Optimal. Leaf size=34 \[ -\frac {2}{3 a^2 x^3}-\frac {2}{3} a \left (\frac {1}{a^2 x^2}+1\right )^{3/2}-\frac {1}{x} \]
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Rubi [A] time = 0.20, antiderivative size = 54, normalized size of antiderivative = 1.59, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6338, 6715, 2117} \[ -\frac {1}{6} a \left (\sqrt {\frac {1}{a^2 x^2}+1}+\frac {1}{a x}\right )^3-\frac {1}{2} a \sqrt {\frac {1}{a^2 x^2}+1}-\frac {1}{2 x} \]
Antiderivative was successfully verified.
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Rule 2117
Rule 6338
Rule 6715
Rubi steps
\begin {align*} \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^2} \, dx &=\int \frac {\left (\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )^2}{x^2} \, dx\\ &=-\operatorname {Subst}\left (\int \left (\frac {x}{a}+\sqrt {1+\frac {x^2}{a^2}}\right )^2 \, dx,x,\frac {1}{x}\right )\\ &=-\left (\frac {1}{2} a \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )\right )\\ &=-\frac {1}{2} a \sqrt {1+\frac {1}{a^2 x^2}}-\frac {1}{6} a \left (\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )^3-\frac {1}{2 x}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 46, normalized size = 1.35 \[ -\frac {3 a^2 x^2+2 a x \sqrt {\frac {1}{a^2 x^2}+1} \left (a^2 x^2+1\right )+2}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 57, normalized size = 1.68 \[ -\frac {2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 2 \, {\left (a^{3} x^{3} + a x\right )} \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + 2}{3 \, a^{2} x^{3}} \]
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 = 63, normalized size = 1.85 \[ \frac {-\frac {a^{2}}{x}-\frac {1}{3 x^{3}}}{a^{2}}-\frac {2 \sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, \left (a^{2} x^{2}+1\right )}{3 a \,x^{2}}-\frac {1}{3 x^{3} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 28, normalized size = 0.82 \[ -\frac {2}{3} \, a {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - \frac {1}{x} - \frac {2}{3 \, a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.21, size = 51, normalized size = 1.50 \[ -\frac {\frac {2}{3\,a^2}+\frac {2\,x\,\sqrt {\frac {1}{a^2\,x^2}+1}}{3\,a}}{x^3}-\frac {\frac {2\,a\,x\,\sqrt {\frac {1}{a^2\,x^2}+1}}{3}+1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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