Optimal. Leaf size=53 \[ -\frac {1}{x \sqrt {\sqrt {x^2+1}+x}}+\tan ^{-1}\left (\sqrt {\sqrt {x^2+1}+x}\right )+\tanh ^{-1}\left (\sqrt {\sqrt {x^2+1}+x}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2119, 457, 329, 212, 206, 203} \begin {gather*} -\frac {1}{x \sqrt {\sqrt {x^2+1}+x}}+\tan ^{-1}\left (\sqrt {\sqrt {x^2+1}+x}\right )+\tanh ^{-1}\left (\sqrt {\sqrt {x^2+1}+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 329
Rule 457
Rule 2119
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {x+\sqrt {1+x^2}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (-1+x^2\right )^2} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {1}{x \sqrt {x+\sqrt {1+x^2}}}-\operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {1}{x \sqrt {x+\sqrt {1+x^2}}}-2 \operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=-\frac {1}{x \sqrt {x+\sqrt {1+x^2}}}+\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=-\frac {1}{x \sqrt {x+\sqrt {1+x^2}}}+\tan ^{-1}\left (\sqrt {x+\sqrt {1+x^2}}\right )+\tanh ^{-1}\left (\sqrt {x+\sqrt {1+x^2}}\right )\\ \end {align*}
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Mathematica [C] time = 14.56, size = 983, normalized size = 18.55 \begin {gather*} \frac {159120 \left (x+\sqrt {x^2+1}\right )^{19/2} \left (x^2+\sqrt {x^2+1} x+1\right ) \left (\frac {16}{585} \, _4F_3\left (\frac {5}{4},2,2,2;1,1,\frac {17}{4};\left (x+\sqrt {x^2+1}\right )^2\right ) \left (\left (x+\sqrt {x^2+1}\right )^3+x+\sqrt {x^2+1}\right )^2+\frac {681 \left (x+\sqrt {x^2+1}\right )^6-1483 \left (x+\sqrt {x^2+1}\right )^4-6769 \left (x+\sqrt {x^2+1}\right )^2+5 \left (\left (x+\sqrt {x^2+1}\right )^8-248 \left (x+\sqrt {x^2+1}\right )^6+102 \left (x+\sqrt {x^2+1}\right )^4+1208 \left (x+\sqrt {x^2+1}\right )^2+729\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\left (x+\sqrt {x^2+1}\right )^2\right )-3645}{640 \left (x+\sqrt {x^2+1}\right )^4}\right )}{x \left (\frac {x}{\sqrt {x^2+1}}+1\right ) \left (\left (x+\sqrt {x^2+1}\right )^2+1\right ) \left (1989 \left (54400 x^{10}+54400 \sqrt {x^2+1} x^9+107808 x^8+80608 \sqrt {x^2+1} x^7+90944 x^6+57440 \sqrt {x^2+1} x^5+42462 x^4+20418 \sqrt {x^2+1} x^3+6909 x^2+10 \left (256 x^{11}+256 \sqrt {x^2+1} x^{10}-7296 x^9-7424 \sqrt {x^2+1} x^8-15296 x^7-11552 \sqrt {x^2+1} x^6-12608 x^5-7776 \sqrt {x^2+1} x^4-5590 x^3-2672 \sqrt {x^2+1} x^2-971 x-182 \sqrt {x^2+1}\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\left (x+\sqrt {x^2+1}\right )^2\right ) x+967 \sqrt {x^2+1} x-140\right )+4352 x \left (26624 x^{13}+26624 \sqrt {x^2+1} x^{12}+89088 x^{11}+75776 \sqrt {x^2+1} x^{10}+115200 x^9+80640 \sqrt {x^2+1} x^8+71936 x^7+39424 \sqrt {x^2+1} x^6+22008 x^5+8680 \sqrt {x^2+1} x^4+2916 x^3+696 \sqrt {x^2+1} x^2+112 x+9 \sqrt {x^2+1}\right ) \, _4F_3\left (\frac {5}{4},2,2,2;1,1,\frac {17}{4};\left (x+\sqrt {x^2+1}\right )^2\right )+40960 x \left (8192 x^{15}+8192 \sqrt {x^2+1} x^{14}+32768 x^{13}+28672 \sqrt {x^2+1} x^{12}+52736 x^{11}+39424 \sqrt {x^2+1} x^{10}+43520 x^9+26880 \sqrt {x^2+1} x^8+19392 x^7+9408 \sqrt {x^2+1} x^6+4480 x^5+1568 \sqrt {x^2+1} x^4+462 x^3+98 \sqrt {x^2+1} x^2+14 x+\sqrt {x^2+1}\right ) \, _4F_3\left (\frac {9}{4},3,3,3;2,2,\frac {21}{4};\left (x+\sqrt {x^2+1}\right )^2\right )\right )}-\frac {\left (\left (x+\sqrt {x^2+1}\right )^2-1\right ) \left (-2 \tan ^{-1}\left (\sqrt {x+\sqrt {x^2+1}}\right )-2 \tanh ^{-1}\left (\sqrt {x+\sqrt {x^2+1}}\right )+2 \sqrt {x+\sqrt {x^2+1}}\right )}{2 x \left (\frac {x}{\sqrt {x^2+1}}+1\right ) \left (x+\sqrt {x^2+1}\right ) \left (1-\frac {\left (x+\sqrt {x^2+1}\right )^2-1}{2 \left (x+\sqrt {x^2+1}\right )^2}\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.12, size = 53, normalized size = 1.00 \begin {gather*} -\frac {1}{x \sqrt {x+\sqrt {1+x^2}}}+\tan ^{-1}\left (\sqrt {x+\sqrt {1+x^2}}\right )+\tanh ^{-1}\left (\sqrt {x+\sqrt {1+x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 78, normalized size = 1.47 \begin {gather*} \frac {2 \, x \arctan \left (\sqrt {x + \sqrt {x^{2} + 1}}\right ) + x \log \left (\sqrt {x + \sqrt {x^{2} + 1}} + 1\right ) - x \log \left (\sqrt {x + \sqrt {x^{2} + 1}} - 1\right ) + 2 \, \sqrt {x + \sqrt {x^{2} + 1}} {\left (x - \sqrt {x^{2} + 1}\right )}}{2 \, x} \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 {1}{\sqrt {x + \sqrt {x^{2} + 1}} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 22, normalized size = 0.42
method | result | size |
meijerg | \(-\frac {\sqrt {2}\, \hypergeom \left (\left [\frac {1}{4}, \frac {3}{4}, \frac {3}{4}\right ], \left [\frac {3}{2}, \frac {7}{4}\right ], -\frac {1}{x^{2}}\right )}{3 x^{\frac {3}{2}}}\) | \(22\) |
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 {1}{\sqrt {x + \sqrt {x^{2} + 1}} x^{2}}\,{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 = -0.02 \begin {gather*} \int \frac {1}{x^2\,\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 [C] time = 1.76, size = 44, normalized size = 0.83 \begin {gather*} - \frac {\Gamma \left (\frac {1}{4}\right ) \Gamma ^{2}\left (\frac {3}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4}, \frac {3}{4} \\ \frac {3}{2}, \frac {7}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{2}}} \right )}}{4 \pi x^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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