Optimal. Leaf size=86 \[ \frac{1}{2} \tanh ^{-1}\left (\frac{x+4}{2 \sqrt{x^2+2 x+4}}\right )-\frac{\tanh ^{-1}\left (\frac{2 x+5}{\sqrt{7} \sqrt{x^2+2 x+4}}\right )}{2 \sqrt{7}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^2+2 x+4}}{\sqrt{3}}\right )}{2 \sqrt{3}} \]
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Rubi [A] time = 0.278194, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {1593, 6725, 724, 206, 1033, 688, 207} \[ \frac{1}{2} \tanh ^{-1}\left (\frac{x+4}{2 \sqrt{x^2+2 x+4}}\right )-\frac{\tanh ^{-1}\left (\frac{2 x+5}{\sqrt{7} \sqrt{x^2+2 x+4}}\right )}{2 \sqrt{7}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^2+2 x+4}}{\sqrt{3}}\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 6725
Rule 724
Rule 206
Rule 1033
Rule 688
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{4+2 x+x^2} \left (-x+x^3\right )} \, dx &=\int \frac{1}{x \left (-1+x^2\right ) \sqrt{4+2 x+x^2}} \, dx\\ &=\int \left (-\frac{1}{x \sqrt{4+2 x+x^2}}+\frac{x}{\left (-1+x^2\right ) \sqrt{4+2 x+x^2}}\right ) \, dx\\ &=-\int \frac{1}{x \sqrt{4+2 x+x^2}} \, dx+\int \frac{x}{\left (-1+x^2\right ) \sqrt{4+2 x+x^2}} \, dx\\ &=\frac{1}{2} \int \frac{1}{(-1+x) \sqrt{4+2 x+x^2}} \, dx+\frac{1}{2} \int \frac{1}{(1+x) \sqrt{4+2 x+x^2}} \, dx+2 \operatorname{Subst}\left (\int \frac{1}{16-x^2} \, dx,x,\frac{8+2 x}{\sqrt{4+2 x+x^2}}\right )\\ &=\frac{1}{2} \tanh ^{-1}\left (\frac{4+x}{2 \sqrt{4+2 x+x^2}}\right )+2 \operatorname{Subst}\left (\int \frac{1}{-12+4 x^2} \, dx,x,\sqrt{4+2 x+x^2}\right )-\operatorname{Subst}\left (\int \frac{1}{28-x^2} \, dx,x,\frac{10+4 x}{\sqrt{4+2 x+x^2}}\right )\\ &=\frac{1}{2} \tanh ^{-1}\left (\frac{4+x}{2 \sqrt{4+2 x+x^2}}\right )-\frac{\tanh ^{-1}\left (\frac{10+4 x}{2 \sqrt{7} \sqrt{4+2 x+x^2}}\right )}{2 \sqrt{7}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{4+2 x+x^2}}{\sqrt{3}}\right )}{2 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0602414, size = 83, normalized size = 0.97 \[ \frac{1}{42} \left (21 \tanh ^{-1}\left (\frac{x+4}{2 \sqrt{x^2+2 x+4}}\right )-3 \sqrt{7} \tanh ^{-1}\left (\frac{2 x+5}{\sqrt{7} \sqrt{x^2+2 x+4}}\right )-7 \sqrt{3} \tanh ^{-1}\left (\frac{\sqrt{(x+1)^2+3}}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 69, normalized size = 0.8 \begin{align*}{\frac{1}{2}{\it Artanh} \left ({\frac{8+2\,x}{4}{\frac{1}{\sqrt{{x}^{2}+2\,x+4}}}} \right ) }-{\frac{\sqrt{3}}{6}{\it Artanh} \left ({\sqrt{3}{\frac{1}{\sqrt{ \left ( 1+x \right ) ^{2}+3}}}} \right ) }-{\frac{\sqrt{7}}{14}{\it Artanh} \left ({\frac{ \left ( 10+4\,x \right ) \sqrt{7}}{14}{\frac{1}{\sqrt{ \left ( -1+x \right ) ^{2}+3+4\,x}}}} \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}{{\left (x^{3} - x\right )} \sqrt{x^{2} + 2 \, x + 4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16378, size = 319, normalized size = 3.71 \begin{align*} \frac{1}{14} \, \sqrt{7} \log \left (\frac{\sqrt{7}{\left (2 \, x + 5\right )} + \sqrt{x^{2} + 2 \, x + 4}{\left (2 \, \sqrt{7} - 7\right )} - 4 \, x - 10}{x - 1}\right ) + \frac{1}{6} \, \sqrt{3} \log \left (-\frac{\sqrt{3} - \sqrt{x^{2} + 2 \, x + 4}}{x + 1}\right ) + \frac{1}{2} \, \log \left (-x + \sqrt{x^{2} + 2 \, x + 4} + 2\right ) - \frac{1}{2} \, \log \left (-x + \sqrt{x^{2} + 2 \, x + 4} - 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{1}{x \left (x - 1\right ) \left (x + 1\right ) \sqrt{x^{2} + 2 x + 4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15977, size = 198, normalized size = 2.3 \begin{align*} \frac{1}{14} \, \sqrt{7} \log \left (\frac{{\left | -2 \, x - 2 \, \sqrt{7} + 2 \, \sqrt{x^{2} + 2 \, x + 4} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt{7} + 2 \, \sqrt{x^{2} + 2 \, x + 4} + 2 \right |}}\right ) + \frac{1}{6} \, \sqrt{3} \log \left (-\frac{{\left | -2 \, x - 2 \, \sqrt{3} + 2 \, \sqrt{x^{2} + 2 \, x + 4} - 2 \right |}}{2 \,{\left (x - \sqrt{3} - \sqrt{x^{2} + 2 \, x + 4} + 1\right )}}\right ) + \frac{1}{2} \, \log \left ({\left | -x + \sqrt{x^{2} + 2 \, x + 4} + 2 \right |}\right ) - \frac{1}{2} \, \log \left ({\left | -x + \sqrt{x^{2} + 2 \, x + 4} - 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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