Optimal. Leaf size=83 \[ -\frac {\sqrt {x+\sqrt {1+x}}}{x}-\frac {1}{4} \tan ^{-1}\left (\frac {3+\sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )+\frac {3}{4} \tanh ^{-1}\left (\frac {1-3 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {1028, 1047,
738, 212, 210} \begin {gather*} -\frac {1}{4} \text {ArcTan}\left (\frac {\sqrt {x+1}+3}{2 \sqrt {x+\sqrt {x+1}}}\right )-\frac {\sqrt {x+\sqrt {x+1}}}{x}+\frac {3}{4} \tanh ^{-1}\left (\frac {1-3 \sqrt {x+1}}{2 \sqrt {x+\sqrt {x+1}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 738
Rule 1028
Rule 1047
Rubi steps
\begin {align*} \int \frac {\sqrt {x+\sqrt {1+x}}}{x^2} \, dx &=2 \text {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{\left (-1+x^2\right )^2} \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {\sqrt {x+\sqrt {1+x}}}{x}+\text {Subst}\left (\int \frac {\frac {1}{2}+x}{\left (-1+x^2\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {\sqrt {x+\sqrt {1+x}}}{x}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )+\frac {3}{4} \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {\sqrt {x+\sqrt {1+x}}}{x}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {-3-\sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )-\frac {3}{2} \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+3 \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )\\ &=-\frac {\sqrt {x+\sqrt {1+x}}}{x}-\frac {1}{4} \tan ^{-1}\left (\frac {3+\sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )+\frac {3}{4} \tanh ^{-1}\left (\frac {1-3 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 77, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {x+\sqrt {1+x}}}{x}-\frac {1}{2} \tan ^{-1}\left (1+\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}\right )-\frac {3}{2} \tanh ^{-1}\left (1-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs.
\(2(59)=118\).
time = 0.02, size = 298, normalized size = 3.59
method | result | size |
derivativedivides | \(-\frac {\left (\left (-1+\sqrt {1+x}\right )^{2}-2+3 \sqrt {1+x}\right )^{\frac {3}{2}}}{2 \left (-1+\sqrt {1+x}\right )}+\frac {3 \sqrt {\left (-1+\sqrt {1+x}\right )^{2}-2+3 \sqrt {1+x}}}{4}+\frac {\ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {\left (-1+\sqrt {1+x}\right )^{2}-2+3 \sqrt {1+x}}\right )}{2}-\frac {3 \arctanh \left (\frac {-1+3 \sqrt {1+x}}{2 \sqrt {\left (-1+\sqrt {1+x}\right )^{2}-2+3 \sqrt {1+x}}}\right )}{4}+\frac {\left (1+2 \sqrt {1+x}\right ) \sqrt {\left (-1+\sqrt {1+x}\right )^{2}-2+3 \sqrt {1+x}}}{4}-\frac {\left (\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2\right )^{\frac {3}{2}}}{2 \left (1+\sqrt {1+x}\right )}-\frac {\sqrt {\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2}}{4}-\frac {\ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2}\right )}{2}+\frac {\arctan \left (\frac {-3-\sqrt {1+x}}{2 \sqrt {\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2}}\right )}{4}+\frac {\left (1+2 \sqrt {1+x}\right ) \sqrt {\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2}}{4}\) | \(298\) |
default | \(-\frac {\left (\left (-1+\sqrt {1+x}\right )^{2}-2+3 \sqrt {1+x}\right )^{\frac {3}{2}}}{2 \left (-1+\sqrt {1+x}\right )}+\frac {3 \sqrt {\left (-1+\sqrt {1+x}\right )^{2}-2+3 \sqrt {1+x}}}{4}+\frac {\ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {\left (-1+\sqrt {1+x}\right )^{2}-2+3 \sqrt {1+x}}\right )}{2}-\frac {3 \arctanh \left (\frac {-1+3 \sqrt {1+x}}{2 \sqrt {\left (-1+\sqrt {1+x}\right )^{2}-2+3 \sqrt {1+x}}}\right )}{4}+\frac {\left (1+2 \sqrt {1+x}\right ) \sqrt {\left (-1+\sqrt {1+x}\right )^{2}-2+3 \sqrt {1+x}}}{4}-\frac {\left (\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2\right )^{\frac {3}{2}}}{2 \left (1+\sqrt {1+x}\right )}-\frac {\sqrt {\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2}}{4}-\frac {\ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2}\right )}{2}+\frac {\arctan \left (\frac {-3-\sqrt {1+x}}{2 \sqrt {\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2}}\right )}{4}+\frac {\left (1+2 \sqrt {1+x}\right ) \sqrt {\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2}}{4}\) | \(298\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.67, size = 81, normalized size = 0.98 \begin {gather*} \frac {x \arctan \left (\frac {2 \, \sqrt {x + \sqrt {x + 1}} {\left (\sqrt {x + 1} - 3\right )}}{x - 8}\right ) + 3 \, x \log \left (\frac {2 \, \sqrt {x + \sqrt {x + 1}} {\left (\sqrt {x + 1} + 1\right )} - 3 \, x - 2 \, \sqrt {x + 1} - 2}{x}\right ) - 4 \, \sqrt {x + \sqrt {x + 1}}}{4 \, 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 {\sqrt {x + \sqrt {x + 1}}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 188 vs.
\(2 (59) = 118\).
time = 0.62, size = 188, normalized size = 2.27 \begin {gather*} -\frac {2 \, {\left (\sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1}\right )}^{3} - 3 \, {\left (\sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1}\right )}^{2} - \sqrt {x + \sqrt {x + 1}} + \sqrt {x + 1} + 1}{{\left (\sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1}\right )}^{4} - 2 \, {\left (\sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1}\right )}^{2} + 4 \, \sqrt {x + \sqrt {x + 1}} - 4 \, \sqrt {x + 1}} + \frac {1}{2} \, \arctan \left (\sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1} - 1\right ) - \frac {3}{4} \, \log \left ({\left | \sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1} + 2 \right |}\right ) + \frac {3}{4} \, \log \left ({\left | \sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1} \right |}\right ) \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+\sqrt {x+1}}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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