Optimal. Leaf size=144 \[ \frac {3 (x+2)}{2 \sqrt {\frac {2 x}{x^2+1}+1}}-\frac {x^2+1}{2 (x+1) \sqrt {\frac {2 x}{x^2+1}+1}}-\frac {3 (x+1) \sinh ^{-1}(x)}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}}-\frac {9 (x+1) \tanh ^{-1}\left (\frac {1-x}{\sqrt {2} \sqrt {x^2+1}}\right )}{2 \sqrt {2} \sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}} \]
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Rubi [A] time = 0.08, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6723, 970, 733, 813, 844, 215, 725, 206} \[ \frac {3 (x+2)}{2 \sqrt {\frac {2 x}{x^2+1}+1}}-\frac {x^2+1}{2 (x+1) \sqrt {\frac {2 x}{x^2+1}+1}}-\frac {3 (x+1) \sinh ^{-1}(x)}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}}-\frac {9 (x+1) \tanh ^{-1}\left (\frac {1-x}{\sqrt {2} \sqrt {x^2+1}}\right )}{2 \sqrt {2} \sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 215
Rule 725
Rule 733
Rule 813
Rule 844
Rule 970
Rule 6723
Rubi steps
\begin {align*} \int \frac {1}{\left (1+\frac {2 x}{1+x^2}\right )^{3/2}} \, dx &=\frac {\sqrt {1+2 x+x^2} \int \frac {\left (1+x^2\right )^{3/2}}{\left (1+2 x+x^2\right )^{3/2}} \, dx}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ &=\frac {(4 (2+2 x)) \int \frac {\left (1+x^2\right )^{3/2}}{(2+2 x)^3} \, dx}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ &=-\frac {1+x^2}{2 (1+x) \sqrt {1+\frac {2 x}{1+x^2}}}+\frac {(3 (2+2 x)) \int \frac {x \sqrt {1+x^2}}{(2+2 x)^2} \, dx}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ &=\frac {3 (2+x)}{2 \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {1+x^2}{2 (1+x) \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {(3 (2+2 x)) \int \frac {-4+8 x}{(2+2 x) \sqrt {1+x^2}} \, dx}{8 \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ &=\frac {3 (2+x)}{2 \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {1+x^2}{2 (1+x) \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {(3 (2+2 x)) \int \frac {1}{\sqrt {1+x^2}} \, dx}{2 \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}+\frac {(9 (2+2 x)) \int \frac {1}{(2+2 x) \sqrt {1+x^2}} \, dx}{2 \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ &=\frac {3 (2+x)}{2 \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {1+x^2}{2 (1+x) \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {3 (1+x) \sinh ^{-1}(x)}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {(9 (2+2 x)) \operatorname {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,\frac {2-2 x}{\sqrt {1+x^2}}\right )}{2 \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ &=\frac {3 (2+x)}{2 \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {1+x^2}{2 (1+x) \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {3 (1+x) \sinh ^{-1}(x)}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {9 (1+x) \tanh ^{-1}\left (\frac {1-x}{\sqrt {2} \sqrt {1+x^2}}\right )}{2 \sqrt {2} \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 95, normalized size = 0.66 \[ \frac {(x+1) \left (2 \sqrt {x^2+1} \left (2 x^2+9 x+5\right )+9 \sqrt {2} (x+1)^2 \tanh ^{-1}\left (\frac {x-1}{\sqrt {2} \sqrt {x^2+1}}\right )-12 (x+1)^2 \sinh ^{-1}(x)\right )}{4 \left (\frac {(x+1)^2}{x^2+1}\right )^{3/2} \left (x^2+1\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 205, normalized size = 1.42 \[ \frac {10 \, x^{3} + 9 \, \sqrt {2} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (-\frac {x^{2} + \sqrt {2} {\left (x^{2} - 1\right )} + {\left (2 \, x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + 2\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} - 1}{x^{2} + 2 \, x + 1}\right ) + 30 \, x^{2} + 12 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (-\frac {x^{2} - {\left (x^{2} + 1\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} + x}{x + 1}\right ) + 2 \, {\left (2 \, x^{4} + 9 \, x^{3} + 7 \, x^{2} + 9 \, x + 5\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} + 30 \, x + 10}{4 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} \]
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 \[ \int \frac {1}{{\left (\frac {2 \, x}{x^{2} + 1} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 218, normalized size = 1.51 \[ \frac {\left (x +1\right ) \left (-\left (x^{2}+1\right )^{\frac {3}{2}} x^{3}-6 \sqrt {x^{2}+1}\, x^{3}-24 x^{2} \arcsinh \relax (x )+18 \sqrt {2}\, x^{2} \arctanh \left (\frac {\left (x -1\right ) \sqrt {2}}{2 \sqrt {x^{2}+1}}\right )+\left (x^{2}+1\right )^{\frac {3}{2}} x^{2}+6 \sqrt {x^{2}+1}\, x^{2}-48 x \arcsinh \relax (x )+36 \sqrt {2}\, x \arctanh \left (\frac {\left (x -1\right ) \sqrt {2}}{2 \sqrt {x^{2}+1}}\right )+\left (x^{2}+1\right )^{\frac {5}{2}} x +5 \left (x^{2}+1\right )^{\frac {3}{2}} x +30 \sqrt {x^{2}+1}\, x -24 \arcsinh \relax (x )+18 \sqrt {2}\, \arctanh \left (\frac {\left (x -1\right ) \sqrt {2}}{2 \sqrt {x^{2}+1}}\right )-\left (x^{2}+1\right )^{\frac {5}{2}}+3 \left (x^{2}+1\right )^{\frac {3}{2}}+18 \sqrt {x^{2}+1}\right )}{8 \left (\frac {x^{2}+2 x +1}{x^{2}+1}\right )^{\frac {3}{2}} \left (x^{2}+1\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {1}{{\left (\frac {2 \, x}{x^{2} + 1} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (\frac {2\,x}{x^2+1}+1\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (\frac {2 x}{x^{2} + 1} + 1\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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