Optimal. Leaf size=47 \[ -\frac {2 \sqrt {\frac {1}{a^2 x^2}+1}}{a}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1}{a^2 x^2}+1}\right )}{a}-\frac {2}{a^2 x}+x \]
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Rubi [A] time = 0.07, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6333, 6742, 266, 50, 63, 208} \[ -\frac {2 \sqrt {\frac {1}{a^2 x^2}+1}}{a}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1}{a^2 x^2}+1}\right )}{a}-\frac {2}{a^2 x}+x \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 266
Rule 6333
Rule 6742
Rubi steps
\begin {align*} \int e^{2 \text {csch}^{-1}(a x)} \, dx &=\int \left (\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )^2 \, dx\\ &=\int \left (1+\frac {2}{a^2 x^2}+\frac {2 \sqrt {1+\frac {1}{a^2 x^2}}}{a x}\right ) \, dx\\ &=-\frac {2}{a^2 x}+x+\frac {2 \int \frac {\sqrt {1+\frac {1}{a^2 x^2}}}{x} \, dx}{a}\\ &=-\frac {2}{a^2 x}+x-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a^2}}}{x} \, dx,x,\frac {1}{x^2}\right )}{a}\\ &=-\frac {2 \sqrt {1+\frac {1}{a^2 x^2}}}{a}-\frac {2}{a^2 x}+x-\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a}\\ &=-\frac {2 \sqrt {1+\frac {1}{a^2 x^2}}}{a}-\frac {2}{a^2 x}+x-(2 a) \operatorname {Subst}\left (\int \frac {1}{-a^2+a^2 x^2} \, dx,x,\sqrt {1+\frac {1}{a^2 x^2}}\right )\\ &=-\frac {2 \sqrt {1+\frac {1}{a^2 x^2}}}{a}-\frac {2}{a^2 x}+x+\frac {2 \tanh ^{-1}\left (\sqrt {1+\frac {1}{a^2 x^2}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 52, normalized size = 1.11 \[ -\frac {2 \sqrt {\frac {1}{a^2 x^2}+1}}{a}+\frac {2 \log \left (a x \left (\sqrt {\frac {1}{a^2 x^2}+1}+1\right )\right )}{a}-\frac {2}{a^2 x}+x \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.94, size = 73, normalized size = 1.55 \[ \frac {a^{2} x^{2} - 2 \, a x \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x\right ) - 2 \, a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - 2 \, a x - 2}{a^{2} x} \]
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 = 0.05, size = 112, normalized size = 2.38 \[ x -\frac {2}{a^{2} x}+\frac {2 \sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, \left (-a^{2} \left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}}+\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, x^{2} a^{2}+\ln \left (x +\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\right ) x \right )}{a \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 59, normalized size = 1.26 \[ x - \frac {2 \, \sqrt {\frac {1}{a^{2} x^{2}} + 1} - \log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right ) + \log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right )}{a} - \frac {2}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.26, size = 47, normalized size = 1.00 \[ x-\frac {2\,\sqrt {\frac {1}{a^2\,x^2}+1}}{a}-\frac {2}{a^2\,x}-\frac {\mathrm {atan}\left (\sqrt {\frac {1}{a^2\,x^2}+1}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.63, size = 49, normalized size = 1.04 \[ x - \frac {2 x}{\sqrt {a^{2} x^{2} + 1}} + \frac {2 \operatorname {asinh}{\left (a x \right )}}{a} - \frac {2}{a^{2} x} - \frac {2}{a^{2} x \sqrt {a^{2} x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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