Optimal. Leaf size=45 \[ \frac {4 \sqrt {1-a^2 x^2}}{a x+1}-\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\sin ^{-1}(a x) \]
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Rubi [A] time = 0.71, antiderivative size = 45, 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, 216, 266, 63, 208, 651} \[ \frac {4 \sqrt {1-a^2 x^2}}{a x+1}-\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 651
Rule 6124
Rule 6742
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{x} \, dx &=\int \frac {(1-a x)^2}{x (1+a x) \sqrt {1-a^2 x^2}} \, dx\\ &=\int \left (\frac {a}{\sqrt {1-a^2 x^2}}+\frac {1}{x \sqrt {1-a^2 x^2}}-\frac {4 a}{(1+a x) \sqrt {1-a^2 x^2}}\right ) \, dx\\ &=a \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx-(4 a) \int \frac {1}{(1+a x) \sqrt {1-a^2 x^2}} \, dx+\int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {4 \sqrt {1-a^2 x^2}}{1+a x}+\sin ^{-1}(a x)+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac {4 \sqrt {1-a^2 x^2}}{1+a x}+\sin ^{-1}(a x)-\frac {\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^2}\\ &=\frac {4 \sqrt {1-a^2 x^2}}{1+a x}+\sin ^{-1}(a x)-\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 49, normalized size = 1.09 \[ \frac {4 \sqrt {1-a^2 x^2}}{a x+1}-\log \left (\sqrt {1-a^2 x^2}+1\right )+\sin ^{-1}(a x)+\log (x) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.56, size = 82, normalized size = 1.82 \[ \frac {4 \, a x - 2 \, {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (a x + 1\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + 4 \, \sqrt {-a^{2} x^{2} + 1} + 4}{a x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 86, normalized size = 1.91 \[ \frac {a \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} - \frac {a \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {8 \, a}{{\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 200, normalized size = 4.44 \[ \frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{3} \left (x +\frac {1}{a}\right )^{3}}+\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{2} \left (x +\frac {1}{a}\right )^{2}}+\frac {2 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x +\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.80, size = 80, normalized size = 1.78 \[ \frac {a\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )-\frac {4\,a\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]
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 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x \left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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