Optimal. Leaf size=21 \[ \frac {1}{2} x \sqrt {1+x^2}+\frac {1}{2} \sinh ^{-1}(x) \]
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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {26, 201, 221}
\begin {gather*} \frac {1}{2} \sqrt {x^2+1} x+\frac {1}{2} \sinh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 26
Rule 201
Rule 221
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^4}}{\sqrt {1-x^2}} \, dx &=\int \sqrt {1+x^2} \, dx\\ &=\frac {1}{2} x \sqrt {1+x^2}+\frac {1}{2} \int \frac {1}{\sqrt {1+x^2}} \, dx\\ &=\frac {1}{2} x \sqrt {1+x^2}+\frac {1}{2} \sinh ^{-1}(x)\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(21)=42\).
time = 0.62, size = 70, normalized size = 3.33 \begin {gather*} \frac {1}{2} \left (\frac {x \sqrt {1-x^4}}{\sqrt {1-x^2}}+\log \left (1-x^2\right )-\log \left (-x+x^3+\sqrt {1-x^2} \sqrt {1-x^4}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(46\) vs.
\(2(15)=30\).
time = 0.70, size = 47, normalized size = 2.24
method | result | size |
default | \(-\frac {\sqrt {-x^{4}+1}\, \sqrt {-x^{2}+1}\, \left (x \sqrt {x^{2}+1}+\arcsinh \left (x \right )\right )}{2 \left (x^{2}-1\right ) \sqrt {x^{2}+1}}\) | \(47\) |
risch | \(-\frac {x \sqrt {x^{2}+1}\, \sqrt {\frac {\left (-x^{2}+1\right ) \left (-x^{4}+1\right )}{\left (x^{2}-1\right )^{2}}}\, \left (x^{2}-1\right )}{2 \sqrt {-x^{4}+1}\, \sqrt {-x^{2}+1}}-\frac {\arcsinh \left (x \right ) \sqrt {\frac {\left (-x^{2}+1\right ) \left (-x^{4}+1\right )}{\left (x^{2}-1\right )^{2}}}\, \left (x^{2}-1\right )}{2 \sqrt {-x^{4}+1}\, \sqrt {-x^{2}+1}}\) | \(110\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 120 vs.
\(2 (15) = 30\).
time = 0.33, size = 120, normalized size = 5.71 \begin {gather*} -\frac {2 \, \sqrt {-x^{4} + 1} \sqrt {-x^{2} + 1} x + {\left (x^{2} - 1\right )} \log \left (\frac {x^{3} + \sqrt {-x^{4} + 1} \sqrt {-x^{2} + 1} - x}{x^{3} - x}\right ) - {\left (x^{2} - 1\right )} \log \left (-\frac {x^{3} - \sqrt {-x^{4} + 1} \sqrt {-x^{2} + 1} - x}{x^{3} - x}\right )}{4 \, {\left (x^{2} - 1\right )}} \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 {- \left (x - 1\right ) \left (x + 1\right ) \left (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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.05 \begin {gather*} \int \frac {\sqrt {1-x^4}}{\sqrt {1-x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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