Optimal. Leaf size=32 \[ \frac {x \log \left (\sqrt {x^2-1}+x\right )}{\sqrt {x^2+1}}-\frac {1}{2} \cosh ^{-1}\left (x^2\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {191, 2554, 276, 52} \[ \frac {x \log \left (\sqrt {x^2-1}+x\right )}{\sqrt {x^2+1}}-\frac {1}{2} \cosh ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
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Rule 52
Rule 191
Rule 276
Rule 2554
Rubi steps
\begin {align*} \int \frac {\log \left (x+\sqrt {-1+x^2}\right )}{\left (1+x^2\right )^{3/2}} \, dx &=\frac {x \log \left (x+\sqrt {-1+x^2}\right )}{\sqrt {1+x^2}}-\int \frac {x}{\sqrt {-1+x^2} \sqrt {1+x^2}} \, dx\\ &=\frac {x \log \left (x+\sqrt {-1+x^2}\right )}{\sqrt {1+x^2}}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,x^2\right )\\ &=-\frac {1}{2} \cosh ^{-1}\left (x^2\right )+\frac {x \log \left (x+\sqrt {-1+x^2}\right )}{\sqrt {1+x^2}}\\ \end {align*}
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Mathematica [B] time = 0.09, size = 89, normalized size = 2.78 \[ \frac {4 x \log \left (\sqrt {x^2-1}+x\right )+\frac {\sqrt {x^2-1} \left (x^2+1\right ) \left (\log \left (1-\frac {x^2}{\sqrt {x^4-1}}\right )-\log \left (\frac {x^2}{\sqrt {x^4-1}}+1\right )\right )}{\sqrt {x^4-1}}}{4 \sqrt {x^2+1}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 58, normalized size = 1.81 \[ \frac {2 \, \sqrt {x^{2} + 1} x \log \left (x + \sqrt {x^{2} - 1}\right ) + {\left (x^{2} + 1\right )} \log \left (-x^{2} + \sqrt {x^{2} + 1} \sqrt {x^{2} - 1}\right )}{2 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 36, normalized size = 1.12 \[ \frac {x \log \left (x + \sqrt {x^{2} - 1}\right )}{\sqrt {x^{2} + 1}} + \frac {1}{2} \, \log \left (x^{2} - \sqrt {x^{4} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (x +\sqrt {x^{2}-1}\right )}{\left (x^{2}+1\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\log \left (x + \sqrt {x^{2} - 1}\right )}{{\left (x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\ln \left (x+\sqrt {x^2-1}\right )}{{\left (x^2+1\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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