Optimal. Leaf size=38 \[ -\sqrt {\frac {1}{a^2 x^2}+1}+\tanh ^{-1}\left (\sqrt {\frac {1}{a^2 x^2}+1}\right )-\frac {1}{a x} \]
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Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6336, 30, 266, 50, 63, 208} \[ -\sqrt {\frac {1}{a^2 x^2}+1}+\tanh ^{-1}\left (\sqrt {\frac {1}{a^2 x^2}+1}\right )-\frac {1}{a x} \]
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}(a x)}}{x} \, dx &=\frac {\int \frac {1}{x^2} \, dx}{a}+\int \frac {\sqrt {1+\frac {1}{a^2 x^2}}}{x} \, dx\\ &=-\frac {1}{a x}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a^2}}}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=-\sqrt {1+\frac {1}{a^2 x^2}}-\frac {1}{a x}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )\\ &=-\sqrt {1+\frac {1}{a^2 x^2}}-\frac {1}{a x}-a^2 \operatorname {Subst}\left (\int \frac {1}{-a^2+a^2 x^2} \, dx,x,\sqrt {1+\frac {1}{a^2 x^2}}\right )\\ &=-\sqrt {1+\frac {1}{a^2 x^2}}-\frac {1}{a x}+\tanh ^{-1}\left (\sqrt {1+\frac {1}{a^2 x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 1.11 \[ -\sqrt {\frac {1}{a^2 x^2}+1}+\log \left (x \left (\sqrt {\frac {1}{a^2 x^2}+1}+1\right )\right )-\frac {1}{a x} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.59, size = 64, normalized size = 1.68 \[ -\frac {a x \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x\right ) + a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + a x + 1}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 107, normalized size = 2.82 \[ \frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, \left (-a^{2} \left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}}+\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, x^{2} a^{2}+\ln \left (x +\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\right ) x \right )}{\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}}-\frac {1}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 54, normalized size = 1.42 \[ -\sqrt {\frac {1}{a^{2} x^{2}} + 1} - \frac {1}{a x} + \frac {1}{2} \, \log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.46, size = 34, normalized size = 0.89 \[ \mathrm {atanh}\left (\sqrt {\frac {1}{a^2\,x^2}+1}\right )-\sqrt {\frac {1}{a^2\,x^2}+1}-\frac {1}{a\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.81, size = 41, normalized size = 1.08 \[ - \frac {a x}{\sqrt {a^{2} x^{2} + 1}} + \operatorname {asinh}{\left (a x \right )} - \frac {1}{a x} - \frac {1}{a x \sqrt {a^{2} x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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