Optimal. Leaf size=119 \[ \frac {1}{192} \sqrt {x^2-1} \sqrt {x^2+\sqrt {x^2-1} x} \left (56 x^2-39\right )+\frac {13 \log \left (\sqrt {x^2-1}-\sqrt {2} \sqrt {x^2+\sqrt {x^2-1} x}+x\right )}{64 \sqrt {2}}+\frac {1}{192} \left (13 x-8 x^3\right ) \sqrt {x^2+\sqrt {x^2-1} x} \]
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Rubi [F] time = 0.33, 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 x \sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int x \sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}} \, dx &=\int x \sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}} \, dx\\ \end {align*}
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Mathematica [A] time = 2.12, size = 238, normalized size = 2.00 \begin {gather*} \frac {\sqrt {x^2-1} \left (\sqrt {x^2-1}+x\right )^4 \left (\sqrt {2} \sqrt {x \left (\sqrt {x^2-1}+x\right )} \left (192 x^6-360 x^4+212 x^2+104 \sqrt {x^2-1} x+192 \sqrt {x^2-1} x^5-264 \sqrt {x^2-1} x^3-39\right )-39 \left (4 x^3+4 \sqrt {x^2-1} x^2-\sqrt {x^2-1}-3 x\right ) \sinh ^{-1}\left (\sqrt {x^2-1}+x\right )\right )}{192 \sqrt {2} \left (64 x^8-144 x^6+104 x^4-25 x^2-7 \sqrt {x^2-1} x+64 \sqrt {x^2-1} x^7-112 \sqrt {x^2-1} x^5+56 \sqrt {x^2-1} x^3+1\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.58, size = 119, normalized size = 1.00 \begin {gather*} \frac {1}{192} \sqrt {-1+x^2} \left (-39+56 x^2\right ) \sqrt {x^2+x \sqrt {-1+x^2}}+\frac {1}{192} \left (13 x-8 x^3\right ) \sqrt {x^2+x \sqrt {-1+x^2}}+\frac {13 \log \left (x+\sqrt {-1+x^2}-\sqrt {2} \sqrt {x^2+x \sqrt {-1+x^2}}\right )}{64 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 100, normalized size = 0.84 \begin {gather*} -\frac {1}{192} \, {\left (8 \, x^{3} - {\left (56 \, x^{2} - 39\right )} \sqrt {x^{2} - 1} - 13 \, x\right )} \sqrt {x^{2} + \sqrt {x^{2} - 1} x} + \frac {13}{256} \, \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 \sqrt {x^{2} + \sqrt {x^{2} - 1} x} \sqrt {x^{2} - 1} x\,{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 \[\int x \sqrt {x^{2}-1}\, \sqrt {x^{2}+x \sqrt {x^{2}-1}}\, dx\]
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 \sqrt {x^{2} + \sqrt {x^{2} - 1} x} \sqrt {x^{2} - 1} x\,{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 x\,\sqrt {x^2-1}\,\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \sqrt {x \left (x + \sqrt {x^{2} - 1}\right )} \sqrt {\left (x - 1\right ) \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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