Optimal. Leaf size=153 \[ \frac {-12 x^3-3 x^2+\sqrt {x^2+1} \left (-12 x^2+\left (8 x^2+12 x-4\right ) \sqrt {\sqrt {x^2+1}+x}-3 x\right )+\left (8 x^3+12 x^2+4\right ) \sqrt {\sqrt {x^2+1}+x}-6 x}{24 \sqrt {x^2+1} x+12 \left (2 x^2+1\right )}-\tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}-1}{\sqrt {\sqrt {x^2+1}+x}+1}\right ) \]
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Rubi [A] time = 0.22, antiderivative size = 109, normalized size of antiderivative = 0.71, number of steps used = 13, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6742, 195, 215, 2117, 14, 2119, 448, 2122, 270} \begin {gather*} \frac {x^2}{4}-\frac {1}{4} \sqrt {x^2+1} x+\frac {1}{6} \left (\sqrt {x^2+1}+x\right )^{3/2}+\frac {1}{2} \sqrt {\sqrt {x^2+1}+x}-\frac {1}{2 \sqrt {\sqrt {x^2+1}+x}}-\frac {1}{6 \left (\sqrt {x^2+1}+x\right )^{3/2}}-\frac {x}{2}-\frac {1}{4} \sinh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 195
Rule 215
Rule 270
Rule 448
Rule 2117
Rule 2119
Rule 2122
Rule 6742
Rubi steps
\begin {align*} \int \frac {x}{1+\sqrt {x+\sqrt {1+x^2}}} \, dx &=\int \left (-\frac {1}{2}+\frac {x}{2}-\frac {\sqrt {1+x^2}}{2}+\frac {1}{2} \sqrt {x+\sqrt {1+x^2}}-\frac {1}{2} x \sqrt {x+\sqrt {1+x^2}}+\frac {1}{2} \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}\right ) \, dx\\ &=-\frac {x}{2}+\frac {x^2}{4}-\frac {1}{2} \int \sqrt {1+x^2} \, dx+\frac {1}{2} \int \sqrt {x+\sqrt {1+x^2}} \, dx-\frac {1}{2} \int x \sqrt {x+\sqrt {1+x^2}} \, dx+\frac {1}{2} \int \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}} \, dx\\ &=-\frac {x}{2}+\frac {x^2}{4}-\frac {1}{4} x \sqrt {1+x^2}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {\left (-1+x^2\right ) \left (1+x^2\right )}{x^{5/2}} \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{8} \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^{5/2}} \, dx,x,x+\sqrt {1+x^2}\right )-\frac {1}{4} \int \frac {1}{\sqrt {1+x^2}} \, dx+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1+x^2}{x^{3/2}} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {x}{2}+\frac {x^2}{4}-\frac {1}{4} x \sqrt {1+x^2}-\frac {1}{4} \sinh ^{-1}(x)-\frac {1}{8} \operatorname {Subst}\left (\int \left (-\frac {1}{x^{5/2}}+x^{3/2}\right ) \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{8} \operatorname {Subst}\left (\int \left (\frac {1}{x^{5/2}}+\frac {2}{\sqrt {x}}+x^{3/2}\right ) \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{4} \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/2}}+\sqrt {x}\right ) \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {x}{2}+\frac {x^2}{4}-\frac {1}{4} x \sqrt {1+x^2}-\frac {1}{6 \left (x+\sqrt {1+x^2}\right )^{3/2}}-\frac {1}{2 \sqrt {x+\sqrt {1+x^2}}}+\frac {1}{2} \sqrt {x+\sqrt {1+x^2}}+\frac {1}{6} \left (x+\sqrt {1+x^2}\right )^{3/2}-\frac {1}{4} \sinh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 1.06, size = 217, normalized size = 1.42 \begin {gather*} \frac {1}{60} \left (15 x^2-15 \sqrt {x^2+1} x-20 \left (\sqrt {x^2+1}-2 x\right ) \sqrt {\sqrt {x^2+1}+x}-\frac {4 \sqrt {x^2+1} \left (6 x^4+6 x^2+3 \sqrt {x^2+1} x+6 \sqrt {x^2+1} x^3+2\right )}{\sqrt {\sqrt {x^2+1}+x} \left (x^2+\sqrt {x^2+1} x+1\right )}+\frac {4 \sqrt {\sqrt {x^2+1}+x} \left (6 x^4+21 x^2+18 \sqrt {x^2+1} x+6 \sqrt {x^2+1} x^3+7\right )}{2 x^2+2 \sqrt {x^2+1} x+1}-30 x-15 \sinh ^{-1}(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.27, size = 153, normalized size = 1.00 \begin {gather*} \frac {-6 x-3 x^2-12 x^3+\left (4+12 x^2+8 x^3\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-3 x-12 x^2+\left (-4+12 x+8 x^2\right ) \sqrt {x+\sqrt {1+x^2}}\right )}{24 x \sqrt {1+x^2}+12 \left (1+2 x^2\right )}-\tanh ^{-1}\left (\frac {-1+\sqrt {x+\sqrt {1+x^2}}}{1+\sqrt {x+\sqrt {1+x^2}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 66, normalized size = 0.43 \begin {gather*} \frac {1}{4} \, x^{2} - \frac {1}{3} \, {\left (x^{2} - \sqrt {x^{2} + 1} {\left (x - 1\right )} - 2 \, x - 1\right )} \sqrt {x + \sqrt {x^{2} + 1}} - \frac {1}{4} \, \sqrt {x^{2} + 1} x - \frac {1}{2} \, x - \frac {1}{2} \, \log \left (\sqrt {x + \sqrt {x^{2} + 1}}\right ) \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*} \int \frac {x}{\sqrt {x + \sqrt {x^{2} + 1}} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x}{1+\sqrt {x +\sqrt {x^{2}+1}}}\, dx\]
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*} \int \frac {x}{\sqrt {x + \sqrt {x^{2} + 1}} + 1}\,{d x} \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 = -0.01 \begin {gather*} \int \frac {x}{\sqrt {x+\sqrt {x^2+1}}+1} \,d x \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 {x}{\sqrt {x + \sqrt {x^{2} + 1}} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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