Optimal. Leaf size=116 \[ -\frac {a^2}{2 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}-\frac {a^2}{2 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}+\frac {a^2}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}-\frac {2 a^2}{3 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^3} \]
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Rubi [A] time = 0.42, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6337, 1612} \[ -\frac {a^2}{2 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}-\frac {a^2}{2 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}+\frac {a^2}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}-\frac {2 a^2}{3 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^3} \]
Antiderivative was successfully verified.
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Rule 1612
Rule 6337
Rubi steps
\begin {align*} \int \frac {e^{-\text {sech}^{-1}(a x)}}{x^3} \, dx &=\int \frac {1}{x^3 \left (\frac {1}{a x}+\sqrt {\frac {1-a x}{1+a x}}+\frac {\sqrt {\frac {1-a x}{1+a x}}}{a x}\right )} \, dx\\ &=-\left (\left (4 a^2\right ) \operatorname {Subst}\left (\int \frac {x \left (1+x^2\right )}{(-1+x)^2 (1+x)^4} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\right )\\ &=-\left (\left (4 a^2\right ) \operatorname {Subst}\left (\int \left (\frac {1}{8 (-1+x)^2}-\frac {1}{2 (1+x)^4}+\frac {1}{2 (1+x)^3}-\frac {1}{8 (1+x)^2}\right ) \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\right )\\ &=-\frac {a^2}{2 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {2 a^2}{3 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^3}+\frac {a^2}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}-\frac {a^2}{2 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 43, normalized size = 0.37 \[ -\frac {(a x-1) \sqrt {\frac {1-a x}{a x+1}} (a x+1)^2+1}{3 a x^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.79, size = 52, normalized size = 0.45 \[ -\frac {{\left (a^{3} x^{3} - a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1}{3 \, a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right ) x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.14, size = 58, normalized size = 0.50 \[ \frac {\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {x}{3}-\frac {a\,x^2}{3}+\frac {1}{3\,a}-\frac {a^2\,x^3}{3}\right )}{x^3\,\sqrt {\frac {1}{a\,x}+1}}-\frac {1}{3\,a\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \int \frac {1}{a x^{3} \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}} + x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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