Optimal. Leaf size=40 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {1-c^4 x^4}}{c \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{c} \]
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Rubi [A]
time = 0.05, antiderivative size = 44, normalized size of antiderivative = 1.10, number of steps
used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1462, 1266,
862, 65, 214} \begin {gather*} -\frac {x \sqrt {\frac {1}{c^2 x^2}+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{\sqrt {c^2 x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 862
Rule 1266
Rule 1462
Rubi steps
\begin {align*} \int \frac {\sqrt {1+\frac {1}{c^2 x^2}}}{\sqrt {1-c^4 x^4}} \, dx &=\frac {\left (\sqrt {1+\frac {1}{c^2 x^2}} x\right ) \int \frac {\sqrt {1+c^2 x^2}}{x \sqrt {1-c^4 x^4}} \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {\left (\sqrt {1+\frac {1}{c^2 x^2}} x\right ) \text {Subst}\left (\int \frac {\sqrt {1+c^2 x}}{x \sqrt {1-c^4 x^2}} \, dx,x,x^2\right )}{2 \sqrt {1+c^2 x^2}}\\ &=\frac {\left (\sqrt {1+\frac {1}{c^2 x^2}} x\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{2 \sqrt {1+c^2 x^2}}\\ &=-\frac {\left (\sqrt {1+\frac {1}{c^2 x^2}} x\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{c^2 \sqrt {1+c^2 x^2}}\\ &=-\frac {\sqrt {1+\frac {1}{c^2 x^2}} x \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{\sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.47, size = 58, normalized size = 1.45 \begin {gather*} -\frac {\sqrt {1+\frac {1}{c^2 x^2}} x \tanh ^{-1}\left (\frac {\sqrt {1-c^4 x^4}}{\sqrt {1+c^2 x^2}}\right )}{\sqrt {1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.13, size = 101, normalized size = 2.52
method | result | size |
default | \(-\frac {\sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x \sqrt {-c^{4} x^{4}+1}\, \mathrm {csgn}\left (\frac {1}{c}\right ) \ln \left (\frac {2 \,\mathrm {csgn}\left (\frac {1}{c}\right ) c \sqrt {-\frac {c^{2} x^{2}-1}{c^{2}}}+2}{c^{2} x}\right )}{\left (c^{2} x^{2}+1\right ) \sqrt {-\frac {c^{2} x^{2}-1}{c^{2}}}\, c}\) | \(101\) |
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.
\(2 (36) = 72\).
time = 0.37, size = 120, normalized size = 3.00 \begin {gather*} -\frac {\log \left (\frac {c^{2} x^{2} + \sqrt {-c^{4} x^{4} + 1} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c^{2} x^{2} + 1}\right ) - \log \left (-\frac {c^{2} x^{2} - \sqrt {-c^{4} x^{4} + 1} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c^{2} x^{2} + 1}\right )}{2 \, c} \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 {1 + \frac {1}{c^{2} x^{2}}}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right ) \left (c^{2} x^{2} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.06, size = 42, normalized size = 1.05 \begin {gather*} -\frac {{\left (\log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - \log \left (-\sqrt {-c^{2} x^{2} + 1} + 1\right )\right )} {\left | c \right |}}{2 \, c^{2}} \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 = -0.02 \begin {gather*} \int \frac {\sqrt {\frac {1}{c^2\,x^2}+1}}{\sqrt {1-c^4\,x^4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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