Optimal. Leaf size=43 \[ \frac {2 x \sqrt {\frac {1}{a^2 x^2}+1}}{a}+\frac {2 \log (x)}{a^2}-\frac {2 \text {csch}^{-1}(a x)}{a^2}+\frac {x^2}{2} \]
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Rubi [A] time = 0.16, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6338, 6742, 242, 277, 215} \[ \frac {2 x \sqrt {\frac {1}{a^2 x^2}+1}}{a}+\frac {2 \log (x)}{a^2}-\frac {2 \text {csch}^{-1}(a x)}{a^2}+\frac {x^2}{2} \]
Antiderivative was successfully verified.
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Rule 215
Rule 242
Rule 277
Rule 6338
Rule 6742
Rubi steps
\begin {align*} \int e^{2 \text {csch}^{-1}(a x)} x \, dx &=\int \left (\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )^2 x \, dx\\ &=\int \left (\frac {2 \sqrt {1+\frac {1}{a^2 x^2}}}{a}+\frac {2}{a^2 x}+x\right ) \, dx\\ &=\frac {x^2}{2}+\frac {2 \log (x)}{a^2}+\frac {2 \int \sqrt {1+\frac {1}{a^2 x^2}} \, dx}{a}\\ &=\frac {x^2}{2}+\frac {2 \log (x)}{a^2}-\frac {2 \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x^2}{a^2}}}{x^2} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {2 \sqrt {1+\frac {1}{a^2 x^2}} x}{a}+\frac {x^2}{2}+\frac {2 \log (x)}{a^2}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^3}\\ &=\frac {2 \sqrt {1+\frac {1}{a^2 x^2}} x}{a}+\frac {x^2}{2}-\frac {2 \text {csch}^{-1}(a x)}{a^2}+\frac {2 \log (x)}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 44, normalized size = 1.02 \[ \frac {a x \left (4 \sqrt {\frac {1}{a^2 x^2}+1}+a x\right )-4 \sinh ^{-1}\left (\frac {1}{a x}\right )+4 \log (x)}{2 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.61, size = 99, normalized size = 2.30 \[ \frac {a^{2} x^{2} + 4 \, a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - 4 \, \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) + 4 \, \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) + 4 \, \log \relax (x)}{2 \, a^{2}} \]
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 = 0.06, size = 120, normalized size = 2.79 \[ \frac {x^{2}}{2}+\frac {2 \ln \relax (x )}{a^{2}}+\frac {2 \sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, x \left (\sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}-\ln \left (\frac {2 \sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+2}{a^{2} x}\right )\right )}{a^{3} \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, \sqrt {\frac {1}{a^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 75, normalized size = 1.74 \[ \frac {1}{2} \, x^{2} + \frac {2 \, x \sqrt {\frac {1}{a^{2} x^{2}} + 1} - \frac {\log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right )}{a} + \frac {\log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right )}{a}}{a} + \frac {2 \, \log \relax (x)}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.20, size = 52, normalized size = 1.21 \[ \frac {x^2}{2}-\frac {2\,\ln \left (\frac {1}{x}\right )}{a^2}+\frac {2\,x\,\sqrt {\frac {1}{a^2\,x^2}+1}}{a}-\frac {2\,\mathrm {asinh}\left (\frac {\sqrt {\frac {1}{a^2}}}{x}\right )}{a^3\,\sqrt {\frac {1}{a^2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.79, size = 63, normalized size = 1.47 \[ \frac {x^{2}}{2} + \frac {2 x}{a \sqrt {1 + \frac {1}{a^{2} x^{2}}}} + \frac {2 \log {\relax (x )}}{a^{2}} - \frac {2 \operatorname {asinh}{\left (\frac {1}{a x} \right )}}{a^{2}} + \frac {2}{a^{3} x \sqrt {1 + \frac {1}{a^{2} x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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