Optimal. Leaf size=46 \[ -\frac {1}{2} \sqrt {\frac {1}{a^2 x^4}+1}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {\frac {1}{a^2 x^4}+1}\right )-\frac {1}{2 a x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6336, 30, 266, 50, 63, 208} \[ -\frac {1}{2} \sqrt {\frac {1}{a^2 x^4}+1}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {\frac {1}{a^2 x^4}+1}\right )-\frac {1}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 50
Rule 63
Rule 208
Rule 266
Rule 6336
Rubi steps
\begin {align*} \int \frac {e^{\text {csch}^{-1}\left (a x^2\right )}}{x} \, dx &=\frac {\int \frac {1}{x^3} \, dx}{a}+\int \frac {\sqrt {1+\frac {1}{a^2 x^4}}}{x} \, dx\\ &=-\frac {1}{2 a x^2}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a^2}}}{x} \, dx,x,\frac {1}{x^4}\right )\\ &=-\frac {1}{2} \sqrt {1+\frac {1}{a^2 x^4}}-\frac {1}{2 a x^2}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a^2}}} \, dx,x,\frac {1}{x^4}\right )\\ &=-\frac {1}{2} \sqrt {1+\frac {1}{a^2 x^4}}-\frac {1}{2 a x^2}-\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {1}{-a^2+a^2 x^2} \, dx,x,\sqrt {1+\frac {1}{a^2 x^4}}\right )\\ &=-\frac {1}{2} \sqrt {1+\frac {1}{a^2 x^4}}-\frac {1}{2 a x^2}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a^2 x^4}}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 22, normalized size = 0.48 \[ \tanh ^{-1}\left (e^{\text {csch}^{-1}\left (a x^2\right )}\right )-\frac {1}{2} e^{\text {csch}^{-1}\left (a x^2\right )} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.87, size = 74, normalized size = 1.61 \[ -\frac {a x^{2} \log \left (a x^{2} \sqrt {\frac {a^{2} x^{4} + 1}{a^{2} x^{4}}} - a x^{2}\right ) + a x^{2} \sqrt {\frac {a^{2} x^{4} + 1}{a^{2} x^{4}}} + a x^{2} + 1}{2 \, a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 66, normalized size = 1.43 \[ -\frac {a^{2} \log \left (-x^{2} {\left | a \right |} + \sqrt {a^{2} x^{4} + 1}\right ) - \frac {2 \, a^{2}}{{\left (x^{2} {\left | a \right |} - \sqrt {a^{2} x^{4} + 1}\right )}^{2} - 1} + \frac {a}{x^{2}}}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 86, normalized size = 1.87 \[ -\frac {\sqrt {\frac {a^{2} x^{4}+1}{a^{2} x^{4}}}\, \left (-\ln \left (x^{2}+\sqrt {\frac {a^{2} x^{4}+1}{a^{2}}}\right ) x^{2}+\sqrt {\frac {a^{2} x^{4}+1}{a^{2}}}\right )}{2 \sqrt {\frac {a^{2} x^{4}+1}{a^{2}}}}-\frac {1}{2 a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 54, normalized size = 1.17 \[ -\frac {1}{2} \, \sqrt {\frac {1}{a^{2} x^{4}} + 1} - \frac {1}{2 \, a x^{2}} + \frac {1}{4} \, \log \left (\sqrt {\frac {1}{a^{2} x^{4}} + 1} + 1\right ) - \frac {1}{4} \, \log \left (\sqrt {\frac {1}{a^{2} x^{4}} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.38, size = 36, normalized size = 0.78 \[ \frac {\mathrm {atanh}\left (\sqrt {\frac {1}{a^2\,x^4}+1}\right )}{2}-\frac {\sqrt {\frac {1}{a^2\,x^4}+1}}{2}-\frac {1}{2\,a\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.69, size = 54, normalized size = 1.17 \[ - \frac {a x^{2}}{2 \sqrt {a^{2} x^{4} + 1}} + \frac {\operatorname {asinh}{\left (a x^{2} \right )}}{2} - \frac {1}{2 a x^{2}} - \frac {1}{2 a x^{2} \sqrt {a^{2} x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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