Optimal. Leaf size=90 \[ \frac {4 a^2 \sqrt {1-a^2 x^2}}{a x+1}+\frac {3 a \sqrt {1-a^2 x^2}}{x}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}-\frac {9}{2} a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
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Rubi [A] time = 0.74, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6124, 6742, 266, 51, 63, 208, 264, 651} \[ \frac {4 a^2 \sqrt {1-a^2 x^2}}{a x+1}+\frac {3 a \sqrt {1-a^2 x^2}}{x}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}-\frac {9}{2} a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 51
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^3} \, dx &=\int \frac {(1-a x)^2}{x^3 (1+a x) \sqrt {1-a^2 x^2}} \, dx\\ &=\int \left (\frac {1}{x^3 \sqrt {1-a^2 x^2}}-\frac {3 a}{x^2 \sqrt {1-a^2 x^2}}+\frac {4 a^2}{x \sqrt {1-a^2 x^2}}-\frac {4 a^3}{(1+a x) \sqrt {1-a^2 x^2}}\right ) \, dx\\ &=-\left ((3 a) \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx\right )+\left (4 a^2\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx-\left (4 a^3\right ) \int \frac {1}{(1+a x) \sqrt {1-a^2 x^2}} \, dx+\int \frac {1}{x^3 \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3 a \sqrt {1-a^2 x^2}}{x}+\frac {4 a^2 \sqrt {1-a^2 x^2}}{1+a x}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-a^2 x}} \, dx,x,x^2\right )+\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-a^2 x^2}}{2 x^2}+\frac {3 a \sqrt {1-a^2 x^2}}{x}+\frac {4 a^2 \sqrt {1-a^2 x^2}}{1+a x}-4 \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )+\frac {1}{4} a^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-a^2 x^2}}{2 x^2}+\frac {3 a \sqrt {1-a^2 x^2}}{x}+\frac {4 a^2 \sqrt {1-a^2 x^2}}{1+a x}-4 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )\\ &=-\frac {\sqrt {1-a^2 x^2}}{2 x^2}+\frac {3 a \sqrt {1-a^2 x^2}}{x}+\frac {4 a^2 \sqrt {1-a^2 x^2}}{1+a x}-\frac {9}{2} a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 75, normalized size = 0.83 \[ \sqrt {1-a^2 x^2} \left (\frac {4 a^2}{a x+1}+\frac {3 a}{x}-\frac {1}{2 x^2}\right )-\frac {9}{2} a^2 \log \left (\sqrt {1-a^2 x^2}+1\right )+\frac {9}{2} a^2 \log (x) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.47, size = 93, normalized size = 1.03 \[ \frac {8 \, a^{3} x^{3} + 8 \, a^{2} x^{2} + 9 \, {\left (a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (14 \, a^{2} x^{2} + 5 \, a x - 1\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, {\left (a x^{3} + x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 214, normalized size = 2.38 \[ \frac {{\left (a^{3} - \frac {11 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a}{x} - \frac {76 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a x^{2}}\right )} a^{4} x^{2}}{8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} - \frac {9 \, a^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} + \frac {\frac {12 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a {\left | a \right |}}{x} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 319, normalized size = 3.54 \[ \frac {3 a^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2}+\frac {9 a^{2} \sqrt {-a^{2} x^{2}+1}}{2}-\frac {9 a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {3 a \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}+3 a^{3} x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}+\frac {9 a^{3} x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {9 a^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{2 x^{2}}+\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{3}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{\left (x +\frac {1}{a}\right )^{2}}-3 a^{2} \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}-\frac {9 a^{3} \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x}{2}-\frac {9 a^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.81, size = 105, normalized size = 1.17 \[ \frac {3\,a\,\sqrt {1-a^2\,x^2}}{x}-\frac {\sqrt {1-a^2\,x^2}}{2\,x^2}-\frac {4\,a^3\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}+\frac {a^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{2} \]
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 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{3} \left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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