Optimal. Leaf size=116 \[ -\frac {a^2}{2 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {2 a^2}{3 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^3}+\frac {a^2}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}-\frac {a^2}{2 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )} \]
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Rubi [A]
time = 0.28, 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 = {6472, 1626}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 1626
Rule 6472
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 ) \text {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 ) \text {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 = 0.04, size = 43, normalized size = 0.37 \begin {gather*} -\frac {1+(-1+a x) \sqrt {\frac {1-a x}{1+a x}} (1+a x)^2}{3 a x^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 52, normalized size = 0.45 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} a \int \frac {1}{a x^{3} \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}} + x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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