Optimal. Leaf size=83 \[ -\frac {x \log \left (x^2-\sqrt {x^2}+1\right )}{6 \sqrt {x^2}}+\frac {x \log \left (\sqrt {x^2}+1\right )}{3 \sqrt {x^2}}-\frac {x \tan ^{-1}\left (\frac {1-2 \sqrt {x^2}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt {x^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {254, 200, 31, 634, 618, 204, 628} \[ -\frac {x \log \left (x^2-\sqrt {x^2}+1\right )}{6 \sqrt {x^2}}+\frac {x \log \left (\sqrt {x^2}+1\right )}{3 \sqrt {x^2}}-\frac {x \tan ^{-1}\left (\frac {1-2 \sqrt {x^2}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt {x^2}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 254
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{1+\left (x^2\right )^{3/2}} \, dx &=\frac {x \operatorname {Subst}\left (\int \frac {1}{1+x^3} \, dx,x,\sqrt {x^2}\right )}{\sqrt {x^2}}\\ &=\frac {x \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt {x^2}\right )}{3 \sqrt {x^2}}+\frac {x \operatorname {Subst}\left (\int \frac {2-x}{1-x+x^2} \, dx,x,\sqrt {x^2}\right )}{3 \sqrt {x^2}}\\ &=\frac {x \log \left (1+\sqrt {x^2}\right )}{3 \sqrt {x^2}}-\frac {x \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\sqrt {x^2}\right )}{6 \sqrt {x^2}}+\frac {x \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt {x^2}\right )}{2 \sqrt {x^2}}\\ &=-\frac {x \log \left (1+x^2-\sqrt {x^2}\right )}{6 \sqrt {x^2}}+\frac {x \log \left (1+\sqrt {x^2}\right )}{3 \sqrt {x^2}}-\frac {x \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt {x^2}\right )}{\sqrt {x^2}}\\ &=-\frac {x \tan ^{-1}\left (\frac {1-2 \sqrt {x^2}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt {x^2}}-\frac {x \log \left (1+x^2-\sqrt {x^2}\right )}{6 \sqrt {x^2}}+\frac {x \log \left (1+\sqrt {x^2}\right )}{3 \sqrt {x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 67, normalized size = 0.81 \[ \frac {x \left (-\log \left (x^2-\sqrt {x^2}+1\right )+2 \log \left (\sqrt {x^2}+1\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt {x^2}-1}{\sqrt {3}}\right )\right )}{6 \sqrt {x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 48, normalized size = 0.58 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x^{2}} - \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{6} \, \log \left (x^{2} - \sqrt {x^{2}} + 1\right ) + \frac {1}{3} \, \log \left (\sqrt {x^{2}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.20, size = 108, normalized size = 1.30 \[ -\frac {\sqrt {3} {\left (-i \, \sqrt {3} - 1\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {1}{\mathrm {sgn}\relax (x)}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {1}{\mathrm {sgn}\relax (x)}\right )^{\frac {1}{3}}}\right )}{6 \, \mathrm {sgn}\relax (x)^{\frac {1}{3}}} - \frac {1}{9} i \, \pi \mathrm {sgn}\relax (x) - \frac {{\left (-i \, \sqrt {3} - 1\right )} \log \left (x^{2} + x \left (-\frac {1}{\mathrm {sgn}\relax (x)}\right )^{\frac {1}{3}} + \left (-\frac {1}{\mathrm {sgn}\relax (x)}\right )^{\frac {2}{3}}\right )}{12 \, \mathrm {sgn}\relax (x)^{\frac {1}{3}}} - \frac {1}{3} \, \left (-\frac {1}{\mathrm {sgn}\relax (x)}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {1}{\mathrm {sgn}\relax (x)}\right )^{\frac {1}{3}} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 108, normalized size = 1.30 \[ \frac {\left (-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {x^{3}}{\left (x^{2}\right )^{\frac {3}{2}}}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {x^{3}}{\left (x^{2}\right )^{\frac {3}{2}}}\right )^{\frac {1}{3}}}\right )+2 \ln \left (x +\left (\frac {x^{3}}{\left (x^{2}\right )^{\frac {3}{2}}}\right )^{\frac {1}{3}}\right )-\ln \left (x^{2}-\left (\frac {x^{3}}{\left (x^{2}\right )^{\frac {3}{2}}}\right )^{\frac {1}{3}} x +\left (\frac {x^{3}}{\left (x^{2}\right )^{\frac {3}{2}}}\right )^{\frac {2}{3}}\right )\right ) x^{3}}{6 \left (x^{2}\right )^{\frac {3}{2}} \left (\frac {x^{3}}{\left (x^{2}\right )^{\frac {3}{2}}}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 34, normalized size = 0.41 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{6} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{3} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (x^2\right )}^{3/2}+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 41, normalized size = 0.49 \[ \frac {\log {\left (x + 1 \right )}}{3} - \frac {\log {\left (x^{2} - x + 1 \right )}}{6} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} - \frac {\sqrt {3}}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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