Optimal. Leaf size=79 \[ \frac {2 \left (28 x^6+147 x^4+112 x^2+9\right )}{35 \left (\sqrt {x^2+1}+x\right )^{7/2}}+\frac {2 \sqrt {x^2+1} \left (4 x^5+19 x^3+7 x\right )}{5 \left (\sqrt {x^2+1}+x\right )^{7/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 77, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2122, 270} \begin {gather*} \frac {1}{20} \left (\sqrt {x^2+1}+x\right )^{5/2}+\frac {3}{4} \sqrt {\sqrt {x^2+1}+x}-\frac {1}{4 \left (\sqrt {x^2+1}+x\right )^{3/2}}-\frac {1}{28 \left (\sqrt {x^2+1}+x\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 2122
Rubi steps
\begin {align*} \int \frac {1+x^2}{\sqrt {x+\sqrt {1+x^2}}} \, dx &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^{9/2}} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=\frac {1}{8} \operatorname {Subst}\left (\int \left (\frac {1}{x^{9/2}}+\frac {3}{x^{5/2}}+\frac {3}{\sqrt {x}}+x^{3/2}\right ) \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {1}{28 \left (x+\sqrt {1+x^2}\right )^{7/2}}-\frac {1}{4 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\frac {3}{4} \sqrt {x+\sqrt {1+x^2}}+\frac {1}{20} \left (x+\sqrt {1+x^2}\right )^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 66, normalized size = 0.84 \begin {gather*} \frac {7 \left (\sqrt {x^2+1}+x\right )^6+105 \left (\sqrt {x^2+1}+x\right )^4-35 \left (\sqrt {x^2+1}+x\right )^2-5}{140 \left (\sqrt {x^2+1}+x\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 79, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {1+x^2} \left (7 x+19 x^3+4 x^5\right )}{5 \left (x+\sqrt {1+x^2}\right )^{7/2}}+\frac {2 \left (9+112 x^2+147 x^4+28 x^6\right )}{35 \left (x+\sqrt {1+x^2}\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 43, normalized size = 0.54 \begin {gather*} -\frac {2}{35} \, {\left (5 \, x^{4} + 12 \, x^{2} - {\left (5 \, x^{3} + 13 \, x\right )} \sqrt {x^{2} + 1} - 9\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 {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 [C] time = 0.06, size = 84, normalized size = 1.06
method | result | size |
meijerg | \(-\frac {-\frac {32 \sqrt {\pi }\, \sqrt {2}\, \cosh \left (\frac {3 \arcsinh \left (\frac {1}{x}\right )}{2}\right )}{3 x^{\frac {3}{2}}}-\frac {8 \sqrt {\pi }\, \sqrt {2}\, x^{\frac {3}{2}} \left (-\frac {4}{3 x^{4}}-\frac {2}{3 x^{2}}+\frac {2}{3}\right ) \sinh \left (\frac {3 \arcsinh \left (\frac {1}{x}\right )}{2}\right )}{\sqrt {1+\frac {1}{x^{2}}}}}{8 \sqrt {\pi }}+\frac {\sqrt {2}\, x^{\frac {5}{2}} \hypergeom \left (\left [-\frac {5}{4}, \frac {1}{4}, \frac {3}{4}\right ], \left [-\frac {1}{4}, \frac {3}{2}\right ], -\frac {1}{x^{2}}\right )}{5}\) | \(84\) |
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^{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 {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 = 0.62, size = 92, normalized size = 1.16 \begin {gather*} \frac {12 x^{3}}{35 \sqrt {x + \sqrt {x^{2} + 1}}} + \frac {2 x^{2} \sqrt {x^{2} + 1}}{35 \sqrt {x + \sqrt {x^{2} + 1}}} + \frac {44 x}{35 \sqrt {x + \sqrt {x^{2} + 1}}} + \frac {18 \sqrt {x^{2} + 1}}{35 \sqrt {x + \sqrt {x^{2} + 1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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