Optimal. Leaf size=304 \[ \frac {1}{3} \left (\sqrt {x^2+1}+x\right )^{3/2}-\frac {1}{\sqrt {\sqrt {x^2+1}+x}}-\sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {\sqrt {2}-1}}\right )+\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {1+\sqrt {2}}}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\frac {\sqrt {x^2+1}}{\sqrt {2}}+\frac {x}{\sqrt {2}}-\frac {1}{\sqrt {2}}}{\sqrt {\sqrt {x^2+1}+x}}\right )+\sqrt {\sqrt {2}-1} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {\sqrt {2}-1}}\right )-\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {1+\sqrt {2}}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\frac {\sqrt {x^2+1}}{\sqrt {2}}+\frac {x}{\sqrt {2}}+\frac {1}{\sqrt {2}}}{\sqrt {\sqrt {x^2+1}+x}}\right ) \]
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Rubi [A] time = 0.76, antiderivative size = 343, normalized size of antiderivative = 1.13, number of steps used = 34, number of rules used = 18, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6725, 2117, 14, 2119, 1628, 826, 1166, 204, 206, 207, 203, 2122, 329, 297, 1162, 617, 1165, 628} \begin {gather*} \frac {1}{3} \left (\sqrt {x^2+1}+x\right )^{3/2}-\frac {1}{\sqrt {\sqrt {x^2+1}+x}}-\frac {\log \left (\sqrt {x^2+1}-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+x+1\right )}{\sqrt {2}}+\frac {\log \left (\sqrt {x^2+1}+\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+x+1\right )}{\sqrt {2}}+\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )-\sqrt {\sqrt {2}-1} \tan ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )+\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}\right )-\sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+1\right )-\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )+\sqrt {\sqrt {2}-1} \tanh ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 203
Rule 204
Rule 206
Rule 207
Rule 297
Rule 329
Rule 617
Rule 628
Rule 826
Rule 1162
Rule 1165
Rule 1166
Rule 1628
Rule 2117
Rule 2119
Rule 2122
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}}{-1+x^4} \, dx &=\int \left (\sqrt {x+\sqrt {1+x^2}}+\frac {2 \sqrt {x+\sqrt {1+x^2}}}{-1+x^4}\right ) \, dx\\ &=2 \int \frac {\sqrt {x+\sqrt {1+x^2}}}{-1+x^4} \, dx+\int \sqrt {x+\sqrt {1+x^2}} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x^2}{x^{3/2}} \, dx,x,x+\sqrt {1+x^2}\right )+2 \int \left (-\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1-x^2\right )}-\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1+x^2\right )}\right ) \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/2}}+\sqrt {x}\right ) \, dx,x,x+\sqrt {1+x^2}\right )-\int \frac {\sqrt {x+\sqrt {1+x^2}}}{1-x^2} \, dx-\int \frac {\sqrt {x+\sqrt {1+x^2}}}{1+x^2} \, dx\\ &=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-2 \operatorname {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,x+\sqrt {1+x^2}\right )-\int \left (\frac {\sqrt {x+\sqrt {1+x^2}}}{2 (1-x)}+\frac {\sqrt {x+\sqrt {1+x^2}}}{2 (1+x)}\right ) \, dx\\ &=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-\frac {1}{2} \int \frac {\sqrt {x+\sqrt {1+x^2}}}{1-x} \, dx-\frac {1}{2} \int \frac {\sqrt {x+\sqrt {1+x^2}}}{1+x} \, dx-4 \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (1+2 x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (-1+2 x+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+2 \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-2 \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {x}}+\frac {2 (1+x)}{\sqrt {x} \left (1+2 x-x^2\right )}\right ) \, dx,x,x+\sqrt {1+x^2}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {x}}+\frac {2 (1-x)}{\sqrt {x} \left (-1+2 x+x^2\right )}\right ) \, dx,x,x+\sqrt {1+x^2}\right )-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )}{\sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )}{\sqrt {2}}-\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-\frac {\log \left (1+x+\sqrt {1+x^2}-\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )}{\sqrt {2}}+\frac {\log \left (1+x+\sqrt {1+x^2}+\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )}{\sqrt {2}}-\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )+\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )-\operatorname {Subst}\left (\int \frac {1+x}{\sqrt {x} \left (1+2 x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )-\operatorname {Subst}\left (\int \frac {1-x}{\sqrt {x} \left (-1+2 x+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}+\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )-\sqrt {2} \tan ^{-1}\left (1+\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )-\frac {\log \left (1+x+\sqrt {1+x^2}-\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )}{\sqrt {2}}+\frac {\log \left (1+x+\sqrt {1+x^2}+\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )}{\sqrt {2}}-2 \operatorname {Subst}\left (\int \frac {1+x^2}{1+2 x^2-x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-2 \operatorname {Subst}\left (\int \frac {1-x^2}{-1+2 x^2+x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}+\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )-\sqrt {2} \tan ^{-1}\left (1+\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )-\frac {\log \left (1+x+\sqrt {1+x^2}-\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )}{\sqrt {2}}+\frac {\log \left (1+x+\sqrt {1+x^2}+\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )}{\sqrt {2}}-\left (-1-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2}+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\left (1-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\left (-1+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2}+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\left (1+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-\sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )+\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )+\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )-\sqrt {2} \tan ^{-1}\left (1+\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )+\sqrt {-1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )-\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )-\frac {\log \left (1+x+\sqrt {1+x^2}-\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )}{\sqrt {2}}+\frac {\log \left (1+x+\sqrt {1+x^2}+\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 1.08, size = 379, normalized size = 1.25 \begin {gather*} \frac {1}{3} \left (\sqrt {x^2+1}+x\right )^{3/2}-\frac {1}{\sqrt {\sqrt {x^2+1}+x}}-\frac {\log \left (\sqrt {x^2+1}-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+x+1\right )}{\sqrt {2}}+\frac {\log \left (\sqrt {x^2+1}+\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+x+1\right )}{\sqrt {2}}+\frac {\left (\sqrt {2}-2\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {\sqrt {2}-1}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\left (2+\sqrt {2}\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {1+\sqrt {2}}}\right )}{\sqrt {2 \left (1+\sqrt {2}\right )}}+\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}\right )-\sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+1\right )-\frac {\left (\sqrt {2}-2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {\sqrt {2}-1}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}-\frac {\left (2+\sqrt {2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {1+\sqrt {2}}}\right )}{\sqrt {2 \left (1+\sqrt {2}\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.60, size = 304, normalized size = 1.00 \begin {gather*} -\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}+\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )-\sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )-\sqrt {2} \tan ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {x}{\sqrt {2}}+\frac {\sqrt {1+x^2}}{\sqrt {2}}}{\sqrt {x+\sqrt {1+x^2}}}\right )-\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )+\sqrt {-1+\sqrt {2}} \tanh ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\frac {1}{\sqrt {2}}+\frac {x}{\sqrt {2}}+\frac {\sqrt {1+x^2}}{\sqrt {2}}}{\sqrt {x+\sqrt {1+x^2}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 463, normalized size = 1.52 \begin {gather*} \frac {2}{3} \, {\left (2 \, x - \sqrt {x^{2} + 1}\right )} \sqrt {x + \sqrt {x^{2} + 1}} + 2 \, \sqrt {\sqrt {2} - 1} \arctan \left (\sqrt {x + \sqrt {2} + \sqrt {x^{2} + 1} - 1} {\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} - \sqrt {x + \sqrt {x^{2} + 1}} {\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1}\right ) - 2 \, \sqrt {\sqrt {2} + 1} \arctan \left (\sqrt {x + \sqrt {2} + \sqrt {x^{2} + 1} + 1} \sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )} - \sqrt {x + \sqrt {x^{2} + 1}} \sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )}\right ) + 2 \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} \sqrt {x + \sqrt {x^{2} + 1}} + x + \sqrt {x^{2} + 1} + 1} - \sqrt {2} \sqrt {x + \sqrt {x^{2} + 1}} - 1\right ) + 2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} \sqrt {x + \sqrt {x^{2} + 1}} + 4 \, x + 4 \, \sqrt {x^{2} + 1} + 4} - \sqrt {2} \sqrt {x + \sqrt {x^{2} + 1}} + 1\right ) + \frac {1}{2} \, \sqrt {2} \log \left (4 \, \sqrt {2} \sqrt {x + \sqrt {x^{2} + 1}} + 4 \, x + 4 \, \sqrt {x^{2} + 1} + 4\right ) - \frac {1}{2} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {x + \sqrt {x^{2} + 1}} + 4 \, x + 4 \, \sqrt {x^{2} + 1} + 4\right ) - \frac {1}{2} \, \sqrt {\sqrt {2} + 1} \log \left (\sqrt {x + \sqrt {x^{2} + 1}} + \sqrt {\sqrt {2} + 1}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} + 1} \log \left (\sqrt {x + \sqrt {x^{2} + 1}} - \sqrt {\sqrt {2} + 1}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (\sqrt {x + \sqrt {x^{2} + 1}} + \sqrt {\sqrt {2} - 1}\right ) - \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (\sqrt {x + \sqrt {x^{2} + 1}} - \sqrt {\sqrt {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 {{\left (x^{4} + 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}}{x^{4} - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}+1\right ) \sqrt {x +\sqrt {x^{2}+1}}}{x^{4}-1}\, dx\]
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 {{\left (x^{4} + 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}}{x^{4} - 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 = -0.00 \begin {gather*} \int \frac {\left (x^4+1\right )\,\sqrt {x+\sqrt {x^2+1}}}{x^4-1} \,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 {\sqrt {x + \sqrt {x^{2} + 1}} \left (x^{4} + 1\right )}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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