Optimal. Leaf size=38 \[ -\frac{\sqrt{\frac{1}{a^2 x^2}+1}}{a x}-\frac{1}{a^2 x^2}-\text{csch}^{-1}(a x)+\log (x) \]
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Rubi [A] time = 0.214397, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6338, 6742, 335, 195, 215} \[ -\frac{\sqrt{\frac{1}{a^2 x^2}+1}}{a x}-\frac{1}{a^2 x^2}-\text{csch}^{-1}(a x)+\log (x) \]
Antiderivative was successfully verified.
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Rule 6338
Rule 6742
Rule 335
Rule 195
Rule 215
Rubi steps
\begin{align*} \int \frac{e^{2 \text{csch}^{-1}(a x)}}{x} \, dx &=\int \frac{\left (\sqrt{1+\frac{1}{a^2 x^2}}+\frac{1}{a x}\right )^2}{x} \, dx\\ &=\int \left (\frac{2}{a^2 x^3}+\frac{2 \sqrt{1+\frac{1}{a^2 x^2}}}{a x^2}+\frac{1}{x}\right ) \, dx\\ &=-\frac{1}{a^2 x^2}+\log (x)+\frac{2 \int \frac{\sqrt{1+\frac{1}{a^2 x^2}}}{x^2} \, dx}{a}\\ &=-\frac{1}{a^2 x^2}+\log (x)-\frac{2 \operatorname{Subst}\left (\int \sqrt{1+\frac{x^2}{a^2}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{1}{a^2 x^2}-\frac{\sqrt{1+\frac{1}{a^2 x^2}}}{a x}+\log (x)-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{1}{a^2 x^2}-\frac{\sqrt{1+\frac{1}{a^2 x^2}}}{a x}-\text{csch}^{-1}(a x)+\log (x)\\ \end{align*}
Mathematica [A] time = 0.0571885, size = 39, normalized size = 1.03 \[ -\frac{a x \sqrt{\frac{1}{a^2 x^2}+1}+1}{a^2 x^2}-\sinh ^{-1}\left (\frac{1}{a x}\right )+\log (x) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.177, size = 150, normalized size = 4. \begin{align*} \ln \left ( x \right ) -{\frac{1}{{a}^{2}{x}^{2}}}-{\frac{1}{ax}\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{{a}^{-2}}-\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}\sqrt{{a}^{-2}}{x}^{2}{a}^{2}+\ln \left ( 2\,{\frac{1}{{a}^{2}x} \left ( \sqrt{{a}^{-2}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}{a}^{2}+1 \right ) } \right ){x}^{2} \right ){\frac{1}{\sqrt{{a}^{-2}}}}{\frac{1}{\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01631, size = 126, normalized size = 3.32 \begin{align*} -\frac{\frac{2 \, a^{2} x \sqrt{\frac{1}{a^{2} x^{2}} + 1}}{a^{2} x^{2}{\left (\frac{1}{a^{2} x^{2}} + 1\right )} - 1} + a \log \left (a x \sqrt{\frac{1}{a^{2} x^{2}} + 1} + 1\right ) - a \log \left (a x \sqrt{\frac{1}{a^{2} x^{2}} + 1} - 1\right )}{2 \, a} - \frac{1}{a^{2} x^{2}} + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.59509, size = 252, normalized size = 6.63 \begin{align*} -\frac{a^{2} x^{2} \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) - a^{2} x^{2} \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) - a^{2} x^{2} \log \left (x\right ) + a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} + 1}{a^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.73874, size = 34, normalized size = 0.89 \begin{align*} \log{\left (x \right )} - \operatorname{asinh}{\left (\frac{1}{a x} \right )} - \frac{\sqrt{1 + \frac{1}{a^{2} x^{2}}}}{a x} - \frac{1}{a^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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