Optimal. Leaf size=73 \[ -\frac{a \sqrt{\frac{1}{a^2 x^2}+1}}{4 x}-\frac{\sqrt{\frac{1}{a^2 x^2}+1}}{2 a x^3}-\frac{1}{2 a^2 x^4}+\frac{1}{4} a^2 \text{csch}^{-1}(a x)-\frac{1}{2 x^2} \]
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Rubi [A] time = 0.233842, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6338, 6742, 335, 279, 321, 215} \[ -\frac{a \sqrt{\frac{1}{a^2 x^2}+1}}{4 x}-\frac{\sqrt{\frac{1}{a^2 x^2}+1}}{2 a x^3}-\frac{1}{2 a^2 x^4}+\frac{1}{4} a^2 \text{csch}^{-1}(a x)-\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 6338
Rule 6742
Rule 335
Rule 279
Rule 321
Rule 215
Rubi steps
\begin{align*} \int \frac{e^{2 \text{csch}^{-1}(a x)}}{x^3} \, dx &=\int \frac{\left (\sqrt{1+\frac{1}{a^2 x^2}}+\frac{1}{a x}\right )^2}{x^3} \, dx\\ &=\int \left (\frac{2}{a^2 x^5}+\frac{2 \sqrt{1+\frac{1}{a^2 x^2}}}{a x^4}+\frac{1}{x^3}\right ) \, dx\\ &=-\frac{1}{2 a^2 x^4}-\frac{1}{2 x^2}+\frac{2 \int \frac{\sqrt{1+\frac{1}{a^2 x^2}}}{x^4} \, dx}{a}\\ &=-\frac{1}{2 a^2 x^4}-\frac{1}{2 x^2}-\frac{2 \operatorname{Subst}\left (\int x^2 \sqrt{1+\frac{x^2}{a^2}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{1}{2 a^2 x^4}-\frac{\sqrt{1+\frac{1}{a^2 x^2}}}{2 a x^3}-\frac{1}{2 x^2}-\frac{\operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{1}{2 a^2 x^4}-\frac{\sqrt{1+\frac{1}{a^2 x^2}}}{2 a x^3}-\frac{1}{2 x^2}-\frac{a \sqrt{1+\frac{1}{a^2 x^2}}}{4 x}+\frac{1}{4} a \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{2 a^2 x^4}-\frac{\sqrt{1+\frac{1}{a^2 x^2}}}{2 a x^3}-\frac{1}{2 x^2}-\frac{a \sqrt{1+\frac{1}{a^2 x^2}}}{4 x}+\frac{1}{4} a^2 \text{csch}^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0486543, size = 73, normalized size = 1. \[ \left (-\frac{1}{2 a x^3}-\frac{a}{4 x}\right ) \sqrt{\frac{a^2 x^2+1}{a^2 x^2}}-\frac{1}{2 a^2 x^4}+\frac{1}{4} a^2 \sinh ^{-1}\left (\frac{1}{a x}\right )-\frac{1}{2 x^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.21, size = 176, normalized size = 2.4 \begin{align*} -{\frac{1}{2\,{a}^{2}{x}^{4}}}-{\frac{1}{2\,{x}^{2}}}+{\frac{a}{4\,{x}^{3}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}} \left ( \left ({\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}} \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{-2}}{x}^{2}{a}^{2}-\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}\sqrt{{a}^{-2}}{x}^{4}{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}^{4}-2\, \left ({\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}} \right ) ^{3/2}\sqrt{{a}^{-2}} \right ){\frac{1}{\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}}}{\frac{1}{\sqrt{{a}^{-2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01003, size = 188, normalized size = 2.58 \begin{align*} \frac{a^{3} \log \left (a x \sqrt{\frac{1}{a^{2} x^{2}} + 1} + 1\right ) - a^{3} \log \left (a x \sqrt{\frac{1}{a^{2} x^{2}} + 1} - 1\right ) - \frac{2 \,{\left (a^{6} x^{3}{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + a^{4} x \sqrt{\frac{1}{a^{2} x^{2}} + 1}\right )}}{a^{4} x^{4}{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{2} - 2 \, a^{2} x^{2}{\left (\frac{1}{a^{2} x^{2}} + 1\right )} + 1}}{8 \, a} - \frac{1}{2 \, x^{2}} - \frac{1}{2 \, a^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.52876, size = 269, normalized size = 3.68 \begin{align*} \frac{a^{4} x^{4} \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) - a^{4} x^{4} \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) - 2 \, a^{2} x^{2} -{\left (a^{3} x^{3} + 2 \, a x\right )} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - 2}{4 \, a^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.10566, size = 92, normalized size = 1.26 \begin{align*} \frac{a^{2} \operatorname{asinh}{\left (\frac{1}{a x} \right )}}{4} - \frac{a}{4 x \sqrt{1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{2 x^{2}} - \frac{3}{4 a x^{3} \sqrt{1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{2 a^{2} x^{4}} - \frac{1}{2 a^{3} x^{5} \sqrt{1 + \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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