Optimal. Leaf size=153 \[ \frac {1}{4} \sqrt {x^2+\sqrt {x^2-1} x} (-x-4)+\frac {3}{4} \sqrt {x^2-1} \sqrt {x^2+\sqrt {x^2-1} x}-\frac {3 \log \left (\sqrt {x^2-1}-\sqrt {2} \sqrt {x^2+\sqrt {x^2-1} x}+x\right )}{4 \sqrt {2}}-\sqrt {2} \tanh ^{-1}\left (\sqrt {x^2-1}-\sqrt {2} \sqrt {x^2+\sqrt {x^2-1} x}+x\right ) \]
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Rubi [F] time = 0.37, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{1+x} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{1+x} \, dx &=\int \frac {\sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{1+x} \, dx\\ \end {align*}
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Mathematica [A] time = 0.73, size = 150, normalized size = 0.98 \begin {gather*} \frac {\sqrt {x^2-1} \left (\sqrt {2} \sqrt {x \left (\sqrt {x^2-1}+x\right )} \left (2 x^2+2 \left (\sqrt {x^2-1}-2\right ) x-4 \sqrt {x^2-1}-3\right )+3 \left (\sqrt {x^2-1}+x\right ) \sinh ^{-1}\left (\sqrt {x^2-1}+x\right )+4 \left (\sqrt {x^2-1}+x\right ) \tanh ^{-1}\left (\sqrt {\left (\sqrt {x^2-1}+x\right )^2+1}\right )\right )}{4 \sqrt {2} \left (x^2+\sqrt {x^2-1} x-1\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.44, size = 153, normalized size = 1.00 \begin {gather*} \frac {1}{4} (-4-x) \sqrt {x^2+x \sqrt {-1+x^2}}+\frac {3}{4} \sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}-\sqrt {2} \tanh ^{-1}\left (x+\sqrt {-1+x^2}-\sqrt {2} \sqrt {x^2+x \sqrt {-1+x^2}}\right )-\frac {3 \log \left (x+\sqrt {-1+x^2}-\sqrt {2} \sqrt {x^2+x \sqrt {-1+x^2}}\right )}{4 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 3.76, size = 153, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \, \sqrt {x^{2} + \sqrt {x^{2} - 1} x} {\left (x - 3 \, \sqrt {x^{2} - 1} + 4\right )} + \frac {1}{4} \, \sqrt {2} \log \left (4 \, x^{2} - 2 \, {\left (2 \, \sqrt {2} \sqrt {x^{2} - 1} x - \sqrt {2} {\left (2 \, x^{2} - 1\right )}\right )} \sqrt {x^{2} + \sqrt {x^{2} - 1} x} - 4 \, \sqrt {x^{2} - 1} x - 1\right ) + \frac {3}{16} \, \sqrt {2} \log \left (-4 \, x^{2} - 2 \, \sqrt {x^{2} + \sqrt {x^{2} - 1} x} {\left (\sqrt {2} x + \sqrt {2} \sqrt {x^{2} - 1}\right )} - 4 \, \sqrt {x^{2} - 1} x + 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 {\sqrt {x^{2} + \sqrt {x^{2} - 1} x} \sqrt {x^{2} - 1}}{x + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}-1}\, \sqrt {x^{2}+x \sqrt {x^{2}-1}}}{1+x}\, dx\]
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 {\sqrt {x^{2} + \sqrt {x^{2} - 1} x} \sqrt {x^{2} - 1}}{x + 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}+x^2}}{x+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 \left (x + \sqrt {x^{2} - 1}\right )} \sqrt {\left (x - 1\right ) \left (x + 1\right )}}{x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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