Optimal. Leaf size=67 \[ \frac {2 \left (10 x^4-5 x^2-7\right )}{15 \left (\sqrt {x^2+1}+x\right )^{5/2}}+\frac {4 \sqrt {x^2+1} \left (x^3-x\right )}{3 \left (\sqrt {x^2+1}+x\right )^{5/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 56, normalized size of antiderivative = 0.84, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2122, 270} \begin {gather*} \frac {1}{6} \left (\sqrt {x^2+1}+x\right )^{3/2}-\frac {1}{\sqrt {\sqrt {x^2+1}+x}}-\frac {1}{10 \left (\sqrt {x^2+1}+x\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 2122
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^2}}{\sqrt {x+\sqrt {1+x^2}}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^{7/2}} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (\frac {1}{x^{7/2}}+\frac {2}{x^{3/2}}+\sqrt {x}\right ) \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {1}{10 \left (x+\sqrt {1+x^2}\right )^{5/2}}-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{6} \left (x+\sqrt {1+x^2}\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 0.76 \begin {gather*} \frac {5 \left (\sqrt {x^2+1}+x\right )^4-30 \left (\sqrt {x^2+1}+x\right )^2-3}{30 \left (\sqrt {x^2+1}+x\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 67, normalized size = 1.00 \begin {gather*} \frac {2 \left (10 x^4-5 x^2-7\right )}{15 \left (\sqrt {x^2+1}+x\right )^{5/2}}+\frac {4 \sqrt {x^2+1} \left (x^3-x\right )}{3 \left (\sqrt {x^2+1}+x\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 38, normalized size = 0.57 \begin {gather*} \frac {2}{15} \, {\left (3 \, x^{3} - {\left (3 \, x^{2} + 7\right )} \sqrt {x^{2} + 1} + 11 \, x\right )} \sqrt {x + \sqrt {x^{2} + 1}} \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 {\sqrt {x^{2} + 1}}{\sqrt {x + \sqrt {x^{2} + 1}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2}+1}}{\sqrt {x +\sqrt {x^{2}+1}}}\, dx \end {gather*}
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 {\sqrt {x^{2} + 1}}{\sqrt {x + \sqrt {x^{2} + 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 {\sqrt {x^2+1}}{\sqrt {x+\sqrt {x^2+1}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.51, size = 63, normalized size = 0.94 \begin {gather*} \frac {2 x^{2}}{15 \sqrt {x + \sqrt {x^{2} + 1}}} + \frac {8 x \sqrt {x^{2} + 1}}{15 \sqrt {x + \sqrt {x^{2} + 1}}} - \frac {14}{15 \sqrt {x + \sqrt {x^{2} + 1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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