x + 1 \right |}}\right ) + \frac{1}{4} \, \sqrt{x^{2} + 2} - \frac{1}{16} \, \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{2} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.5118, size = 190, normalized size = 3.39 \begin{align*} \frac{1}{16} \, \sqrt{33} \log \left (-\frac{\sqrt{33}{\left (x - 8\right )} + \sqrt{x^{2} + 2}{\left (\sqrt{33} + 33\right )} + x - 8}{4 \, x + 1}\right ) + \frac{1}{4} \, \sqrt{x^{2} + 2} + \frac{1}{16} \, \log \left (-x + \sqrt{x^{2} + 2}\right ) \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{\sqrt{x^{2} + 2}}{4 x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.48047, size = 96, normalized size = 1.71 \begin{align*} \frac{1}{16} \, \sqrt{33} \log \left (\frac{{\left | -4 \, x - \sqrt{33} + 4 \, \sqrt{x^{2} + 2} - 1 \right |}}{{\left | -4 \, x + \sqrt{33} + 4 \, \sqrt{x^{2} + 2} - 1 \right |}}\right ) + \frac{1}{4} \, \sqrt{x^{2} + 2} + \frac{1}{16} \, \log \left (-x + \sqrt{x^{2} + 2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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