Optimal. Leaf size=62 \[ -\frac {4 a \sqrt {1-a^2 x^2}}{a x+1}-\frac {\sqrt {1-a^2 x^2}}{x}+3 a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
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Rubi [A] time = 0.69, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6124, 6742, 264, 266, 63, 208, 651} \[ -\frac {4 a \sqrt {1-a^2 x^2}}{a x+1}-\frac {\sqrt {1-a^2 x^2}}{x}+3 a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 264
Rule 266
Rule 651
Rule 6124
Rule 6742
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{x^2} \, dx &=\int \frac {(1-a x)^2}{x^2 (1+a x) \sqrt {1-a^2 x^2}} \, dx\\ &=\int \left (\frac {1}{x^2 \sqrt {1-a^2 x^2}}-\frac {3 a}{x \sqrt {1-a^2 x^2}}+\frac {4 a^2}{(1+a x) \sqrt {1-a^2 x^2}}\right ) \, dx\\ &=-\left ((3 a) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\right )+\left (4 a^2\right ) \int \frac {1}{(1+a x) \sqrt {1-a^2 x^2}} \, dx+\int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{x}-\frac {4 a \sqrt {1-a^2 x^2}}{1+a x}-\frac {1}{2} (3 a) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-a^2 x^2}}{x}-\frac {4 a \sqrt {1-a^2 x^2}}{1+a x}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a}\\ &=-\frac {\sqrt {1-a^2 x^2}}{x}-\frac {4 a \sqrt {1-a^2 x^2}}{1+a x}+3 a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 57, normalized size = 0.92 \[ \sqrt {1-a^2 x^2} \left (-\frac {4 a}{a x+1}-\frac {1}{x}\right )+3 a \log \left (\sqrt {1-a^2 x^2}+1\right )-3 a \log (x) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.43, size = 75, normalized size = 1.21 \[ -\frac {4 \, a^{2} x^{2} + 4 \, a x + 3 \, {\left (a^{2} x^{2} + a x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (5 \, a x + 1\right )}}{a x^{2} + x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 150, normalized size = 2.42 \[ \frac {3 \, a^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} + \frac {{\left (a^{2} + \frac {17 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{x}\right )} a^{2} x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{2 \, x {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 261, normalized size = 4.21 \[ -a \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}-3 a \sqrt {-a^{2} x^{2}+1}+3 a \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}-a^{2} x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}-\frac {3 a^{2} x \sqrt {-a^{2} x^{2}+1}}{2}-\frac {3 a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{2} \left (x +\frac {1}{a}\right )^{3}}+a \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}+\frac {3 a^{2} \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x}{2}+\frac {3 a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.80, size = 81, normalized size = 1.31 \[ 3\,a\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )-\frac {\sqrt {1-a^2\,x^2}}{x}+\frac {4\,a^2\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]
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 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{2} \left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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